phd mathematics entrance exam

The Qualifying Exam

The qualifying exam is designed to measure the breadth of students’ knowledge in mathematics. While some students are able to pass the qualifying exam in one try, passing the exam early is mainly an indication that a student has attended an undergraduate university with a broad undergraduate program in mathematics. It is not a good predictor of the quality of the eventual PhD dissertation.

Students are required to take the qualifying examination at the beginning in the first term. The exam may prove a useful diagnostic in helping to identify areas in which a student’s knowledge is weak. There is no stigma attached to taking the exam several times, but students are expected to pass the examination by the second year in residence in order to begin more specialized study leading to research work.

The department runs tutorials and offers several introductory graduate courses (e.g. Math 212a, 213a, 230a, 231a, and 232a) to help students acquire the necessary broad basic background in mathematics to pass the exam.

The exam consists of three, three-hour papers held on consecutive afternoons. Each paper has six questions, one each on the subjects: Algebra, Algebraic Geometry, Algebraic Topology, Differential Geometry, Real Analysis and Complex Analysis. Each question carries 10 points. In order to pass each subject, students must obtain at least 20 of the 30 points in that subject. Students are considered to have passed the qualifying exam when they have passed in all six subjects (120 of 180 points) in one sitting, or they have passed at least four subjects in one sitting and obtained an A or A- grade in the basic graduate courses corresponding to the subject(s) not passed. Students are expected take the recommended course(s) at the first opportunity.

Once students have passed the qualifying exam, they no longer need to take math courses for a letter grade and may elect to receive the grade (EXC) excused. Students should inform the instructor at the beginning of the term if they elect to take (EXC) as a grade.

PhD Qualifying Exams

Current Requirement: To qualify for the Ph.D. in Mathematics, students must pass two examinations: one in algebra and one in real analysis. 

Requirement for students starting in Autumn 2023 and later:  To qualify for the Ph.D. in Mathematics, students must choose and pass examinations in two of the following four areas: (i) algebra, (ii) real analysis, (iii) geometry and topology, (iv) applied mathematics. 

The exams each consist of two parts. Students are given three hours for each part.

Topics Covered on the Exams:

Algebra Syllabus

Applied Mathematics Syllabus

 Geometry and Topology Syllabus

Real Analysis Syllabus

Past and Practice Qualifying Exams

Timeline for Completion:

Current Requirement: Students must pass both qualifying exams by the autumn of their second year. Ordinarily first-year students take courses in algebra and real analysis throughout the year to prepare them for the exams. The exams are then taken at the beginning of Spring Quarter. A student who does not pass one or more of the exams at that time is given a second chance in Autumn. 

Because some students have already taken graduate courses as undergraduates, incoming graduate students are allowed to take either or both of the exams in the autumn. If they pass either or both of the exams, they thereby fulfill the requirement in those subjects. However, they are in no way penalized for failing either of the exams.

Requirement for students starting in Autumn 2023 and later: Students must choose and pass two out of the four qualifying exams by the autumn of their second year. Students take courses in algebra, real analysis, geometry and topology, and applied math in the autumn and winter quarters of their first year to prepare them for the exams. The exams are taken during the first week of Spring Quarter. A student who does not pass one or more of the exams at that time is given a second chance in Autumn. 

Because some students have already taken graduate courses as undergraduates, incoming graduate students are allowed to take any of the exams in the autumn. If they pass any of the exams, they thereby fulfill the requirement in those subjects. However, they are in no way penalized for failing any of the exams.

Exam Schedule

Unless otherwise noted, the exams will be held each year according to the following schedule:

Autumn Quarter:  The exams are held during the week prior to the first week of the quarter. Spring Quarter:  The exams are held during the first week of the quarter.

The exams are held over two three-hour blocks. The morning block is 9:30am-12:30pm and the afternoon block is 2:00-5:00pm.

For the start date of the current or future years’ quarters please see the  Academic Calendar

Upcoming Exam Dates

Spring 2024.

The exams will be held on the following dates: 

Monday, April 1st: Analysis, Room 384H

Wednesday, April 3rd: Algebra, Room 384I

Thursday, April 4th: Geometry & Topology, Room 384I

Friday, April 5th: Applied Math, Room 384I

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Ph.D. Program in Mathematics

Degree requirements.

A candidate for the Ph.D. degree in mathematics must fulfill a number of different departmental requirements .

NYU Shanghai Ph.D. Track

The Ph.D. program also offers students the opportunity to pursue their study and research with Mathematics faculty based at NYU Shanghai. With this opportunity, students generally complete their coursework in New York City before moving full-time to Shanghai for their dissertation research. For more information, please visit the  NYU Shanghai Ph.D. page .

Sample course schedules (Years 1 and 2) for students with a primary interest in:

Applied Math (Math Biology, Scientific Computing, Physical Applied Math, etc.)

Additional information for students interested in studying applied math is available here .

Probability

PDE/Analysis

The Written Comprehensive Examination

The examination tests the basic knowledge required for any serious mathematical study. It consists of the three following sections: Advanced Calculus, Complex Variables, and Linear Algebra. The examination is given on three consecutive days, twice a year, in early September and early January. Each section is allotted three hours and is written at the level of a good undergraduate course. Samples of previous examinations are available in the departmental office. Cooperative preparation is encouraged, as it is for all examinations. In the fall term, the Department offers a workshop, taught by an advanced Teaching Assistant, to help students prepare for the written examinations.

Entering students with a solid preparation are encouraged to consider taking the examination in their first year of full-time study. All students must take the examinations in order to be allowed to register for coursework beyond 36 points of credit; it is recommended that students attempt to take the examinations well before this deadline. Graduate Assistants are required to take the examinations during their first year of study.

For further details, consult the page on the written comprehensive exams .

The Oral Preliminary Examination

This examination is usually (but not invariably) taken after two years of full-time study. The purpose of the examination is to determine if the candidate has acquired sufficient mathematical knowledge and maturity to commence a dissertation. The phrase "mathematical knowledge" is intended to convey rather broad acquaintance with the basic facts of mathematical life, with emphasis on a good understanding of the simplest interesting examples. In particular, highly technical or abstract material is inappropriate, as is the rote reproduction of information. What the examiners look for is something a little different and less easy to quantify. It is conveyed in part by the word "maturity." This means some idea of how mathematics hangs together; the ability to think a little on one's feet; some appreciation of what is natural and important, and what is artificial. The point is that the ability to do successful research depends on more than formal learning, and it is part of the examiners' task to assess these less tangible aspects of the candidate's preparation.

The orals are comprised of a general section and a special section, each lasting one hour, and are conducted by two different panels of three faculty members. The examination takes place three times a year: fall, mid-winter and late spring. Cooperative preparation of often helpful and is encouraged. The general section consists of five topics, one of which may be chosen freely. The other four topics are determined by field of interest, but often turn out to be standard: complex variables, real variables, ordinary differential equations, and partial differential equations. Here, the level of knowledge that is expected is equivalent to that of a one or two term course of the kind Courant normally presents. A brochure containing the most common questions on the general oral examination, edited by Courant students, is available at the Department Office.

The special section is usually devoted to a single topic at a more advanced level and extent of knowledge. The precise content is negotiated with the candidate's faculty advisor. Normally, the chosen topic will have a direct bearing on the candidate's Ph.D. dissertation.

All students must take the oral examinations in order to be allowed to register for coursework beyond 60 points of credit. It is recommended that students attempt the examinations well before this deadline.

The Dissertation Defense

The oral defense is the final examination on the student's dissertation. The defense is conducted by a panel of five faculty members (including the student's advisor) and generally lasts one to two hours. The candidate presents his/her work to a mixed audience, some expert in the student's topic, some not. Often, this presentation is followed by a question-and-answer period and mutual discussion of related material and directions for future work.

Summer Internships and Employment

The Department encourages Ph.D. students at any stage of their studies, including the very early stage, to seek summer employment opportunities at various government and industry facilities. In the past few years, Courant students have taken summer internships at the National Institute of Health, Los Alamos National Laboratory, Woods Hole Oceanographic Institution, Lawrence Livermore National Laboratory and NASA, as well as Wall Street firms. Such opportunities can greatly expand students' understanding of the mathematical sciences, offer them possible areas of interest for thesis research, and enhance their career options. The Director of Graduate Studies and members of the faculty (and in particular the students' academic advisors) can assist students in finding appropriate summer employment.

Mentoring and Grievance Policy

For detailed information, consult the page on the Mentoring and Grievance Policy .

Visiting Doctoral Students

Information about spending a term at the Courant Institute's Department of Mathematics as a visiting doctoral student is available on the Visitor Programs  page.

Ph.D. Program Overview

Description.

The graduate program in the field of mathematics at Cornell leads to the Ph.D. degree, which takes most students five to six years of graduate study to complete. One feature that makes the program at Cornell particularly attractive is the broad range of  interests of the faculty . The department has outstanding groups in the areas of algebra, algebraic geometry,  analysis, applied mathematics, combinatorics, dynamical systems, geometry, logic, Lie groups, number theory, probability, and topology. The field also maintains close ties with distinguished graduate programs in the fields of  applied mathematics ,  computer science ,  operations research , and  statistics .

Core Courses

A normal course load for a beginning graduate student is three courses per term. 

There are no qualifying exams, but the program requires that all students pass four courses to be selected from the six core courses. First-year students are allowed to place out of some (possibly, all) of the core courses. In order to place out of a course, students should contact the faculty member who is teaching the course during the current academic year, and that faculty member will make a decision. The minimum passing grade for the core courses is B-; no grade is assigned for placing out of a core course.

At least two core courses should be taken (or placed out) by the end of the first year. At least four core courses should be taken (or placed out) by the end of the second year (cumulative). These time requirements can be waived for students with health problems or other significant non-academic problems. They can be also waived for students who take time-consuming courses in another area (for example, CS) and who have strong support from a faculty; requests from such students should be made before the beginning of the spring semester. 

The core courses  are distributed among three main areas: analysis, algebra and topology/geometry. A student must pass at least one course from each group. All entering graduate students are encouraged to eventually take all six core courses with the option of an S/U grade for two of them. 

The six core courses are:

MATH 6110, Real Analysis

MATH 6120, Complex Analysis

MATH 6310, Algebra 1

MATH 6320, Algebra 2

MATH 6510, Introductory Algebraic Topology

MATH 6520, Differentiable Manifolds.

Students who are not ready to take some of the core courses may take MATH 4130-4140, Introduction to Analysis, and/or MATH 4330-4340, Introduction to Algebra, which are the honors versions of our core undergraduate courses.

"What is...?" Seminar

The "What Is...?" Seminar is a series of talks given by faculty in the graduate field of Mathematics. Speakers are selected by an organizing committee of graduate students. The goal of the seminar is to aid students in finding advisors.

Schedule for the "What Is...?" seminar

Special Committee

The Cornell Graduate School requires that every student selects a special committee (in particular, a thesis adviser, who is the chair or the committee) by the end of the third semester.

The emphasis in the Graduate School at Cornell is on individualized instruction and training for independent investigation. There are very few formal requirements and each student develops a program in conjunction with his or her special committee, which consists of three faculty members, some of which may be chosen from outside the field of mathematics. 

Entering students are not assigned special committees. Such students may contact any of the members on the Advising Committee if they have questions or need advice.

Current Advising Committee

Analysis / Probability / Dynamical Systems / Logic: Lionel Levine Geometry / Topology / Combinatorics: Kathryn Mann Probability / Statistics:  Philippe Sosoe Applied Mathematics Liaison: Richard Rand

Admission to Candidacy

To be admitted formally to candidacy for the Ph.D. degree, the student must pass the oral admission to candidacy examination or A exam. This must be completed before the beginning of the student's fourth year. Upon passing the A exam, the student will be awarded (at his/her request) an M.S. degree without thesis.

The admission to candidacy examination is given to determine if the student is “ready to begin work on a thesis.” The content and methods of examination are agreed on by the student and his/her special committee before the examination. The student must be prepared to answer questions on the proposed area of research, and to pass the exam, he/she must demonstrate expertise beyond just mastery of basic mathematics covered in the core graduate courses. 

To receive an advanced degree a student must fulfill the residence requirements of the Graduate School. One unit of residence is granted for successful completion of one semester of full-time study, as judged by the chair of the special committee. The Ph.D. program requires a minimum of six residence units. This is not a difficult requirement to satisfy since the program generally takes five to six years to complete. A student who has done graduate work at another institution may petition to transfer residence credit but may not receive more than two such credits.

The candidate must write a thesis that represents creative work and contains original results in that area. The research is carried on independently by the candidate under the supervision of the chairperson of the special committee. By the time of the oral admission to candidacy examination, the candidate should have selected as chairperson of the committee the faculty member who will supervise the research. When the thesis is completed, the student presents his/her results at the thesis defense or B Exam. All doctoral students take a Final Examination (the B Exam, which is the oral defense of the dissertation) upon completion of all requirements for the degree, no earlier than one month before completion of the minimum registration requirement.

Masters Degree in the Minor Field

Ph.D. students in the field of mathematics may earn a Special Master's of Science in Computer Science. Interested students must apply to the Graduate School using a form available for this purpose. To be eligible for this degree, the student must have a member representing the minor field on the special committee and pass the A-exam in the major field. The rules and the specific requirements for each master's program are explained on the referenced page.

Cornell will award at most one master's degree to any student. In particular, a student awarded a master's degree in a minor field will not be eligible for a master's degree in the major field.

Graduate Student Funding

Funding commitments made at the time of admission to the Ph.D. program are typically for a period of five years. Support in the sixth year is available by application, as needed. Support in the seventh year is only available by request from an advisor, and dependent on the availability of teaching lines. Following a policy from the Cornell Graduate School, students who require more than seven years to complete their degree shall not be funded as teaching assistants after the 14th semester.

Special Requests

Students who have special requests should first discuss them with their Ph.D. advisor (or with a field member with whom they work, if they don't have an advisor yet). If the advisor (or field faculty) supports the request, then it should be sent to the Director of Graduate Studies.  

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Ph.D. Admissions

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The application window for Fall 2024 is closed.

The Mathematics Department offers two programs to obtain a Ph.D. Applicants can pursue a Ph.D. in  Applied & Interdisciplinary Mathematics or Mathematics.  Please use the "Programs" link at the left to explore our offerings.

• Three Letters of Recommendation  (May submit up to 5 letters, but only 3 are required.)
• Curriculum Vitae or Resume
• Academic Statement of Purpose (concise - no limit)
• Personal Statement (500 word limit)
• TOEFL iBT Special Home Edition and IELTS Indicator online test are accepted.
• Exemption rules:   https://rackham.umich.edu/admissions/applying/tests/
• List of  International English Exclusive Institutions  approved by Rackham

GRE General Test scores are no longer included in the admission process in accordance with a  policy of the Rackham Graduate School .

GRE Mathematics Subject Test scores are strictly optional. However, if an applicant chooses so, they may submit them as a combined pdf file with their transcript or personal statement.

Application Timeline

The Mathematics Department's graduate programs only accept applications for Fall semesters. 

General Requirements for Admission

A student must have completed a bachelor's degree at an accredited college or university by the time of entry in order to be considered for admission.

Applied & Interdisciplinary Mathematics (AIM) Ph.D. Admissions Requirements

Successful AIM Ph.D. applicants will demonstrate an interest in an interdisciplinary area of applied mathematics in addition to substantial mathematical ability. Two types of students are generally considered for admission to the AIM Ph.D. program:

• Mathematics majors with excellent grades in mathematics courses and excellent letters of recommendation. The admissions committee will also take into account other scholarly activities such as summer research experience, published papers, or courses in other fields.
• Non-mathematics majors from the physical, life, or engineering sciences, or from other appropriate areas of study. Such students are expected to have completed at least two upper division mathematics courses, and/or have substantial exposure to mathematics in other courses, and may submit a GRE mathematics subject test score. Other experience in working with mathematics (for instance, summer research positions) will also be taken into account, as well as grade point average and letters of recommendation.

Mathematics Ph.D.  Admission Requirements The undergraduate major need not be mathematics, but a student should have mastered material roughly equivalent to the undergraduate mathematics major at The University of Michigan including:

• three semesters of calculus
• one or two semesters of differential equations
• one semester courses in modern algebra, linear algebra, geometry or topology
• advanced calculus of one and several variables

In addition, a student should have completed at least three additional mathematics courses and at least two courses in related fields such as statistics, computer science, or the physical sciences. Students with strong records in less comprehensive programs will be considered for admission but if admitted should expect to spend the first one or two semesters in graduate school completing their undergraduate preparation in mathematics. Based on historical data, we expect that successful applicants to the Ph.D. program will have an overall GPA of at least 3.3 on a 4.0 scale.

Application Requirement Details

GRE, TOEFL, and IELTS Tests

• GRE General test scores are not required. 
• GRE Mathematics Subject Test scores may be submitted as a combined pdf file with your transcript or personal statement. (Optional)
• TOEFL or IELTS exam is required for students whose native language is not English
• TOEFL and IELTS exams should not be older than two years as of the Admission Deadline.
• Minimum TOEFL and IELTS scores must meet Rackham's requirements here .
• TOEFL or IELTS Exemptions are only given per Rackham's rules here .
• ETS school code for the University of Michigan Rackham Graduate School is 1839

Letters of Recommendation

Letters of recommendation play an especially crucial role in the admission process. At least three letters are required, and up to five may be submitted. Applicants should choose as recommenders people who know their strengths and weaknesses relevant to graduate study in mathematics. The most useful letters are those which list in some detail the accomplishments of the student and make direct comparisons with other students who have succeeded at major U.S. graduate schools. International students already in the U.S. should submit letters from their U.S. institution, whenever possible.  Please register your recommenders for the electronic Letters of Recommendation when using the Online Application.  Letters received after the application deadline will be accepted, but should be received within 1 week of that deadline.

Those students who will have completed a Master's degree in Mathematics by the time they begin studies at the University of Michigan must apply to the Ph.D. program. Others may apply to either program. 

Academic Statement of Purpose

Focus your academic statement on your mathematical interests, research experience, published papers, math camps, teaching & tutoring experience etc. Be sure to mention any specific faculty with whom you wish to work.

Personal Statement:

Focus your personal statement on what makes you unique, any struggles you have experienced and overcome, and why you feel U-M Math is the right place for you.  Be sure to include any hardships you have experiencedand how you overcame them. These could be financial, familial, or personal.

Transcript Submission:

The Mathematics Admissions Committee will review uploaded transcripts with university logos during the application process.  While these are considered "unofficial" transcripts because they have been opened from their original sealed envelopes, they are acceptable.  If an applicant receives an offer of admission, an official transcript in a sealed envelope will need to be mailed from the institution directly to the Rackham Graduate School.

Please submit your most current transcript with your online application by the due date.  If you would like the Admissions Committee to see your Fall term scores, you may email them to [email protected] after the due date, and they will be included with your application.

Additonal Application Materials: If you have additional materials you would like to submit with your application, you may email them to [email protected].  Be sure to include your name and umid number in the email and attach files in pdf format.

Note:  All credentials submitted for admission consideration become the property of the University of Michigan and will not be returned in original or copy form.

Additional Information:  Please visit the admissions page of the Rackham Graduate School for additional information regarding admission including: minimum graduate school requirements, residency, and application fees.    Unfortunately, application fee waivers are not available for international students.

Financial Support for Ph.D. Students

Ph.D. Programs

Most students enrolled in the Ph.D. program in Mathematics are granted full financial support including an annual stipend, tuition waiver, and health insurance for a period of five years, subject to satisfactory progress. The Department offers aid in the form of Graduate Student Instructorships, Research Assistantships, and Fellowships.

All entering Ph.D. students will be considered for Graduate Student Instructorships, which normally require four classroom hours of teaching per week plus additional office hours during the Fall and Winter terms. The stipend for such an appointment in 2021-2022 is $11,598 per term. In addition, Graduate Student Instructors receive a full tuition waiver. Teaching duties may involve teaching a section of a first-year calculus or pre-calculus course or serving as an instructor for recitation sections attached to a faculty lecture in multivariable calculus or elementary differential equations. The Department of Mathematics has many fellowship opportunities, including the Copeland, Glover, Rainich, and Shields Fellowships which may provide a stipend, tuition waiver and in some cases a reduced teaching load. Other fellowships administered by the Rackham Graduate School can be found at their  Fellowships office .  The University of Michigan is part of the CIC consortium, which also awards fellowships to outstanding underrepresented applicants. Also available are prestigious Rackham Science Award’s given out by the Rackham Graduate School.

After Admission

All new Graduate Student Instructors are required to attend an orientation and training program which is held the week before classes begin. New Graduate Student Instructors whose Undergraduate Degree is not from an English speaking University must pass an English Evaluation which tests the specific oral skills needed for classroom teaching and are required to attend a three-week cultural orientation program starting in July.

Research Assistantships are awarded mainly to senior Ph.D. students to relieve them of teaching duties during the final part of their dissertation research. Students at this point may also compete for Rackham Dissertation Fellowships, which provide full support for one year, or Research Partnerships. A small number of positions as paper-graders for the larger advanced courses is available each term.

Some additional funds are often available for support during the summer. More advanced students who are actively involved in research may be supported from NSF grants awarded to faculty members. For other students there is a limited number of Departmental fellowships and a few teaching positions are available. No advanced graduate courses are offered in either the Spring or Summer half-terms and students are encouraged to spend some of their summers attending workshops, doing research, working in government, or seeking internships in industry.

Marjorie Lee Browne (MLB) Scholars Program - an MS bridge to PhD program for diverse students

The Department of Mathematics at the University of Michigan is pleased to offer the Marjorie Lee Browne (MLB) Scholars Program. The program is named for Dr. Marjorie Lee Browne, who in 1949 became the first African-American woman to earn a Ph.D. in Mathematics at the University of Michigan. The MLB Scholars Program is an enhanced option for the M.S. degree in either Mathematics or Applied and Interdisciplinary Mathematics that is designed to give students professional knowledge of pure or applied mathematics in order to prepare them for continuing toward a Ph.D.  Please see this Marjorie Lee Brown Scholars webpage for eligibility and details.

If you have any questions regarding the application process, please contact the Department of Mathematics at [email protected].

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MS Advanced/Ph.D. Entrance Exams in Mathematics

This exam has three sections. There are nine questions, three from each section and students are expected to answer a total of six questions (two from each section).

i. Group Theory

Basic properties of groups, subgroups, normal subgroups, quotient groups, group homomorphisms, fundamental isomorphism theorems, cyclic groups, permutation groups, direct sums, automorphism groups, group actions, Sylow theorems and their applications, simple groups, solvable groups.

ii. Field and Galois Theory

Field extensions, splitting fields, cyclotomic fields, Galois group of a polynomial, fundamental theorem of Galois theory, applications of Galois theory, solvability by radicals.

iii. Ring and Module Theory

Basic properties of rings, subrings, ideals, quotient rings, ring homomorphisms, fundamental isomorphism theorems for rings, principal ideal domains, unique factorization domains, Euclidean domains, polynomial rings, direct sums of rings, Artinian and Noetherian rings, rings of fractions.

Basic properties of modules, submodules, quotient modules, module homomorphisms, fundamental isomorphism theorems for modules, modules of fractions (localization), length and composition series, simple modules, free and projective modules, direct sums and products of modules, chain conditions (Artinian and Noetherian modules), exact sequences of modules, and diagram chasing.

References: Abstract Algebra, 3rd edition, by David S. Dummit and Richard M. Foote (main text) Chapters 1-4, 6.1, 7-9, 10.1-10.3, 10.5, 13, 14, 15.1, 15.4 Algebra, 3rd edition, Graduate Texts in Mathematics, by Serge Lang (supplementary text) Basic Algebra I, 2nd edition, by Nathan Jacobson (supplementary text)

Examiners: Ela Celikbas , Olgur Celikbas

II. Real Analysis

This exam covers Lebesgue Measure and associated topics through the L p spaces. Questions from advanced calculus also form a significant part of the exam.

Advanced Calculus

• Concept of limit for sequences and functions
• Including sup/inf and limsup/liminf
• Axiom of real numbers, and its equivalent properties
• Sets and properties
• Open sets, closed sets, compact sets
• Continuity, uniform continuity, absolute continuity
• Continuous functions on bounded closed intervals
• Infinite series of numbers and functions
• Uniform convergence, pointwise convergence, power series
• Basic theory of differentiation and Riemann integration

Real Analysis

• Lebesgue measure and measurable functions
• Littlewood’s three principles
• The development of the Lebesgue integral
• Convergence theorems for integrals of sequences of functions
• Monotone functions and functions of bounded total variation
• Differentiation and absolute continuity, the fundamental theorem of calculus
• Contrasting Riemann and Lebesgue integration
• L p(E) spaces and their properties
• Minkowski and Holder inequalities, completeness, approximation by step functions and continuous functions

References: Elementary Analysis: The Theory of Calculus (2nd Ed) by Kenneth A. Ross Real Analysis (3rd Ed. Chap. 1-6 or 4th Ed. Chap. 1-8) by H.L. Royden and P.M. Fitzpatrick

Examiners: Adrian Tudorascu , Qingtian Zhang

III. Differential Equations

This exam covers intermediate ordinary differential equations at about the level of Brauer and Nohel. Below is a list of topics covered on the exam.

• Linear homogeneous systems with constant coefficients
• The general solution, the fundamental matrix, the matrix exponential, phase diagrams in 2-dimensions, stability of equilibrium solutions.
• General linear systems
• The fundamental matrix, nonhomogeneous systems.
• Linearization about critical points of nonlinear autonomous systems
• Phase plane portraits in 2-dimensions.
• Theory of Existence, uniqueness, and continuity
• Stability analysis by perturbation (or linearization) and Liapunov method
• Existence and stability of periodic solutions in 2-dimensions
• Poincare-Bendixson Theorem. Floquet theory.
• Bifurcation analysis and center, stable, unstable manifolds
• Saddle-node, transcritical, pitchfork.

References: Nonlinear Ordinary Differential Equations, Third Edition, by D. W. Jordan and P. Smith Oxford Applied Math and Computing Science Series (1999) Chapters 1, 2, 4, 8-12, and Appendix A. Qualitative Theory of Ordinary Differential Equations by F. Brauer and J. Nohel W. B. Benjamin, Inc, New York, (1969) Chapters 1-6 Differential Equations and Dynamical Systems by Lawrence Perko. Springer-Verlag Chapters 1, 2 (except2.13), 3 (3.1 to 3.9), 4 (4.1 to 4.4).

Examiners: Casian Pantea , Harumi Hattori

IV. Topology

• Methods of defining topological spaces
• Bases and subbases, local bases
• Continuous functions
• Various equivalent conditions for continuity
• Techniques of producing new spaces
• Subspaces, product spaces
• Separation axioms
• T0, T1, Hausdorff, regular, T3, completely regular, Tychonoff, normal, T4
• Compact spaces, connected spaces
• Statement and applications of Tychonoff theorem
• Metric spaces
• Completeness, compactness in the context of metric spaces

References: Topology (2nd edition) by J. Munkres Chapters I-7 General Topology by S. Willard Sections 2–8, 10, 13–17, 24 and 26 General Topology by J. Kelley All Chapters except 2, 6 and Appendix

Examiners:   Dylan Wilson , Jerzy Wojciechowski

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The PhD Oral Exam

The oral exam (also known as the PhD preliminary exam, as opposed to our prelims which is also known as the Graduate Preliminary Exam) offers an opportunity to determine whether a candidate is ready for and likely to succeed in a research career. In order to pass, the student needs to demonstrate three criteria: 

ability to master core material (relevant to planned research, and tested mostly by the major part of the exam);

ability to quickly pick up new material from an unrelated area (which is important if and when new developments necessitate such expansion; tested mostly by the minor part of the exam); and

ability to work productively with a potential advisor.

The Ph.D. Oral Exam Committee: This committee is appointed by the Graduate Group Chair after consultation with the student. It consists of three or more faculty members, at least two of whom are tenured members of the department. The syllabi for the exam are agreed upon by the committee and the student, with the approval of the Graduate Group Chair. The Graduate Group Chair notifies the mathematics faculty of the exam (date, time, place, committee members, reading lists). All faculty are to be explicitly invited, and each faculty member who attends the exam is a full voting member of the Exam Committee.

Scheduling the Ph.D. Oral Exam: To take this exam, the student should be in good standing. The student also should have taken at least six graduate mathematics courses (including those taken elsewhere). The student discusses in advance the topics, syllabi and the composition of the oral exam committee with the Graduate Group Chair, whose written approval is needed. The exam is to be taken by the end of the student's second academic year in the program. A one or two semester extension may be requested from the Graduate Group Chair in the case of students who took mostly 500-level courses in their first year.

If the Ph.D. Oral Exam is not passed on the first try, it may be taken just once more, and this must occur before the end of the following semester. Passing the Ph.D. Oral Exam on the second try, at the latest, is a requirement for remaining in the Ph.D. program.

With the approval of the oral exam committee and the graduate chair, the minor part of the exam can be substituted by a written thesis. In exceptional cases, as discussed below, it might be possible instead to substitute a thesis for the major part of the exam (but not for both parts). In either case, this thesis should be comparable, in quality, length and style, to a Masters thesis. It is typically expository in nature, but may also be a research paper or collection of papers. Committee members should approve such a substitution if they are convinced that writing it will adequately demonstrate that the student has accomplished the three criteria above. Thus, a thesis written in lieu of the minor should demonstrate the candidate’s ability to learn and assemble all ingredients needed for a particular problem or set of problems with which the student was not very familiar previously. A thesis written in lieu of the major should demonstrate broad understanding of the basics of the subject as well as ability to apply them to specific problems. But in this situation it is also crucial to establish the student’s ability to work productively and effectively with the potential advisor, so sufficient interactions between them must take place to establish this. Permission to substitute a thesis for the major part of the exam will therefore be given only in exceptional circumstances.

General Information

Thank you for your interest in the Applied Mathematics/Mathematics PhD Programs! The Math Department admits new graduate students to the Fall Semester only.  The graduate admissions application for Fall 2024 admission is available as of Wednesday, September 13th, 2023. The deadline to apply is Monday, December 11th, 2023, at 8:59pm Pacific Time.  Access the Graduate Division's Online Application . Please read the information on Graduate Division requirements and information required to complete the application. Copies and "unofficial" transcripts are accepted via the online application system if the name of your institution and your name are on the transcript. Please  do not  mail original transcripts for the review process. We require three letters of recommendation, which should be submitted online. We do not accept mailed letters of recommendation. Letters should be received online by the admissions deadline. To learn more about the program, here is a recording of a recent presentation we held at the 2022 Graduate Diversity Admissions Fair .

Admission to the Ph.D. Program

The Department of Mathematics offers 2 Ph.D. degrees, one in Mathematics and one in Applied Mathematics. Applicants for admission to either Ph.D. program are expected to have preparation comparable to the undergraduate major at Berkeley in Mathematics or in Applied Mathematics. These majors consist of 2 full years of lower-division work (covering calculus, linear algebra, differential equations, and multivariable calculus), followed by 8 one-semester courses including real analysis, complex analysis, abstract algebra, and linear algebra. These eight courses may include some mathematically based courses offered by other departments (e.g., Physics, Engineering, Computer Science, or Economics).

Applicants for admission are considered by the department's Graduate Admissions Committee. The number of students that can be admitted each year is determined by the Graduate Division. In making admissions decisions, the committee conducts a comprehensive review taking into consideration letters of recommendation, level of mathematics preparation, performance in courses, broader impacts (community contributions), research experience, etc.

A Note on the GRE Exams

Due to the COVID-19 pandemic, the  GRE exams will be optional for the Fall 2024 graduate admissions application.  You may still submit GRE scores; however, an absence of GRE scores will not negatively impact your application. For those who choose to take the General GRE and/or the Mathematics Subject GRE exams: to ensure timely reception of official score reports, we recommend you take both exams no later than October; official score reports are due by the admissions deadline. The Educational Testing Service will send your scores to the institutions you specify when you take the exams, please consult their website to see how long it takes for ETS to transmit score reports to institutions. Additional information about the GRE exams, and how to register, can be obtained on the web .

A Note on GPAs

If your undergraduate institution does not use a 4-point grading scale please do not convert your GPA to match the 4.0 scale. Instead, leave the section blank, or write in 0.00. The faculty review committee will be able to see your true academic performance from your uploaded unofficial transcripts. For the "Advanced GPA," our program does not require a GPA calculation spreadsheet.

International Students

All applicants from countries in which the official language is not English are required to submit official evidence of English language proficiency. Please refer to the Graduate Division's admissions page for information on the English language requirement .

For international applicants, we are pleased to offer a limited number of need based  fee  waivers each year to eligible applicants. The waiver is available in the program page of the online application and must be submitted by December 1st. 

The M.A. in Math

The Department of Mathematics no longer accepts applications for the Master's in Mathematics Program. If you are interested in graduate study at UC Berkeley, please apply for the Applied Mathematics or Mathematics Ph.D. program. The MA application is only available to current Ph.D. students at Berkeley interested in a simultaneous MA in Mathematics.

We expect to make final decisions by the end of February or mid-March.

Email Math Graduate Admissions

After reading the above material and our  FAQ  page, if you have further questions on admission to our program you can send an email to  [email protected] . 

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Mathematics - phd, admission requirements.

Terms and Deadlines

Degree and GPA Requirements

Additional Standards for Non-Native English Speakers

Additional standards for international applicants.

For the 2024-2025 academic year

See 2023-2024 requirements instead

Fall 2024 quarter (beginning in September)

Priority deadline: February 1, 2024

Final submission deadline: June 3, 2024

International submission deadline: May 6, 2024

Winter 2025 quarter (beginning in January)

Final submission deadline: November 15, 2024

International submission deadline: September 9, 2024

Priority deadline: Applications will be considered after the Priority deadline provided space is available.

Final submission deadline: Applicants cannot submit applications after the final submission deadline.

Degrees and GPA Requirements

Bachelors degree: All graduate applicants must hold an earned baccalaureate from a regionally accredited college or university or the recognized equivalent from an international institution.

Grade point average: The minimum undergraduate GPA for admission consideration for graduate study at the University of Denver is a cumulative 2.5 on a 4.0 scale or a 2.5 on a 4.0 scale for the last 60 semester credits or 90 quarter credits (approximately two years of work) for the baccalaureate degree. An earned master’s degree or higher from a regionally accredited institution supersedes the minimum standards for the baccalaureate. For applicants with graduate coursework but who have not earned a master’s degree or higher, the GPA from the graduate work may be used to meet the requirement. The minimum GPA is a cumulative 3.0 on a 4.0 scale for all graduate coursework undertaken.

Program GPA requirement: The minimum undergraduate GPA for admission consideration for this program is a cumulative 2.5 on a 4.0 scale

Official scores from the Test of English as a Foreign Language (TOEFL), International English Language Testing System (IELTS), C1 Advanced or Duolingo English Test are required of all graduate applicants, regardless of citizenship status, whose native language is not English or who have been educated in countries where English is not the native language. Your TOEFL/IELTS/C1 Advanced/Duolingo English Test scores are valid for two years from the test date.

The minimum TOEFL/IELTS/C1 Advanced/Duolingo English Test score requirements for this degree program are:

Minimum TOEFL Score (Internet-based test): 80

Minimum IELTS Score: 6.5

Minimum C1 Advanced Score: 176

Minimum Duolingo English Test Score: 115

English Conditional Acceptance Offered: No, this program does not offer English Conditional Admission.

Read the English Language Proficiency policy for more details.

Read the Required Tests for GTA Eligibility policy for more details.

Per Student & Exchange Visitor Program (SEVP) regulation, international applicants must meet all standards for admission before an I-20 or DS-2019 is issued, [per U.S. Federal Register: 8 CFR § 214.3(k)] or is academically eligible for admission and is admitted [per 22 C.F.R. §62]. Read the Additional Standards For International Applicants policy for more details.

Application Materials

Transcripts, standardized test scores, letters of recommendation.

Required Essays and Statements

We require a scanned copy of your transcripts from every college or university you have attended. Scanned copies must be clearly legible and sized to print on standard 8½-by-11-inch paper. Transcripts that do not show degrees awarded must also be accompanied by a scanned copy of the diploma or degree certificate. If your academic transcripts were issued in a language other than English, both the original documents and certified English translations are required.

Transcripts and proof of degree documents for postsecondary degrees earned from institutions outside of the United States will be released to a third-party international credential evaluator to assess U.S. education system equivalencies. Beginning July 2023, a non-refundable fee for this service will be required before the application is processed.

Upon admission to the University of Denver, official transcripts will be required from each institution attended.

GRE scores are optional for admission to this program. Applications submitted without scores will receive full consideration. Every application undergoes a comprehensive evaluation, including a careful review of all application materials. If you choose to submit test scores, you may upload your Test Taker Score Report PDF, which is considered unofficial. Official scores must be received directly from the appropriate testing agency upon admission to the University of Denver. The ETS institution code to submit GRE scores to the University of Denver is 4842.

University Standardized Test Policy

Three (3) letters of recommendation are required.  Letters should be submitted by recommenders through the online application.

Essays and Statements

Personal statement instructions.

A personal statement of at least 300 words is required. The statement should include information concerning your life, education, practical experience, special interests and specific purpose in applying to the University of Denver’s Department of Mathematics.

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Financial Aid Information

Start your application.

Your submitted materials will be reviewed once all materials and application fees have been received.

Our program can only consider your application for admission if our Office of Graduate Education has received all your online materials and supplemental materials by our application deadline.

Application Fee: $65.00 Application Fee

International Degree Evaluation Fee: $50.00 Evaluation Fee for degrees (bachelor's or higher) earned from institutions outside the United States.

Applicants should complete their Free Application for Federal Student Aid (FAFSA) by February 15. Visit the Office of Financial Aid for additional information.

IIT Bombay Department of Mathematics

The m.sc mathematics programme, ph.d admissions.

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Admission to Ph.D programme in Mathematics and Statistics

2016 ). With state-of-the art infrastructural facilities and a sound research base, IITB Mathematics offers a Ph.D. programme in a wide variety of areas. The program leading to the Ph.D. degree involves a course credit requirement and a research project leading to thesis submission. All that you need to know about admissions to our PhD program can be found in the information brochure for PhD admissions, which can be found at the following link. ----- A special welcome to all who wish to pursue a career in Mathematics and Statistics research. The Department of Mathematics, IITB offers a Ph.D. program in a wide variety of areas. To know more about the research interests of faculty members, please visit the page here . The program leading to the Ph.D. degree involves a course credit requirement, clearing of qualifier examinations and a research project leading to thesis submission. For more details, follow one of the links below

A special welcome to all who wish to pursue a career in Mathematics and Statistics research. The Department of Mathematics, IITB offers Ph.D. program in the areas of Mathematics and Statistics. Admission to the PhD program is based on a written test and interview. There are separate written tests and interviews for students in Mathematics and Statistics. The syllabus is given below. Students are required to choose one option specifying either Mathematics or Statistics.

To know more about the research interests of faculty members, please visit the page here . The program leading to the Ph.D. degree involves a course credit requirement, clearing of qualifier examinations and a research project leading to thesis submission. For more details, follow one of the links below

Syllabus for Entrance exam can be found here Past question papers can be found here

Syllabus for Mathematics Entrance Exam

Syllabus for Statistics Entrance Exam

Past Question Papers

PhD Admissions

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The Faculty of Mathematics offers three doctoral (PhD) and one MPhil research programmes.

Select a course below to visit the University’s Course Directory where you can read about the structure of the programmes, fees and maintenance costs, entry requirements and key deadlines.

Research Areas and Potential Supervisors

Determining whether your interests and ambitions align with our research and expertise is a vital part of the application and admissions process. When we receive your formal application, we will consider the information you provide on your research interests carefully, alongside other factors such as your academic suitability and potential, how you compare to other applicants in the field, and whether we have a suitable academic supervisor with the capacity to take on new students.

We are committed to widening participation in mathematical research at Cambridge. We welcome and encourage applications from people from groups underrepresented in postgraduate study.

Before making an application to study with us we recommend you:

• Investigate our areas of research and consider how they fit with your interests and ambitions.

A list of broad research areas is provided below, together with links to further information. Your interests may span more than one area. On your application form you will be asked to indicate at least one broad area of interest. This is to help us direct your application to the most suitable group of people to review it.

• Identify 2 or 3 appropriate supervisor(s) with whom you might work.

The information linked below will take you to lists of supervisors working in each broad research area, with an indication of their availability. You are encouraged to make informal contact with potential supervisors prior to making an application. Initial contact should be made by email. In your email we recommend you provide a concise explanation of your areas of interest, how your research interests align with the supervisor(s) research, and that you highlight any relevant work you have done in this area. We recommend that you attach an up-to-date CV. The purpose of this contact is to enquire on supervisor capacity and willingness to supervise, and to see if there is a good fit between your interests and theirs.

If you haven’t had a response to an informal enquiry, you are still welcome to apply and list the individual concerned on your application form, although you may also wish to consider other options.

• Give some thought to your intended research and why you want to study with us.

On your application form you will be asked to submit a short research summary, details of your research experience and your reasons for applying to undertake a PhD/MPhil with us. Whilst you are not expected to submit a detailed research proposal at any stage of the process, we do want to know that you have considered the areas of research that you wish to pursue.

Research areas

Click on a research area to find out more about available supervisors and their research:

Please note that a  large majority of the successful applicants for PhD studentships with  the High Energy Physics, and General Relativity & Cosmology (GR) groups   will have taken Part III of the Mathematical Tripos.

Funding Opportunities

Each Department works hard to secure funding for as many offer holders as possible, either from within its own funds, in collaboration with funding partners, or via the University Postgraduate Funding Competition. However, funding is not guaranteed via these routes, and you should investigate funding opportunities early in the process to be sure that you can meet advertised deadlines.

All application deadlines are 23:59pm (midnight) UK time on the stated date. So that your application can be given full consideration please apply by the following deadlines:

Note for PhD applicants:

We will accept applications for an October start up until the general University deadline in May, but your chances of obtaining funding are significantly reduced. In addition, space limitations may mean that late applications cannot be considered (i.e., the most appropriate supervisor may already have committed to taking other students).

Only in exceptional circumstances will we consider admission to a later start date in the academic year (i.e., January or April). If you intend to apply for a later start date please contact us at [email protected] so we can advise you on the feasibility of your plan.

Note for MPhil applicants:

We will accept applications until the general University deadline in February, but you will not be considered for funding. In addition, space limitations may mean that late applications cannot be considered (i.e., the most appropriate supervisor may already have committed to taking other students).

Most interviews are expected to take place in the second half of January.

The purpose of the interview is to try to ascertain the extent of the applicant's relevant knowledge and experience, and to gauge whether their interests and abilities align with the research of the potential supervisor and/or research group. It will most likely consist of a discussion of your background and motivations for applying to the course, as well as some questions on relevant topics.

Not all applicants will be selected for interview.

If you are selected for interview, you will be contacted by email at the address you provided on your application. The email should confirm:

• the location of the interview (it may be in-person or on-line dependent upon interviewer availability, your distance from Cambridge, as well as individual preferences),
• the interview format and whether you should prepare anything specific in advance,
• the approximate duration of the interview,
• who you will be meeting.

Prior to interview you may declare a disability, serious health problem or caring responsibility which may require reasonable adjustments for the interview to be made.

Due to interviewer availability and the tight admissions timetable, we can usually only rearrange the time and date of your interview under exceptional circumstances.

Decision timeline

Both DAMTP and DPMMS make most of their PhD/MPhil admissions decisions for October entry in January and early February, and you should not expect to receive a decision on your application before mid-February (even if you apply much earlier). We expect to have made decisions on all applications by mid-July. The Department makes every effort to take decisions on applications at the earliest opportunity. In some cases, however, it may take some time for a decision to be made. Applications may need to be viewed by several potential supervisors before a final decision can be reached.

To consider your application formally we must receive a complete application form, together with all supporting documents, by the deadline.

Communication of outcomes

You will be notified of the formal outcome of your application via the Applicant Portal.

Following an interview, you can normally expect to receive notification of the outcome within a week or two.

If you are successful, the University’s Postgraduate Admissions Office will issue a formal offer of admission which will outline all your conditions. As processing times can vary, we may also contact you informally to notify you of our decision.

We do not provide formal feedback to applicants who are unsuccessful at either the application or interview stage.

Take a look at our frequently asked questions for PhD applicants.

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PhD Written Entrance Exam

List of examination subjects for applicants to the phd program.

• The test for subject 0 (BMA) consists of 10 multiple choice questions. No explanation is needed in this part.
• Tests for subjects 1-11 consist of 4-5 questions. Each subject will be marked out of 100. Here you have to show your work for full mark – results without explanation may earn you very few marks.
• Here is a sample exam from 2018.
• Here is the detailed list of conditions of the exam together with topics for the subjects .
• While preparing for the exam, you may also find helpful the table containing the description of BSc courses .

The stucture of the written entrance exam

When applying for PhD studies, applicants must choose a wider area of studies in which they would want to pursue research. Note that under certain conditions, changes are possible later, during the studies. However, for the purpose of the entrance examination one such area should be chosen from the list below (see below ). Once the choice is made, the tests of the entrance examination should be chosen according to the following rules:

• Each applicant has to write the test from 0. (BMA) BASIC MATHEMATICS.
• Applicants have to write the mandatory test for the chosen area (subjects 1-7).
• Applicants have to choose one test marked as elective for the chosen area (subjects 1-11).
• Applicants should choose one test from the remaining subjects (not excluding the remaining elective subjects).
• Thus altogether each exam will consist of four different subjects.
• The tests will be sent electronically to the applicant and the scanned solutions should be sent back within 6 hours to the address given on the cover sheet of the exam. In order to assess your work accurately, please, make sure that your solution pages are numbered and each page contains your name and country plus the subject and the number of problem .
• In case the applicant is not sure about the chosen research topic, a secondary choice may be taken into account by choosing for free subject the mandatory subject of the second choice.

Areas and the corresponding mandatory and elective subjects for the entrance exam

PhD Mathematics: Admission, Top Colleges, Entrance Exams, Syllabus, Jobs and Salary

Doctor of Philosophy in Mathematics or PhD Mathematics is a 3-6 years doctorate course for candidates who want to pursue research in various domains of Mathematics.

In PhD Mathematics some research domains are Algebra, Geometry, Partial Differential Equations, Mathematical Biology, Mathematical Finance, Numerical Analysis and Scientific Computing, Topology and Differential Geometry etc.

The basic eligibility for PhD in Mathematics is Masters in Mathematics with minimum of 60% marks from a recognized university. As per latest UGC guidelines now a student can join PhD after four years Bachelor’s degree with minimum of 75% marks in aggregate or its equivalent grade.

Admission to PhD in Mathematics is either based on entrance or performance in various entrance exams namely CSIR NET, NBHM and GATE etc.

There are verities of job profiles available after completing PhD Mathematics such as Scientist, Professor or Lecturer, Researcher, finance companies etc. The average salary offered to candidates holding PhD Mathematics ranges between INR 3-20 Lakhs per annum in various sectors.

PhD Mathematics: Course Highlights

About phd mathematics.

PhD in Mathematics is a doctoral degree in Mathematics related domains that spans over a period of 3-6 years. It is a detailed academic study and research on the chosen topic.

Some topics for PhD in Mathematics are –

• Algebraic geometry
• Commutative Algebra
• Complex Analysis & Operator Theory
• Computational Acoustics and Electromagnetics
• Computational Fluid Dynamics
• Differential Equations
• Functional Analysis & Operator Theory
• Harmonic Analysis
• Homological Algebra
• Image Processing
• Mathematical Biology
• Number Theory & Arithmetic Geometry
• Numerical Analysis and Scientific Computing
• Operator Algebra
• Representation Theory
• Set Theory and Logic
• Several Complex Variables
• Topology and Geometry

PhD Mathematics Top Institutes

Some of the top institutes in India providing PhD in Mathematics are:

PhD Mathematics: Distance Education

PhD Mathematics distance education is not valid as per the UGC Guidelines. You can go for part time PhD if you are employed. There are many institutes including the top ones which offer part time PhD programs.

PhD Mathematics Syllabus

Student admitted into PhD in Mathematics course has to do coursework before starting their research works.

Some Common Papers in PhD Mathematics Coursework

• Research Methodology
• Research and Technical Communication
• Review of Literature in the Area of Research
• Departmental Elective (Depends on institutes)
• Research field related Subjects (As per your field of research)

After Coursework candidates has to do research work on their area of interest.

PhD Mathematics Admission Process

Most of the universities conduct their own PhD entrance exam or they shortlist candidates based on performance in entrance examinations namely CSIR NET, NBHM, GATE etc.

Shortlisted candidates have to appear for interview for PhD Mathematics selection.

Steps for Merit-Based Admission or Entrance Exempted Category:

• Colleges publish notice of admission on their website.
• Candidates satisfying the minimum eligibility criteria must visit the respective college website and fill the application form online.
• Colleges declare and publish merit lists based on cut-off marks in various National Exams like CSIR NET, NBHM, and GATE etc. on their website.
• Shortlisted candidates would have a personal round of interviews based on subject specialization.
• Candidates who clear the interview must visit the respective college and complete the necessary formalities for final admission.

Steps for Entrance Based Admission:

• The university/college publish dates of entrance examination, application form submission dates, dates of admit card download on their respective websites.
• Candidates satisfying minimum eligibility criteria must fill up the online application form of the entrance exam by sharing essential information followed by paying the requisite application fee.
• The exam conducting body publishes the list of successful candidates with their respective marks on the website.
• Successful candidates are called for interview for further selection process.
• After that they publish list for final admission.

Eligibility

Every institute/university may have different eligibility criteria, some common are

• Candidates who have completed their post-graduation in Mathematics or relevant subjects from a recognized university.
• Candidates with M.Phil. in Mathematics or related subjects are eligible to apply.
• Candidates must hold a minimum of 60% aggregate marks in the qualifying examination.

Every institute/university releases their eligibility criteria when the admission notification is released.

PhD Mathematics Entrance Exams

Some of the important PhD in Mathematics Entrance Exams are –

PhD Mathematics Admission Notifications

Latest PhD Mathematics Admission Notifications

Uttaranchal University PhD Admission 2023 (Winter): Dates, Application Form

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Check More – PhD Mathematics Admission Notifications

Career Options after PhD Mathematics

After PhD in Mathematics candidate can work as Scientist, Professor or Lecturer, Researcher, in finance companies.

You can consider the following profiles after pursuing a PhD in Mathematics:

• Scientist 
• Professor 
• Insurance Manager
• Finance Manager
• Scientific Data Entry Specialist
• Research & Development Manager

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Ph.D Mathematics: Course Details, Eligibility, Admission, Fees

Updated on - Jan 4, 2023

The PhD Mathematics is a three-to-five-year doctoral course that deals with the doctoral practice of mathematics in logical interpretation, analysis, theory, formulation, statistical, and topology. Graduates work as Mathematics Experts, Subject Matter Experts, Lecturers among many other job profiles.

PhD in Mathematics Course Details

About phd mathematics.

PhD Mathematics full form is Doctor of Philosophy in Mathematics, a speciality that focuses on the structure in mathematics and propagates new theories and laws and data for the development of mathematics. The course falls under the umbrella of  PhD.

According to Wikipedia, “Mathematics is an area of knowledge, which includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis).” 

Table of Contents

Eligibility Criteria for PhD Mathematics

• PhD Mathematics Admission Process

Who Should Pursue PhD Mathematics?

Types of phd mathematics.

• Popular PhD Mathematics Entrance Exams

Study PhD Mathematics in India

• Top PhD Mathematics Colleges
• Study PhD Abroad
• Fees Structure for PhD Mathematics
• Why Choose PhD Mathematics
• Syllabus and Subjects for PhD Mathematics

Preparation Tips for PhD Mathematics

• PhD Mathematics Salary

Career Options After PhD Mathematics

• Skills to Excel

Eligibility for a PhD Mathematics course in India varies by university. Candidates for admission to the PhD Mathematics course must have a 55% in their Master's degree in Mathematics or Physical Science and a bachelor's degree in Mathematics from a recognized university. Some universities hold entrance exams. There is no upper age limit for candidates for PhD Mathematics. The following are the general admission requirements for the course.

How To Get Admission For PhD Mathematics?

Candidates for PhD Mathematics admission must meet the university's eligibility requirements, which vary by course. Admissions procedures differ depending on the colleges to which applicants apply. The procedure, on the other hand, follows a predetermined pattern.

How to Apply?

The PhD Mathematics course duration is 3-5 years. Before applying for the PhD Program, aspirants may be required to submit a Letter Of Recommendation, a Statement Of Purpose, and other documents specified by the university. They can then apply for a doctoral degree at the university of their choice.

Selection Process

Admission is primarily based on merit, determined by the candidate's performance in state, national, or institutional entrance exams. Some institutes provide the second round of counselling consisting of a group discussion followed by a personal interview. After completing all of these rounds, a final merit list is created and released, confirming a candidate's admission to the institution.

PhD Mathematics should be pursued by students who are interested in the subject of Mathematics and want to pursue it at a research-level. The degree is a research-based degree, and hence the aspirants must be interested in doing research. In addition, students can opt for any suitable specialization that matches their interests to make a bright career in their preferred profession.

When To Do PhD Mathematics?

Students must pursue a PhD Mathematics after they have completed their Master’s degree. After the students have completed their M.Phil, M.Sc, or MA in Math or related subjects, the decision must be taken.

Aspirants can pursue the PhD in Mathematics course in multidisciplinary and multiple ways. Students can pursue it traditionally like a Full-time course, or they can pursue it part-time or distance learning.

Full Time PhD Mathematics

A full-time PhD course is a three to five-year-long course where the students will have to attend regular classes. The classes are conducted in person, and the students will have to attend college to finish these classes.

Part-Time PhD Mathematics

A part-time PhD Mathematics course is a doctorate course for students who can't commit to attending classes regularly. In this format, the students can study according to their schedules.

Popular Entrance Examination For PhD Mathematics

An entrance exam is required for candidates for the PhD Mathematics course. In addition to these exams, other universities may hold their own set of entrance exams for admission to the PhD Mathematics course. Among these are:

A Quick Glance At The PhD Mathematics Entrance Exam

The entrance exam is the only way to gain admission to prestigious universities. The top PhD Mathematics colleges in India hold entrance exams to determine the final merit list. The entrance exam is used to determine a candidate's merit for admission. The university entrance exam cut-off varies by university and is available online, along with the PhD Mathematics course details. The following is a summary of the PhD Mathematics entrance exam:

• Typically, the entrance exam is taken offline.
• The exam will take approximately 3 hours to complete.
• The questions are objective in nature.
• The purpose of the exam is to assess qualitative ability, data interpretation, and knowledge of the subject.

PhD Mathematics colleges exist in India in abundance for the students to choose from. Aspirants can apply to India's best PhD Mathematics colleges based on their specialization and merit. However, the admission process for these colleges remains remarkably consistent, and therefore, students must ensure that the students research them.

Top 10 PhD Mathematics Colleges

Numerous colleges in India offer PhD Mathematics. In addition to private institutions, India has a plethora of government-run PhD Mathematics colleges. Students can choose a college based on their academic standing and specialization needs. The following are the top ten PhD Mathematics Colleges in India:

Top PhD Mathematics Colleges in Karnataka

The table below shows the top PhD Mathematics colleges in Karnataka:

Top PhD Mathematics Colleges in West Bengal

The table below shows the top PhD Mathematics colleges in West Bengal:

Top PhD Mathematics Colleges in Maharashtra

The table below shows the top PhD Mathematics colleges in Maharashtra:

Top PhD Mathematics Colleges in Orissa

The table below shows the top PhD Mathematics colleges in Orissa:

Study PhD Mathematics Abroad

Aspirants can choose to pursue a PhD Mathematics course abroad if they so choose. The PhD Mathematics fees abroad will vary depending on the location, university and ranking. Below, are some of the top universities for PhD Mathematics from various countries abroad:

Top PhD Mathematics Colleges in the USA

The table below shows the top PhD Mathematics colleges in the USA:

Top PhD Mathematics Colleges in the UK

The table below shows the top PhD Mathematics colleges in the UK:

Top PhD Mathematics Colleges in Canada

The table below shows the top PhD Mathematics colleges in Canada:

Top PhD Mathematics Colleges in Australia

The table below shows the top M.Phil colleges in Australia:

Fee Structure For PhD Mathematics

In India, there are approximately 5,000 PhD Mathematics colleges. In India, the average fee structure for PhD Mathematics is 50,000 - 1.9 LPA. The course fee for a PhD Mathematics varies across the country, depending on the infrastructure and faculty availability. Government colleges in India charge lower PhD mathematics fees than private colleges. The following are some of the top PhD Mathematics Colleges and PhD Mathematics course fees:

Syllabus and Subjects For PhD Mathematics 

The PhD Mathematics course lasts 3-5 years and can be completed semester by semester or annually, depending on the university. The PhD Mathematics syllabus prepares students to formulate, research, and find new mathematical patterns. PhD Mathematics is a doctoral program that prepares students for careers in mathematics through research. The PhD Mathematics syllabus is intended to place a strong emphasis on:

• Differential Equation
• Mathematical Finance
• Differential Geometry Mechanics

Read More: PhD Mathematics Syllabus and Subjects

Why Choose PhD Mathematics?

There are numerous opportunities available in research institutes and universities for candidates interested in a research career. And for those interested in teaching, there are numerous well-paying teaching positions available in private engineering institutions. To assist aspirants in understanding the answers to these questions, we have created the following three pointers:

What is PhD Mathematics All About?

PhD Mathematics is the doctoral program in which students are exposed to mathematical logic and analysis, statistical, topology, and stochastic processes, emphasising the practical. A PhD in Mathematics opens the door to a plethora of career opportunities in the corporate and other sectors of the economy, not just in education.

What Does a PhD Mathematics Graduate Do?

Mathematics is a highly sought-after and difficult specialization in subject matter that necessitates a high level of expertise. PhD in Mathematics teaches students how to research, formulate research methodologies and find solutions.

Accountants:  Accountants are professionals in charge of administrative tasks such as creating forms and the upkeep of schedules. Accounting firms prepare financial statements and tax returns for clients and represent them in financial matters.

Reasons Why PhD Mathematics Can Fetch You a Rewarding Career?

PhD Mathematics are affiliated researchers who find solutions to the existing mathematical questions or try to find the next to the best solution to the question.

Research Skills Development:  The syllabus prepares students to keep up with the expanding frontiers of knowledge and to conduct research relevant to the country's current social and economic goals.

Opportunities to diverse sectors:  Mathematicians can find work in various fields, including banking, the stock exchange, defence, etc. 

Pursuit of Excellence:  Mathematics is a pursuit of excellence, and mathematicians researchers work more to understand the complexities of mathematics and create more solving patterns through research.

Read More: PhD Mathematics Job Opportunities and Scope

PhD Mathematics Course Comparison

PhD Mathematics is a doctoral degree with a prime focus on Mathematics as a discipline and field of study. Based on their preferences, aspirants can pursue a Doctor of Philosophy or PhD course and carve their paths as researchers and field specialists. Below is the comparison of PhD Mathematics with other PhD Courses.

PhD Mathematics vs PhD Sociology

The table below is a comparison between PhD Mathematics and PhD Sociology:

Read More: PhD Sociology

The following are some preparation tips for students to complete the course:

Consult a Syllabus Advisor: Examine your comprehension of the syllabus and, if necessary, seek assistance from seniors or advisors.

Allow One Month For Planning: Allow a month for proper preparation. Make a list of what you need to study and practice before you begin working.

Make a Schedule: Finish your preparations ahead of time to allow for content revision.

Salary for PhD Mathematics

The average PhD Mathematics salary in India for freshers is INR 2.5 LPA, according to Glassdoor. The type of practice, performance, expertise, and other relevant factors influence salary and annual earnings.

Read More : PhD Mathematics Salary

A graduate of the PhD Mathematics program has numerous career options. PhD Mathematics graduates primarily work in government and international organizations. They are sought in a variety of fields, including:

• Academic Institutes
• Banking Sector
• Stock Exchanges
• Math Research Centers
• Military Schools
• Defense Services
• Trade & Transportation Centers

Skills Required For PhD Mathematics

An aspirant who wishes to pursue a PhD Mathematics must have the necessary skills to progress in the course. The course is practical and requires prior knowledge. Here is a list of a few skills an applicant should have:

• Researching
• Mathematical skills
• Interpretation skills
• Qualitative Understanding
• Quantitative Data Interpretation

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PhD in Mathematics

Minimum Eligibility

A minimum of 17 years of education with a Master's degree in Mathematics/Computer Science/Statistics/Operations Research/Physics or an equivalent degree with 55% marks in the aggregate or equivalent grade.

Entrance Test and Interview. A minimum of 50% marks will have to be secured separately, in both the Entrance Test and the Interview, in order to be eligible for admission. The weightage of the Entrance Test will be 50% and the weightage of the Interview will be 50%.

Format of the Entrance Test Paper

• The duration of the Entrance Test will be 2 hours.
• The question paper will consist of 70 multiple choice questions.
• There will be no negative marking.
• Candidates up to five times the number of seats available will be short-listed for interview on the basis of their performance in the Entrance Test, subject to a minimum cut off.
• A final merit list will be drawn by adding together the marks of the Entrance Test and the interview.
• Calculators are not allowed. However, Log Tables may be used.
• The questions may be asked in the Entrance Test in the following areas:

Analysis:  Real functions; limit, continuity, differentiability; sequences; series; uniform convergence; functions of complex variables; analytic functions, complex integration; singularities, power and Laurent series; metric spaces; stereographic projection; topology, compactness, connectedness; normed linear spaces, inner product spaces; dual spaces, linear operators; Lebesgue measure and integration; convergence theorems.

Algebra: Basic theory of matrices and determinants; eigenvalues and eigenvectors; Groups and their elementary properties; subgroups, normal subgroups, cyclic groups, permutation groups; Lagrange's theorem; quotient groups, homomorphism of groups; Cauchy Theorem and p-groups; the structure of groups; Sylow's theorems and their applications; rings, integral domains and fields; ring homomorphism and ideals; polynomial rings and irreducibility criteria; vector space, vector subspace, linear independence of vectors, basis and dimensions of a vector space, inner product spaces, orthonormal basis; Gram-Schmidt process, linear transformations.

Differential Equations:  First order ordinary differential equations (ODEs); solution of first order initial value problems; singular solution of first order ODEs; system of linear first order ODEs; method of solution of dx/P=dy/Q=dz/R; orthogonal trajectory; solution of Pfaffian differential equations in three variables; linear second order ODEs; Sturm-Liouville problems; Laplace transformation of ODEs; series solutions; Cauchy problem for first order partial differential equations (PDEs); method of characteristics; second order linear PDEs in two variables and their classification; separation of variables; solution of Laplace, wave and diffusion equations; Fourier transform and Laplace transform of PDEs.

Numerical Analysis:  Numerical solution of algebraic and transcendental equations; direct and iterative methods for system of linear equations; matrix eigenvalue problems; interpolation and approximations; numerical differentiation and integration; composite numerical integration; double numerical integration; numerical solution for initial value problems; finite difference and finite element methods for boundary value problems.

Probability and Statistics:  Axiomatic approach of probability; random variables; expectation, moments generating functions, density and distribution functions; conditional expectation.

Linear Programming:  Linear programming problem and its formulation; graphical method, simplex method; artificial starting solution; sensitivity analysis; duality and post-optimality analysis.

Preference will be given to those opting for the research in the following areas:

• Numerical Methods
• Computational Fluid Dynamics
• Graph Theory
• Function Spaces
• Fourier Analysis
• Parallel Algorithms
• Scientific Computing
• Optimization Techniques
• Swarm Intelligence
• Mathematical Biology
• Nonlinear Dynamical System
• Bio-informatics
• Optimization
• Graph theory
• Differential Equations
• Numerical Analysis
• Mathematical Modeling

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PhD in Mathematics Course, Admission 2024, Eligibility, Syllabus, Fees, Career

PhD mathematics is a doctoral degree programme of a minimum of three years of duration and maximum of six years that deals with various mathematics concepts such as the study of space, structure, change, and quantity. The mathematics degree programme comes to its completion after the students submit their ‘thesis’ and get it approved by their supervisor.

Highlights - PhD Mathematics

Phd in mathematics eligibility criteria, phd mathematics admission process, top ph.d in mathematics entrance examinations, phd mathematics cutoff, skills required for ph.d in mathematics, syllabus of ph.d in mathematics, ph.d in mathematics fee structure, scope of ph.d mathematics, career options after ph.d in mathematics, benefits of studying phd mathematics, salary after phd in mathematics, list of top 10 ph.d. mathematics colleges in india with fees, top private ph.d mathematics colleges in india with fees, top government ph.d in mathematics colleges in india with fees.

The eligibility criteria to pursue a Phd in Mathematics depends on the institute, but generally candidates require a Bachelor’s degree in Mathematics or any other related discipline. Many career options are available after graduating in a Ph.D Mathematics course such as Mathematician, Statistician and Mathematics Lecturer.

Students applying for the Ph.D in Mathematics course must know the top colleges offering Ph.D in Mathematics including Acharya Institute of Technology, Aditanar College of Arts and Science, and Agra College. Admission depends on the institute, generally students need to appear for entrance exams such as UGC NET to pursue Ph.D in their desired field.

Generally, the minimum duration for this programme is three years but students can take up to a maximum of six years to complete the course. Candidates with a master’s degree in mathematics (MSc mathematics), physical sciences or M.Tech candidates can choose to pursue a PhD in mathematics. After its completion, candidates can find various options of employment in various sectors.

The eligibility criteria for PhD mathematics is quite subjective and depends on the chosen institute. Tier-1 institutes generally require students to have strong academic backgrounds as well as some work experience in teaching or research. Students are required to meet the eligibility criteria and ensure their admission. Some of the most common eligibility requirements for PhD mathematics are mentioned below:

• Candidates must have completed their Master’s degree in mathematics or physical sciences or a related field. Some of the popular master’s degree programs are MSc statistics, MSc mathematics, MSc chemistry, MA Economics, or MTech.
• Candidates must have obtained a minimum of 60 per cent aggregate score in class 12, bachelor’s, and master’s degree programmes.
• Candidates must appear for the relevant entrance exam such as UGC NET and score a valid cut-off.

The admission process for PhD mathematics is quite simple and follows certain steps that students need to complete. The PhD mathematics admission process is both merit-based as well as entrance based. Eligible candidates must carefully follow the complete admission process. Candidates can find the complete step-by-step information on admission procedure for PhD mathematics mentioned below:

Merit based admission:

• Many institutes offer admission to the Ph.D in Mathematics course, the admission to the programme is based on the score of the previous course of study. Some universities may also conduct personal interviews, group discussions or aptitude tests.
• Many institutes also offer admission to Ph.D. mathematics degree programmes based on their scores of previous courses of study. In addition to this, these universities might conduct a personal interview to screen the most deserving candidates.

Entrance Exam Based:

• Many institutes also offer the admission based on the entrance examination, one of the popular entrance exams for Ph.D degree is UGC NET, some universities may conduct their own entrance exams.
• The universities offering the Ph.D in Mathematics course provide admissions on the basis of entrance exams. The exam is conducted on the basis of mathematics subject and includes Multiple Choice Questions from different topics from Mathematics.
• Many institutes also offer mathematics degree programmes and generally provide admission based on entrance exams. The entrance examination is conducted on the mathematics subject and includes multiple choice questions from various chapters such as advanced algebra, discrete mathematics, analysis, and more.
• The universities offering Ph.D. mathematics degree programmes shortlist candidates based on their scores in the entrance examinations.
• One of the popular entrance examinations for admission to top PhD mathematics universities are UGC NET.

Various colleges in India provide admission to PhD mathematics degree programmes only through an entrance examination. Not many universities conduct their own entrance test and rely on central or state-level entrance tests. Mentioned below is the entrance examination accepted by most of the universities.

UGC NET : UGC-NET exam is conducted by the National Testing Agency (NTA) for admission to top Ph.D. degree programmes across the country. Candidates who qualify for the exam with high scores are offered junior fellowship awards (JFA) and assistant professorships across various universities in India. The relevant UGC-NET information is listed below:

• UGC NET Admit Card
• UGC NET Syllabus
• UGC NET Exam Pattern
• UGC NET Result
• UGC NET Cutoff
• UGC NET Preparation Tips
• UGC NET Eligibility
• UGC NET Application

The cutoff for PhD mathematics is different for every university since every university has a different number of seats available and different levels of popularity among students. The cutoff is defined for a course and a university and the minimum score that students need to obtain in their recent qualifying exam.

Candidates interested in pursuing Ph.D in Mathematics must have excellent research skills, and interest and aptitude for Mathematics and its application. Apart from these skills, candidates need many skills that will help the candidate perform well both academically and professionally. Mentioned below are the skills required for a Phd in Mathematics.

• Problem-Solving
• Good Communication
• Good Observation skills
• Quantitative Aptitude

The syllabus for PhD mathematics is subjective in nature and depends on the desired institute. PhD mathematics syllabus comprises both theoretical as well as practical subjects and a final thesis. Candidates can find the complete PhD mathematics syllabus according to the year and specialisations mentioned in the table below is the MDU Rohtak curriculum of Ph.D in Mathematics.

The fee structure for PhD mathematics is variable and depends on various factors such as on the type of institute, location of the company and the status of the institute.Candidates’ academic scores also play a huge role in deciding the fee. The average fees of a Ph.D Mathematics course ranges from Rs. 2,480 to Rs. 3.97 Lakhs.

The scope of PhD mathematics is quite huge and there are ample opportunities in both private as well as government sectors. After the completion of PhD mathematics degree programmes, candidates are offered with lucrative job opportunities in the academic sector as well as in corporate, depending on the experience of the graduate.

The employment opportunities after completing PhD mathematics are huge and various sectors are open for PhD graduates to choose from. In the case of academics, candidates can choose to work as an assistant professor or professor in a private or government college within their specialisation.

Economist : An economist is a professional who studies different economic equations and analyses the consumption and production of a particular state in order to decide their economic strengths and weaknesses. After a certain result is derived, an economist comes with better solutions and equations for how to develop better economic trades.

Statistician : A statistician is a professional who is responsible for analysing and studying large amounts of data such as bar graphs, histograms, and others to make economic, strategic, or operational decisions. A statistician derives the concepts from a huge amount of data and delivers it to the clients so that the final decision can be made.

Scientist : A scientist is a professional who is responsible for conducting various research and experiments around mathematics and related fields to derive new equations and technical solutions. He or she works under both theoretical and practical mathematics.

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PhD mathematics is one of the highly regarded and sought after degree programmes when it comes to doctoral studies. The degree programme not only offers a thorough understanding of one of the most complex subjects but also a wide range of career options to choose from. Students pursuing PhD mathematics work in various mathematical fields such as mathematical physics, dynamical systems, and numerical analysis.

The employment opportunities are also strong in the case of PhD mathematics. Candidates can find senior profile employment in the banks, colleges, investment banking, Indian civil service, commerce industry, inventory management, business industry, research and development firms, insurance agencies, and inventory management.

The salary after completion of a Ph.D. in Mathematics depends on various factors, such as the location of the company, type of company, skills, and experience of the candidate, and the job profile applied by the candidates. Mentioned in the table below are the average salaries of some of the top career options in the field of Ph.D. in Mathematics.

Source: AmbitionBox

There are various PhD mathematics degree colleges in India and the top PhD mathematics colleges in India are the ones with quality education, research proposals, and good industrial exposure. Candidates applying to these colleges must meet the eligibility requirements and carefully follow the admission process. Some of the top PhD Mathematics colleges in India are mentioned below:

Private colleges in India offering admission to top PhD mathematics degree programmes are gaining a lot of popularity among students due to attractive fellowship awards, and research scope offered by them. The top private PhD Mathematics colleges in India are mentioned below:

The government colleges offering admission to PhD mathematics degree programmes are top in students’ list when it comes to PhD. The government colleges not only offer fellowships and scholarships but also offer a lot of research facilities in the PhD mathematics field. Some of the top government PhD mathematics colleges in India are mentioned below:

Ph.D in Mathematics offers the students with various career options and wide scope, students after completing their Ph.D Mathematics course work can join both private and government sectors and make a handsome salary. Some of the top career options include Statistician, Economist and Scientist.

Frequently Asked Question (FAQs)

PhD mathematics is a doctoral degree programme which deals with various mathematics concepts. 

The minimum duration of getting a PhD in mathematics is three years whereas the maximum is six years.

Yes, after the completion of PhD mathematics degree programmes, candidates are offered with lucrative job opportunities in the academic sector as well as in corporate roles. 

In the government sector, the average salary for a professor or scientist is around Rs. 80,000 to Rs. 95,000 per month. In the private sector, the salary bar ranges from Rs. 65,000 to Rs. 1.5 lakh per month and depends on the job profile.

Some of the popular job profiles are statistician, economist, demographer, scientist, research associate, and more.

How much does it cost to do a PhD mathematics?

The eligibility criteria for a PhD mathematics degree programme is that students must have completed a master’s degree in mathematics.

Ernst & Young, Tiger Analytics, TCS Innovation Lab, and Weizmann Institute are some of the companies that hire PhD in mathematics graduates.

UGC-NET, and GATE are some of the top entrance examinations for admission to top PhD mathematics degree programmes. 

The PhD in mathematics is generally considered a tough degree programme. Some of the core subjects of PhD in mathematics are mathematical physics, dynamical systems, biomathematics group, numerical analysis, fluid dynamics, and more.

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Data Administrator

Database professionals use software to store and organise data such as financial information, and customer shipping records. Individuals who opt for a career as data administrators ensure that data is available for users and secured from unauthorised sales. DB administrators may work in various types of industries. It may involve computer systems design, service firms, insurance companies, banks and hospitals.

Bio Medical Engineer

The field of biomedical engineering opens up a universe of expert chances. An Individual in the biomedical engineering career path work in the field of engineering as well as medicine, in order to find out solutions to common problems of the two fields. The biomedical engineering job opportunities are to collaborate with doctors and researchers to develop medical systems, equipment, or devices that can solve clinical problems. Here we will be discussing jobs after biomedical engineering, how to get a job in biomedical engineering, biomedical engineering scope, and salary. 

Ethical Hacker

A career as ethical hacker involves various challenges and provides lucrative opportunities in the digital era where every giant business and startup owns its cyberspace on the world wide web. Individuals in the ethical hacker career path try to find the vulnerabilities in the cyber system to get its authority. If he or she succeeds in it then he or she gets its illegal authority. Individuals in the ethical hacker career path then steal information or delete the file that could affect the business, functioning, or services of the organization.

GIS officer work on various GIS software to conduct a study and gather spatial and non-spatial information. GIS experts update the GIS data and maintain it. The databases include aerial or satellite imagery, latitudinal and longitudinal coordinates, and manually digitized images of maps. In a career as GIS expert, one is responsible for creating online and mobile maps.

Data Analyst

The invention of the database has given fresh breath to the people involved in the data analytics career path. Analysis refers to splitting up a whole into its individual components for individual analysis. Data analysis is a method through which raw data are processed and transformed into information that would be beneficial for user strategic thinking.

Data are collected and examined to respond to questions, evaluate hypotheses or contradict theories. It is a tool for analyzing, transforming, modeling, and arranging data with useful knowledge, to assist in decision-making and methods, encompassing various strategies, and is used in different fields of business, research, and social science.

Geothermal Engineer

Individuals who opt for a career as geothermal engineers are the professionals involved in the processing of geothermal energy. The responsibilities of geothermal engineers may vary depending on the workplace location. Those who work in fields design facilities to process and distribute geothermal energy. They oversee the functioning of machinery used in the field.

Database Architect

If you are intrigued by the programming world and are interested in developing communications networks then a career as database architect may be a good option for you. Data architect roles and responsibilities include building design models for data communication networks. Wide Area Networks (WANs), local area networks (LANs), and intranets are included in the database networks. It is expected that database architects will have in-depth knowledge of a company's business to develop a network to fulfil the requirements of the organisation. Stay tuned as we look at the larger picture and give you more information on what is db architecture, why you should pursue database architecture, what to expect from such a degree and what your job opportunities will be after graduation. Here, we will be discussing how to become a data architect. Students can visit NIT Trichy , IIT Kharagpur , JMI New Delhi . 

Remote Sensing Technician

Individuals who opt for a career as a remote sensing technician possess unique personalities. Remote sensing analysts seem to be rational human beings, they are strong, independent, persistent, sincere, realistic and resourceful. Some of them are analytical as well, which means they are intelligent, introspective and inquisitive. 

Remote sensing scientists use remote sensing technology to support scientists in fields such as community planning, flight planning or the management of natural resources. Analysing data collected from aircraft, satellites or ground-based platforms using statistical analysis software, image analysis software or Geographic Information Systems (GIS) is a significant part of their work. Do you want to learn how to become remote sensing technician? There's no need to be concerned; we've devised a simple remote sensing technician career path for you. Scroll through the pages and read.

Budget Analyst

Budget analysis, in a nutshell, entails thoroughly analyzing the details of a financial budget. The budget analysis aims to better understand and manage revenue. Budget analysts assist in the achievement of financial targets, the preservation of profitability, and the pursuit of long-term growth for a business. Budget analysts generally have a bachelor's degree in accounting, finance, economics, or a closely related field. Knowledge of Financial Management is of prime importance in this career.

Underwriter

An underwriter is a person who assesses and evaluates the risk of insurance in his or her field like mortgage, loan, health policy, investment, and so on and so forth. The underwriter career path does involve risks as analysing the risks means finding out if there is a way for the insurance underwriter jobs to recover the money from its clients. If the risk turns out to be too much for the company then in the future it is an underwriter who will be held accountable for it. Therefore, one must carry out his or her job with a lot of attention and diligence.

Finance Executive

Product manager.

A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.  

Operations Manager

Individuals in the operations manager jobs are responsible for ensuring the efficiency of each department to acquire its optimal goal. They plan the use of resources and distribution of materials. The operations manager's job description includes managing budgets, negotiating contracts, and performing administrative tasks.

Stock Analyst

Individuals who opt for a career as a stock analyst examine the company's investments makes decisions and keep track of financial securities. The nature of such investments will differ from one business to the next. Individuals in the stock analyst career use data mining to forecast a company's profits and revenues, advise clients on whether to buy or sell, participate in seminars, and discussing financial matters with executives and evaluate annual reports.

A Researcher is a professional who is responsible for collecting data and information by reviewing the literature and conducting experiments and surveys. He or she uses various methodological processes to provide accurate data and information that is utilised by academicians and other industry professionals. Here, we will discuss what is a researcher, the researcher's salary, types of researchers.

Welding Engineer

Welding Engineer Job Description: A Welding Engineer work involves managing welding projects and supervising welding teams. He or she is responsible for reviewing welding procedures, processes and documentation. A career as Welding Engineer involves conducting failure analyses and causes on welding issues. 

Transportation Planner

A career as Transportation Planner requires technical application of science and technology in engineering, particularly the concepts, equipment and technologies involved in the production of products and services. In fields like land use, infrastructure review, ecological standards and street design, he or she considers issues of health, environment and performance. A Transportation Planner assigns resources for implementing and designing programmes. He or she is responsible for assessing needs, preparing plans and forecasts and compliance with regulations.

Environmental Engineer

Individuals who opt for a career as an environmental engineer are construction professionals who utilise the skills and knowledge of biology, soil science, chemistry and the concept of engineering to design and develop projects that serve as solutions to various environmental problems. 

Safety Manager

A Safety Manager is a professional responsible for employee’s safety at work. He or she plans, implements and oversees the company’s employee safety. A Safety Manager ensures compliance and adherence to Occupational Health and Safety (OHS) guidelines.

Conservation Architect

A Conservation Architect is a professional responsible for conserving and restoring buildings or monuments having a historic value. He or she applies techniques to document and stabilise the object’s state without any further damage. A Conservation Architect restores the monuments and heritage buildings to bring them back to their original state.

Structural Engineer

A Structural Engineer designs buildings, bridges, and other related structures. He or she analyzes the structures and makes sure the structures are strong enough to be used by the people. A career as a Structural Engineer requires working in the construction process. It comes under the civil engineering discipline. A Structure Engineer creates structural models with the help of computer-aided design software. 

Highway Engineer

Highway Engineer Job Description:  A Highway Engineer is a civil engineer who specialises in planning and building thousands of miles of roads that support connectivity and allow transportation across the country. He or she ensures that traffic management schemes are effectively planned concerning economic sustainability and successful implementation.

Field Surveyor

Are you searching for a Field Surveyor Job Description? A Field Surveyor is a professional responsible for conducting field surveys for various places or geographical conditions. He or she collects the required data and information as per the instructions given by senior officials. 

Orthotist and Prosthetist

Orthotists and Prosthetists are professionals who provide aid to patients with disabilities. They fix them to artificial limbs (prosthetics) and help them to regain stability. There are times when people lose their limbs in an accident. In some other occasions, they are born without a limb or orthopaedic impairment. Orthotists and prosthetists play a crucial role in their lives with fixing them to assistive devices and provide mobility.

Pathologist

A career in pathology in India is filled with several responsibilities as it is a medical branch and affects human lives. The demand for pathologists has been increasing over the past few years as people are getting more aware of different diseases. Not only that, but an increase in population and lifestyle changes have also contributed to the increase in a pathologist’s demand. The pathology careers provide an extremely huge number of opportunities and if you want to be a part of the medical field you can consider being a pathologist. If you want to know more about a career in pathology in India then continue reading this article.

Veterinary Doctor

Speech therapist, gynaecologist.

Gynaecology can be defined as the study of the female body. The job outlook for gynaecology is excellent since there is evergreen demand for one because of their responsibility of dealing with not only women’s health but also fertility and pregnancy issues. Although most women prefer to have a women obstetrician gynaecologist as their doctor, men also explore a career as a gynaecologist and there are ample amounts of male doctors in the field who are gynaecologists and aid women during delivery and childbirth. 

Audiologist

The audiologist career involves audiology professionals who are responsible to treat hearing loss and proactively preventing the relevant damage. Individuals who opt for a career as an audiologist use various testing strategies with the aim to determine if someone has a normal sensitivity to sounds or not. After the identification of hearing loss, a hearing doctor is required to determine which sections of the hearing are affected, to what extent they are affected, and where the wound causing the hearing loss is found. As soon as the hearing loss is identified, the patients are provided with recommendations for interventions and rehabilitation such as hearing aids, cochlear implants, and appropriate medical referrals. While audiology is a branch of science that studies and researches hearing, balance, and related disorders.

An oncologist is a specialised doctor responsible for providing medical care to patients diagnosed with cancer. He or she uses several therapies to control the cancer and its effect on the human body such as chemotherapy, immunotherapy, radiation therapy and biopsy. An oncologist designs a treatment plan based on a pathology report after diagnosing the type of cancer and where it is spreading inside the body.

Are you searching for an ‘Anatomist job description’? An Anatomist is a research professional who applies the laws of biological science to determine the ability of bodies of various living organisms including animals and humans to regenerate the damaged or destroyed organs. If you want to know what does an anatomist do, then read the entire article, where we will answer all your questions.

For an individual who opts for a career as an actor, the primary responsibility is to completely speak to the character he or she is playing and to persuade the crowd that the character is genuine by connecting with them and bringing them into the story. This applies to significant roles and littler parts, as all roles join to make an effective creation. Here in this article, we will discuss how to become an actor in India, actor exams, actor salary in India, and actor jobs. 

Individuals who opt for a career as acrobats create and direct original routines for themselves, in addition to developing interpretations of existing routines. The work of circus acrobats can be seen in a variety of performance settings, including circus, reality shows, sports events like the Olympics, movies and commercials. Individuals who opt for a career as acrobats must be prepared to face rejections and intermittent periods of work. The creativity of acrobats may extend to other aspects of the performance. For example, acrobats in the circus may work with gym trainers, celebrities or collaborate with other professionals to enhance such performance elements as costume and or maybe at the teaching end of the career.

Video Game Designer

Career as a video game designer is filled with excitement as well as responsibilities. A video game designer is someone who is involved in the process of creating a game from day one. He or she is responsible for fulfilling duties like designing the character of the game, the several levels involved, plot, art and similar other elements. Individuals who opt for a career as a video game designer may also write the codes for the game using different programming languages.

Depending on the video game designer job description and experience they may also have to lead a team and do the early testing of the game in order to suggest changes and find loopholes.

Radio Jockey

Radio Jockey is an exciting, promising career and a great challenge for music lovers. If you are really interested in a career as radio jockey, then it is very important for an RJ to have an automatic, fun, and friendly personality. If you want to get a job done in this field, a strong command of the language and a good voice are always good things. Apart from this, in order to be a good radio jockey, you will also listen to good radio jockeys so that you can understand their style and later make your own by practicing.

A career as radio jockey has a lot to offer to deserving candidates. If you want to know more about a career as radio jockey, and how to become a radio jockey then continue reading the article.

Choreographer

The word “choreography" actually comes from Greek words that mean “dance writing." Individuals who opt for a career as a choreographer create and direct original dances, in addition to developing interpretations of existing dances. A Choreographer dances and utilises his or her creativity in other aspects of dance performance. For example, he or she may work with the music director to select music or collaborate with other famous choreographers to enhance such performance elements as lighting, costume and set design.

Social Media Manager

A career as social media manager involves implementing the company’s or brand’s marketing plan across all social media channels. Social media managers help in building or improving a brand’s or a company’s website traffic, build brand awareness, create and implement marketing and brand strategy. Social media managers are key to important social communication as well.

Photographer

Photography is considered both a science and an art, an artistic means of expression in which the camera replaces the pen. In a career as a photographer, an individual is hired to capture the moments of public and private events, such as press conferences or weddings, or may also work inside a studio, where people go to get their picture clicked. Photography is divided into many streams each generating numerous career opportunities in photography. With the boom in advertising, media, and the fashion industry, photography has emerged as a lucrative and thrilling career option for many Indian youths.

An individual who is pursuing a career as a producer is responsible for managing the business aspects of production. They are involved in each aspect of production from its inception to deception. Famous movie producers review the script, recommend changes and visualise the story. 

They are responsible for overseeing the finance involved in the project and distributing the film for broadcasting on various platforms. A career as a producer is quite fulfilling as well as exhaustive in terms of playing different roles in order for a production to be successful. Famous movie producers are responsible for hiring creative and technical personnel on contract basis.

Copy Writer

In a career as a copywriter, one has to consult with the client and understand the brief well. A career as a copywriter has a lot to offer to deserving candidates. Several new mediums of advertising are opening therefore making it a lucrative career choice. Students can pursue various copywriter courses such as Journalism , Advertising , Marketing Management . Here, we have discussed how to become a freelance copywriter, copywriter career path, how to become a copywriter in India, and copywriting career outlook. 

In a career as a vlogger, one generally works for himself or herself. However, once an individual has gained viewership there are several brands and companies that approach them for paid collaboration. It is one of those fields where an individual can earn well while following his or her passion. 

Ever since internet costs got reduced the viewership for these types of content has increased on a large scale. Therefore, a career as a vlogger has a lot to offer. If you want to know more about the Vlogger eligibility, roles and responsibilities then continue reading the article. 

For publishing books, newspapers, magazines and digital material, editorial and commercial strategies are set by publishers. Individuals in publishing career paths make choices about the markets their businesses will reach and the type of content that their audience will be served. Individuals in book publisher careers collaborate with editorial staff, designers, authors, and freelance contributors who develop and manage the creation of content.

Careers in journalism are filled with excitement as well as responsibilities. One cannot afford to miss out on the details. As it is the small details that provide insights into a story. Depending on those insights a journalist goes about writing a news article. A journalism career can be stressful at times but if you are someone who is passionate about it then it is the right choice for you. If you want to know more about the media field and journalist career then continue reading this article.

Individuals in the editor career path is an unsung hero of the news industry who polishes the language of the news stories provided by stringers, reporters, copywriters and content writers and also news agencies. Individuals who opt for a career as an editor make it more persuasive, concise and clear for readers. In this article, we will discuss the details of the editor's career path such as how to become an editor in India, editor salary in India and editor skills and qualities.

Individuals who opt for a career as a reporter may often be at work on national holidays and festivities. He or she pitches various story ideas and covers news stories in risky situations. Students can pursue a BMC (Bachelor of Mass Communication) , B.M.M. (Bachelor of Mass Media) , or  MAJMC (MA in Journalism and Mass Communication) to become a reporter. While we sit at home reporters travel to locations to collect information that carries a news value.  

Corporate Executive

Are you searching for a Corporate Executive job description? A Corporate Executive role comes with administrative duties. He or she provides support to the leadership of the organisation. A Corporate Executive fulfils the business purpose and ensures its financial stability. In this article, we are going to discuss how to become corporate executive.

Multimedia Specialist

A multimedia specialist is a media professional who creates, audio, videos, graphic image files, computer animations for multimedia applications. He or she is responsible for planning, producing, and maintaining websites and applications. 

Quality Controller

A quality controller plays a crucial role in an organisation. He or she is responsible for performing quality checks on manufactured products. He or she identifies the defects in a product and rejects the product. 

A quality controller records detailed information about products with defects and sends it to the supervisor or plant manager to take necessary actions to improve the production process.

Production Manager

A QA Lead is in charge of the QA Team. The role of QA Lead comes with the responsibility of assessing services and products in order to determine that he or she meets the quality standards. He or she develops, implements and manages test plans. 

Process Development Engineer

The Process Development Engineers design, implement, manufacture, mine, and other production systems using technical knowledge and expertise in the industry. They use computer modeling software to test technologies and machinery. An individual who is opting career as Process Development Engineer is responsible for developing cost-effective and efficient processes. They also monitor the production process and ensure it functions smoothly and efficiently.

AWS Solution Architect

An AWS Solution Architect is someone who specializes in developing and implementing cloud computing systems. He or she has a good understanding of the various aspects of cloud computing and can confidently deploy and manage their systems. He or she troubleshoots the issues and evaluates the risk from the third party. 

Azure Administrator

An Azure Administrator is a professional responsible for implementing, monitoring, and maintaining Azure Solutions. He or she manages cloud infrastructure service instances and various cloud servers as well as sets up public and private cloud systems. 

Computer Programmer

Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.

Information Security Manager

Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack 

ITSM Manager

Automation test engineer.

An Automation Test Engineer job involves executing automated test scripts. He or she identifies the project’s problems and troubleshoots them. The role involves documenting the defect using management tools. He or she works with the application team in order to resolve any issues arising during the testing process. 

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