Mathematics

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.

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

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 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.

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.

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.

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.

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.

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.

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.

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

Continuation of MATH G4151x (see Fall listing).

Continuation of MATH G4152x (see fall listing).

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

Continuation of MATH G4175x (see Fall listing).

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

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

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

Continuation of MATH G4307x (see Fall listing).

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.

Continuation of MATH G4343x (see Fall listing).

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.

Continuation of Mathematics G4401x (see Fall listing).

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.

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.

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.

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.

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 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.

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.

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.

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

Continuation of MATH G6151x (see Fall listing).

Continuation of MATH G6152x (see fall listing).

Continuation of MATH G6175x (see Fall listing).

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.

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

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

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

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

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.

Continuation of MATH G6307x (see Fall listing).

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.

Continuation of MATH G6343x (see Fall listing).

Continuation of Mathematics G6402x (see Fall listing).

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.

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.

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.

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

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

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.

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.

The 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.  

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.

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.

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.

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.

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.

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

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.

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.

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.

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.

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.