# Mathematics

The Department of Mathematics offers courses in calculus, algebra, geometry, differential equations, linear algebra, topology, number theory, and knot theory.

For questions about specific courses, contact the department.

## Mathematics Library

A comprehensive mathematics reference library is situated on the main floor of the Mathematics building.

## Courses for First-Year Students

The systematic study of mathematics begins with one of the following: Calculus I, II, III, IV (Mathematics V1101, V1102, V1201, V1202); Honors mathematics A, B (Mathematics V1207, V1208). The calculus sequence is a standard course in differential and integral calculus; it is intended for students who need calculus primarily for its applications.

Students who have no previous experience with calculus or who do not feel able to start with a second course in it should begin with Calculus I. Students who are not adequately prepared for calculus are strongly advised to begin with Mathematics W1003.

The two-term honors mathematics sequence is designed for students with strong mathematical talent and motivation. Honors Math A-B is aimed at students with a strong grasp of one-variable calculus and a high degree of mathematical sophistication. It covers linear algebra as well as several-variables calculus, and prepares students for the more advanced courses offered by the department.

Students who wish to transfer from one division of calculus to another are allowed to do so beyond the date specified in the academic calendar. They are considered to be adjusting their level, not changing their programs, but they must make the change official through the Registrar.

For questions about specific courses, contact the department.

### Courses

Intended as an enrichment to the mathemathics curriculum of the first years, this course introduces a variety of mathematical topics (such as three dimensional geometry, probability, number theory) that are often not discussed until later, and explains some current applications of mathematics in the sciences, technology and economics.

##### Course Number

MATH 2001##### Points

1##### Prerequisite

some calculus or the instructor's permission.Honors-level introductory course in enumerative combinatorics. Pigeonhole principle, binomial coefficients, permutations and combinations. Polya enumeration, inclusion-exclusion principle, generating functions and recurrence relations.

##### Course Number

MATH 2006##### Points

3Surveys the field of quantitative investment strategies from a "buy side" perspective, through the eyes of portfolio managers, analysts and investors. Financial modeling there often involves avoiding complexity in favor of simplicity and practical compromise. All necessary material scattered in finance, computer science and statistics is combined into a project-based curriculum, which give students hands-on experience to solve real world problems in portfolio management. Students will work with market and historical data to develop and test trading and risk management strategies. Programming projects are required to complete this course.

##### Course Number

MATH 4073##### Points

3##### Prerequisite

knowledge of statistics basics and programming skills in any programming language.Course covers modern statistical and physical methods of analysis and prediction of financial price data. Methods from statistics, physics and econometrics will be presented with the goal to create and analyze different quantitative investment models.

##### Course Number

MATH 4075##### Points

3Risk/return tradeoff, diversification and their role in the modern portfolio theory, their consequences for asset allocation, portfilio optimization. Capitol Asset Pricing Model, Modern Portfolio Theory, Factor Models, Equities Valuation, definition and treatment of futures, options and fixed income securities will be covered.

##### Course Number

MATH 4076##### Points

3The course will cover practical issues such as: how to select an investment universe and instruments, derive long term risk/return forecasts, create tactical models, construct and implement an efficient portfolio,to take into account constraints and transaction costs, measure and manage portfolio risk, and analyze the performance of the total portfolio.

##### Course Number

MATH 4078##### Points

3Nonlinear Option Pricing is a major and popular theme of research today in quantitative finance, covering a wide variety of topics such as American option pricing, uncertain volatility, uncertain mortality, different rates for borrowing and lending, calibration of models to market smiles, credit valuation adjustment (CVA), transaction costs, illiquid markets, super-replication under delta and gamma constraints, etc. The objective of this course is twofold: (1) introduce some nonlinear aspects of quantitative finance, and (2) present and compare various numerical methods for solving high-dimensional nonlinear problems arising in option pricing.

##### Course Number

MATH 4079##### Points

3##### Prerequisite

familiarity with Brownian motion, ItÃ´'s formula, stochastic differential equations, and Black-Scholes option pricing.The hedge fund industry has continued to grow after the financial crisis, and hedge funds are increasingly important as an investable asset class for institutional investors as well as wealthy individuals. This course will cover hedge funds from the point of view of portfolio managers and investors. We will analyze a number of hedge fund trading strategies, including fixed income arbitrage, global macro, and various equities strategies, with a strong focus on quantitative strategies. We distinguish hedge fund managers from other asset managers, and discuss issues such as fees and incentives, liquidity, performance evaluation, and risk management. We also discuss career development in the hedge fund context.

##### Course Number

MATH 4080##### Points

3The course will introduce the notions of financial risk management, review the structure of the markets and the contracts traded, introduce risk measures such as VaR, PFE and EE, overview regulation of financial markets, and study a number of risk management failures. After successfully completing the course, the student will understand the basics of computing parametric VaR, historical VaR, Monte Carlo VaR, cedit exposures and CVA and the issues and computations associated with managing market risk and credit risk. The student will be familiar with the different categories of financial risk, current regulatory practices, and the events of financial crises, especially the most recent one.

##### Course Number

MATH 4082##### Points

3##### Prerequisite

student expected to be mathematically mature and familiar with probability and statistics, arbitrage pricing theory, and stochastic processes.The course covers the fundamentals of fixed income portfolio management. Its goal is to help the students develop concepts and tools for valuation and hedging of fixed income securities within a fixed set of parameters. There will be an emphasis on understanding how an investment professional manages a portfolio given a budget and a set of limits.

##### Course Number

MATH 4083##### Points

3##### Prerequisite

comfortable with algebra, calculus, probability, statistics, and stochastic calculus.*This is a course for Ph.D. students, and for majors in Mathematics. *Measure theory; elements of probability; elements of Fourier analysis; Brownian motion.

##### Course Number

MATH 4151##### Points

4.5##### Prerequisite

the instructor's written permission.Continuation of *MATH G4151x* (see Fall listing).

##### Course Number

MATH 4152##### Points

4.5Continuation of *MATH G4152x* (see fall listing).

##### Course Number

MATH 4153##### Points

4.5##### Prerequisite

<i>MATH G4151</i> Analysis & Probability I.Topics include holomorphic functions; analytic continuation; Riemann surfaces; theta functions and modular forms.

##### Course Number

MATH 4175Continuation of *MATH G4175x* (see Fall listing).

##### Course Number

MATH 4176##### Points

4.5Commutative rings; modules; localization; primary decoposition; integral extensions; Noetherian and Artinian rings; Nullstellensatz; Dedekind domains; dimension theory; regular local rings.

##### Course Number

MATH 4261##### Points

4.5Affine and projective varieties; schemes; morphisms; sheaves; divisors; cohomology theory; curves; Riemann-Roch theorem.

##### Course Number

MATH 4263##### Points

4.5Topics include homology and homotopy theory; covering spaces; homology with local coefficients; cohomology; Chech cohomology.

##### Course Number

MATH 4307Continuation of *MATH G4307x* (see Fall listing).

##### Course Number

MATH 4308##### Points

4.5Topics include basic notions of groups with algebraic and geometric examples; symmetry; Lie algebras and groups; representations of finite and compact Lie groups; finite groups and counting principles; maximal tori of a compact Lie group.

##### Course Number

MATH 4343Continuation of *MATH G4343x* (see Fall listing).

##### Course Number

MATH 4344Manifold theory; differential forms, tensors and curvature; homology and cohomology; Lie groups and Lie algebras; fiber bundles; homotopy theory and defects in quantum field theory; geometry and string theory.

##### Course Number

MATH 4402##### Points

4.5Continuation of Mathematics G4401x (see Fall listing).

##### Course Number

MATH 4403##### Points

4.5Review of the basic numerical methods for partial differential equations, variational inequalities and free-boundary problems. Numerical methods for solving stochastic differential equations; random number generation, Monte Carlo techniques for evaluating path-integrals, numerical techniques for the valuation of American, path-dependent and barrier options.

##### Course Number

MATH 5030##### Points

3##### Prerequisite

some familiarity with the basic principles of partial differential equations, probability and stochastic processes, and of mathematical finance as provided, e.g., in <i>MATH W5010</i>.Surveys the field of quantitative investment strategies from a "buy side" perspective, through the eyes of portfolio managers, analysts and investors. Financial modeling there often involves avoiding complexity in favor of simplicity and practical compromise. All necessary material scattered in finance, computer science and statistics is combined into a project-based curriculum, which give students hands-on experience to solve real world problems in portfolio management. Students will work with market and historical data to develop and test trading and risk management strategies. Programming projects are required to complete this course.

##### Course Number

MATH 5220##### Points

3##### Prerequisite

knowledge of statistics basics and programming skills in any programming language.

##### Course Number

MATH 5240##### Points

3Risk/return tradeoff, diversification and their role in the modern portfolio theory, their consequences for asset allocation, portfilio optimization. Capitol Asset Pricing Model, Modern Portfolio Theory, Factor Models, Equities Valuation, definition and treatment of futures, options and fixed income securities will be covered.

##### Course Number

MATH 5280##### Points

3The course will introduce the notions of financial risk management, review the structure of the markets and the contracts traded, introduce risk measures such as VaR, PFE and EE, overview regulation of financial markets, and study a number of risk management failures. After successfully completing the course, the student will understand the basics of computing parametric VaR, historical VaR, Monte Carlo VaR, cedit exposures and CVA and the issues and computations associated with managing market risk and credit risk. The student will be familiar with the different categories of financial risk, current regulatory practices, and the events of financial crises, especially the most recent one.

##### Course Number

MATH 5320##### Points

3##### Prerequisite

The student is expected to be mathematically mature and to be familiar with probability and statistics, arbitrage pricing theory, and stochastic processes.The course covers the fundamentals of fixed income portfolio management. Its goal is to help the students develop concepts and tools for valuation and hedging of fixed income securities within a fixed set of parameters. There will be an emphasis on understanding how an investment professional manages a portfolio given a budget and a set of limits.

##### Course Number

MATH 5340##### Points

3##### Prerequisite

Students should be comfortable with algebra, calculus, probability, statistics, and stochastic calculus.Course covers modern statistical and physical methods of analysis and prediction of financial price data. Methods from statistics, physics and econometrics will be presented with the goal to create and analyze different quantitative investment models.

##### Course Number

MATH 5360##### Points

3The course will cover practical issues such as: how to select an investment universe and instruments, derive long term risk/return forecasts, create tactical models, construct and implement an efficient portfolio,to take into account constraints and transaction costs, measure and manage portfolio risk, and analyze the performance of the total portfolio.

##### Course Number

MATH 5380##### Points

3

##### Course Number

MATH 5900##### Points

3

##### Course Number

MATH 5920##### Points

3Review of the basic numerical methods for partial differential equations, variational inequalities and free-boundary problems. Numerical methods for solving stochastic differential equations; random number generation, Monte Carlo techniques for evaluating path-integrals, numerical techniques for the valuation of American, path-dependent and barrier options.

##### Course Number

MATH 6071##### Points

4.5##### Prerequisite

some familiarity with the basic principles of partial differential equations, probability and stochastic processes, and of mathematical finance as provided, e.g., in <i>MATH W4071</i>.Automorphic representations of GL(2). Analytical aspects of the trace formula. Applications to the principle of functoriality and Artin conjecture.

##### Course Number

MATH 6116##### Points

4.5##### Prerequisite

Lie groups and representations (G6343)and elementary Number Theory.Continuation of *MATH G6151x* (see Fall listing).

##### Course Number

MATH 6152##### Points

4.5Continuation of *MATH G6152x* (see fall listing).

##### Course Number

MATH 6153##### Points

4.5##### Prerequisite

<i>MATH G6151</i> Analysis & Probability I.Continuation of *MATH G6175x* (see Fall listing).

##### Course Number

MATH 6176##### Points

4.5A Self-contained introduction to the theory of soliton equations with an emphasis on its applications to algebraic-geometry. Topics include: 1. General features of the soliton systems. Lax representation. Zero-curvature equations. Integrals of motion. Hierarchies of commuting flows. Discrete and finite-dimensional integrable systems. 2. Algebraic-geometrical integration theory. Spectral curves. Baker-Akhiezer functions. Theta-functional formulae. 3. Hamiltonian theory of soliton equations. 4. Commuting differential operators and holomorphic vector bundles on the spectral curve. Hitchin-type systems. 5. Characterization of the Jacobians (Riemann-Schottky problem) and Prym varieties via soliton equations. 6. Perturbation theory of soliton equations and its applications.

##### Course Number

MATH 6200##### Points

4.5Topics of linear and non-linear partial differential equations of second order, with particular emphasis to Elliptic and Parabolic equations and modern approaches.

##### Course Number

MATH 6209##### Points

4.5Topics of linear and non-linear partial differential equations of second order, with particular emphasis to Elliptic and Parabolic equations and modern approaches.

##### Course Number

MATH 6210##### Points

4.5##### Prerequisite

<i>MATH G6209</i>.p-adic modular forms. p-adic Eisenstein series. Generalizations.

##### Course Number

MATH 6248##### Points

4.5

##### Course Number

MATH 6262##### Points

4.5Affine and projective varieties; schemes; morphisms; sheaves; divisors; cohomology theory; curves; Riemann-Roch theorem.

##### Course Number

MATH 6263##### Points

4.5Introduction to categorification, with one of the goals to understand the interplay between link homology and derived representation theory of nonsemisimple rings. Categorification of quantum groups and its applications to representation theory and link homology will be covered in the second half of the course.

##### Course Number

MATH 6306##### Points

4.5Continuation of *MATH G6307x* (see Fall listing).

##### Course Number

MATH 6308##### Points

4.5A one semester course covering Perelman's recent proof of the Poincare Conjecture using the Ricci flow on the space of metrics. The course will begin with a brief outline of Thurston's Geometrization Conjecture for 3-manifolds, and a brief introduction to the basics of Ricci flow as developed by Hamilton. The course will concentrate on the parts of Perelman's two papers and the Colding-Minicozzi paper needed to prove the Poincare Conjecture.

##### Course Number

MATH 6325##### Points

4.5##### Prerequisite

first year graduate course in modern geometry. First year course in analysis helpful but not required.Continuation of *MATH G6343x* (see Fall listing).

##### Course Number

MATH 6344Continuation of Mathematics G6402x (see Fall listing).

##### Course Number

MATH 6403##### Points

4.5Analytic and geometric methods in the study of partial differential equations, in particular maximum principles, Harnack inequalities, isoperimetric inequalities, formation and singularities. Emphasis on non-linear heat equations and geometric evolution equations.

##### Course Number

MATH 6428##### Points

4.5Analytic and geometric methods in the study of partial differential equations, in particular maximum principles, Harnack inequalities, isoperimetric inequalities, formation and singularities. Emphasis on non-linear heat equations and geometric evolution equations.

##### Course Number

MATH 6429##### Points

4.5##### Prerequisite

<i>MATH G6428</i>.

##### Course Number

MATH 6657##### Points

4.5

##### Course Number

MATH 8110##### Points

4.5

##### Course Number

MATH 8190##### Points

4.5A Self-contained introduction to the theory of soliton equations with an emphasis on its applications to algebraic-geometry. Topics include: 1. General features of the soliton systems. Lax representation. Zero-curvature equations. Integrals of motion. Hierarchies of commuting flows. Discrete and finite-dimensional integrable systems. 2. Algebraic-geometrical integration theory. Spectral curves. Baker-Akhiezer functions. Theta-functional formulae. 3. Hamiltonian theory of soliton equations. 4. Commuting differential operators and holomorphic vector bundles on the spectral curve. Hitchin-type systems. 5. Characterization of the Jacobians (Riemann-Schottky problem) and Prym varieties via soliton equations. 6. Perturbation theory of soliton equations and its applications.

##### Course Number

MATH 8200##### Points

4.5Topics of linear and non-linear partial differential equations of second order, with particular emphasis to Elliptic and Parabolic equations and modern approaches.

##### Course Number

MATH 8209##### Points

4.5##### Course Number

MATH 8210##### Points

4.5##### Prerequisite

<i>MATH G8209</i>.

##### Course Number

MATH 8250##### Points

4.5

##### Course Number

MATH 8273##### Points

4.5Introduction to categorification, with one of the goals to understand the interplay between link homology and derived representation theory of nonsemisimple rings. Categorification of quantum groups and its applications to representation theory and link homology will be covered in the second half of the course.

##### Course Number

MATH 8306##### Points

4.5

##### Course Number

MATH 8313##### Points

4.5

##### Course Number

MATH 8400##### Points

4.5The first part of the course is devoted to Segal's axiomatic approach to Conformal Field Theory (CFT). The second part deals with the simplest CFTs, described by vertex operator algebras. In the third part some advanced topics were covered, including the relationship of CFT with quantum groups and chiral de Rham complex with applications to mirror symmetry.

##### Course Number

MATH 8440##### Points

4.5

##### Course Number

MATH 8480##### Points

4.5

##### Course Number

MATH 8507##### Points

4.5The course will cover various topics in number theory located at the interface of p-adic Hodge theory, p-adic geometry, and the p-adic Langlands program.

##### Course Number

MATH 8674##### Points

4.5

##### Course Number

MATH 8675##### Points

4.5

##### Course Number

MATH 8811##### Points

4.5Columbia College students do not receive any credit for this course and must see their CSA advising dean. 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.

##### Course Number

MATH 1003##### Points

3##### Prerequisite

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.#### Spring 2020

##### Times/Location

Tu Th 11:40a - 12:55p407 MATHEMATICS BUILDING

##### Section/Call Number

002/12023##### Enrollment

10 of 30##### Instructor

Yier Lin#### Spring 2020

##### Times/Location

M W 6:10p - 7:25p302 BARNARD HALL

##### Section/Call Number

003/00593##### Enrollment

26 of 36##### Instructor

Lindsay Piechnik#### Fall 2020

##### Times/Location

M W 6:10p - 7:25pRoom TBA Building TBA

##### Section/Call Number

001/11290##### Enrollment

14 of 30##### Instructor

Alexander Pieloch#### Fall 2020

##### Times/Location

Tu Th 2:40p - 3:55pONLINE ONLY

##### Section/Call Number

002/11291##### Enrollment

11 of 30##### Instructor

Mrudul ThatteA more extensive treatment of the material in Math V2010, with increased emphasis on proof. Not to be taken in addition to Math V2010 or Math V1207-Math V1208.

##### Course Number

MATH 2020##### Points

3##### Prerequisite

<i>MATH V1201</i>.For specially selected mathematics majors, the opportunity to write a senior thesis on a problem in contemporary mathematics under the supervision of a faculty member.

##### Course Number

MATH 3997##### Points

3##### Prerequisite

the written permission of the faculty member who agrees to act as a supervisor, and the director of undergraduate studies' permission.For specially selected mathematics majors, the opportunity to write a senior thesis on a problem in contemporary mathematics under the supervision of a faculty member.

##### Course Number

MATH 3998##### Points

3##### Prerequisite

the written permission of the faculty member who agrees to act as a supervisor, and the director of undergraduate studies' permission.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.

##### Course Number

MATH 4043##### Points

3##### Prerequisite

<i>MATH W4041-MATH W4042</i> or the equivalent.Categories, functors, natural transformations, adjoint functors, limits and colimits, introduction to higher categories and diagrammatic methods in algebra.

##### Course Number

MATH 4046##### Points

3##### Prerequisite

<i>MATH W4041</i>.The study of algebraic and geometric properties of knots in R^3, including but not limited to knot projections and Reidemeister's 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.

##### Course Number

MATH 4052##### Points

3##### Prerequisite

MATH W4051 Topology and / or MATH W4061 Introduction To Modern Analysis I (or equivalents) \nRecommended (can be taken concurrently): MATH V2010 linear algebra, or equivalentThe mathematics of finance, principally the problem of pricing of derivative securities, developed using only calculus and basic probability. Topics include mathematical models for financial instruments, Brownian motion, normal and lognormal distributions, the BlackûScholes formula, and binomial models.

##### Course Number

MATH 4071##### Points

3##### Prerequisite

<i>MATH V1202, MATH V3027, STAT W4150, SEIOW4150</i>, or their equivalents.Concept of a differentiable manifold. Tangent spaces and vector fields. The inverse function theorem. Transversality and Sard's theorem. Intersection theory. Orientations. Poincare-Hopf theorem. Differential forms and Stoke's theorem.

##### Course Number

MATH 4081##### Points

3##### Prerequisite

<i>MATH W4051</i> or <i>MATH W4061</i> and <i>MATH V2010</i>.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 for undergraduates with no previous formal training in quantum theory. The measurement problem and issues of non-locality will be stressed.

##### Course Number

MATH 4392##### Points

3##### Prerequisite

<i>MATH V1202</i> or the equivalent, <i>MATH V2010</i>, and <i>MATH W4391</i>.Open only to students who need relevant internship/work experience in finance as part of their program of study. Final report on project required. This course may not be taken for pass/fail credit or audited.