applied math phd columbia

Graduate Student Handbook (Coming Soon: New Graduate Student Handbook)

Phd program overview.

The PhD program prepares students for research careers in probability and statistics in academia and industry. Students admitted to the PhD program earn the MA and MPhil along the way. The first year of the program is spent on foundational courses in theoretical statistics, applied statistics, and probability. In the following years, students take advanced topics courses. Research toward the dissertation typically begins in the second year. Students also have opportunities to take part in a wide variety of projects involving applied probability or applications of statistics.

Students are expected to register continuously until they distribute and successfully defend their dissertation. Our core required and elective curricula in Statistics, Probability, and Machine Learning aim to provide our doctoral students with advanced learning that is both broad and focused. We expect our students to make Satisfactory Academic Progress in their advanced learning and research training by meeting the following program milestones through courseworks, independent research, and dissertation research:

By the end of year 1: passing the qualifying exams;

By the end of year 2: fulfilling all course requirements for the MA degree and finding a dissertation advisor;

By the end of year 3: passing the oral exam (dissertation prospectus) and fulfilling all requirements for the MPhil degree

By the end of year 5: distributing and defending the dissertation.

We believe in the Professional Development value of active participation in intellectual exchange and pedagogical practices for future statistical faculty and researchers. Students are required to serve as teaching assistants and present research during their training. In addition, each student is expected to attend seminars regularly and participate in Statistical Practicum activities before graduation.

We provide in the following sections a comprehensive collection of the PhD program requirements and milestones. Also included are policies that outline how these requirements will be enforced with ample flexibility. Questions on these requirements should be directed to ADAA Cindy Meekins at [email protected] and the DGS, Professor John Cunningham at [email protected] .

Applications for Admission

• Our students receive very solid training in all aspects of modern statistics. See Graduate Student Handbook for more information.
• Our students receive Fellowship and full financial support for the entire duration of their PhD. See more details here .
• Our students receive job offers from top academic and non-academic institutions .
• Our students can work with world-class faculty members from Statistics Department or the Data Science Institute .
• Our students have access to high-speed computer clusters for their ambitious, computationally demanding research.
• Our students benefit from a wide range of seminars, workshops, and Boot Camps organized by our department and the data science institute .
• Suggested Prerequisites: A student admitted to the PhD program normally has a background in linear algebra and real analysis, and has taken a few courses in statistics, probability, and programming. Students who are quantitatively trained or have substantial background/experience in other scientific disciplines are also encouraged to apply for admission.
• GRE requirement: Waived for Fall 2024.
• Language requirement: The English Proficiency Test requirement (TOEFL) is a Provost's requirement that cannot be waived.
• The Columbia GSAS minimum requirements for TOEFL and IELTS are: 100 (IBT), 600 (PBT) TOEFL, or 7.5 IELTS. To see if this requirement can be waived for you, please check the frequently asked questions below.
• Deadline: Jan 8, 2024 .
• Application process: Please apply by completing the Application for Admission to the Columbia University Graduate School of Arts & Sciences .
• Timeline: P.hD students begin the program in September only.  Admissions decisions are made in mid-March of each year for the Fall semester.

Frequently Asked Questions

• What is the application deadline? What is the deadline for financial aid? Our application deadline is January 5, 2024 .
• Can I meet with you in person or talk to you on the phone? Unfortunately given the high number of applications we receive, we are unable to meet or speak with our applicants.
• What are the required application materials? Specific admission requirements for our programs can be found here .
• Due to financial hardship, I cannot pay the application fee, can I still apply to your program? Yes. Many of our prospective students are eligible for fee waivers. The Graduate School of Arts and Sciences offers a variety of application fee waivers . If you have further questions regarding the waiver please contact  gsas-admissions@ columbia.edu .
• How many students do you admit each year? It varies year to year. We finalize our numbers between December - early February.
• What is the distribution of students currently enrolled in your program? (their background, GPA, standard tests, etc)? Unfortunately, we are unable to share this information.
• How many accepted students receive financial aid? All students in the PhD program receive, for up to five years, a funding package consisting of tuition, fees, and a stipend. These fellowships are awarded in recognition of academic achievement and in expectation of scholarly success; they are contingent upon the student remaining in good academic standing. Summer support, while not guaranteed, is generally provided. Teaching and research experience are considered important aspects of the training of graduate students. Thus, graduate fellowships include some teaching and research apprenticeship. PhD students are given funds to purchase a laptop PC, and additional computing resources are supplied for research projects as necessary. The Department also subsidizes travel expenses for up to two scientific meetings and/or conferences per year for those students selected to present. Additional matching funds from the Graduate School Arts and Sciences are available to students who have passed the oral qualifying exam.
• Can I contact the department with specific scores and get feedback on my competitiveness for the program? We receive more than 450 applications a year and there are many students in our applicant pool who are qualified for our program. However, we can only admit a few top students. Before seeing the entire applicant pool, we cannot comment on admission probabilities.
• What is the minimum GPA for admissions? While we don’t have a GPA threshold, we will carefully review applicants’ transcripts and grades obtained in individual courses.
• Is there a minimum GRE requirement? No. The general GRE exam is waived for the Fall 2024 admissions cycle. 
• Can I upload a copy of my GRE score to the application? Yes, but make sure you arrange for ETS to send the official score to the Graduate School of Arts and Sciences.
• Is the GRE math subject exam required? No, we do not require the GRE math subject exam.
• What is the minimum TOEFL or IELTS  requirement? The Columbia Graduate School of Arts and Sciences minimum requirements for TOEFL and IELTS are: 100 (IBT), 600 (PBT) TOEFL, or 7.5 IELTS
•  I took the TOEFL and IELTS more than two years ago; is my score valid? Scores more than two years old are not accepted. Applicants are strongly urged to make arrangements to take these examinations early in the fall and before completing their application.
• I am an international student and earned a master’s degree from a US university. Can I obtain a TOEFL or IELTS waiver? You may only request a waiver of the English proficiency requirement from the Graduate School of Arts and Sciences by submitting the English Proficiency Waiver Request form and if you meet any of the criteria described here . If you have further questions regarding the waiver please contact  gsas-admissions@ columbia.edu .
• My transcript is not in English. What should I do? You have to submit a notarized translated copy along with the original transcript.

Can I apply to more than one PhD program? You may not submit more than one PhD application to the Graduate School of Arts and Sciences. However, you may elect to have your application reviewed by a second program or department within the Graduate School of Arts and Sciences if you are not offered admission by your first-choice program. Please see the application instructions for a more detailed explanation of this policy and the various restrictions that apply to a second choice. You may apply concurrently to a program housed at the Graduate School of Arts and Sciences and to programs housed at other divisions of the University. However, since the Graduate School of Arts and Sciences does not share application materials with other divisions, you must complete the application requirements for each school.

How do I apply to a dual- or joint-degree program? The Graduate School of Arts and Sciences refers to these programs as dual-degree programs. Applicants must complete the application requirements for both schools. Application materials are not shared between schools. Students can only apply to an established dual-degree program and may not create their own.

With the sole exception of approved dual-degree programs , students may not pursue a degree in more than one Columbia program concurrently, and may not be registered in more than one degree program at any institution in the same semester. Enrollment in another degree program at Columbia or elsewhere while enrolled in a Graduate School of Arts and Sciences master's or doctoral program is strictly prohibited by the Graduate School. Violation of this policy will lead to the rescission of an offer of admission, or termination for a current student.

When will I receive a decision on my application? Notification of decisions for all PhD applicants generally takes place by the end of March.

Notification of MA decisions varies by department and application deadlines. Some MA decisions are sent out in early spring; others may be released as late as mid-August.

Can I apply to both MA Statistics and PhD statistics simultaneously?  For any given entry term, applicants may elect to apply to up to two programs—either one PhD program and one MA program, or two MA programs—by submitting a single (combined) application to the Graduate School of Arts and Sciences.  Applicants who attempt to submit more than one Graduate School of Arts and Sciences application for the same entry term will be required to withdraw one of the applications.

The Graduate School of Arts and Sciences permits applicants to be reviewed by a second program if they do not receive an offer of admission from their first-choice program, with the following restrictions:

• This option is only available for fall-term applicants.
• Applicants will be able to view and opt for a second choice (if applicable) after selecting their first choice. Applicants should not submit a second application. (Note: Selecting a second choice will not affect the consideration of your application by your first choice.)
• Applicants must upload a separate Statement of Purpose and submit any additional supporting materials required by the second program. Transcripts, letters, and test scores should only be submitted once.
• An application will be forwarded to the second-choice program only after the first-choice program has completed its review and rendered its decision. An application file will not be reviewed concurrently by both programs.
• Programs may stop considering second-choice applications at any time during the season; Graduate School of Arts and Sciences cannot guarantee that your application will receive a second review.
• What is the mailing address for your PhD admission office? Students are encouraged to apply online . Please note: Materials should not be mailed to the Graduate School of Arts and Sciences unless specifically requested by the Office of Admissions. Unofficial transcripts and other supplemental application materials should be uploaded through the online application system. Graduate School of Arts and Sciences Office of Admissions Columbia University  107 Low Library, MC 4303 535 West 116th Street  New York, NY 10027
• How many years does it take to pursue a PhD degree in your program? Our students usually graduate in 4‐6 years.
• Can the PhD be pursued part-time? No, all of our students are full-time students. We do not offer a part-time option.
• One of the requirements is to have knowledge of linear algebra (through the level of MATH V2020 at Columbia) and advanced calculus (through the level of MATH V1201). I studied these topics; how do I know if I meet the knowledge content requirement? We interview our top candidates and based on the information on your transcripts and your grades, if we are not sure about what you covered in your courses we will ask you during the interview.
• Can I contact faculty members to learn more about their research and hopefully gain their support? Yes, you are more than welcome to contact faculty members and discuss your research interests with them. However, please note that all the applications are processed by a central admission committee, and individual faculty members cannot and will not guarantee admission to our program.
• How do I find out which professors are taking on new students to mentor this year?  Applications are evaluated through a central admissions committee. Openings in individual faculty groups are not considered during the admissions process. Therefore, we suggest contacting the faculty members you would like to work with and asking if they are planning to take on new students.

For more information please contact us at [email protected] .

For more information please contact us at  [email protected]

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Applied Mathematics

Departmental Undergraduate Office: 410 Mathematics; 212-854-2432 http://www.math.columbia.edu/

Director of Undergraduate Studies: Prof. Mu-Tao Wang, 514 Mathematics; 212-854-3052; [email protected]  

Calculus Director: Prof. George Dragomir, 525 Mathematics; 212-854-2849; [email protected]

Computer Science-Mathematics Adviser: Computer Science : Dr. Jae Woo Lee, 715 CEPSR; 212-939-7066;  [email protected] Mathematics : Prof. Chiu-Chu Melissa Liu, 623 Mathematics; 212-854-2499; c [email protected]

Economics-Mathematics Advisers: Mathematics : Prof. Julien Dubedat, 601 Mathematics; 212-854-8806; [email protected] Economics : Dr. Susan Elmes, 1006 International Affairs Building; 212-854-9124; [email protected]

Mathematics-Statistics Advisers: Mathematics : Prof. Julien Dubedat, 601 Mathematics; 212-854-8806; [email protected] Statistics : Ronald Neath, 612 Watson; 212-853-1398; [email protected] Statistics : Gabriel Young, 610 Watson; 212-853-1395; [email protected]

The major in mathematics is an introduction to some of the highlights of the development of theoretical mathematics over the past four hundred years from a modern perspective. This study is also applied to many problems, both internal to mathematics and arising in other disciplines such as physics, cryptography, and finance.

Majors begin by taking either Honors mathematics or the calculus sequence. Students who do not take MATH UN1207 HONORS MATHEMATICS A  and  MATH UN1208 HONORS MATHEMATICS B  normally take MATH UN2010 LINEAR ALGEBRA  in the second year. Following this, majors begin to learn some aspects of the main branches of modern mathematics: algebra, analysis, and geometry; as well as some of their subdivisions and hybrids (e.g., number theory, differential geometry, and complex analysis). As the courses become more advanced, they also become more theoretical and proof-oriented and less computational.

Aside from the courses offered by the Mathematics Department, cognate courses in areas such as astronomy, chemistry, physics, probability, logic, economics, and computer science can be used toward the major. A cognate course must be a  2000 -level (or higher) course and must be approved by the director of undergraduate studies. In general, a course not taught by the Mathematics Department is a cognate course for the mathematics major if either (a) it has at least two semesters of calculus as a stated prerequisite, or (b) the subject matter in the course is mathematics beyond an elementary level, such as  PHIL UN3411 SYMBOLIC LOGIC , in the Philosophy Department, or  COMS W3203 DISCRETE MATHEMATICS , in the Computer Science Department.

Another requirement for majors is participation in an undergraduate seminar, usually in the junior or senior year. Applied math majors must take the undergraduate seminar in both the junior and senior year. In these seminars, students gain experience in learning an advanced topic and lecturing on it. In order to be eligible for departmental honors, majors must write a senior thesis.

Courses for First-Year Students

The systematic study of mathematics begins with one of the following three alternative calculus and linear algebra sequences:

Credit is allowed for only one calculus and linear algebra sequence.

Calculus I, II  is a standard course in single-variable differential and integral calculus; Calculus III, IV is a standard course in multivariable differential and integral calculus; Accelerated Multivariable Calculus is an accelerated course in multivariable differential and integral calculus.

While Calculus II is no longer a prerequisite for Calculus III , students are strongly urged to take it before taking Calculus III . In particular, students thinking of majoring or concentrating in mathematics or one of the joint majors involving mathematics should take Calculus II before taking Calculus III . Note that Calculus II is a prerequisite for Accelerated Multivariable Calculus,  and both Calculus II and Calculus III are prerequisites for Calculus IV .

The third sequence,  Honors Mathematics A- B , is for exceptionally well-qualified students who have strong Advanced Placement scores. It covers multivariable calculus ( MATH UN1201 CALCULUS III - MATH UN1202 CALCULUS IV ) and linear algebra ( MATH UN2010 LINEAR ALGEBRA ), with an emphasis on theory.

Advanced Placement

The department grants 3 credits for a score of 4 or 5 on the AP Calculus AB exam provided students complete MATH UN1102 CALCULUS II or MATH UN1201 CALCULUS III  with a grade of C or better. The department grants 3 credits for a score of 4 on the AP Calculus BC exam provided students complete MATH UN1102 CALCULUS II or  MATH UN1201 CALCULUS III with a grade of C or better. The department grants 6 credits for a score of 5 on the AP Calculus BC exam provided students complete MATH UN1201 CALCULUS III or  MATH UN1205 ACCELERATED MULTIVARIABLE CALC   MATH UN1207 HONORS MATHEMATICS A with a grade of C or better. Students can receive credit for only one calculus sequence.

Placement in the Calculus Sequences

Students who have essentially mastered a precalculus course and those who have a score of 3 or less on an Advanced Placement (AP) exam (either AB or BC) should begin their study of calculus with MATH UN1101 CALCULUS I .

Calculus II and III

Students with a score of 4 or 5 on the AB exam, 4 on the BC exam, or those with no AP score but with a grade of A in a full year of high school calculus may begin with either MATH UN1102 CALCULUS II  or MATH UN1201 CALCULUS III .  Note that such students who decide to start with Calculus III may still need to take Calculus II since it is a requirement or prerequisite for other courses. In particular, they MUST take  Calculus II before going on to MATH UN1202 CALCULUS IV . Students with a score of 5 on the BC exam may begin with Calculus III and do not need to take Calculus II .

Those with a score of 4 or 5 on the AB exam or 4 on the BC exam may receive 3 points of AP credit upon completion of Calculus II with a grade of C or higher. Those students with a score of 5 on the BC exam may receive 6 points of AP credit upon completion of Calculus III with a grade of C or higher.

Accelerated Multivariable Calculus

Students with a score of 5 on the AP BC exam or 7 on the IB HL exam may begin with MATH UN1205 ACCELERATED MULTIVARIABLE CALC . Upon completion of this course with a grade of C or higher, they may receive 6 points of AP credit.

Honors Mathematics A

Students who want a proof-oriented theoretical sequence and have a score of 5 on the BC exam may begin with MATH UN1207 HONORS MATHEMATICS A , which is especially designed for mathematics majors. Upon completion of this course with a grade of C or higher, they may receive 6 points of AP credit.

Transfers Inside the Calculus Sequences

Students who wish to transfer from one calculus course to another are allowed to do so beyond the date specified on the Academic Calendar. They are considered to be adjusting their level, not changing their program. However, students must obtain the approval of the new instructor and their advising dean prior to reporting to the Office of the Registrar.

No course with a grade of D or lower can count toward the major, interdepartmental major, or concentration. Students who are doing a double major cannot double count courses for their majors.

Senior Thesis and Departmental Honors

In order to be eligible for departmental honors, majors must write a senior thesis. Normally no more than 10% of graduating majors receive departmental honors in a given academic year.

A Senior Thesis in Mathematics is an original presentation of a subject in pure or applied mathematics from sources in the published literature. The thesis must demonstrate significant independent work of the author. A thesis is expected to be between 20 and 50 pages with complete references and must have a substantial expository component to be well received.

A student who is interested in writing a senior thesis needs to identify a faculty member in the Department of Mathematics as an advisor, determine an appropriate topic, and receive the written approval from the faculty advisor and the DUS. The research of the thesis is conducted primarily during the fall term and the final paper is submitted to the DUS by the end of March.

Students must register for MATH UN3994 Senior Thesis in Mathematics I (4 credits) in the fall semester of their senior year. An optional continuation course MATH UN3995 Senior Thesis in Mathematics II (2 credits) is available during the spring. The second term of this sequence may not be taken without the first. Registration for the spring continuation course has no impact on the timeline or outcome of the final paper. Sections of Senior Thesis in Mathematics I and II do NOT count towards the major requirements, with the exception of an advanced written approval by the DUS.

• David A. Bayer (Barnard)
• Andrew Blumberg
• Simon Brendle
• Ivan Corwin
• Panagiota Daskalopoulos
• Aise Johan de Jong (Department Chair)
• Daniela De Silva (Barnard Chair)
• Julien Dubedat
• Robert Friedman
• Dorian Goldfeld
• Brian Greene
• Richard Hamilton
• Michael Harris
• Ioannis Karatzas
• Mikhail Khovanov
• Alisa Knizel (Barnard)
• Chiu-Chu Liu
• Dusa McDuff (Barnard)
• Andrei Okounkov
• D. H. Phong
• Ovidiu Savin
• Michael Thaddeus
• Mu-Tao Wang

Associate Professors

• Amol Aggarwal
• Lindsay Piechnik (Barnard)

Assistant Professors

• Elena Giorgi
• Francesco Lin
• Giulia Sacca

J.F. Ritt Assistant Professors

• Rostislav Akhmechet
• Konstantin Aleshkin
• Shaoyun Bai
• Jeanne Boursier
• Marco Castronovo
• Nathan Chen
• Sam Collingbourne
• Andres Fernandez-Herrero
• James Hotchkiss
• Yoonjoo Kim
• Xi Sisi Shen

Senior Lecturers in Discipline

• Lars Nielsen
• Mikhail Smirnov

Lecturers in Discipline

• George Dragomir
• Mrudul Thatte
• Profs. Aggarwal, Bayer, Corwin, Daskalopoulos, Fernandez-Herrero, Khovanov, Li, Sacca, Sawin, Woit  (Fall 2023)
• Profs. Aggarwal, Bayer, Daskalopoulos, Giorgi, Hotchkiss, Khovanov, Li, Thaddeus  (Spring 2024)

Major in Mathematics

The major requires 40-42 points as follows:

UN2015 (Linear Algebra and Probability) does NOT replace UN2010 (Linear Algebra) as prerequisite requirements of math courses. Students will not receive full credit for both courses UN2010 and UN2015. 

Students who are not contemplating graduate study in mathematics may replace one or both of the two terms of MATH GU4061 - MATH GU4062 by one or two of the following courses: MATH UN2500 ANALYSIS AND OPTIMIZATION ,  MATH UN3007 COMPLEX VARIABLES , MATH UN3028 PARTIAL DIFFERENTIAL EQUATIONS , or MATH GU4032 FOURIER ANALYSIS .

Only one Undergraduate Seminar may count towards the major requirements.

Additional courses may be selected only with prior written approval from the Director of Undergraduate Studies.

The program of study should be planned with a departmental adviser before the end of the sophomore year. Majors who are planning on graduate studies in mathematics are urged to obtain a reading knowledge of one of the following languages: French, German, or Russian.

Majors are offered the opportunity to write an honors senior thesis under the guidance of a faculty member. Interested students should contact the director of undergraduate studies.

Major in Applied Mathematics

The major requires 37-41 points as follows:

Major in Computer Science–Mathematics

The goal of this interdepartmental major is to provide substantial background in each of these two disciplines, focusing on some of the parts of each which are closest to the other. Students intending to pursue a Ph.D. program in either discipline are urged to take additional courses, in consultation with their advisers.

The major requires 20 points in computer science, 19-21 points in mathematics, and two 3-point electives in either computer science or mathematics.

Major in Economics-Mathematics

For a description of the joint major in economics-mathematics, see the  Economics  section of this bulletin.

Major in Mathematics-Statistics

The program is designed to prepare the student for: (1) a career in industries such as finance and insurance that require a high level of mathematical sophistication and a substantial knowledge of probability and statistics, and (2) graduate study in quantitative disciplines. Students choose electives in finance, actuarial science, operations research, or other quantitative fields to complement requirements in mathematics, statistics, and computer science.

Students interested in modeling applications are recommended to take MATH UN2030 ORDINARY DIFFERENTIAL EQUATIONS and MATH UN3028 PARTIAL DIFFERENTIAL EQUATIONS .

Students interested in finance are recommended to take MATH GR5010 INTRO TO THE MATH OF FINANCE , STAT GU4261 STATISTICAL METHODS IN FINANCE , and STAT GU4221 TIME SERIES ANALYSIS .

Students interested in graduate study in mathematics or in statistics are recommended to take MATH GU4061 INTRO MODERN ANALYSIS I and MATH GU4062 INTRO MODERN ANALYSIS II .

Students preparing for a career in actuarial science are encouraged to replace STAT GU4205 LINEAR REGRESSION MODELS with STAT GU4282 Linear Regression and Time Series Methods , and to take among their electives STAT GU4281 Theory of Interest .

Concentration in Mathematics

The concentration requires the following:

For mathematics courses taken in other departments, consult with the director of undergraduate studies.

 Any course given by the Mathematics department fulfills the General Studies quantitative reasoning requirement when passed with a satisfactory letter grade.

MATH UN1003 COLLEGE ALGEBRA-ANLYTC GEOMTRY. 3.00 points .

Prerequisites: score of 550 on the mathematics portion of the SAT completed within the last year, or the appropriate grade on the General Studies Mathematics Placement Examination. For students who wish to study calculus but do not know analytic geometry. Algebra review, graphs and functions, polynomial functions, rational functions, conic sections, systems of equations in two variables, exponential and logarithmic functions, trigonometric functions and trigonometric identities, applications of trigonometry, sequences, series, and limits

MATH UN1101 CALCULUS I. 3.00 points .

Prerequisites: (see Courses for First-Year Students). Functions, limits, derivatives, introduction to integrals, or an understanding of pre-calculus will be assumed. (SC)

MATH UN1102 CALCULUS II. 3.00 points .

Prerequisites: MATH UN1101 or the equivalent. Prerequisites: MATH UN1101 or the equivalent. Methods of integration, applications of the integral, Taylors theorem, infinite series. (SC)

MATH UN1201 CALCULUS III. 3.00 points .

Prerequisites: MATH UN1101 or the equivalent Prerequisites: MATH UN1101 or the equivalent Vectors in dimensions 2 and 3, complex numbers and the complex exponential function with applications to differential equations, Cramers rule, vector-valued functions of one variable, scalar-valued functions of several variables, partial derivatives, gradients, surfaces, optimization, the method of Lagrange multipliers. (SC)

MATH UN1202 CALCULUS IV. 3.00 points .

Prerequisites: MATH UN1102 and MATH UN1201 or the equivalent Prerequisites: MATH UN1102 and MATH UN1201 or the equivalent Multiple integrals, Taylor's formula in several variables, line and surface integrals, calculus of vector fields, Fourier series. (SC)

MATH UN1205 ACCELERATED MULTIVARIABLE CALC. 4.00 points .

Prerequisites: ( MATH UN1101 and MATH UN1102 ) Prerequisites: ( MATH UN1101 and MATH UN1102 ) Vectors in dimensions 2 and 3, vector-valued functions of one variable, scalar-valued functions of several variables, partial derivatives, gradients, optimization, Lagrange multipliers, double and triple integrals, line and surface integrals, vector calculus. This course is an accelerated version of MATH UN1201 - MATH UN1202 . Students taking this course may not receive credit for MATH UN1201 and MATH UN1202

MATH UN1207 HONORS MATHEMATICS A. 4.00 points .

Prerequisites: (see Courses for First-Year Students). The second term of this course may not be taken without the first. Multivariable calculus and linear algebra from a rigorous point of view. Recommended for mathematics majors. Fulfills the linear algebra requirement for the major. (SC)

MATH UN1208 HONORS MATHEMATICS B. 4.00 points .

Prerequisites: (see Courses for First-Year Students). Prerequisites: (see Courses for First-Year Students). The second term of this course may not be taken without the first. Multivariable calculus and linear algebra from a rigorous point of view. Recommended for mathematics majors. Fulfills the linear algebra requirement for the major. (SC)

MATH UN2000 INTRO TO HIGHER MATHEMATICS. 3.00 points .

Introduction to understanding and writing mathematical proofs. Emphasis on precise thinking and the presentation of mathematical results, both in oral and in written form. Intended for students who are considering majoring in mathematics but wish additional training. CC/GS: Partial Fulfillment of Science Requirement. BC: Fulfillment of General Education Requirement: Quantitative and Deductive Reasoning (QUA)

MATH UN2005 INTRODUCTION TO MATHEMATICS PROOFS. 0.00 points .

This is a seminar course that covers the basics of mathematical proofs and in particular the epsilon-delta argument in single variable calculus. Students who have little experience with mathematical proofs are strongly encouraged to take this course concurrently with Honors Math, Into to Modern Algebra, or Intro to Modern Analysis

MATH BC2006 COMBINATORICS. 3.00 points .

MATH UN2010 LINEAR ALGEBRA. 3.00 points .

Matrices, vector spaces, linear transformations, eigenvalues and eigenvectors, canonical forms, applications. (SC)

MATH UN2015 Linear Algebra and Probability. 3.00 points .

Linear algebra with a focus on probability and statistics. The course covers the standard linear algebra topics: systems of linear equations, matrices, determinants, vector spaces, bases, dimension, eigenvalues and eigenvectors, the Spectral Theorem and singular value decompositions. It also teaches applications of linear algebra to probability, statistics and dynamical systems giving a background sufficient for higher level courses in probability and statistics. The topics covered in the probability theory part include conditional probability, discrete and continuous random variables, probability distributions and the limit theorems, as well as Markov chains, curve fitting, regression, and pattern analysis. The course contains applications to life sciences, chemistry, and environmental life sciences. No a prior i background in the life sciences is assumed. This course is best suited for students who wish to focus on applications and practical approaches to problem solving. It is recommended to students majoring in engineering, technology, life sciences, social sciences, and economics. Math majors, joint majors, and math concentrators must take MATH UN2010 Linear Algebra, which focuses on linear algebra concepts and foundations that are needed for upper-level math courses. MATH UN2015 (Linear Algebra and Probability) does NOT replace MATH UN2010 (Linear Algebra) as prerequisite requirements of math courses. Students may not receive full credit for both courses MATH UN2010 and MATH UN2015

MATH UN2030 ORDINARY DIFFERENTIAL EQUATIONS. 3.00 points .

Prerequisites: MATH UN1102 and MATH UN1201 or the equivalent. Prerequisites: MATH UN1102 and MATH UN1201 or the equivalent. Special differential equations of order one. Linear differential equations with constant and variable coefficients. Systems of such equations. Transform and series solution techniques. Emphasis on applications

MATH UN2500 ANALYSIS AND OPTIMIZATION. 3.00 points .

Prerequisites: MATH UN1102 and MATH UN1201 or the equivalent and MATH UN2010 . Prerequisites: MATH UN1102 and MATH UN1201 or the equivalent and MATH UN2010 . Mathematical methods for economics. Quadratic forms, Hessian, implicit functions. Convex sets, convex functions. Optimization, constrained optimization, Kuhn-Tucker conditions. Elements of the calculus of variations and optimal control. (SC)

MATH UN3007 COMPLEX VARIABLES. 3.00 points .

Prerequisites: MATH UN1202 An elementary course in functions of a complex variable. Prerequisites: MATH UN1202 An elementary course in functions of a complex variable. Fundamental properties of the complex numbers, differentiability, Cauchy-Riemann equations. Cauchy integral theorem. Taylor and Laurent series, poles, and essential singularities. Residue theorem and conformal mapping.(SC)

MATH UN3020 NUMBER THEORY AND CRYPTOGRAPHY. 3.00 points .

Prerequisites: one year of calculus. Prerequisites: one year of calculus. Prerequisite: One year of Calculus. Congruences. Primitive roots. Quadratic residues. Contemporary applications

MATH UN3025 MAKING, BREAKING CODES. 3.00 points .

Prerequisites: ( MATH UN1101 and MATH UN1102 and MATH UN1201 ) and and MATH UN2010 . Prerequisites: ( MATH UN1101 and MATH UN1102 and MATH UN1201 ) and and MATH UN2010 . A concrete introduction to abstract algebra. Topics in abstract algebra used in cryptography and coding theory

MATH UN3028 PARTIAL DIFFERENTIAL EQUATIONS. 3.00 points .

Prerequisites: MATH UN3027 and MATH UN2010 or the equivalent Prerequisites: ( MATH UN2010 and MATH UN2030 ) or the equivalent introduction to partial differential equations. First-order equations. Linear second-order equations; separation of variables, solution by series expansions. Boundary value problems

MATH UN3050 DISCRETE TIME MODELS IN FINANC. 3.00 points .

Prerequisites: ( MATH UN1102 and MATH UN1201 ) or ( MATH UN1101 and MATH UN1102 and MATH UN1201 ) and MATH UN2010 Recommended: MATH UN3027 (or MATH UN2030 and SIEO W3600 ). Prerequisites: ( MATH UN1102 and MATH UN1201 ) or ( MATH UN1101 and MATH UN1102 and MATH UN1201 ) and MATH UN2010 Recommended: MATH UN3027 (or MATH UN2030 and SIEO W3600 ). Elementary discrete time methods for pricing financial instruments, such as options. Notions of arbitrage, risk-neutral valuation, hedging, term-structure of interest rates

MATH UN3386 DIFFERENTIAL GEOMETRY. 3.00 points .

Prerequisites: MATH UN1202 or the equivalent.

Local and global differential geometry of submanifolds of Euclidiean 3-space. Frenet formulas for curves. Various types of curvatures for curves and surfaces and their relations. The Gauss-Bonnet theorem.

MATH UN3901 SUPERVISED READINGS I. 1.00-3.00 points .

Prerequisites: The written permission of the faculty member who agrees to act as sponsor (sponsorship limited to full-time instructors on the staff list), as well as the permission of the Director of Undergraduate Studies. The written permission must be deposited with the Director of Undergraduate Studies before registration is completed. Guided reading and study in mathematics. A student who wishes to undertake individual study under this program must present a specific project to a member of the staff and secure his or her willingness to act as sponsor. Written reports and periodic conferences with the instructor. Supervising Readings do NOT count towards major requirements, with the exception of an advanced written approval by the DUS

MATH UN3902 SUPERVISED READINGS II. 1.00-3.00 points .

MATH UN3951 UNDERGRADUATE SEMINARS I. 3.00 points .

Prerequisites: Two years of calculus, at least one year of additional mathematics courses, and the director of undergraduate studies' permission. Prerequisites: Two years of calculus, at least one year of additional mathematics courses, and the director of undergraduate studies permission. The subject matter is announced at the start of registration and is different in each section. Each student prepares talks to be given to the seminar, under the supervision of a faculty member or senior teaching fellow

MATH UN3952 UNDERGRADUATE SEMINARS II. 3.00 points .

Prerequisites: two years of calculus, at least one year of additional mathematics courses, and the director of undergraduate studies' permission. Prerequisites: two years of calculus, at least one year of additional mathematics courses, and the director of undergraduate studies' permission. The subject matter is announced at the start of registration and is different in each section. Each student prepares talks to be given to the seminar, under the supervision of a faculty member or senior teaching fellow. Prerequisite: two years of calculus, at least one year of additional mathematics courses, and the director of undergraduate studies' permission

MATH UN3994 SENIOR THESIS IN MATHEMATICS I. 4.00 points .

Majors in Mathematics are offered the opportunity to write an honors senior thesis under the guidance of a faculty member. Interested students should contact a faculty member to determine an appropriate topic, and receive written approval from the faculty advisor and the Director of Undergraduate Studies (faculty sponsorship is limited to full-time instructors on the staff list). Research is conducted primarily during the fall term; the final paper is submitted to the Director of Undergraduate Studies during the subsequent spring term. MATH UN3994 SENIOR THESIS IN MATHEMATICS I must be taken in the fall term, during which period the student conducts primary research on the agreed topic. An optional continuation course MATH UN3995 SENIOR THESIS IN MATHEMATICS II is available during the spring. The second term of this sequence may not be taken without the first. Registration for the spring continuation course has no impact on the timeline or outcome of the final paper. Sections of SENIOR THESIS IN MATHEMATICS I and II do NOT count towards the major requirements, with the exception of an advanced written approval by the DUS

MATH UN3995 SENIOR THESIS IN MATHEMATICS II. 2.00 points .

MATH GU4007 ANALYTIC NUMBER THEORY. 3.00 points .

Prerequisites: MATH UN3007 Prerequisites: MATH UN3007 A one semeser course covering the theory of modular forms, zeta functions, L -functions, and the Riemann hypothesis. Particular topics covered include the Riemann zeta function, the prime number theorem, Dirichlet characters, Dirichlet L-functions, Siegel zeros, prime number theorem for arithmetic progressions, SL (2, Z) and subgroups, quotients of the upper half-plane and cusps, modular forms, Fourier expansions of modular forms, Hecke operators, L-functions of modular forms

MATH GU4032 FOURIER ANALYSIS. 3.00 points .

Prerequisites: three terms of calculus and linear algebra or four terms of calculus. Prerequisites: three terms of calculus and linear algebra or four terms of calculus. Prerequisite: three terms of calculus and linear algebra or four terms of calculus. Fourier series and integrals, discrete analogues, inversion and Poisson summation formulae, convolution. Heisenberg uncertainty principle. Stress on the application of Fourier analysis to a wide range of disciplines

MATH GU4041 INTRO MODERN ALGEBRA I. 3.00 points .

Prerequisites: MATH UN1102 and MATH UN1202 and MATH UN2010 or the equivalent Prerequisites: MATH UN1102 and MATH UN1202 and MATH UN2010 or the equivalent. The second term of this course may not be taken without the first. Groups, homomorphisms, normal subgroups, the isomorphism theorems, symmetric groups, group actions, the Sylow theorems, finitely generated abelian groups

MATH GU4042 INTRO MODERN ALGEBRA II. 3.00 points .

Prerequisites: MATH UN1102 and MATH UN1202 and MATH UN2010 or the equivalent. Prerequisites: MATH UN1102 and MATH UN1202 and MATH UN2010 or the equivalent. The second term of this course may not be taken without the first. Rings, homomorphisms, ideals, integral and Euclidean domains, the division algorithm, principal ideal and unique factorization domains, fields, algebraic and transcendental extensions, splitting fields, finite fields, Galois theory

MATH GU4043 ALGEBRAIC NUMBER THEORY. 3.00 points .

Prerequisites: MATH GU4041 and MATH GU4042 or the equivalent Prerequisites: MATH GU4041 and MATH GU4042 or the equivalent Algebraic number fields, unique factorization of ideals in the ring of algebraic integers in the field into prime ideals. Dirichlet unit theorem, finiteness of the class number, ramification. If time permits, p-adic numbers and Dedekind zeta function

MATH GU4044 REPRESENTATNS OF FINITE GROUPS. 3.00 points .

Prerequisites: MATH UN2010 and MATH GU4041 or the equivalent. Prerequisites: MATH UN2010 and MATH GU4041 or the equivalent. Finite groups acting on finite sets and finite dimensional vector spaces. Group characters. Relations with subgroups and factor groups. Arithmetic properties of character values. Applications to the theory of finite groups: Frobenius groups, Hall subgroups and solvable groups. Characters of the symmetric groups. Spherical functions on finite groups

MATH GU4045 ALGEBRAIC CURVES. 3.00 points .

Prerequisites: ( MATH GU4041 and MATH GU4042 ) and MATH UN3007 Prerequisites: ( MATH GU4041 and MATH GU4042 ) and MATH UN3007 Plane curves, affine and projective varieties, singularities, normalization, Riemann surfaces, divisors, linear systems, Riemann-Roch theorem

MATH GU4051 TOPOLOGY. 3.00 points .

Prerequisites: ( MATH UN1202 and MATH UN2010 ) and rudiments of group theory (e.g., MATH GU4041 ). MATH UN1208 or MATH GU4061 is recommended, but not required. Prerequisites: ( MATH UN1202 and MATH UN2010 ) and rudiments of group theory (e.g. MATH GU4041 ). MATH UN1208 or MATH GU4061 is recommended, but not required. Metric spaces, continuity, compactness, quotient spaces. The fundamental group of topological space. Examples from knot theory and surfaces. Covering spaces

MATH GU4052 INTRODUCTION TO KNOT THEORY. 3.00 points .

CC/GS: Partial Fulfillment of Science Requirement

Prerequisites: MATH GU4051 Topology and / or MATH GU4061 Introduction To Modern Analysis I (or equivalents). Recommended (can be taken concurrently): MATH UN2010 linear algebra, or equivalent. Prerequisites: MATH GU4051 Topology and / or MATH GU4061 Introduction To Modern Analysis I (or equivalents). Recommended (can be taken concurrently): MATH UN2010 linear algebra, or equivalent. The study of algebraic and geometric properties of knots in R＾3, including but not limited to knot projections and Reidemeisters theorm, Seifert surfaces, braids, tangles, knot polynomials, fundamental group of knot complements. Depending on time and student interest, we will discuss more advanced topics like knot concordance, relationship to 3-manifold topology, other algebraic knot invariants

MATH GU4053 INTRO TO ALGEBRAIC TOPOLOGY. 3.00 points .

Prerequisites: MATH UN2010 and MATH GU4041 and MATH GU4051 Prerequisites: MATH UN2010 and MATH GU4041 and MATH GU4051 The study of topological spaces from algebraic properties, including the essentials of homology and the fundamental group. The Brouwer fixed point theorem. The homology of surfaces. Covering spaces

MATH GU4061 INTRO MODERN ANALYSIS I. 3.00 points .

Prerequisites: MATH UN1202 or the equivalent, and MATH UN2010 . The second term of this course may not be taken without the first. Prerequisites: MATH UN1202 or the equivalent, and MATH UN2010 . The second term of this course may not be taken without the first. Real numbers, metric spaces, elements of general topology, sequences and series, continuity, differentiation, integration, uniform convergence, Ascoli-Arzela theorem, Stone-Weierstrass theorem

MATH GU4062 INTRO MODERN ANALYSIS II. 3.00 points .

Prerequisites: MATH UN1202 or the equivalent, and MATH UN2010 . The second term of this course may not be taken without the first. The second term of this course may not be taken without the first. Power series, analytic functions, Implicit function theorem, Fubini theorem, change of variables formula, Lebesgue measure and integration, function spaces

MATH GU4065 HONORS COMPLEX VARIABLES. 3.00 points .

Prerequisites: ( MATH UN1207 and MATH UN1208 ) or MATH GU4061 Prerequisites: ( MATH UN1207 and MATH UN1208 ) or MATH GU4061 A theoretical introduction to analytic functions. Holomorphic functions, harmonic functions, power series, Cauchy-Riemann equations, Cauchy's integral formula, poles, Laurent series, residue theorem. Other topics as time permits: elliptic functions, the gamma and zeta function, the Riemann mapping theorem, Riemann surfaces, Nevanlinna theory

MATH GU4081 INTRO-DIFFERENTIABLE MANIFOLDS. 3.00 points .

Prerequisites: ( MATH GU4051 or MATH GU4061 ) and MATH UN2010 Prerequisites: ( MATH GU4051 or MATH GU4061 ) and MATH UN2010 Concept of a differentiable manifold. Tangent spaces and vector fields. The inverse function theorem. Transversality and Sards theorem. Intersection theory. Orientations. Poincare-Hopf theorem. Differential forms and Stokes theorem

MATH GU4155 PROBABILITY THEORY. 3.00 points .

Prerequisites: MATH GU4061 or MATH UN3007 Prerequisites: MATH GU4061 or MATH UN3007 A rigorous introduction to the concepts and methods of mathematical probability starting with basic notions and making use of combinatorial and analytic techniques. Generating functions. Convergence in probability and in distribution. Discrete probability spaces, recurrence and transience of random walks. Infinite models, proof of the law of large numbers and the central limit theorem. Markov chains

MATH GU4156 ADVANCED PROBABILITY THEORY. 3.00 points .

This course will cover advance topics in probability, including: the theory of martingales in discrete and in continuous time; Brownian motion and its properties, stochastic integration, ordinary and partial stochastic differential equations; Applications to optimal filtering, stopping, control, and finance; Continuous-time Markov chains, systems of interacting particles, relative entropy dissipation, notions of information theory; Electrical networks, random walks on graphs and groups, percolation

MATH GU4391 INTRO TO QUANTUM MECHANICS. 3.00 points .

This course will focus on quantum mechanics, paying attention to both the underlying mathematical structures as well as their physical motivations and consequences. It is meant to be accessible to students with no previous formal training in quantum theory. The role of symmetry, groups and representations will be stressed

MATH GU4392 INTRO TO QUANTUM MECHANICS II. 3.00 points .

Continuation of GU4391. This course will focus on quantum mechanics, paying attention to both the underlying mathematical structures as well as their physical motivations and consequences. It is meant to be accessible to students with no previous formal training in quantum theory. The role of symmetry, groups and representations will be stressed.

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• Our Curriculum

MAFN Faculty

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Graeme S. Baker

Irina Bogacheva

Irina Bogacheva is Director of Research at Millburn Ridgefield and Adjunct Professor within the Department of Mathematics.

Previously, she held senior research positions with Goldman Sachs, Deutsche Asset Management, QS Investors, and Franklin Templeton. Irina’s professional experience includes research and implementation of systematic global macro strategies, strategic and dynamic asset allocation, and active factor equity strategies. She holds Diploma in Mathematics from Moscow State University, Master’s in Economics from New Economic School (Moscow), and an MBA in Analytic Finance from the University of Chicago Booth School of Business, where she also completed Ph.D. coursework in Finance.

Alberto Botter

Adjunct Professor, Department of Mathematics MS, Columbia University Quantitative Methods in Investment Management

Alberto Botter is a Managing Director within the Portfolio Management department at AQR Capital Management. In this role, he oversees the construction, optimization, and management of AQR’s Equities Long-Short and Tax-Managed portfolios. Prior to AQR, Mr. Botter was a quant in the Wealth Strategies Group at Morgan Stanley. Alberto earned a B.S. and an M.S. in economics from the University of Bologna and an M.A. in Mathematics of Finance from Columbia University.

[email protected]

Luca Capriotti

Adjunct Professor, Department of Mathematics Ph.D., International School for Advanced Studies Credit Models, Computational Finance, and Machine Learning

Luca is the Global Head of Quantitative Strategies (QS) Credit at Credit Suisse, where he has worked since 2004. Previous to this role, he was the global head of QS for Credit and Structured Notes; he was the EMEA head and the US head of QS Global Credit Products; he worked in Commodities in New York and London, and he was part of the cross-asset modeling R&D group of QS in the London office.

Luca is also Visiting Professor at the Department of Mathematics at University College London, and an Adjunct Professor at Columbia University. His current research interests are in Credit Models, Computational Finance, and Machine Learning, with a focus on efficient numerical techniques for Derivatives Pricing and Risk Management, and applications of Adjoint Algorithmic Differentiation (AAD), which he has pioneered in Finance and Physics, and for which he holds a US Patent. Luca has published over 70 scientific papers, with the top 3 papers collecting to date over 1000 citations (h factor 27, i10 factor 49).

Prior to working in Finance, Luca was a researcher at the Kavli Institute for Theoretical Physics, Santa Barbara, California, working in High-Temperature Superconductivity and Quantum Monte Carlo methods for Condensed Matter systems. He has been awarded the Director’s fellowship at Los Alamos National Laboratory, and the Wigner Fellowship at Oak Ridge National Laboratory.

Luca holds an M.S. cum laude in Physics from the University of Florence, and an M.Phil. and a Ph.D. cum laude in Condensed Matter Theory, from the International School for Advanced Studies, Trieste.

[email protected]

Alexei Chekhlov

Capital Markets and Investments, Math Methods in Financial Price Analysis

Alexei Chekhlov is an Adjunct Assistant Professor within the Department of Mathematics. He has previously taught graduate courses such as “Mathematical Methods in Financial Price Analysis” and “Capital Markets and Investments.” Additionally, Chekhlov serves as the Head of Research and Partner at Systematic Alpha Management, LLC, and he previously worked as a research associate at Princeton University, where he conducted research on the theory of fluid turbulence. He has published repeatedly on fluid mechanics, the kinetic theory of gases, turbulence, and within the fields of applied mathematics and quantitative finance. Chekhlov earned his Ph.D. in Applied and Computational Mathematics from Princeton University.

[email protected]

Paul-Guillaume Fournié

MSc, Ecole Polytechnique, 2015 MSc, Corps des Mines, 2018

Interest Rates Models

Paul-Guillaume Fournié has been working as an Options Rates Quant at BNP Paribas’ New York office since 2019, heading the team locally since 2022. Covering both Vanilla and Exotic options, his most recent focus has been on the creation and adaptation of interest rates models to SOFR. He holds MSc in Mathematics and Physics from Ecole Polytechnique and MSc in Economics and Public Policy from Corps des Mines. Before switching to finance he occupied several positions in leading French industrial and telecommunications companies and served in France’s Ministry of the Economy.

[email protected]

Tat Sang Fung

Adjunct Professor, Department of Mathematics PhD, Columbia University, 1996 Numerical Methods in Finance and Risk Model Methodologies

Tat Sang Fung is an Adjunct Professor at Columbia University, and he currently teaches “Numerical Methods in Finances,” a graduate course required for students in the Mathematics of Finance program. His areas of expertise include Quantitative Finance and Risk Management Methodology, and Mathematics. Additionally, Fung serves as the Global Head of Risk Model Methodology at Jefferies and has held other senior positions in the finance industry at Finch Lead Inc. and Finastra. Fung earned his Ph.D. in Mathematics from Columbia University.

[email protected] | 212-854-5880 | Website

Julien guyon

Adjunct Professor, Department of Mathematics PhD, École des ponts ParisTech, 2006 Nonlinear Option Pricing

Julien Guyon is a professor of Applied Mathematics at Ecole des Ponts ParisTech, one of the oldest and one of the most prestigious French Grandes Ecoles, where he holds the BNP Paribas Chair Futures of Quantitative Finance. Before joining Ecole des Ponts, Julien worked in the financial industry for 16 years, first in the Global Markets Quantitative Research team at Societe Generale in Paris (2006-2012), then as a senior quantitative analyst in the Quantitative Research group at Bloomberg L.P., New York (2012-2022). Julien was also an adjunct professor in the Department of Mathematics at Columbia University and at the Courant Institute of Mathematical Sciences, NYU, from 2015 to 2022; and previously at Universite Paris Diderot and Ecole des Ponts ParisTech. Julien serves as an Associate Editor of Finance & Stochastics, SIAM Journal on Financial Mathematics, Quantitative Finance, and Journal of Dynamics and Games. He is also a Louis Bachelier Fellow.

Julien co-authored the book Nonlinear Option Pricing (Chapman & Hall, 2014) with Pierre Henry-Labordere. He has published more than 20 articles in peer-reviewed journals (including Finance and Stochastics, SIAM Journal on Financial Mathematics, Quantitative Finance, Risk, Journal of Computational Finance, Annals of Applied Probability, Stochastic Processes and their Applications) and is a regular speaker at international conferences, both academic and professional. His main research interests include volatility and correlation modeling, option pricing, optimal transport, and numerical probabilistic methods.

A big soccer fan, Julien has also published articles on fairness in sports both in academic journals and in top-tier newspapers including The New York Times, The Times, Le Monde, and El Pais. Some of his suggestions for draws and tournament design have been adopted by FIFA and UEFA, including a new, fairer draw method for the FIFA World Cup; a fairer format for the 2026 FIFA World Cup (in progress); a new knockout bracket for the UEFA Euro; and an optimized schedule of the UEFA Champions League. His paper “Risk of collusion: Will groups of 3 ruin the FIFA World Cup?” won the 2nd prize at the 2021 MIT Sloan Sports Analytics Conference, the biggest sports analytics event in the world.

Ioannis Karatzas

Ioannis Karatzas is the Eugene Higgins Professor of Applied Probability in Columbia’s Department of Mathematics, whose research interests include Probability and Mathematical Statistics, Random Processes, Stochastic Analysis, Optimization, and Mathematical Economics and Finance. He has served as the managing editor for the book series Applications of Mathematics and on numerous editorial boards such as “Applied Mathematics & Optimization,” “Stochastics,” the “SIAM Journal on Mathematical Analysis,” and the “SIAM Journal on Control & Optimization.” His book with Steven Shreve, Brownian Motion and Stochastic Calculus, first published in 1987, is the standard reference within the field of Stochastic Analysis. Karatzas earned his Ph.D. from Columbia University and helped build and establish this Mathematics of Finance Master’s program.

[email protected] | Website

David X. Li

Adjunct Professor, Department of Mathematics PhD, University of Waterloo, 1995 Credit analytics, Risk Management, FinTech

David X. Li currently teaches at Shanghai Advanced Institute of Finance, Shanghai Jiaotong University. Previously, he held senior positions at various leading financial institutions for more than two decades in the areas of new product development, risk management, asset/liability management and investment analytics.

David has a PhD degree in statistics from the University of Waterloo, Master’s degrees in economics, finance and actuarial science, and a bachelor’s degree in mathematics. Dr. Li was one of the early practitioners in credit derivatives. His work of using copula functions for credit portfolio modeling has been widely cited by academic researchers, broadly used by practitioners for credit portfolio trading, credit risk management and credit rating, and well covered by media such as Wall Street Journal, Financial Times, Nikkei, CBC News.

[email protected]

Bryan Liang

Adjunct Professor, Department of Mathematics PhD, University of Michigan, 2002 Non-Linear Option Pricing

Bryan Liang is an Adjunct Assistant Professor in the Department of Mathematics, specializing in Derivatives Modelling. Additionally, Liang has extensive experience as a quantitative analyst in the finance industry, currently working as a Quantitative Researcher in the Quantitative Financial Research Group for Bloomberg. Liang earned his Ph.D. in Mathematics from the University of Michigan.

Amal Moussa

Adjunct Professor, Department of Mathematics PhD, Columbia University, 2011 Modeling and Trading Derivatives

Dr. Amal Moussa is a Managing Director at Goldman Sachs, where she leads the Single Stocks Exotic Derivatives Trading desk. Prior to that, Amal held senior-level positions in equity derivatives trading at other leading financial institutions such as J.P. Morgan, UBS, and Citigroup. In addition to her work in Markets, Amal is an Adjunct Professor at Columbia University, where she teaches a graduate course on Modeling and Trading Derivatives in the Mathematics of Finance Masters program.

Amal has a Ph.D. in Statistics, obtained with distinction, from Columbia University. Her thesis “Contagion and Systemic Risk in Financial Networks” shed light on the importance of the network structure in identifying systemic financial institutions and formulating regulatory policies and has been cited by several scholars and industry professionals, including former Federal Reserve president Janet Yellen. She was also awarded the Minghui Yu Teaching Award at Columbia University. Prior to her Ph.D., Amal graduated with a Masters in Mathematical Finance from Sorbonne University (former Paris VI) and a Grande Ecole engineering degree from Télécom Paris.

Amal is a board member of Teach for Lebanon, an NGO working to ensure that all children in Lebanon have access to education regardless of socioeconomic background, and she is an active member of the Women in Trading network at Goldman Sachs.

[email protected]

Adjunct Assistant Professor, Department of Mathematics PhD, Columbia University, 1996 Programming for Quantitative & Computational Finance

Ka Yi Ng is an Adjunct Assistant Professor in the Department of Mathematics with extensive quantitative experience in FinTech. Currently, Ng works at Calypso Technology Inc. and serves as an advisor at Finch Lead Inc. Ng previously worked at Wall Street Systems and ION. At Columbia, her interests include Derivatives and Structured Products Development, and Machine Learning. Ng earned her Ph.D. in Mathematics from Columbia University.

[email protected] | Website

Lars Tyge Nielsen

Director of the Mathematics of Finance MA program PhD, Harvard University, 1985

Lars Tyge Nielsen is the Director of the Mathematics of Finance MA (MAFN) program. He was a Chaired Professor, Associate Dean, and Director of the PhD program at INSEAD, the international business school in Fontainebleau, France and Singapore and now Abu Dhabi. He has held academic appointments at Nankai University, China (Chair Professor), NYU, UT Austin, Copenhagen Business School and the University of Copenhagen. Dr. Nielsen spent 13 years on Wall Street as an executive at various institutions including Morgan Stanley and Goldman Sachs.

[email protected] | 212-854-4306 | Website

Colm O’Cinneide

Adjunct Professor, Department of Mathematics PhD, University of Kentucky, 1983 Multi-Asset Portfolio Management

Colm O’Cinneide is an adjunct professor in the Department of Mathematics. He has worked in quantitative asset allocation and portfolio construction roles for the past 20 years at Deutsche Asset Management, QS Investors, and Franklin Templeton Investments, where he is currently an SVP. He was a partner at QS investors. Prior to this, he worked in academia from 1982 to 2000 and held tenured positions in Mathematical Sciences (Statistics) at the University of Arkansas and Industrial Engineering (Operations Research) at Purdue University. He has 40+ refereed publications related to probability, statistics, numerical analysis, and finance, with 1300+ citations and a track record of National Science Foundation funding. He has a PhD in Statistics from the University of Kentucky.

[email protected]

Inna Okounkova

Editor-in-Chief of the Journal of Investment Consulting, Wealth Management eJournal, and Wealth Management Editor’s Choice eJournal; formerly Director of Investment Strategy at AB Bernstein; prior to that head of Strategic Asset Allocation Portfolio Management and partner at QS Investors, LLC and head of Strategic Asset Allocation Portfolio Management at Deutsche Asset Management; Past President and Director of the Honorary Board of the Society of Quantitative Analysts; MS in Mathematical Economics and ABD from Moscow State University; MBA from the University of Chicago (1999).

[email protected]

Rosanna Pezzo-Brizio

Rosanna Pezzo-Brizio is an Adjunct Professor in the Department of Mathematics, specializing in Fixed Income Portfolio Management. Pezzo-Brizio has a vast array of professional, senior experience, working at Goldman Sachs, Greenwich Capital Markets, Intesa Sanpaolo. Currently, she is the Director of the Investment Consulting Group at New York Life Investments. Pezzo-Brizio holds a Ph.D. in Mathematics of Finance from the University of Brescia. Additionally, she graduated from Columbia University’s Mathematics of Finance program in 1998 as one of the program’s first classes.

[email protected]

Franz Rembart

Adjunct Assistant Professor, Department of Statistics PhD, University of Oxford, 2016 Statistical Inference / Time-Series Modelling

[email protected]

Gordon Ritter

PhD, Harvard University Machine Learning for Finance

Professor Ritter is founder and CIO of Ritter Alpha LP, a registered investment adviser running systematic absolute-return trading strategies across multiple asset classes and geographical regions. Before Ritter Alpha, he was a senior portfolio manager at GSA Capital and a Vice President in the Statistical Arbitrage Group at Highbridge Capital Management (HCM). Gordon completed his PhD in mathematical physics at Harvard University and his Bachelors’ degree with honors in Mathematics from the University of Chicago. While at Harvard, he published several papers in the areas of quantum field theory, differential geometry, quantum computation and abstract algebra. His current research is on portfolio optimization and statistical machine learning. Notable publications include “Optimal turnover, liquidity, and autocorrelation,” with @Bastien Baldacci of @ Université Paris Dauphine – PSL and @Jerome Benveniste of @New York University, Risk, 2022, and “Machine learning for trading,” Risk, 2017. In recognition of the latter publication, Professor Ritter was named Buy-Side Quant of the Year in 2019.

[email protected]

Renzo Silva

Adjunct Assistant Professor, Department of Mathematics MBA, Columbia Business School Machine Learning for Finance

Renzo Silva is an Adjunct Assistant Professor in the Department of Mathematics. In addition to his teaching position, Mr. Silva has extensive professional experience in the Financial Technology industry, and he currently serves as a Software Engineering Manager at Google. Previous experiences include Software Development Manager at Amazon, CTO at P1 Capital, and Managing Director at the New York Stock Exchange. Mr. Silva is a graduate of Columbia University’s Mathematics of Finance MA program, and he also holds an MBA in Finance and Economics from Columbia Business School. His research interests include Artificial Intelligence and Machine Learning, Optimization, Simulation, and Quantum Computing.

[email protected]

Mikhail Smirnov

Mikhail Smirnov is a Senior Lecturer in Discipline in the Department of Mathematics and was Director of the Mathematics of Finance program in 1998-2012. His research interests include Quantitative Portfolio Management, Quantitative Investment Strategies, and Risk Measurement. He holds a Ph.D. from Princeton University.

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Harvey J. Stein

Dr. Harvey J. Stein left Bloomberg in March 2022 after a distinguished 28 3/4 year career where he built quant and engineering teams and introduced a variety of innovations in option pricing, computation and risk analysis. Dr. Stein is well known in the industry, having published and lectured on credit risk modeling, financial regulation, interest rate and FX modeling, CVA calculations, mortgage-backed security valuation, COVID-19 data analysis, and other subjects. Dr. Stein is on the board of directors of the IAQF, a board member of the Rutgers University Mathematical Finance program, an adjunct professor at Columbia University, and an organizer of the IAQF/Thalesians financial seminar series. He’s also worked as a quant researcher on the Bloomberg for President campaign. He received his BA in mathematics from WPI in 1982 and his PhD in mathematics from UC Berkeley in 1991. He recently started a new position as Senior VP, Quant Research at Two Sigma.

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Statistical Interference/ Time-Series Modeling

Yisha Yao is an Assistant Professor with the Department of Statistics. Her research interests span a wide range of modern high-dimensional statistics, including but not limited to nonparametric methods, statistical inferences, mixture models, FDR control, iterative algorithms, tensor analysis, and applications in biomedical sciences. She obtained her PhD in statistics from Rutgers University (New Brunswick) in 2021, and then spent two years at Yale University as a postdoctoral researcher in biostatistics.

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Adjunct Professor, Department of Mathematics AB/SM, Harvard University, 1998 Hedge Funds Strategies and Risk

Eric Yeh is an Adjunct Professor in the Department of Mathematics, specializing in Quantitative Investment Strategies. Yeh’s vast professional experience in the Finance industry includes senior positions at Morgan Stanley, Deutsche Bank, Tower Research Capital, and AllianceBernstein. Currently, he is President of Vermillion Leaf Capital LLC and an advisor to multiple investment managers, including the $100M hedge fund he previously co-founded. Yeh holds an AB in Mathematics and an SM in Computer Science from Harvard University.

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