# Course Descriptions: AY 2014-15

Algebra and Number
Theory

Geometry and Topology

Analysis

Numerical Analysis

Applied Mathematics

Probability and
Statistics

All course descriptions are subject to change

**MATH-GA 2110.001,
2120.001 LINEAR ALGEBRA I, II**

3 points per
term. Fall and spring terms.

Tuesday, 5:10-7:00 (fall); Monday, 5:10-7:00 (spring), F.
Greenleaf.

Fall Term

Prerequisites: Undergraduate linear algebra or permission of the instructor.

Linear spaces, subspaces. Linear dependence, linear independence; span, basis, dimension, isomorphism. Quotient spaces. Linear functionals, Dual spaces. Linear mappings, null space, range, fundamental theorem of linear algebra. Underdetermined systems of linear equations. Composition, inverse, transpose of linear maps, algebra of linear maps. Similarity transformations. Matrices, matrix Multiplication, Matrix Inverse, Matrix Representation of Linear Maps, Determinant, Laplace expansion, Cramer's rule. Eigenvalue problem, eigenvalues and eigenvectors, characteristic polynomial, Cayley-Hamilton theorem. Diagonalization.

Text: Friedberg, S.H., Insel, A.J., & Spence, L.E. (2003).Linear Algebra(4^{th}ed.). Upper Saddle River, NJ: Prentice-Hall/ Pearson Education.

Recommended Text: Lipschutz, S., & Lipson, M. (2012).Schaum's Outlines[Series].Schaum's Outline of Linear Algebra(5^{th}ed.). New York, NY: McGraw-Hill.

Note: Extensive lecture notes keyed to these texts will be issued by the instructor.

Spring Term

Prerequisites: Linear Algebra I or permission of the instructor.

Review of: Trace, determinant, characteristic and minimal polynomial. Eigenvalues, diagonalization, spectral theorem and spectral mapping theorem. When diagonalization fails: nilpotent operators and their structure, generalized eigenspace decomposition, Jordan canonical form. Polar and singular value decompositions. Complexification and diagonalization over R. Matrix norms, series and the matrix exponential map, applications to ODE. Bilinear and quadratic forms and their normal forms. The classical matrix groups: unitary, orthogonal, symplectic. Implicit function theorem, smooth surfaces in R^n and their tangent spaces. Intro to matrix Lie algebras and Lie groups.

Text: Friedberg, S.H., Insel, A.J., & Spence, L.E. (2003).Linear Algebra(4^{th}ed.). Upper Saddle River, NJ: Prentice-Hall/ Pearson Education.

Note: Extensive lecture notes will be issued by the instructor.

Cross-listing: MATH-UA 0141.001, 0142.001.

**MATH-GA 2110.001 LINEAR
ALGEBRA I**

3 points. Spring term.

Tuesday, 5:10-7:00,
Instructor TBA.

Prerequisites: Undergraduate Linear Algebra or permission of the instructor.

Linear spaces, subspaces. Linear dependence, linear independence; span, basis, dimension, isomorphism. Quotient spaces. Linear functionals, dual spaces. Linear mappings, null space, range, fundamental theorem of linear algebra. Underdetermined systems of linear equations. Composition, inverse, transpose of linear maps, algebra of linear maps. Similarity transformations. Matrices, matrix Multiplication, Matrix Inverse, Matrix Representation of Linear Maps determinant, Laplace expansion, Cramer's rule. Eigenvalue problem, eigenvalues and eigenvectors, characteristic polynomial, Cayley-Hamilton theorem. Diagonalization.

Text: Lax, P.D. (2007).Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts[Series, Bk. 78].Linear Algebra and Its Applications(2^{nd}ed.). Hoboken, NJ: John Wiley & Sons/ Wiley-Interscience.

**MATH-GA 2111.001 LINEAR ALGEBRA
(one-term format)**

3 points. Fall term.

Thursday, 9:00-10:50, G. Stadler.

Prerequisites: Undergraduate linear algebra.

Linear algebra is two things in one: a general methodology for solving linear systems, and a beautiful abstract structure underlying much of mathematics and the sciences. This course will try to strike a balance between both. We will follow the book of our own Peter Lax, which does a superb job in describing the mathematical structure of linear algebra, and complement it with applications and computing. The most advanced topics include spectral theory, convexity, duality, and various matrix decompositions.

Text: Lax, P.D. (2007).Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts[Series, Bk. 78].Linear Algebra and Its Applications(2^{nd}ed.). Hoboken, NJ: John Wiley & Sons/ Wiley-Interscience.

Recommended Text: Strang, G. (2005).Linear Algebra and Its Applications(4^{th}ed.). Stamford, CT: Cengage Learning.

**MATH-GA 2130.001, 2140.001 ALGEBRA I, II**

3 points per term. Fall and spring terms.

Thursday, 7:10-9:00, Y.
Tschinkel (fall); Monday, 7:10-9:00, F.
Bogomolov (spring).

__Prerequisites__: Elements of linear algebra and the theory of rings and fields.

Fall TermFall Term

Basic concepts of groups, rings and fields. Symmetry groups, linear groups, Sylow theorems; quotient rings, polynomial rings, ideals, unique factorization, Nullstellensatz; field extensions, finite fields.

Recommended Texts: Artin, M. (2010).Featured Titles for Abstract Algebra[Series].Algebra(2nd ed.). Upper Saddle River, NJ: Pearson.

Chambert-Loir, A. (2004).Undergraduate Texts in Mathematics[Series].A Field Guide to Algebra(2005 ed.). New York, NY: Springer-Verlag.

Serre, J-P. (1996).Graduate Texts in Mathematics[Series, Vol. 7].A Course in Arithmetic(Corr. 3rd printing 1996 ed.). New York, NY: Springer-Verlag.

Spring Term

Representations of finite groups. Characters, orthogonality of characters of irreducible representations, a ring of representations. Induced representations, Artin's theorem, Brauer's theorem. Representations of compact groups and the Peter-Weyl theorem. Lie groups, examples of Lie groups, representations and characters of Lie group. Lie algebras associated with Lie groups. Applications of the group representations in algebra and physics. Elements of algebraic geometry.

Text: Fulton, W., Harris, J. (2008).Graduate Texts in Mathematics/ Readings in Mathematics[Series, Bk.129].Representation Theory: A First Course(Corrected ed.). New York, NY: Springer-Verlag.

Recommended Texts: Lang, S. (2005).Graduate Texts in Mathematics[Series, Bk. 211].Algebra(3^{rd}ed.). New York, NY: Springer-Verlag.

Serre, J.P. (1977).Graduate Texts in Mathematics[Series, Bk. 42].Linear Representations of Finite Groups. New York, NY: Springer-Verlag.

Reid, M. (1989).London Mathematical Society Student Texts[Series].Undergraduate Algebraic Geometry. New York, NY: Cambridge University Press.

James, G., & Liebeck, M. (1993).Cambridge Mathematical Textbooks[Series].Representations and Characters of Groups. New York, NY: Cambridge University Press.

Artin, M. (2010).Algebra(2^{nd}ed.). Upper Saddle River, NJ: Prentice-Hall/ Pearson Education.

Sagan, B.E. (1991).Wadsworth Series in Computer Information Systems[Series].The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions. Pacific Grove, CA: Wadsworth & Brooks/Cole.

Brocker, T., & Dieck, T. (2003).Graduate Texts in Mathematics[Series, Bk. 98].Representations of Compact Lie Groups. New York, NY: Springer-Verlag.

**MATH-GA 2170.001 INTRODUCTION TO CRYPTOGRAPHY**

3 points. Fall term.

Monday, 11:00-12:50, O.Regev.

Prerequisites: Strong mathematical background.

The primary focus of this course will be on definitions and constructions of various cryptographic objects, such as pseudo-random generators, encryption schemes, digital signature schemes, message authentication codes, block ciphers, and others time permitting. We will try to understand what security properties are desirable in such objects, how to properly define these properties, and how to design objects that satisfy them. Once we establish a good definition for a particular object, the emphasis will be on constructing examples that provably satisfy the definition. Thus, a main prerequisite of this course is mathematical maturity and a certain comfort level with proofs. Secondary topics that we will cover only briefly will be current cryptographic practice and the history of cryptography and cryptanalisys.

Cross-listing: CSCI-GA 3210.001.

**MATH-GA 2210.001 ELEMENTARY
NUMBER THEORY**

3 points. Spring term.

Wednesday, 7:10-9:00, S. Marques.

Prerequisites: Undergraduate elementary number theory, abstract algebra, including groups, rings and ideals, fields, and Galois theory (e.g. undergraduate Algebra I and II).

The course is a graduate-level introduction to algebraic number theory, in which we will cover fundamentals of the subject. Topics include: rings of integers, Dedekind domains, factorization of prime ideals, ramification theory, Minkowski's theorum, the theory of the valuation, and more.

Text: Neukirch, J. (1999).Grundlehren der mathematischen Wissenschaften[Series, Book 322]. Algebraic Number Theory. New York, NY: Springer-Verlag.

**MATH-GA 2310.001, 2320.001 TOPOLOGY I, II**

3 points per
term. Fall and spring terms.

Tuesday, 7:10-9:00 (fall); Tuesday, 7:10-9:00 (spring), S.
Cappell.

Fall Term

Prerequisites: Any knowledge of groups, rings, vector spaces and multivariable calculus is helpful. Undergraduate students planning to take this course must have V63.0343 Algebra I or permission of the Department.

After introducing metric and general topological spaces, the emphasis will be on the algebraic topology of manifolds and cell complexes. Elements of algebraic topology to be covered include fundamental groups and covering spaces, homotopy and the degree of maps and its applications. Some differential topology will be introduced including transversality and intersection theory. Some examples will be taken from knot theory.

Recommended Texts: Hatcher, A. (2002).Algebraic Topology. New York, NY: Cambridge University Press.

Munkres, J. (2000).Topology(2^{nd}ed.). Upper Saddle River, NJ: Prentice-Hall/ Pearson Education.

Guillemin, V., and Pollack, A. (1974).Differential Topology. Englewood Cliffs, NJ: Prentice-Hall.

Milnor, J.W. (1997).Princeton Landmarks in Mathematics[Series].Topology from a Differentiable Viewpoint(Rev. ed.). Princeton, NJ: Princeton University Press.

Spring Term

Homology and cohomology from simplicial, singular, cellular, axiomatic and differential form viewpoints. Axiomatic characterizations and applications to geometrical problems of embedding and fixed points. Manifolds and Poincar duality. Products and ring structures. Vector bundles, tangent bundles, DeRham cohomology and differential forms.

**MATH-GA 2350.001, 2360.001 DIFFERENTIAL GEOMETRY I, II**

3 points per
term. Fall and spring terms.

Monday, 1:25-3:15, J. Cheeger.

Fall Term

Differentiable manifolds, tangent bundle, embedding theorems, vector fields and differential forms. Introduction to Riemannian metrics, connections and geodesics.

Text: Lee, J.M. (2009).Graduate Studies in Mathematics[Series, Vol. 107].Manifolds and Differential Geometry. Providence, RI: American Mathematical Society.

Spring Term

Course description not yet available.

**MATH-GA 2400.001 ADVANCED TOPICS IN GEOMETRY (High Dimensional Expanders and Ramanujan Complexes)**

3 points. Fall term.

Monday, 5:10-7:00, A. Lubotzky.

Expander graphs, in general, and Ramanujan graphs, in particular, have played an important role in computer science in the last four decades. In the last decay they also found more and more applications in pure mathematics. Recently a high dimensional theory is starting to emerge. Namely, simplical complexes (and/or hypergraphs) extending the classical graphs. In the course, we will start by going over some of the classical theory of expander graphs; their constructions and applications- in a form that will pave the way to present the high dimensional theory. Along the way we will present various pure math concepts (like buildings etc.) but also applications to computer science (such as error correcting codes etc.). The course should be suitable (and hopefully of interest) for graduate students in either pure math or computer science. A good number of suggestions for future research projects will be presented.

**MATH-GA 2410.001 ADVANCED
TOPICS IN GEOMETRY (Topic TBA)**

3 points.
Spring term.

Thursday, 1:25-3:15, M. Gromov.

Course description not yet available.

**MATH-GA 2410.002 ADVANCED TOPICS IN GEOMETRY (Randomness and Complexity)**

3 points. Spring term.

Wednesday, 1:25-3:15, R. Young.

Prerequisites: Some topology (fundamental groups, etc.) and geometry (Riemannian metrics).

The probabilistic method is a powerful way to construct graphs, manifolds, and spaces with unusual, often unintuitive properties. In fact, many theorems show that unusual properties are in fact overwhelmingly common. Why is randomness so powerful, and why is it that the properties that are the most common are frequently the least intuitive? In this course, we will study applications of randomness and complexity in geometry, see how some of the resulting strange properties can be tamed through embeddings in R^n and other metric spaces, and explore how randomness can expand our geometric intuition.

Examples of topics we may cover include: the Erdos-Renyi random graph; the geometry of random surfaces; randomness in higher dimensions; the geometry of the moduli space of Riemannian metrics; Gromov's systolic inequality; spectral bounds on metrics on the 2-sphere; inequalities for embedded surfaces.

MATH-GA 1002.001 MULTIVARIABLE
ANALYSIS

3 points. Spring term.

Monday, 7:10-9:00, E. Hameiri.

Differentiation and integration for vector-valued functions of one and several variables: curves, surfaces, manifolds, inverse and implicit function theorems, integration on manifolds, Stokes' theorem, applications.

**MATH-GA 1410.001, 1420.001 INTRODUCTION TO
MATHEMATICAL ANALYSIS I, II**

3 points per
term. Fall and spring terms.

Monday, 5:10-7:00, R.
Haslhofer (fall); Thursday, 5:10-7:00, J. Shatah
(spring).

Fall Term

Elements of topology on the real line. Rigorous treatment of limits, continuity, differentiation, and the Riemann integral. Taylor series. Introduction to metric spaces. Pointwise and uniform convergence for sequences and series of functions. Applications.

Recommended Text: Johnsonbaugh, R., & Pfaffenberger, W.E. (2010).Dover Books on Mathematics[Series].Foundations of Mathematical Analysis. Mineola, NY: Dover Publications.

Spring Term

Measure theory and Lebesgue integration on the Euclidean space. Convergence theorems. L^p spaces and Hilbert spaces. Fourier series. Introduction to abstract measure theory and integration.

Recitation/ Problem Session: 7:15-8:30 (following the course in both terms).

**MATH-GA 2430.001 REAL VARIABLES
(one-term format)
**

3 points. per term. Fall term.

Mondays, Wednesdays, 9:35-10:50, P. Deift.

Note: Master's students need permission of course instructor before registering for this course.

Prerequisites: A familiarity with rigorous mathematics, proof writing, and the epsilon-delta approach to analysis, preferably at the level of MATH-GA 1410, 1420 Introduction to Mathematical Analysis I, II.

Measure theory and integration. Lebesgue measure on the line and abstract measure spaces. Absolute continuity, Lebesgue differentiation, and the Radon-Nikodym theorem. Product measures, the Fubini theorem, etc. L^p spaces, Hilbert spaces, and the Riesz representation theorem. Fourier series.

Text:Royden, H.L. (1988).Real Analysis(3^{rd}ed.). Englewood Cliffs, NJ: Prentice-Hall.

Recommended Texts: Kolmogorov, A.N., & Fomin, S.V. (1975).Introductory Real Analysis. Mineola, NY: Dover Publications.

Rudin, W. (1986).International Series in Pure and Applied Mathematics[Series].Real and Complex Analysis(3^{rd}ed.). New York, NY: McGraw-Hill.

Folland, G.B. (1999).Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts[Series, Bk. 40].Real Analysis: Modern Techniques and Their Applications(2^{nd}ed.) . New York, NY: John Wiley & Sons/ Wiley-Interscience.

**MATH-GA 2450.001,
2460.001
COMPLEX VARIABLES I, II**

3 points per term. Fall and spring terms.

Thursday, 5:10-7:00, O. Widlund (fall); Wednesday, 5:10-7:00 (spring), E. Hameiri.

Fall Term

Prerequisites (Complex Variables I): Advanced calculus (or equivalent).

Complex numbers; analytic functions, Cauchy-Riemann equations; linear fractional transformations; construction and geometry of the elementary functions; Green's theorem, Cauchy's theorem; Jordan curve theorem, Cauchy's formula; Taylor's theorem, Laurent expansion; analytic continuation; isolated singularities, Liouville's theorem; Abel's convergence theorem and the Poisson integral formula.

Text: Brown, J., & Churchill, R. (2008).Complex Variables and Applications(8^{th}ed.). New York, NY: McGraw-Hill.

Spring Term

Prerequisites (Complex Variables II): Complex Variables I (or equivalent).

The fundamental theorem of algebra, the argument principle; calculus of residues, Fourier transform; the Gamma and Zeta functions, product expansions; Schwarz principle of reflection and Schwarz-Christoffel transformation; elliptic functions, Riemann surfaces; conformal mapping and univalent functions; maximum principle and Schwarz's lemma; the Riemann mapping theorem.

Text: Ahlfors, L. (1979).International Series in Pure and Applied Mathematics[Series, Bk. 7].Complex Analysis(3^{rd}ed.). New York, NY: McGraw-Hill.

Cross-listings: MATH-UA 0393.001, 0394.001.

**MATH-GA 2451.001 COMPLEX
VARIABLES (one-term format)**

3 points. Fall term.

Tuesday, Thursday, 1:25-2:40, F. Hang.

Note: Master's students need permission of course instructor before registering for this course.

Prerequisites: Complex Variables I (or equivalent) and MATH-GA 1410 Introduction to Math Analysis I.

Complex numbers, the complex plane. Power series, differentiability of convergent power series. Cauchy-Riemann equations, harmonic functions. conformal mapping, linear fractional transformation. Integration, Cauchy integral theorem, Cauchy integral formula. Morera's theorem. Taylor series, residue calculus. Maximum modulus theorem. Poisson formula. Liouville theorem. Rouche's theorem. Weierstrass and Mittag-Leffler representation theorems. Singularities of analytic functions, poles, branch points, essential singularities, branch points. Analytic continuation, monodromy theorem, Schwarz reflection principle. Compactness of families of uniformly bounded analytic functions. Integral representations of special functions. Distribution of function values of entire functions.

Text: Ahlfors, L. (1979).International Series in Pure and Applied Mathematics[Series, Bk. 7].Complex Analysis(3^{rd}ed.). New York, NY: McGraw-Hill.

**MATH-GA 2470.001 ORDINARY
DIFFERENTIAL EQUATIONS**

3 points. Spring term.

Tuesday, 9:00-10:50, N. Masmoudi.

Prerequisites: Undergraduate background in analysis, linear algebra and complex variable..

Existence and uniqueness of initial value problems. Linear equations. Complex analytic equations. Boundary value problems and Sturm-Liouville theory. Comparison theorem for second order equations. Asymptotic behavior of nonlinear systems. Perturbation theory and Poincar-Bendixson theorems.

Recommended Text: Teschl, G. (2012).Graduate Studies in Mathematics[Series, Vol. 140].Ordinary Differential Equations and Dynamical Systems. Providence, RI: AMS Chelsea Publishing/ American Mathematical Society.

**MATH-GA 2490.001, 2500.001 PARTIAL DIFFERENTIAL
EQUATIONS I, II**

3 points per term. Fall and spring terms.

Tuesday, 5:10-7:00, R. Kohn (fall); Tuesday, 9:00-10:50, P. Germain (spring).

Note: Master's students should consult course instructor before registering for PDE II in the spring.

Fall Term

Prerequisites: Knowledge of undergraduate level linear algebra and ODE; also some exposure to complex variables (can be taken concurrently).

A basic introduction to PDEs, designed for a broad range of students whose goals may range from theory to applications. This course emphasizes examples, representation formulas, and properties that can be understood using relatively elementary tools. We will take a broad viewpoint, including how the equations we consider emerge from applications, and how they can be solved numerically. Topics will include: the heat equation; the wave equation; Laplace's equation; conservation laws; and Hamilton-Jacobi equations. Methods introduced through these topics will include: fundamental solutions and Green's functions; energy principles; maximum principles; separation of variables; Duhamel's principle; the method of characteristics; numerical schemes involving finite differences or Galerkin approximation; and many more. For more information (including a tentative semester plan) see the syllabus here.

Recommended Texts: Guenther, R.B., & Lee, J.W. (1996).Partial Differential Equations of Mathematical Physics and Integral Equations. Mineola, NY: Dover Publications.

Evans, L.C. (2010).Graduate Studies in Mathematics[Series, Bk. 19].Partial Differential Equations(2^{nd}ed.). Providence, RI: American Mathematical Society.

Spring Term

Prerequisites: MATH-GA 2490.001 PDE I and MATH-GA 2430.001 Real Variables, or the equivalent.

This course is a continuation of MATH-GA 2490 and is designed for students who are interested in analysis and PDEs. The course gives an introduction to Sobolev spaces, Holder spaces, and the theory of distributions; elliptic equations, harmonic functions, boundary value problems, regularity of solutions, Hilbert space methods, eigenvalue problems; general hyperbolic systems, initial value problems, energy methods; linear evaluation equations, Fourier methods for solving initial value problems, parabolic equations, dispersive equations; nonlinear elliptic problems, variational methods; Navier-Stokes and Euler equations.

Recommended Texts: Garabedian, P.R. (1998).Partial Differential Equations(2^{nd}Rev. ed.). Providence, RI: AMS Chelsea Publishing/ American Mathematical Society.

Evans, L.C. (2010).Graduate Studies in Mathematics[Series, Bk. 19].Partial Differential Equations(2^{nd}ed.). Providence, RI: American Mathematical Society.

John, F. (1995).Applied Mathematical Sciences[Series, Vol. 1].Partial Differential Equations(4th ed.). New York, NY: Springer-Verlag.

**MATH-GA 2550.001 FUNCTIONAL
ANALYSIS**

3 points. Spring term.

Monday, 9:00-10:50, J. Shatah.

Prerequisites: Linear algebra, real variables or the equivalent, and some complex function-theory would be helpful.

The course will concentrate on concrete aspects of the subject and on the spaces most commonly used in practice such as Lp (1? p ? ?), C, C?, and their duals. Working knowledge of Lebesgue measure and integral is expected. Special attention to Hilbert space (L2, Hardy spaces, Sobolev spaces, etc.), to the general spectral theorem there, and to its application to ordinary and partial differential equations. Fourier series and integrals in that setting. Compact operators and Fredholm determinants with an application or two. Introduction to measure/volume in infinite-dimensional spaces (Brownian motion). Some indications about non-linear analysis in an infinite-dimensional setting. General theme: How does ordinary linear algebra and calculus extend to d=? dimensions?

Recommended Texts: Lax, P.D. (2002).Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts[Series, Bk. 55].Functional Analysis(1^{st}ed.). New York, NY: John Wiley & Sons/ Wiley-Interscience.

Reed, M., & Simon, B. (1972).Methods of Modern Mathematical Physics[Series, Vol. 1].Functional Analysis(1^{st}ed.).New York, NY: Academic Press.

**MATH-GA 2563.001 HARMONIC
ANALYSIS**

3 points.
Fall term.

Wednesday, 9:00-10:50, S. Güntürk.

Prerequisites: Real analysis; basic knowledge of complex variables and functional analysis.

Classical Fourier Analysis: Fourier series on the circle, Fourier transform on the Euclidean space, Introduction to Fourier transform on LCA groups. Stationary phase. Topics in real variable methods: maximal functions, Hilbert and Riesz transforms and singular integral operators. Time permitting: Introduction to Littlewood-Paley theory, time-frequency analysis, and wavelet theory.

Recommended Text: Muscalu, C., & Schlag, W. (2013).Cambridge Studies in Advanced Mathematics[Series, Bk. 137].Classical and Multilinear Harmonic Analysis(Vol.1). New York, NY: Cambridge University Press. (Online version available to NYU users through Cambridge University Press.)

**MATH-GA 2610.001 ADVANCED TOPICS IN PDE (Resonances in Partial Differential Equations)**

3 points. Fall term.

Tuesday, 9:00-10:50, J. Shatah.

Prerequisite: Good knowledge of PDEs and Fourier analysis at the level of MATH-GA 2490.001, 2500.001 and 2430.001 classes.

In this class, we will study the effects of resonances on the long-time behavior of solutions to weakly nonlinear systems. To illustrate the type of phenomenon that we will study, consider as an example the nonlinear Schrödinger equation:iuΔ_{t}-u=f(u); u(0) =ϵu. A basic question in the weakly nonlinear regime (ϵ<<1) is the effect of the nonlinearity on wave propagation: Do solutions exist globally? If yes, what is their asymptotic behavior? For what time scale Tϵ can we describe the solutions? Is there a reduced system that approximates the dynamics of solutions?_{0}

It turns out that resonances play a fundamental role in providing answers to these questions for dispersive systems, as was shown for nonlinear Klein-Gordon equations, Schrödinger equations, and water wave systems, to name a few. In this class, we will: 1) Start by reviewing resonances in ODEs and their impact on solutions. We will also review the method of averaging in ODEs and its relation to normal forms. 2) We will define resonances (or time resonances) for PDEs and study some of their impacts on solutions. 3) We will introduce the concept of space resonances and illustrate how they influence the long-time behavior of small solutions. 4) Time permitting, we will study equations with periodic boundary conditions and discuss the relation between resonances and the largebox limit, i.e., letting the period go to ∞.

**MATH-GA 2650.001 ADVANCED TOPICS IN ANALYSIS (Dynamical Systems)**

3 points. Fall term.

Monday, 5:10-7:00, L. Young

Prerequisite: Real analysis at the graduate level.

This is the first semester of a year course on dynamical systems. It is an introductory sequence, requiring no prior knowledge of the subject. In the fall semester, I will cover mostly ergodic theory, a probabilistic approach to dynamical systems. Topics include ergodicity, mixing properties, entropy; ergodic theory of continuous and differentiable maps, Lyapunov exponents etc. In the spring semester, the focus will be on differentiable dynamical systems. Topics include invariant manifolds, hyperbolicity, and various models of chaotic systems.

Recommended Text: Walters, P. (2000).Graduate Texts in Mathematics[Series, Bk. 79].An Introduction to Ergodic Theory. New York, NY: Springer-Verlag.

**MATH-GA 2650.002 ADVANCED TOPICS IN ANALYSIS (Calculus of Variations)**

3 points. Fall term.

Monday, 9:00-10:50, G. Francfort.

Prerequisites: The prerequisites for this course are essentially PDE 2.

This course focuses on that part of the Calculus of Variations which is concerned with the search for minimizers for an integral functional. We do not discuss critical points and barely touch regularity. Also, we do not strive for the most general context, but rather illustrate the various concepts through examples coming mostly from the mechanics of materials.

After a short review of the basic tenet of the direct method, we look at convexity and duality as a paradigm for the existence theory of minimizers, and convexiﬁcation as a paradigm for the lack of existence of such minimizers. We then introduce quasi-convexity and quasi-convexiﬁcation/relaxation in a vectorial context. We prove higher integrability of minimizers in a standard setting. For functionals that are parameter dependent, we discuss Gamma-convergence and introduce homogenization of periodic structures within that context.

We ﬁnally introduce minimizing movements as an alternative to gradient ﬂows for many mechanical systems and illustrate those in a setting — damage — which will use the tools developed in the earlier part of the course.

Recommended Text: Braides, A., & Defranceschi, A. (1999).Oxford Lecture Series in Mathematics and Its Applications[Series, Bk. 12].Homogenization of Multiple Integrals. New York, NY: Oxford University Press.

Other sources will be indicated during class.

**MATH-GA 2660.001 ADVANCED
TOPICS IN ANALYSIS (Introduction to Riemann-Hilbert Theory)**

3 points. Spring term.

Thursday, 1:25-3:15, P. Deift.

Prerequisites: Functional Analysis, Complex Analysis

The course will be in two parts:

1. General theory of Riemann-Hilbert Problems

2. Applications of Riemann-Hilbert theory to problems in integrable systems, random matrix theory, Toeplitz determinants, orthogonal polynomials and also combinatorics.

**MATH-GA 2660.002 ADVANCED TOPICS IN ANALYSIS (Topic TBA)**

Course description not yet available.

**MATH-GA 2660.003 ADVANCED TOPICS IN ANALYSIS (Regularity Theory for Free Boundary Problems)**

3 points. Spring term.

Monday, Wednesday, 9:00-10:50, F. Lin.Prerequisites: Basic Elliptic PDEs, Real Variables, and Sobolev Spaces.

The goal of this course is to present a regularity theory for several classical free boundary problems including obstacle problems (thin or thick obstacles), phase transition problems and some geometric variational problems.

Reference texts to be provided by instructor.

Note: Course will run from 1/26 to 3/12/2015.

**MATH-GA 2660.004 ADVANCED TOPICS IN ANALYSIS (Inequalities)**

3 points. Spring term.

Thursday, 9:00-10:50, N. Masmoudi.Prerequisites: Real Variables, familiarity with PDEs and Sobolev Spaces.

Functional inequalities are one the most important tools in analysis (and in mathematics in general). The goal of this course is to review some on the major inequalities that analysts use every day : Sobolev, Hardy, Moser-Trudinger, Gagliardo-Nirenberg, HLS… We will also look at some of the more recent advances using mass transportation techniques, stability, and some applications of these inequalities in the study of long time behavior of PDEs...Registered students will be asked to make a presentation about a topic of their choice.

**MATH-GA 2010.001,
2020.001
NUMERICAL METHODS I, II**

3 points per term. Fall and spring terms.

Thursday, 5:10-7:00 A. Donev (fall); Thursday, 5:10-7:00, O. Widlund (spring).

Fall Term

Prerequisites: A good background in linear algebra, and experience with writing computer programs (in MATLAB, Python, Fortran, C, C++, or other language). Prior knowledge of MATLAB is not required, but it will be used as the main language for the course. You encouraged but not required to learn and use a compiled language.

This course is part of a two-course series meant to introduce Ph.D. students in mathematics to the fundamentals of numerical mathematics (but any Ph.D. student seriously interested in applied mathematics should take it). It will be a demanding course covering a broad range of topics, both theoretically and with extensive homework assignments. There will be a final take-home exam examining a topic of relevance not covered in the class. Topics covered in the class include floating-point arithmetic, linear systems, eigenvalue problems, interpolation and quadrature (approximation theory), nonlinear systems of equations, optimization, Fourier transforms, and Monte Carlo methods. This course will not cover differential equations, which form the core of the second part of this series, Numerical Methods II.

Recommended Texts: Quarteroni, A., Sacco, R., & Saleri, F. (2006).Texts in Applied Mathematics[Series, Bk. 37].Numerical Mathematics(2nd ed.). New York, NY: Springer-Verlag.

Gander, W., Gander, M.J., & Kwok, F. (2014).Texts in Computation Science and Engineering[Series, Vol. 11].Scientific Computing – An Introduction Using Maple and MATLAB. New York, NY: Springer-Verlag.

Corless, R.M., & Fillion, N. (2014).A Graduate Introduction to Numerical Methods: From the Viewpoint of Backward Error Analysis. New York, NY: Springer-Verlag.

Bau III, D., & Trefethen, L.N. (1997).Numerical Linear Algebra. Philadelphia, PA: Society for Industrial & Applied Mathematics. (Available on reserve at the Courant Library.)

Cross-listing: CSCI-GA 2420.001.

Spring Term

Prerequisites: Numerical linear algebra, elements of ODE and PDE.

This course will cover fundamental methods that are essential for numerical solution of differential equations. It is intended for students familiar with ODE and PDE and interested in numerical computing; computer programming assignments form an essential part of the course. The course will introduce students to numerical methods for (1) nonlinear equations, Newton's method; (2) ordinary differential equations, Runge-Kutta and multistep methods, convergence and stability; (3) finite difference and finite element methods; (4) fast solvers, multigrid method; and (5) parabolic and hyperbolic partial differential equations.

Text: LeVeque, R. (2007).Classics in Applied Mathematics[Series].Finite Difference Methods for Ordinary and Partial Differential Equations.Philadelphia, PA: Society for Industrial and Applied Mathematics.

Cross-listing: CSCI-GA 2421.001.

**MATH-GA 2011.001
ADVANCED TOPICS IN NUMERICAL ANALYSIS (Numerical
Optimization)**

3 points.
Fall term.

Tuesday, 5:10-7:00, M.
Wright.

Prerequisites: Students are expected to be comfortable with numerical linear algebra and multivariate calculus, and to have programming experience (preferably in MATLAB). Students without all elements of this background are likely to have difficulty with the course material.

Many problems in science, engineering, medicine, and business involve optimization, in which we seek to optimize a mathematical measure of goodness subject to constraints. This course will survey widely used methods for continuous optimization, focusing on both theoretical foundations and implementation as software, with particular attention to numerical issues. Topics include linear programming (optimization of a linear function subject to linear constraints), line search and trust region methods for unconstrained optimization, and a selection of approaches (including active-set, sequential quadratic programming, and interior methods) for constrained optimization.

Most homework assignments will include programming and analysis of the numerical results. Students will be asked to submit their code as part of each homework assignment. The instructor will use MATLAB, an interactive software package and programming environment, for her own programs. If you would like to use another language, please obtain advance permission from the instructor. MATLAB may be purchased at the Computer Store, used in a Courant computer lab, or used remotely (with a few solvable complications if you wish to use its graphics capabilities).

For additional information, see the course website.

Recommended Text: Nocedal, J., & Wright, S. J. (2006).Springer Series in Operations Research and Financial Engineering[Series].Numerical Optimization(2nd ed.). New York, NY: Springer-Verlag.

Cross-listing: CSCI-GA 2945.001.

**MATH-GA 2011.002
ADVANCED TOPICS IN NUMERICAL ANALYSIS (Spectral Methods for
ODEs and PDEs)**

3 points.
Fall term.

Tuesday, Thursday, 1:25-3:15, N. Trefethen.

Most of the course content will be based on Prof. Trefethen's textbook,Spectral Methods in MATLAB. The final segment of the course will turn to advanced topics depending on the interests of the students.

This course will be held from 9/2/2014-10/9/2014.: Trefethen, L.N. (2001).

TextSoftware, Environments, Tools[Series, Bk. 10].Spectral Methods in MATLAB. Philadelphia, PA: Society for Industrial & Applied Mathematics.

Cross-listing: CSCI-GA 2945.002.

**MATH-GA 2011.003
ADVANCED TOPICS IN NUMERICAL ANALYSIS (Computational Fluid
Dynamics)**

3 points. Fall term.

Wednesday, 1:25-3:15, A. Donev.

Prerequisites: Foundations of methods for solving ODEs and PDEs will be assumed: Forward and backward Euler method for ODEs, accuracy, stability, stiffness; basic multistep and Runge-Kutta schemes for ODEs; basic elliptic problem (finite difference and finite element methods for the Poisson equations; iterative methods for linear systems); basic parabolic problems (heat equation; spatial discretization; explicit and implicit temporal discretization methods; von Neumann Fourier mode stability analysis; CFL numbers); basic hyperbolic problems (advection equation; finite volume spatial discretization; method of lines; upwinding). For Courant students, this means having taken Numerical Methods II, for example. Basic familiarity with fluid dynamics, and an understanding of at least the incompressible isothermal Navier-Stokes equations will also be assumed.

This course will cover advanced numerical techniques for solving PDEs, with a particular focus on fluid dynamics. This includes advection-diffusion-reaction equations, compressible and incompressible Navier-Stokes equations, and fluid-structure coupling. Basic familiarity with temporal integrators for ODEs (multistep, Runge-Kutta), methods for solving PDEs (finite difference, finite volume, finite elements for parabolic and elliptic problems), iterative solvers for linear systems, and the Navier-Stokes equations will be assumed. Topics covered will include: higher-order spatio-temporal discretizations for advection-diffusion equations; artificial dissipation and dispersion; compressible flow (conservation laws, limiters, shock-capturing methods, boundary layers, turbulence); incompressible flow (projection methods, Stokes solvers, spectral methods); fluid-structure coupling (boundary-integral formulations, immersed boundary methods); geo-physical dynamics (shallow water, wave equations, turbulent flows)

Recommended Text: Hundsdorfer, W., & Verwer, J.G. (2003).Springer Series in Computational Mathematics[Series, Vol. 33].Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations. New York, NY: Springer-Verlag.

Cross-listing: CSCI-GA 2945.003.

**MATH-GA 2012.001 ADVANCED
TOPICS IN NUMERICAL ANALYSIS (Monte Carlo Methods)**

3 points. Spring term.

Tuesday, 5:10-7:00, J. Goodman.

Course description not yet available.

Cross-listing: CSCI-GA 2945.001.

**MATH-GA 2012.002 ADVANCED
TOPICS IN NUMERICAL ANALYSIS (The Immersed Boundary Method
for Fluid-Structure Interaction)**

3 points. Spring term.

Monday, 1:25-3:15, C. Peskin.

Prerequisite: Familiarity with numerical methods and fluid dynamics.

The immersed boundary (IB) method is a general framework for the computer simulation of flows with immersed elastic boundaries and/or complicated geometry. It was originally developed to study the fluid dynamics of heart valves, and it has since been applied to a wide variety of problems in biofluid dynamics, such as wave propagation in the inner ear, blood clotting, fish swimming, and insect flight. Non-biological applications include sails, parachutes, flows of suspensions, and two-fluid or multifluid problems.

Topics to be covered include: mathematical formulation of fluid-structure interaction in Eulerian and Lagrangian variables with the Dirac delta function as the link between these two kinds of variables; discretization of the structure, fluid, and interaction equations, including energy-based discretization of the structure equations, finite-difference discretization of the fluid equations, and IB delta functions with specified mathematical properties; IB methods for immersed boundaries with non-trivial mass and for fluids with non-uniform density and viscosity; IB methods for immersed filaments with bend and twist; a stochastic IB method for thermally fluctuating hydrodynamics within biological cells; an IB method for the bidomain equations of cardiac electrophysiology; and progress on analysis of the IB method.

: Peskin, C.S. (2011).

TextThe Immersed Boundary Method[PDF Lecture Notes]. Retrieved from: http://www.math.nyu.edu/faculty/peskin/ib_lecture_notes/index.html.

These notes will be supplemented by selected publications on the topics of the course.

Cross-listing: CSCI-GA 2945.002.

**MATH-GA 2012.003 ADVANCED TOPICS IN NUMERICAL ANALYSIS (High Performance Computing)**

3 points. Spring term.

Monday, 5:10-7:00, G. Stadler.

Prerequisites: (Serial) programming experience with C/C++ or FORTRAN (C will mostly be used in class), and some familiarity with numerical methods.

This class will be an introduction to the fundamentals of parallel scientific computing. We will establish a basic understanding of modern computer architectures (CPUs and accelerators, memory hierarchies, interconnects) and of parallel approaches to programming these machines (distributed vs. shared memory parallelism: MPI, OpenMP, OpenCL). Issues such as load balancing, communication, and synchronization will be covered and illustrated in the context of parallel numerical algorithms. Since a prerequisite for good parallel performance is good serial performance, this aspect will also be addressed. Along the way you will be exposed to important tools for high performance computing such as debuggers, schedulers, visualization, and version control systems.

This will be a hands-on class, with several parallel (and serial) computing assignments, in which students can and will have to explore material independently. Students will learn some Unix in this course.

Text: Rauber, T., & Runger, G. (2013).Parallel Programming for Multicore and Cluster Systems(2nd ed.). New York, NY: Springer-Verlag.

Cross-listing: CSCI-GA 2945.003.

**MATH-GA 2041.001 COMPUTING IN
FINANCE**

3 points. Fall term.

Thursday, 7:10-9:00, L. Maclin & E. Fishler.

This course will introduce students to the software development process, including applications in financial asset trading, research, hedging, portfolio management, and risk management. Students will use the Java programming language to develop object-oriented software, and will focus on the most broadly important elements of programming - superior design, effective problem solving, and the proper use of data structures and algorithms. Students will work with market and historical data to run simulations and test strategies. The course is designed to give students a feel for the practical considerations of software development and deployment. Several key technologies and recent innovations in financial computing will be presented and discussed.

**MATH-GA 2043.001
SCIENTIFIC COMPUTING**

3 points.
Fall and spring terms.

Thursday, 5:10-7:00, A.
Rangan (fall); Thursday, 7:10-9:00, Instructor TBA
(spring).

Prerequisites: Undergraduate multivariate calculus and linear algebra. Programming experience strongly recommended but not required.

Fall Term

This course is intended to provide a practical introduction to computational problem solving. Topics covered include: the notion of well-conditioned and poorly conditioned problems, with examples drawn from linear algebra; the concepts of forward and backward stability of an algorithm, with examples drawn from floating point arithmetic and linear-algebra; basic techniques for the numerical solution of linear and nonlinear equations, and for numerical optimization, with examples taken from linear algebra and linear programming; principles of numerical interpolation, differentiation and integration, with examples such as splines and quadrature schemes; an introduction to numerical methods for solving ordinary differential equations, with examples such as multistep, Runge Kutta and collocation methods, along with a basic introduction of concepts such as convergence and linear stability; An introduction to basic matrix factorizations, such as the SVD; techniques for computing matrix factorizations, with examples such as the QR method for finding eigenvectors; Basic principles of the discrete/fast Fourier transform, with applications to signal processing, data compression and the solution of differential equations.

This is not a programming course but programming in homework projects with MATLAB/Octave and/or C is an important part of the course work. As many of the class handouts are in the form of MATLAB/Octave scripts, students are strongly encouraged to obtain access to and familiarize themselves with these programming environments.

Recommended Texts: Bau III, D., & Trefethen, L.N. (1997).Numerical Linear Algebra.Philadelphia, PA: Society for Industrial & Applied Mathematics.

Quarteroni, A.M., & Saleri, F. (2006).Texts in Computational Science & Engineering[Series, Bk. 2].Scientific Computing with MATLAB and Octave(2^{nd}ed.). New York, NY: Springer-Verlag.

Otto, S.R., & Denier, J.P. (2005).An Introduction to Programming and Numerical Methods in MATLAB. London: Springer-Verlag London.

Cross-listing: CSCI-GA 2112.001.

Spring Term

A practical introduction to computational problem solving. Conditioning of problems and stability of algorithms; floating point arithmetic; principles of reliable and robust computational software; scientific visualization; applied approximation theory, including numerical interpolation, differentiation and integration; solution of linear and nonlinear systems of equations and optimization; Eigenvalue problems and SVD decomposition; ordinary differential equations; Fourier transforms; Introduction to Monte Carlo simulation.

This is not a programming course but programming in homework projects with MATLAB (Python, Fortran, C/C++, or other language of your choice) is an important part of the course work.

Text: Quarteroni, A.M., & Saleri, F. (2006).Texts in Computational Science & Engineering[Series, Bk. 2].Scientific Computing with MATLAB and Octave(2^{nd}ed.). New York, NY: Springer-Verlag.

Recommended Texts: Otto, S.R., & Denier, J.P. (2005).An Introduction to Programming and Numerical Methods in MATLAB. London: Springer-Verlag London.

O'Leary, D.P. (2008).Scientific Computing with Case Studies.Philadelphia, PA: Society for Industrial and Applied Mathematics.

Cross-listing: CSCI-GA 2112.001.

**MATH-GA 2045.001
COMPUTATIONAL METHODS FOR FINANCE**

3 points.
Fall term.

Tuesday, 7:10-9:00, A.
Hirsa.

Prerequisites: Scientific Computing or Numerical Methods II, Continuous Time Finance, or permission of instructor.

Computational techniques for solving mathematical problems arising in finance. Dynamic programming for decision problems involving Markov chains and stochastic games. Numerical solution of parabolic partial differential equations for option valuation and their relation to tree methods. Stochastic simulation, Monte Carlo, and path generation for stochastic differential equations, including variance reduction techniques, low discrepancy sequences, and sensitivity analysis.

Cross-listing: FINC-GB 7311.010.

**MATH-GA 2046.001 Advanced
Econometric Modeling and Big Data**

3 points. Fall term.

Thursdays, 7:10-9:00, G. Ritter.

Prerequisites: Derivative Securities, Risk & Portfolio Management with Econometrics, and Computing in Finance (or equivalent programming experience).

A rigorous background in Bayesian statistics geared towards applications in finance, including decision theory and the Bayesian approach to modeling, inference, point estimation, and forecasting, sufficient statistics, exponential families and conjugate priors, and the posterior predictive density. A detailed treatment of multivariate regression including Bayesian regression, variable selection techniques, multilevel/hierarchical regression models, and generalized linear models (GLMs). Inference for classical time-series models, state estimation and parameter learning in Hidden Markov Models (HMMs) including the Kalman filter, the Baum-Welch algorithm and more generally, Bayesian networks and belief propagation. Solution techniques including Markov Chain Monte Carlo methods, Gibbs Sampling, the EM algorithm, and variational mean field. Real world examples drawn from finance to include stochastic volatility models, portfolio optimization with transaction costs, risk models, and multivariate forecasting

**MATH-GA 2701.001 METHODS OF
APPLIED MATHEMATICS**

3 points.
Fall term.

Tuesday, 1:25-3:15, A. Rangan.

Prerequisites: Elementary linear algebra and differential equations.

This is a first-year course for all incoming PhD and Master students interested in pursuing research in applied mathematics. It provides a concise and self-contained introduction to advanced mathematical methods, especially in the asymptotic analysis of differential equations. Topics include scaling, perturbation methods, multi-scale asymptotics, transform methods, geometric wave theory, and calculus of variations

Recommended Texts: Barenblatt, G.I. (1996).Cambridge Texts in Applied Mathematics[Series, Bk. 14].Scaling, Self-similarity, and Intermediate Asymptotics: Dimensional Analysis and Intermediate Asymptotics.New York, NY: Cambridge University Press.

Hinch, E.J. (1991).Camridge Texts in Applied Mathematics[Series, Bk. 6].Perturbation Methods. New York, NY: Cambridge University Press.

Bender, C.M., & Orszag, S.A. (1999).Advanced Mathematical Methods for Scientists and Engineers[Series, Vol. 1].Asymptotic Methods and Perturbation Theory.New York, NY: Springer-Verlag.

Whitham, G.B. (1999).Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts[Series Bk. 42].Linear and Nonlinear Waves(Reprint ed.). New York, NY: John Wiley & Sons/ Wiley-Interscience.

Gelfand, I.M., & Fomin, S.V. (2000).Calculus of Variations. Mineola, NY: Dover Publications.

**MATH-GA 2702.001 FLUID DYNAMICS**

3 points.
Fall term.

Wednesday, 1:25-3:15, D. Giannakis.

Prerequisites: Introductory complex variable and partial differential equations.

The course will expose students to basic fluid dynamics from a mathematical and physical perspectives, covering both compressible and incompressible flows. Topics: conservation of mass, momentum, and Energy. Eulerian and Lagrangian formulations. Basic theory of inviscid incompressible and compressible fluids, including the formation of shock waves. Kinematics and dynamics of vorticity and circulation. Special solutions to the Euler equations: potential flows, rotational flows, irrotationall flows and conformal mapping methods. The Navier-Stokes equations, boundary conditions, boundary layer theory. The Stokes Equations.

Text: Childress, S.Courant Lecture Notes in Mathematics[Series, Bk. 19].An Introduction to Theoretical Fluid Mechanics. Providence, RI: American Mathematical Society/ Courant Institute of Mathematical Sciences.

Recommended Text: Acheson, D.J. (1990).Oxford Applied Mathematics & Computing Science Series[Series].Elementary Fluid Dynamics. New York, NY: Oxford University Press.

**MATH-GA 2704.001 APPLIED
STOCHASTIC ANALYSIS**

3 points. Spring term.

Monday, 1:25-3:15, M. Holmes-Cerfon.

Prerequisites: Basic knowledge (e.g. undergraduate) of: probability, linear algebra, ODEs, PDEs, and analysis.

This is a graduate class that will introduce the major topics in stochastic analysis from an applied mathematics perspective. Topics to be covered include Markov chains, stochastic processes, stochastic differential equations, numerical algorithms, and asymptotics. It will pay particular attention to the connection between stochastic processes and PDEs, as well as to physical principles and applications. The class will attempt to strike a balance between rigour and heuristic arguments: it will assume that students have some familiarity with measure theory and analysis and will make occasional reference to these, but many results will be derived through other arguments.

Recommended Texts: Arnold, L. (1974).Stochastic differential equations: Theory and applications.New York: John Wiley & Sons.

Oksendal B. (2010).Universitext[Series].Stochastic Differential Equations: An Introduction with Applications(6th Ed.). New York, NY: Springer-Verlag Berlin Heidelberg.

Koralov, L., & Sinai, Y.G. (2012).Universitext[Series].Theory of Probability and Random Processes(2nd Ed.). New York, NY: Springer-Verlag Berlin Heidelberg.

Karatzas, I., & Shreve, S.E. (1991).Graduate Texts in Mathematics[Series, Vol. 113].Brownian Motion and Stochastic Calculus

(2nd Ed.). New York, NY: Springer Science+Business Media, Inc.

Kloeden, P., & Platen, E. (1992).Applications of Mathematics: Stochastic Modelling and Applied Probability[Series, Bk. 23].Numerical Solution of Stochastic Differential Equations(Corrected 3rd Printing). New York, NY: Springer-Verlag Berlin Heidelberg New York.

Rogers, L.C.G. & Willams, D. (2000).Cambridge Mathematical Library[Series, Bks. 1-2].Diffusions, Markov Processes, and Martingales: Foundations(Vol. 1, 2nd Ed.); andDiffusions, Markov Processes, and Martingales: Ito Calculus(Vol. 2, 2nd Ed.). New York, NY: Cambridge University Press.

Grimmett, G.R., & Stirzaker, D.R. (2001).Probability and Random Processes(3rd ed.). New York, NY: Oxford University Press.

Gardiner, C.W. (2009).Springer Series in Synergetics[Series, Bk. 13].Stochastic Methods: A Handbook for the Natural and Social Sciences(4th Ed.). New York, NY: Springer-Verlag Berlin Heidelberg New York.

Risken, H., & Frank, T. (1996).Springer Series in Synergetics[Series, Bk. 18].The Fokker-Planck Equation: Methods of Solution and Applications(1996 2nd Ed.). New York, NY: Springer-Verlag Berlin Heidelberg New York.

**MATH-GA 2706.001 PARTIAL
DIFFERENTIAL EQUATIONS FOR FINANCE**

3 points. Spring term.

Monday, 5:10-7:00, R. Kohn.

Prerequisites: Stochastic Calculus or equivalent.

An introduction to those aspects of partial differential equations and optimal control most relevant to finance. Linear parabolic PDE and their relations with stochastic differential equations: the forward and backward Kolmogorov equation, exit times, fundamental solutions, boundary value problems, maximum principle. Deterministic and stochastic optimal control: dynamic programming, Hamilton-Jacobi-Bellman equation, verification arguments, optimal stopping. Applications to finance, including portfolio optimization and option pricing -- are distributed throughout the course.

**MATH-GA 2707.001 TIME SERIES
ANALYSIS AND STATISTICAL ARBITRAGE**

3 points. Fall term.

Monday, 7:10-9:00, F. Asl & R. Reider.

Prerequisites: Derivative Securities, Scientific Computing, Computing for Finance, and Stochastic Calculus.

The term "statistical arbitrage" covers any trading strategy that uses statistical tools and time series analysis to identify approximate arbitrage opportunities while evaluating the risks inherent in the trades (considering the transaction costs and other practical aspects). This course starts with a review of Time Series models and addresses econometric aspects of financial markets such as volatility and correlation models. We will review several stochastic volatility models and their estimation and calibration techniques as well as their applications in volatility based trading strategies. We will then focus on statistical arbitrage trading strategies based on cointegration, and review pairs trading strategies. We will present several key concepts of market microstructure, including models of market impact, which will be discussed in the context of developing strategies for optimal execution. We will also present practical constraints in trading strategies and further practical issues in simulation techniques. Finally, we will review several algorithmic trading strategies frequently used by practitioners.

**MATH-GA 2708.001 ALGORITHMIC
TRADING AND QUANTITATIVE STRATEGIES**

3 points. Spring term.

Tuesday, 7:10-9:00, P. Kolm & L. Maclin.

Prerequisites: Computing in Finance, and Risk Portfolio Management with Econometrics, or equivalent.

In this course we develop a quantitative investment and trading framework. In the first part of the course, we study the mechanics of trading in the financial markets, some typical trading strategies, and how to work with and model high frequency data. Then we turn to transaction costs and market impact models, portfolio construction and robust optimization, and optimal betting and execution strategies. In the last part of the course, we focus on simulation techniques, back-testing strategies, and performance measurement. We use advanced econometric tools and model risk mitigation techniques throughout the course. Handouts and/or references will be provided on each topic.

**MATH-GA 2710.001 MECHANICS**

3 points.
Spring term.

Wednesday, 1:25-3:15, A. Cerfon.

This course provides a basic mathematical introduction to solid continuum mechanics, classical Newtonian mechanics, and statistical mechanics. Prior knowledge in physics is not required. Key topics include: nonlinear elasticity; linear elasticity; dimension reduction (plate theory); calculus of variations methods; Hamilton's equations; action minimization; Liouville's theorem; microcanonical and canonical ensemble; entropy.

Recommended Texts: Ciarlet, P.G. (1988).Studies in Mathematics & Its Applications: Mathematical Elasticity[Series, Vol. 1].Three-dimensional Elasticity. New York, NY: Elsevier Science/ North-Holland.

Buhler, O. (2006).Courant Lecture Notes in Mathematics[Series, Bk. 13].A Brief Introduction to Classical, Statistical, and Quantum Mechanics.Providence, RI: American Mathematical Society/ Courant Institute of Mathematical Sciences.

**MATH-GA 2751.001 RISK AND
PORTFOLIO MANAGEMENT WITH ECONOMETRICS**

3 points.
Fall and spring terms.

Tuesday, 7:10-9:00, P. Kolm (fall);
Wednesday, 7:10-9:00, M.
Avellaneda (spring).

: Univariate statistics, multivariate calculus, linear algebra, and basic computing (e.g. familiarity with MATLAB or co-registration in Computing in Finance).Fall Term

Prerequisites

A comprehensive introduction to the theory and practice of portfolio management, the central component of which is risk management. Econometric techniques are surveyed and applied to these disciplines. Topics covered include: factor and principal-component models, CAPM, dynamic asset pricing models, Black-Litterman, forecasting techniques and pitfalls, volatility modeling, regime-switching models, and many facets of risk management, both theory and practice.

Spring Term

Course description not yet available (though generally consistent with the fall term description).

**MATH-GA 2752.001 ACTIVE PORTFOLIO MANAGEMENT**

3 points. Spring term.

Monday, 5:10-7:00, Instructor TBA.

Prerequisites: Risk & Portfolio Management with Econometrics, Computing in Finance.

The first part of the course will cover the theoretical aspects of portfolio construction and optimization. The focus will be on advanced techniques in portfolio construction, addressing the extensions to traditional mean-variance optimization including robust optimization, dynamical programming and Bayesian choice. The second part of the course will focus on the econometric issues associated with portfolio optimization. Issues such as estimation of returns, covariance structure, predictability, and the necessary econometric techniques to succeed in portfolio management will be covered. Readings will be drawn from the literature and extensive class notes.

**MATH-GA 2753.001 ADVANCED RISK MANAGEMENT**

3 points. Spring term.

Monday, 7:10-9:00, K. Abbott.

Prerequisites: Derivative Securities, Computing in Finance or equivalent programming.

The importance of financial risk management has been increasingly recognized over the last several years. This course gives a broad overview of the field, from the perspective of both a risk management department and of a trading desk manager, with an emphasis on the role of financial mathematics and modeling in quantifying risk. The course will discuss how key players such as regulators, risk managers, and senior managers interact with trading. Specific techniques for measuring and managing the risk of trading and investment positions will be discussed for positions in equities, credit, interest rates, foreign exchange, commodities, vanilla options, and exotic options. Students will be trained in developing risk sensitivity reports and using them to explain income, design static and dynamic hedges, and measure value-at-risk and stress tests. Students will create Monte Carlo simulations to determine hedge effectiveness. Extensive use will be made of examples drawn from real trading experience, with a particular emphasis on lessons to be learned from trading disasters.

Text: Allen, S.L. (2003).Wiley Finance[Series, Bk. 119].Financial Risk Management: A Practitioner’s Guide to Managing Market and Credit Risk.Hoboken, NJ: John Wiley & Sons.

**MATH-GA 2755.001 PROJECT AND PRESENTATION (MATH FINANCE)**

3 points. Fall and spring Terms.

Monday, 5:10-7:00 (fall); Wednesday 5:10-7:00 (spring), P. Kolm.

Students in the Mathematics in Finance program conduct research projects individually or in small groups under the supervision of finance professionals. The course culminates in oral and written presentations of the research results.

**MATH-GA 2757.001 REGULATION
AND REGULATORY RISK MODELS**

3 points. Fall term.

Wednesdays, 7:10-9:00, K. Abbott and L. Andersen

Prerequisites: Risk Management, Derivative Securities (or equivalent familiarity with market and credit risk models).

The course is divided into two parts. The first addresses the institutional structure surrounding capital markets regulation. It will cover Basel (1, MRA, 2, 2.5, 3), Dodd-Frank, CCAR and model review. The second part covers the actual models used for the calculation of regulatory capital. These models include the Gaussian copula used for market risk, specific risk models, the Incremental Risk Calculation (single factor Vasicek), the Internal Models Method for credit, and the Comprehensive Risk Measure.

**MATH-GA 2791.001 DERIVATIVE SECURITIES**

3 points. Fall and spring terms.

Wednesday, 7:10-9:00, M. Avellaneda (fall); Monday, 7:10-9:00, B. Flesaker (spring).

An introduction to arbitrage-based pricing of derivative securities. Topics include: arbitrage; risk-neutral valuation; the log-normal hypothesis; binomial trees; the Black-Scholes formula and applications; the Black-Scholes partial differential equation; American options; one-factor interest rate models; swaps, caps, floors, swaptions, and other interest-based derivatives; credit risk and credit derivatives.

Cross-listing: FINC-GB 7312.010.

**MATH-GA 2792.001 CONTINUOUS TIME FINANCE**

3 points. Fall and spring terms.

Monday, 7:10-9:00, P. Carr & A. Javaheri (fall); Wednesday, 7:10-9:00, F. Mercurio & B. Dupire (spring).

Prerequisites: Derivative Securities and Stochastic Calculus, or equivalent.

Fall Term

A second course in arbitrage-based pricing of derivative securities. The Black-Scholes model and its generalizations: equivalent martingale measures; the martingale representation theorem; the market price of risk; applications including change of numeraire and the analysis of quantos. Interest rate models: the Heath-Jarrow-Morton approach and its relation to short-rate models; applications including mortgage- backed securities. The volatility smile/skew and approaches to accounting for it: underlyings with jumps, local volatility models, and stochastic volatility models.

Spring Term

A second course in arbitrage-based pricing of derivative securities. Concerning equity and FX models: we'll discuss numerous approaches that are used in practice in these markets, such as the local volatility model, Heston, SABR, and stochastic local volatility. The discussion will include calibration and hedging issues and the pricing of the most common structured products. Concerning interest rate models: we'll start with a thorough discussion of one-factor short-rate models (Vasicek, CIR, Hull-White) then proceed to more advanced topics such as two-factor Hull-White, forward rate models (HJM) and the LIBOR market model. Throughout, the pricing of specific payoffs will be considered and practical examples and insights will be provided. We'll conclude with a discussion of inflation models.: FINC-GB 7310.010.Cross-listing

**MATH-GA 2796.001 SECURITIZED PRODUCTS AND ENERGY DERIVATIVES**

3 points. Spring term.

Wednesday, 7:10-9:00, R. Sunada-Wong & G. Swindle.

Prerequisites: Basic bond mathematics and bond risk measures (duration and convexity); Derivative Securities, Stochastic Calculus.

The first part of the course will cover the fundamentals and building blocks of understanding how mortgage-backed securities are priced and analyzed. The focus will be on prepayment and interest rate risks, benefits and risks associated with mortgage-backed structured bonds and mortgage derivatives. Credit risks of various types of mortgages will also be discussed. The second part of the course will focus on energy commodities and derivatives, from their basic fundamentals and valuation, to practical issues in managing structured energy portfolios. We develop a risk neutral valuation framework starting from basic GBM and extend this to more sophisticated multifactor models. These approaches are then used for the valuation of common, yet challenging, structures. Particular emphasis is placed on the potential pitfalls of modeling methods and the practical aspects of implementation in production trading platforms. We survey market mechanics and valuation of inventory options and delivery risk in the emissions markets.

Recommended Texts: Hayre, L. (2001).Wiley Finance[Series, Bk. 83].Salomon Smith Barney Guide to Mortgage-backed and Asset-backed Securities.New York, NY: John Wiley & Sons.

Fabozzi, F.J. (ed.). (2005).The Handbook of Mortgage-backed Securities(6th ed.). New York, NY: McGraw-Hill.

Eydeland, A., & Wolyniec, K. (2002).Wiley Finance[Series, Bk. 97].Energy and Power Risk Management: New Developments in Modeling, Pricing, and Hedging.Hoboken, NJ: John Wiley & Sons.

Harris, C. (2006).Wiley Finance[Series, Bk. 328].Electricity Markets: Pricing, Structures and Economics(2nd ed.). Hoboken, NJ: John Wiley & Sons.

**MATH-GA 2797.001 CREDIT MARKETS AND MODELS**

3 points. Fall term.

Wednesday, 7:10-9:00, Instructor TBA.

Prerequisites: Computing in Finance (or equivalent), Derivative Securities (or equivalent), familiarity with analytical methods applied to interest rate derivatives.

This course addresses a number of practical issues concerned with modeling, pricing and risk management of a range of fixed-income securities and structured products exposed to default risk. Emphasis is on developing intuition and practical skills in analyzing pricing and hedging problems. In particular, significant attention is devoted to credit derivatives.

We begin with discussing default mechanism and its mathematical representation. Then we proceed to building risky discount curves from market prices and applying this analytics to pricing corporate bonds, asset swaps, and credit default swaps. Risk management of credit books will be addressed as well. We will next examine pricing and hedging of options on assets exposed to default risk.

After that, we will discuss structural (Merton-style) models that connect corporate debt and equity through the firm’s total asset value. Applications of this approach include the estimation of default probability and credit spread from equity prices and effective hedging of credit curve exposures.

A final segment of the course will focus on credit structured products. We start with cross-currency swaps with a credit overlay. We will next analyze models for pricing portfolio transactions using Merton-style approach. We also will discuss portfolio loss model based on a transition matrix approach. These models will then be applied to the pricing of collateralized debt obligation tranches and pricing counterparty credit risk taking wrong-way exposure into account.

Recommended Texts: O’Kane, D. (2008).Wiley Finance[Series, Bk. 545].Modeling Single-name and Multi-name Credit Derivatives.Hoboken, NJ: John Wiley & Sons.

Hull, J. (2008).Options, Futures, & Other Derivatives(7th ed.). Upper Saddle River, NJ: Prentice-Hall/ Pearson Education.

**MATH-GA 2798.001 INTEREST RATE AND FX MODELS**

3 points. Spring term.

Thursday, 5:10-7:00, L. Andersen & A. Gunstensen.

Prerequisites: Derivative Securities, Stochastic Calculus, and Computing in Finance (or equivalent familiarity with financial models, stochastic methods, and computing skills).

The course is divided into two parts. The first addresses the fixed-income models most frequently used in the finance industry, and their applications to the pricing and hedging of interest-based derivatives. The second part covers the foreign exchange derivatives markets, with a focus on vanilla options and first-generation (flow) exotics. Throughout both parts, the emphasis is on practical aspects of modeling, and the significance of the models for the valuation and risk management of widely-used derivative instruments.

**MATH-GA 2830.001 ADVANCED TOPICS IN APPLIED MATH (Filtering Turbulent Signals in Complex Systems)**

3 points. Fall term.

Thursday, 3:15-5:00, A. Majda.

An important emerging scientific issue in many practical problems ranging from climate and weather prediction to material science involves the real time filtering through observations of noisy turbulent signals for complex dynamical systems with many degrees of freedom as well as the statistical accuracy of various strategies in this context. This course is an introduction to the mathematical theories and ideas which are currently being developed at CIMS to address these issues. These ideas blend classical stability analysis for PDE’s and their finite difference approximations, suitable versions of Kalman filtering, and stochastic models form turbulence theory. The course will be an elementary introduction to these topics filling in the necessary background beginning with elementary scalar and low dimensional models with eventual applications to fully turbulent and chaotic, linear and nonlinear, large dimensional systems.

Text: Majda, A.J., & Harlim, J. (2012).Filtering Complex Turbulent Systems.New York, NY: Cambridge University Press. (Online version available through the NYU Library.)

**MATH-GA 2830.002 ADVANCED TOPICS IN APPLIED MATHEMATICS (Metastability and Reactive Events)**

3 points. Fall term.

Wednesday, 1:25-3:15, E. Vanden Eijnden.

Dynamics in nature often proceed in the form of rare reactive events: The system under study spends very long periods of time around specific states and only very rarely transitions from one such metastable state to another. Conformation changes of macromolecules, chemical reactions in solution, nucleation events during phase transitions, thermally induced magnetization reversal in micromagnets, etc. are just a few examples of such reactive events. One can often think of the dynamics of these systems as a navigation over a potential or free energy landscape, under the action of small amplitude noise. In the simplest situations the metastable states are then regions around the local minima on this landscape, and transition events between these regions are rare because the noise has to push the system over the barriers separating them. This is the picture underlying classical tools such as transition state theory or Kramers reaction rate theory, and it can be made mathematically precise within the framework of large deviation theory. In complex high dimensional systems, this picture can however be naive because entropic (i.e. volume) effects start to play an important role: Local features of the energy, such as the location of its minima or saddle points, may have much less of an impact on its dynamics than global features such as the width of low lying basins on the landscape: in these situations a more general framework for the description of metastability is required. This class will review recent advances that have been made in this context. Topics will include: spectral definition of metastability, transition state theory (TST), large deviation theory (LDT), potential theoretic approach, transition path theory (TPT), asymptotic solutions based on TPT, numerical aspects, metastability and data assimilation.

**MATH-GA 2830.003 ADVANCED TOPICS IN APPLIED MATHEMATICS (Information Theory and Predictability)**

3 points. Fall term.

Monday, 1:25-3:15, R. Kleeman.

Prerequisites: Undergraduate introduction to probability and undergraduate calculus.

Information theory is a branch of probability theory which has seen application in fields as diverse as computer science, dynamical systems, financial mathematics and complexity. In this seminar course we shall provide a comprehensive introduction to the theory and explain the many applications.

In the second part of the course we shall apply the theory to the study of predictability in dynamical systems with a particular focus on the atmosphere and ocean. Recent results which have placed the subject of statistical predictability on a more rigorous footing will be emphasized and the connection with practical prediction in real systems such as the weather carefully explained.

Texts: Lecture notes (to be provided in class).

Recommended Text: Cover, T.M., & Thomas, J.A. (2006).Wiley Series in Telecommunications and Signal Processing[Series].Elements of Information Theory(2nd ed.). Hoboken, NJ: John Wiley & Sons.

**MATH-GA 2830.004 ADVANCED TOPICS IN APPLIED MATHEMATICS (The Physics and Mathematics of Active Matter)**

3 points. Fall term.

Wednesday, 9:00-10:50, M. Shelley & J. Palacci.

Prerequisites: Graduate-level Fluid Mechanics or Statistical Mechanics or Electromagnetics.

One of the most fascinating aspects of Nature is the emergence of complex and collective phenomena in systems freed from the constraints of equilibrium: birds flock together while fish spontaneously form mesmerizing schools that exhibit collective behavior. These structures have no equilibrium counterparts, are ubiquitous, and are observed over a broad range of length scales: self-organized structures within a cell, the dynamics of bacterial colonies, the flow of crowds… The organization principles for such systems are mainly unknown and constitute a new frontier of physics, mathematical modeling, and analysis.

In this special topics course on the Physics and Mathematics of Active Matter, we will present recent experimental and theoretical tools developed to study and understand these systems. Focusing mostly on microscopic systems, we will discuss the fundamental constraints of locomotion in the so-called low Reynolds regime, as well as strategies to overcome them, as observed in biological (e.g. bacterial) systems, or developed for the design of artificial systems such as self-propelled microrobots. Following this, we will develop the mathematical framework for the hydrodynamic description of the active-particle systems, such as the suspensions of motile bacteria, or of biopolymers and motor-proteins.

This course will cover some of the following themes: life at low Reynolds, time reversibility of the Stokes equation, and the design of active particles; Statistical Mechanics and hydrodynamics of active suspensions; Interfacial Transport: Osmosis and Phoresis; and active processes within the cell.

**MATH-GA 2852.001 ADVANCED TOPICS IN MATH BIOLOGY (Stochastic Problems in Cellular, Molecular and Neural Biology)**

3 points. Spring term.

Thursday, 1:25-3:15, D. Tranchina.

Prerequisites: Elementary background in ODEs, PDEs, probability theory, Fourier transforms.

A variety of topics of current interest in biology and neural science will be addressed. Topics include: (1) Stochastic gene expression: analytical modeling of stochastic messenger RNA synthesis and degradation; discrete and continuous models; master equation; generating function; steady-state distributions; temporal evolution of the distributions; stochastic protein product. (2) Stochastic cell divisions and population growth: mean growth rate; age distributions. (3) Single-photon responses of retinal rods; statistical measures of variability; reproducibility of the single-photon response; explicit biochemical kinetic models; model testing with Monte Carlo simulations. (4) Stochastic switching between bistable percepts, e.g. 2 very different auditory percepts induced by ABA tone sequence. (5) Optimal filtering of photon noise in vision. (6) Stochastic behavior of neurons in the central nervous system: models for synaptic noise; spike train statistics and renewal theory. (7) Probability density methods or large-scale modeling of neural networks: partial differential-integral equations; Fokker-Plank approximation; applications to modeling visual cortex.

Cross-listing: BIOL-GA 2852.001.

**MATH-GA 2855.001**

**ADVANCED TOPICS IN MATH PHYSIOLOGY (Mathematical Aspects of Neurophysiology)**

3 points. Fall term.

Monday, 1:25-3:15, C. Peskin.

Prerequisites: Familiarity with differential equations and probability, as these subjects are used in applications. Despite the official title "Advanced Topics," this course should be accessible to first-year graduate students with strong undergraduate physics and/or applied mathematics background.

The emphasis of this course is on fundamental mechanisms at the neuron level, i.e., on the building blocks for neural networks. Topics include membrane channels (current-voltage relations and gating, including the probabilistic analysis of patch-clamp data), Hodgkin-Huxley equations (their physical basis, mathematical structure, asymptotic solution, and numerical solution on the tree-like structure of a neuron), synaptic transmission (including the stochastic process of vesicle release), short-term synaptic depression as a mechanism of automatic gain control, and the analysis of neuronal spike trains (including the technique of reverse correlation). Both asymptotic and numerical methods, and also stochastic processes, will be introduced and explained throughout the course, which can therefore serve as an applied introduction to these subjects. Students will have the opportunity to work individually or in teams on computing projects related to the course material, and presentation of the results of such projects to the class is encouraged.

Text: Peskin, C.S. (2000). Mathematical Aspects of Neurophysiology [PDF Lecture Notes]. Retrieved from: http://www.math.nyu.edu/faculty/peskin/neuronotes/index.html.

Recommended Texts: Koch, C. (2004).Computational Neuroscience[Series].Information Processing in Single Neurons. New York, NY: Oxford University Press.

Gerstner, W., & Kistler, W.M. (2002).Spiking Neuron Models: Single Neurons, Populations, Plasticity. New York, NY: Cambridge University Press.

Cross-listing: BIOL-GA 2855.001.

**MATH-GA 3001.001 GEOPHYSICAL FLUID DYNAMICS**

3 points. Fall term.

Tuesday, 9:00-10:50, O. Pauluis.

This course serves as an introduction to the fundamentals of geophysical fluid dynamics. No prior knowledge of fluid dynamics will be assumed, but the course will move quickly into the subtopic of rapidly rotating, stratified flows. Topics to be covered include (but are not limited to): the advective derivative, momentum conservation and continuity, the rotating Navier-Stokes equations and non-dimensional parameters, equations of state and thermodynamics of Newtonian fluids, atmospheric and oceanic basic states, the fundamental balances (thermal wind, geostrophic and hydrostatic), the rotating shallow water model, vorticity and potential vorticity, inertia-gravity waves, geostrophic adjustment, the quasi-geostrophic approximation and other small-Rossby number limits, Rossby waves, baroclinic and barotropic instabilities, Rayleigh and Charney-Stern theorems, geostrophic turbulence. Students will be assigned bi-weekly homework assignments and some computer exercises, and will be expected to complete a final project. This course will be supplemented with out-of-class instruction.

Recommended Texts: Vallis, G.K. (2006).Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-scale Circulation. New York, NY: Cambrdige University Press.

Salmon, R. (1998).Lectures on Geophysical Fluid Dynamics. New York, NY: Oxford University Press.

Pedlosky, J. (1992).Geophysical Fluid Dynamics(2nd ed.). New York, NY: Springer-Verlag.

**MATH-GA 3003.001 OCEAN DYNAMICS**

3 points. Spring term.

Tuesday, 1:25-3:15, O. Buhler.

Course description not yet available.

**MATH-GA 3010.001 ADVANCED TOPICS IN ATMOSPHERE-OCEAN SCIENCE (Environmental Fluid Dynamics Lab)**

3 points. Fall term.

Thursday, 1:25-3:15, D. Holland.

Registration subject to approval by course instructor.

Description available from course instructor.

**MATH-GA 2901.001 BASIC PROBABILITY**

3 points. Fall and spring terms.

Wednesday, 5:10-7:00, P. Bourgade (fall); Wednesday, 7:10-9:00, R. Kleeman (spring).

Prerequisites: Calculus through partial derivatives and multiple integrals; no previous knowledge of probability is required.

Fall Term

The course introduces the basic concepts and methods of probability that are now widely used in scientific research. A feature of this course is that such concepts, usually presented in advanced mathematical settings, are here described in a more elementary framework requiring only calculus through partial derivatives and multiple integrals. Topics include: probability spaces, random variables, distributions, law of large numbers, central limit theorem, random walk, branching processes, Markov chains in discrete and continuous time, diffusion processes including Brownian motion, martingales. Suggested readings on reserve.

Recommended Text: Grimmett, G.R., & Stirzaker, D.R. (2001).Probability and Random Processes(3rd ed.). New York, NY: Oxford University Press.

Spring Term

The course introduces the basic concepts and methods of probability that are now widely used in scientific research. A feature of this course is that such concepts, usually presented in advanced mathematical settings, are here described in a more elementary framework requiring no knowledge of measure theory; nevertheless, analytical thinking and mathematical rigor will still be required at the level of a graduate course (undergraduate courses on relevant topics include MATH-UA 233, 234 and 235). Topics include: probability spaces, random variables, distributions, expectations and variances, law of large numbers, central limit theorem, Markov chains, random walk, diffusion processes including Brownian motion, and martingales.

Texts: Grimmett, G.R., & Stirzaker, D.R. (2001).Probability and Random Processes(3rd ed.). New York, NY: Oxford University Press.

Grinstead, C.M., & Snell, J.L. (1997).Introduction to Probability(2nd Rev. ed.). Providence, RI: American Mathematical Society.

**MATH-GA 2902.001 STOCHASTIC CALCULUS**

3 points. Fall and spring terms .

Monday, 7:10-9:00, J. Goodman (fall); Thursday, 7:10-9:00, A. Kuptsov (spring).

Prerequisites: MATH-GA 2901 Basic Probability or equivalent.

Review of basic probability and useful tools. Bernoulli trials and random walk. Law of large numbers and central limit theorem. Conditional expectation and martingales. Brownian motion and its simplest properties. Diffusion in general: forward and backward Kolmogorov equations, stochastic differential equations and the Ito calculus. Feynman-Kac and Cameron-Martin Formulas. Applications as time permits.

Optional Problem Session: Wednesday, 5:30-7:00 (fall); Monday, 5:30-6:30 (spring).

Text: Durrett, R. (1996).Probability and Stochastics Series[Series, Bk. 6].Stochastic Calculus: A Practical Introduction. New York, NY: CRC Press.

**MATH-GA 2911.001, 2912.001 PROBABILITY: LIMIT THEOREMS I, II**

3 points per term. Fall and spring terms.

Wednesday, 1:25-3:15 (fall); Wednesday, 9:00-10:50 (spring), Y. Bakhtin.

Prerequisites: A first course in probability, familiarity with Lebesgue integral, or MATH-GA 2430 Real Variables as mandatory co-requisite.

Fall Term

Probability, independence, laws of large numbers, limit theorems including the central limit theorem. Markov chains (discrete time). Martingales, Doob inequality, and martingale convergence theorems. Ergodic theorem.

Text: Varadhan, S.R.S. (2001).Courant Lecture Series in Mathematics[Series, Bk. 7].Probability Theory. Providence, RI: American Mathematical Society/ Courant Institute of Mathematics.

Spring Term

Independent increment processes, including Poisson processes and Brownian motion. Markov chains (continuous time). Stochastic differential equations and diffusions, Markov processes, semi-groups, generators and connection with partial differential equations.

Recommended Text: Varadhan, S.R.S. (2007).Courant Lecture Series in Mathematics[Series, Bk. 16].Stochastic Processes. Providence, RI: American Mathematical Society/ Courant Institute of Mathematics.

**MATH-GA 2931.001 ADVANCED TOPICS IN PROBABILITY (Large Deviations)**

3 points. Fall term.

Tuesday, Thursday, 5:10-7:00, O. Zeitouni.

Prerequisites: A graduate level course in probability.

The class will cover the theory of large deviations and its applications. The first half of the course will be devoted to finite dimensional examples and elements of the general theory, while the second half will focus on examples in infinite dimensions. The student's interests will guide the examples treated.

This course will be held from 9/9-10/30/2014.: Dembo, A., & Zeitouni, O. (1998).

Recommended TextStochastic Modelling and Applied Probability[Series, Bk. 38].Large Deviations Techniques and Applications(2nd ed.). New York, NY: Springer-Verlag.

**MATH-GA 2932.001 ADVANCED TOPICS IN PROBABILITY (Markov Chain Analysis)**

3 points. Spring term.

Thursday, 9:00-10:50, E. Lubetzky.

Prerequisites: Basic experience in Probability Theory.

The study of Markov chains has surged in the last few decades, driven by applications both in theoretical mathematics and computer science and in applied areas such as statistical physics, mathematical biology, economics and statistics. Nowadays, Markov chains are considered to be one of the most important objects in probability theory, and by now there are many methods for analyzing its rate of convergence to equilibrium, ranging from coupling techniques to chain comparisons to log-Sobolev inequalities, to name a few. We will survey these methods and highlight their applications, with a special emphasis on Markov chains that model the evolution of classical interacting particle systems such as the Ising model.

Texts: Levin, D.A., Peres, Y., & Wilmer, E.L. (2008).Markov Chains and Mixing Times. Providence, RI: American Mathematical Society.

Saloff-Coste, L. (1997).Lecture Notes in Mathematics[Series, Vol. 1665]. “Lectures on Finite Markov Chains.” InLectures on Probability Theory and Statistics. New York, NY: Springer-Verlag.

Aldous, D., & Fill, J.A. (2002).Reversible Markov Chains and Random Walks on Graphs[Unfinished monograph, recompiled 2014].

**MATH-GA 2932.002 ADVANCED TOPICS IN PROBABILITY (Random Matrices)**

3 points. Spring term.

Wednesday, 1:25-3:15, P. Bourgade.: Basic knowledge of linear algebra, probability theory and stochastic calculus are required.

Prerequisites

This course will introduce techniques to understand the spectrum and eigenvectors of large random self-adjoint matrices, on both global and local scales. Topics include Gaussian and circular ensembles, Dyson's Brownian motion, determinantal processes, bulk and edge scaling limits, universality for random matrices.

*Revised November 2014*