# Mathematics

# 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:

**Departmental Office:** 410 Mathematics

212-854-2432

my@math.columbia.edu

Office Hours: Monday–Friday, 9 a.m.–5 p.m.

## 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.

**Directory of Classes**

The course information displayed on this page relies on an external system and may be incomplete. Please visit Mathematics on the Directory of Classes for complete course information.

After finding your course in the Directory of Classes, click on the section number to open an expanded view. The "Open To" field will indicate whether the course is open to School of Professional Studies students. If School of Professional Studies is not included in the field, students may still be able to cross-register for the course by obtaining permission after being admitted to an academic program.

**MATH BC2001 Perspectives in Mathematics. ***1 point*.

Prerequisites: some calculus or the instructor's permission.

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.

Fall 2018: MATH BC2001 | |||||

Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |
---|---|---|---|---|---|

MATH 2001 | 001/07748 | W 6:10pm - 7:25pm 202 Milbank Hall |
Dusa McDuff | 1 | 19/35 |

**MATH BC2006 Combinatorics. ***3 points*.

Corequisites: *MATH V2010* is helpful as a corequisite, but not required.

Honors-level introductory course in enumerative combinatorics. Pigeonhole principle, binomial coefficients, permutations and combinations. Polya enumeration, inclusion-exclusion principle, generating functions and recurrence relations.

**MATH G4073 Quantitative Methods In Investment Management. ***3 points*.

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

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.

**MATH G4075 Mathematical Methods in Financial Price Analysis. ***3 points*.

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.

**MATH G4076 Capital Markets and Investments-Quantitative Approach. ***3 points*.

Risk/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.

**MATH G4078 Theory and Practice of Multi-Asset Portfolio Management. ***3 points*.

The 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.

**MATH G4079 Non-Linear Option Pricing. ***3 points*.

Prerequisites: familiarity with Brownian motion, ItÃ´'s formula, stochastic differential equations, and Black-Scholes option pricing.

Nonlinear 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.

**MATH G4080 Hedge Funds Strategies & Risk. ***3 points*.

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.

**MATH G4082 Risk Management and Regulation. ***3 points*.

Prerequisites: student expected to be mathematically mature and familiar with probability and statistics, arbitrage pricing theory, and stochastic processes.

The 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.

**MATH G4083 Fixed Income Portfolio Management. ***3 points*.

Prerequisites: comfortable with algebra, calculus, probability, statistics, and stochastic calculus.

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.

**MATH G4151 Analysis and Probability. ***4.5 points*.

Prerequisites: the instructor's written permission.

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

**MATH G4152 Analysis, II. ***4.5 points*.

Continuation of *MATH G4151x* (see Fall listing).

**MATH G4153 Probability, II. ***4.5 points*.

Prerequisites: *MATH G4151* Analysis & Probability I.

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

**MATH G4175 Complex Analysis and Riemann Surfaces.. ***0 points*.

Topics include holomorphic functions; analytic continuation; Riemann surfaces; theta functions and modular forms.

**MATH G4176 Complex Analysis and Riemann Surfaces.. ***4.5 points*.

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

**MATH G4261 Commutative Algebra. ***4.5 points*.

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

**MATH G4263 Algebraic Geometry. ***4.5 points*.

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

**MATH G4307 Algebraic Topology. ***0 points*.

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

**MATH G4308 Algebraic Topology. ***4.5 points*.

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

**MATH G4343 Lie Groups and Representations. ***0 points*.

Topics 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.

**MATH G4344 Lie Groups and Representations. ***0 points*.

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

**MATH G4402 Modern Geometry. ***4.5 points*.

Manifold 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.

**MATH G4403 Modern Geometry. ***4.5 points*.

Continuation of Mathematics G4401x (see Fall listing).

**MATH G5240 Computational Finance. ***3 points*.

**Not offered during 2018-19 academic year.**

**MATH G5900 Problem Sem in Financial Math. ***3 points*.

**Not offered during 2018-19 academic year.**

**MATH G5920 Problem Sem in Financial Math. ***3 points*.

**Not offered during 2018-19 academic year.**

**MATH G6071 Numerical Methods In Finance. ***4.5 points*.

Prerequisites: some familiarity with the basic principles of partial differential equations, probability and stochastic processes, and of mathematical finance as provided, e.g., in *MATH W4071*.

Review 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.

**MATH G6116 The Selberg Trace Formula I. ***4.5 points*.

**Not offered during 2018-19 academic year.**

Prerequisites: Lie groups and representations (G6343)and elementary Number Theory.

Automorphic representations of GL(2). Analytical aspects of the trace formula. Applications to the principle of functoriality and Artin conjecture.

**MATH G6152 Analysis, II. ***4.5 points*.

Continuation of *MATH G6151x* (see Fall listing).

**MATH G6200 Soliton Equations (Integrable Systems). ***4.5 points*.

A 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.

**MATH G6209 Topics in Geometric Analysis. ***4.5 points*.

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

**MATH G6210 Partial Differential Equations. ***4.5 points*.

Prerequisites: *MATH G6209*.

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

**MATH G6248 p-adic Eisenstein series. ***4.5 points*.

**Not offered during 2018-19 academic year.**

p-adic modular forms. p-adic Eisenstein series. Generalizations.

**MATH G6262 Arith & Algebraic Geometry. ***4.5 points*.

**MATH G6263 Topics in Algebraic Geometry. ***4.5 points*.

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

**MATH G6306 Categorification. ***4.5 points*.

**Not offered during 2018-19 academic year.**

Introduction 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.

**MATH G6325 Topics In Geometric Topology. ***4.5 points*.

**Not offered during 2018-19 academic year.**

Prerequisites: first year graduate course in modern geometry. First year course in analysis helpful but not required.

A 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.

**MATH G6428 Topics in Partial Differential Equations I. ***4.5 points*.

Analytic 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.

**MATH G6429 Topics in Partial Differential Equations II. ***4.5 points*.

Prerequisites: *MATH G6428*.

Analytic 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.

**MATH G6657 Algebraic Number Theory. ***4.5 points*.

**MATH G8110 Planar Lattice Models. ***4.5 points*.

**Not offered during 2018-19 academic year.**

**MATH G8190 Obstacle Problems. ***4.5 points*.

**Not offered during 2018-19 academic year.**

**MATH G8200 Soliton Equations (Integrable Systems). ***4.5 points*.

A 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.

**MATH G8209 Topics in Geometric Analysis. ***4.5 points*.

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

**MATH G8250 Topics in Representation Theory. ***4.5 points*.

**Not offered during 2018-19 academic year.**

**MATH G8273 Spec Holonomy & Calibrat. ***4.5 points*.

**Not offered during 2018-19 academic year.**

**MATH G8306 Categorification. ***4.5 points*.

**Not offered during 2018-19 academic year.**

Introduction 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.

**MATH G8313 Topics in Complex Manifolds. ***4.5 points*.

**Not offered during 2018-19 academic year.**

**MATH G8400 Instanton Counting. ***4.5 points*.

**Not offered during 2018-19 academic year.**

**MATH G8440 Conformal Field Theory. ***4.5 points*.

**Not offered during 2018-19 academic year.**

The 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.

**MATH G8480 Gromov-Witten Theory. ***4.5 points*.

**Not offered during 2018-19 academic year.**

**MATH G8507 Topics in Topology. ***4.5 points*.

**Not offered during 2018-19 academic year.**

**MATH GR8675 Topics in Number Theory. ***4.5 points*.

**Not offered during 2018-19 academic year.**

Spring 2018: MATH GR8675 | |||||

Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |
---|---|---|---|---|---|

MATH 8675 | 001/69234 | M W 1:10pm - 2:25pm 407 Mathematics Building |
Eric Urban | 4.5 | 5/30 |

**MATH GR8811 Symplectic Geometry. ***4.5 points*.

**Not offered during 2018-19 academic year.**

Spring 2018: MATH GR8811 | |||||

Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |
---|---|---|---|---|---|

MATH 8811 | 001/21213 | T Th 4:10pm - 5:25pm 507 Mathematics Building |
Yoel Groman | 4.5 | 6/19 |

**MATH UN1003 College Algebra and Analytic Geometry. ***3 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.

Columbia 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.

Spring 2018: MATH UN1003 | |||||

Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |
---|---|---|---|---|---|

MATH 1003 | 001/23487 | M W 6:10pm - 8:00pm 407 Mathematics Building |
Wenhua Yu | 3 | 9/30 |

MATH 1003 | 002/68477 | T Th 12:10pm - 2:00pm C01 Knox Hall |
Daniel Gulotta | 3 | 16/30 |

Fall 2018: MATH UN1003 | |||||

Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |

MATH 1003 | 001/13703 | M W 6:10pm - 8:00pm 407 Mathematics Building |
Dmitrii Pirozhkov | 3 | 34/36 |

MATH 1003 | 002/24228 | T Th 12:10pm - 2:00pm 407 Mathematics Building |
Clara Dolfen | 3 | 31/30 |

**MATH V2020 Honors Linear Algebra. ***3 points*.

CC/GS: Partial Fulfillment of Science Requirement**Not offered during 2018-19 academic year.**

Prerequisites: *MATH V1201*.

A 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.

**MATH W4046 Introduction to Category Theory. ***3 points*.

CC/GS: Partial Fulfillment of Science Requirement**Not offered during 2018-19 academic year.**

Prerequisites: *MATH W4041*.

Categories, functors, natural transformations, adjoint functors, limits and colimits, introduction to higher categories and diagrammatic methods in algebra.

**MATH W4051 Topology. ***3 points*.

CC/GS: Partial Fulfillment of Science Requirement, BC: Fulfillment of General Education Requirement: Quantitative and Deductive Reasoning (QUA).

Prerequisites: *MATH V1202, MATH V2010,* and rudiments of group theory (e.g., *MATH W4041*). *MATH V1208* or *MATH W4061* 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 W4052 Introduction to Knot Theory. ***3 points*.

CC/GS: Partial Fulfillment of Science Requirement

Prerequisites: MATH W4051 Topology and / or MATH W4061 Introduction To Modern Analysis I (or equivalents)
\nRecommended (can be taken concurrently): MATH V2010 linear algebra, or equivalent

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.

**MATH W4071 Introduction to the Mathematics of Finance. ***3 points*.

CC/GS: Partial Fulfillment of Science Requirement, BC: Fulfillment of General Education Requirement: Quantitative and Deductive Reasoning (QUA).

Prerequisites: *MATH V1202, MATH V3027, STAT W4150, SEIOW4150*, or their equivalents.

The 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.

**MATH W4391 Intro to Quantum Mechanics: An Introduction for Mathematicians and Physicists I. ***3 points*.

CC/GS: Partial Fulfillment of Science Requirement**Not offered during 2018-19 academic year.**

Prerequisites: *MATH V1202* or the equivalent and *MATH V2010*.

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.

**MATH W4392 Quantum Mechanics: An Introduction for Mathematicians and Physicists II. ***3 points*.

**Not offered during 2018-19 academic year.**

Prerequisites: *MATH V1202* or the equivalent, *MATH V2010*, and *MATH W4391*.

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.

**MATH W4777 . ***3 points*.

Prerequisites: the instructor's written approval and adequate preparation in mathematical finance.

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.