# Course Descriptions: AY 2015-16

Algebra and Number
Theory

Geometry and Topology

Analysis

Numerical Analysis

Applied Mathematics

Probability and
Statistics

All course descriptions are subject to change

** ALGEBRA AND NUMBER
THEORY**

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

3 points per term. Fall and spring terms.

Tuesday, 5:10-7:00 R. Kleeman Monday, 5:10-7:00, D. Cai (spring).

**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.

**Plus: Extensive instructor’s class notes.**

**MATH-GA 2110.001 LINEAR ALGEBRA I**

3 points. Spring term.

Tuesday, 5:10-7:00, S. Marques.

**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, E. Tabak.

**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, R. Young (spring).

**Prerequisites**: Elements of linear
algebra and the theory of rings and fields.

**Fall 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 2210.001
INTRODUCTION TO NUMBER THEORY I**

3 points. Spring term.

Wednesday, 7:10-9:00, A. Pirutka.

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

This course is a graduate level introduction to algebraic number theory, in which we will cover fundamentals of the subject. Topics include: the theory of the valuation (p-adic numbers, completion, local fields, henselian fields, ramification theory, Galois theory of valuations) and Riemann-Roch theory.

For additional information, see the course website.

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

** GEOMETRY AND TOPOLOGY**

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

3 points per term. Fall and spring terms.

Thursday, 5:10-7:00, S. Cappell
(fall); Tuesday, 7:10-9:00, S.
Cappell, (spring).

**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.

Wednesday, 1:25-3:15, A. Pirutka (fall); Wednesday, 1:25-3:15, J. Cheeger (spring).

**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**

Differential Geometry II will focus on Riemannian geometry. Topics to be covered may include: second variation of arc length, Rauch comparison theorem and applications, Toponogov's theorem, invariant metrics on Lie groups, Morse theory, cut locus, the sphere theorem, complete manifolds of nonnegative curvature.

3 points. Fall term.

TBA, M. Gromov.

This course will cover 3 topics:

1. Smale-Anosov dynamical systems: This will include basic stability and ergodicity theorems for Anosov and more genral Smale systems, including geodesic flows on manifolds with negative curvatures. The course will also prove basic results on Markov partitions and Markov codings.

2. Spaces with negative curvatures: The course will start with the classical material on Riemannian manifolds, then will prove basic properties of singular spaces of Busemann and Alexandrov with negative curvatures and explain the main constructions of these spaces.

3. Hyperbolic groups: The course will define these groups, prove their basic properties and discuss various classes of examples.

Texts: TBA.

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

3 points. Spring term.

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

**Prerequisites**: TBA.

Texts: TBA.

** ANALYSIS**

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Descriptions for All Courses)

MATH-GA 1002.001 MULTIVARIABLE ANALYSIS

3 points. Spring term.

Monday, 7:10-9:00, Y. Chen.

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,
S. Serfaty (fall); Thursday, 5:10-7:00, TBA (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.

Monday, Wednesday, 9:35-10:50, T. Austin.

*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**: Main Text: Folland's "Real Analysis: Modern Techniques and Their Applications"

Secondary Text: Bass' "Real Analysis for Graduate Students"

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

3 points per term. Fall and spring terms.

Tuesday, 7:10-9:00, F. Hang
(fall); Wednesday, 7:00-9:00 (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.

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

3 points. Fall term.

Tuesday, Thursday, 1:25-2:40, A.
Cerfon.

*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, TBA.

**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.

Monday, 11:00-12:50, J. Shatah (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 theTH-GA 2430.001
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. Fall term.

Thursday, 9:00-10:50, F. Lin.

**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. Spring term.

Monday, 9:00-10:50, F. Hang.

**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 2620.001
ADVANCED TOPICS IN PDE (Topic TBA)**

3 points. Spring term.

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

**Prerequisites**: TBA.

Texts: TBA.

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

3 points. Fall term.

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

**Prerequisites**: 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 (for Fall semester)**: 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 (TBA)**

3 points. Fall term.

TBA, S. Gunturk.

**Prerequisites**: TBA.

Texts: TBA.

**MATH-GA 2660.001
ADVANCED TOPICS IN ANALYSIS (Dynamical Systems)**

3 points. Spring term.

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

**Prerequisites**: 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.

Texts: TBA.

**MATH-GA 2660.003
ADVANCED TOPICS IN ANALYSIS (TBA)**

3 points. Spring term.

Tuesday, Thursday, 1:25-3:15, F. Lin.

**Prerequisites**: TBA.

Texts: TBA.

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

3 points. Spring term.

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

**Prerequisites**: TBA.

Texts: TBA.

** NUMERICAL ANALYSIS**

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Descriptions for All Courses)

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

3 points per term. Fall and spring terms.

Thursday, 5:10-7:00, G. Stadler
(fall); Tuesday, 5:10-7:00, L. Greengard (spring).

**Fall Term**

**Prerequisites**:A good background
in linear algebra, and some experience with writing
computer programs (in MATLAB, Python or another
language). MATLAB will be used as the main
language for the course. Alternatively, you can
also use Python for the homework assignments.
You are encouraged but not required to learn
and use a compiled language.

This course is part of a two-course series meant to introduce graduate 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. There will be extensive homework assignments involving a mix of theory and computational experiments, and an in-class final. Topics covered in the class include floating-point arithmetic, solving large linear systems, eigenvalue problems, interpolation and quadrature (approximation theory), nonlinear systems of equations, linear and nonlinear least squares, nonlinear optimization, and Fourier transforms. This course will not cover differential equations, which form the core of the second part of this series, Numerical Methods II.

**Recommended Texts** (Springer books
are available online from the NYU network):

Deuflhard, P. & Hohmann, A. (2003).
*Numerical Analysis in Modern Scientific Computing*.
*Texts in Applied Mathematiks* [Series, Bk. 43].
New York, NY: Springer-Verlag.

**Further Reading** (available on
reserve at the Courant Library):

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

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

**If you want to brush up your MATLAB**:

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

Moler, C. (2004).
*Numerical Computing with Matlab*. SIAM.
Available online.

**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 (Computational Fluid Dynamics)**

3 points. Fall term.

Thursday, 9:00-10:50, A. Donev.

**Prerequisites**: TBA.

Texts: TBA.

**Cross-listing: **CSCI-GA 2945.001

**MATH-GA 2011.002
ADVANCED TOPICS IN NUMERICAL ANALYSIS (Numerical Optimization)**

3 points. Fall term.

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

**Prerequisites**: TBA.

Texts: TBA.

**Cross-listing: **CSCI-GA 2945.002

**MATH-GA 2011.003
ADVANCED TOPICS IN NUMERICAL ANALYSIS (TBA)**

3 points. Fall term.

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

**Prerequisites**: TBA.

The course is intended for Courant Institute graduate students who are either mathematics students on the applied half of the spectrum or computer science students with an interest in numerical computation. ODEs, the starting point of analysis and dynamics, are one of the two foundation stones of applied mathematics (the other is linear algebra). This course will deepen your appreciation of them fundamentally.

This course will be based on a new textbook that I am 2/3 of the way through writing, Exploring ODEs, with coauthors Toby Driscoll and Asgeir Birkisson. The book aims to be interesting and informative for every applied or numerical mathematician. Its unusual missions are to be numerically-enabled, but not a numerical book and mathematically mature, but not technically advanced. To get an idea, including the table of contents, take a look at the talk I gave about the book project posted at https://people.maths.ox.ac.uk/trefethen/talks.html.

The course will be assessed by homework assignments. These will be computational (working in Chebfun), but will be about behavior of ODEs, not about algorithms or numerical analysis.

**Cross-listing: **CSCI-GA 2945.003

**MATH-GA 2011.004
ADVANCED TOPICS IN NUMERICAL ANALYSIS (The Finite Element Method)**

3 points. Fall term.

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

This course will provide an introduction to the finite element method and its theoretical foundation.

Self-adjoint elliptic problem and calculus of variation. Sobolev spaces, Poincare's and Friedrichs' inequalities. Triangulations of bounded domains in two and three dimensions. Lagrange and Hermite finite elements, which are conforming in H^1. The biharmonic problem and H^2 conforming elements.

Error bounds for basic finite element methods: Cea's lemma and the result of Aubin and Nitsche. Nonconforming finite elements: the lemmas due to Strang. Isoparametric elements and spectral elements. Compressible and almost incompressible elasticity. Mixed methods and the inf-sup condition due to Babuska and Brezzi. Incompressible Stokes. Raviart-Thomas elements and other elements conforming in H(div). Nedelec elements and other conforming elements in H(curl). Solving the large linear and non-linear systems of algebraic equations resulting from finite element approximations. An introduction to domain decomposition and multigrid algorithms.

**Cross-listing: **CSCI-GA 2945.004

**MATH-GA 2012.001
ADVANCED TOPICS IN NUMERICAL ANALYSIS (High-Performance Computing)**

3 points. Spring term.

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

**Prerequisites**: TBA.

Texts: TBA.

**MATH-GA 2012.002
ADVANCED TOPICS IN NUMERICAL ANALYSIS (Monte Carlo)**

3 points. Spring term.

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

**Prerequisites**: TBA.

Texts: TBA.

**MATH-GA 2041.001
COMPUTING IN FINANCE**

3 points. Fall term.

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

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 term.

Thursday, 5:10-7:00, A. Rangan.

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

This course is intended to provide a practical introduction to 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.

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

3 points. Fall term.

Tuesday, 7:10-9:00, TBA.

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

The classical curriculum of mathematical finance programs generally covers the link between linear parabolic partial differential equations (PDEs) and stochastic differential equations (SDEs), resulting from Feynmam-Kac's formula. However, the challenges faced by today's practitioners mostly involve nonlinear PDEs. The aim of this course is to provide the students with the mathematical tools and computational methods required to tackle these issues, and illustrate the methods with practical case studies 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), portfolio optimization, transaction costs, illiquid markets, super-replication under delta and gamma constraints, etc.

We will strive to make this course reasonably comprehensive, and to find the right balance between ideas, mathematical theory, and numerical implementations. We will spend some time on the theory: optimal stopping, stochastic control, backward stochastic differential equations (BSDEs), McKean SDEs, branching diffusions. But the main focus will deliberately be on ideas and numerical examples, which we believe help a lot in understanding the tools and building intuition.

Recommended text: Guyon, J. and Henry-Labordère, P.: Nonlinear Option Pricing, Chapman & Hall/CRC Financial Mathematics Series, 2014.

**MATH-GA 2046.001
ADVANCED ECONOMETRIC MODELING AND BIG DATA**

3 points. Fall term.

Thursday, 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 2048.001
SCIENTIFIC COMPUTING IN FINANCE**

3 points. Spring term.

Wednesday, 5:10-7:00, TBA.

**Prerequisites**: Risk and Portfolio Management with Econometrics, Derivative Securities, and Computing in Finance

This is a version of the course Scientific Computing (MATH-GA 2043.001) designed for applications in quantitative finance. It covers software and algorithmic tools necessary to practical numerical calculation for modern quantitative finance. Specific material includes IEEE arithmetic, sources of error in scientific computing, numerical linear algebra (emphasizing PCA/SVD and conditioning), interpolation and curve building with application to bootstrapping, optimization methods, Monte Carlo methods, and the solution of differential equations

**Please Note: Students may not receive credit for both MATH-GA 2043.001 and MATH-GA 2048.001**

** APPLIED MATHEMATICS**

(Show Detailed
Descriptions for All Courses)

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

3 points. Fall term.

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

**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,
O. Buhler.

**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.); and *
Diffusions, 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 2707.001
TIME SERIES ANALYSIS AND STATISTICAL ARBITRAGE**

3 points. Fall term.

Monday, 7:10-9:00, 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, TBA.

**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, R. Kohn.

This course provides a basic mathematical
introduction to classical Newtonian mechanics, the
mechanics of rigid bodies, and quantum mechanics.
Prior knowledge in physics is not required. Key topics
include: variational formulation of classical
mechanics, Lagrangians, conserved quantities,
constrained motion; Hamilton’s equations, phase space
evolution and Poincaré sections, Lyapunov exponents,
Liouville’s theorem ;lLinear stability of fixed
points, integrable systems, homoclinic tangle,
Poincaré -Birkhoff theorem; Moments of inertia, Euler
angles, motion of rigid bodies, spin-orbit coupling;
representation of quantum states, correspondence
principle, observables and their spectrum, Schrödinger
equation, Ehrenfest’s equations, uncertainty
principle, spin angular momentum, quantum description
of the hydrogen atom.

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).

**Fall Term**

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

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**

Risk Management is arguably one of the most important tools for managing a trading book and quantifying the effects of leverage and diversification (or lack thereof).

This course is an introduction to risk-management techniques for portfolios of (i) equities and delta-1 securities and futures (ii) equity derivatives (iii) fixed income securities and derivatives, including credit derivatives, and (iv) mortgage-backed securities.

A systematic approach to the subject is adopted, based on selection of risk factors, econometric analysis, extreme-value theory for tail estimation, correlation analysis, and copulas to estimate joint factor distributions. We will cover the construction of risk-measures (e,g. VaR and Expected Shortfall) and historical back-testing of portfolios. We also review current risk-models and practices used by large financial institutions and clearinghouses.

If time permits, the course will also cover models for managing the liquidity risk of portfolios of financial instruments.

**MATH-GA 2752.001 ACTIVE
PORTFOLIO MANAGEMENT**

3 points. Spring term.

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

**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.

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

**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, TBA
(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.

**MATH-GA 2792.001
CONTINUOUS TIME FINANCE**

3 points. Fall and spring terms.

Monday, 7:10-9:00, A. Javaheri (fall);
Wednesday, 7:10-9:00, TBA (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.

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

3 points. Spring term.

Thursday, 7:10-9:00, TBA.

**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 of Securitized Products, emphasizing Residential Mortgages and Mortgage-Backed Securities (MBS). We will build pricing models that generate cash flows taking into account interest rates and prepayments. The first part of the course will also review subprime mortgages, CDO’s, Commercial Mortgage Backed Securities (CMBS), Auto Asset Backed Securities (ABS), Credit Card ABS, and CLO’s, and will discuss drivers of the financial crisis and model risk.

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.
(2007). *Wiley Finance* [Series, Bk. 83].
*Salomon Smith Barney Guide to Mortgage-backed and
Asset-backed Securities*. New York, NY:
John Wiley & Sons.

Swindle, G. (2014).
*Valuation and Risk Management in Energy Markets*.
New York, NY: Cambridge University Press

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, TBA.

When a corporation borrows money there is a risk that it will not fulfill its obligation to repay the lenders in the future; this is credit risk. This course develops mathematical tools and models that are useful in analyzing, valuing and managing credit risk, both in its original form as found embedded in bonds and loans, and in derived forms as it exists in derivatives like asset swaps, credit default swaps (CDS), CDS indices and options, and tranched portfolio products like synthetic CDOs. We will discuss alternative notions of credit spread, and their dynamics and relationship to fundamental quantities like probability of default and loss given default. The consideration of portfolio products will require the introduction of notions of credit correlation including, but not limited to, the Gaussian default time copula.

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

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

3 points. Spring term.

Thursday, 5:10-7:00, TBA.

**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 (TURBULENT DYNAMICAL SYSTEMS)**

3 points. Fall term.

Thursday, 1:15-3:15, A. Majda.

The course should be interesting for graduate students, and postdocs in pure and applied mathematics, physics, engineering, and climate, atmosphere, ocean science interested in turbulent dynamical systems as well as other complex systems.

Turbulent dynamical systems are ubiquitous complex systems in geoscience and engineering and are characterized by a large dimensional phase space and a large dimension of strong instabilities which transfer energy throughout the system. They also occur in neural and material sciences. Key mathematical issues are their basic mathematical structural properties and qualitative features, their statistical prediction and uncertainty quantification (UQ), their data assimilation, and coping with the inevitable model errors that arise in approximating such complex systems. These model errors arise through both the curse of small ensemble size for large systems and the lack of physical understanding. This is a research expository course on the applied mathematics of turbulent dynamical systems through the paradigm of modern applied mathematics involving the blending of rigorous mathematical theory, qualitative and quantitative modelling, and novel numerical procedures driven by the goal of understanding physical phenomena which are of central importance. The contents include the general mathematical framework and theory, instructive qualitative models, and concrete models from climate atmosphere ocean science. New statistical energy principles for general turbulent dynamical systems are discussed with applications, linear statistical response theory combined with information theory to cope with model errors, reduced low order models, and recent mathematical strategies for UQ in turbulent dynamical systems. Also recent mathematical strategies for online data assimilation of turbulent dynamical systems as well as rigorous results are briefly surveyed. Accessible open problems are often mentioned. This research expository book is the first of its kind to discuss these important issues from a modern applied mathematics perspective.

Texts: Chapter 1, 2, 4, 6, 7, 8 from Nonlinear Dynamics
and Statistical Theories for Basic Geophysical Flow, by A. J.
Majda and X. Wang, Cambridge Press 2006;

Introduction to Turbulent Dynamical Systems for Complex Systems,
Frontiers in Applied Dynamical Systems, by A. J. Majda, 85 pages, Springer 2016.

**MATH-GA 2830.002
ADVANCED TOPICS IN APPLIED MATHEMATICS (Optimization-Based Data Analysis)**

3 points. Fall term.

Monday, 1:25-3:15, C. Fernandez-Granda.

**Prerequisite**: TBA.

Texts: TBA.

**MATH-GA 2840.001
ADVANCED TOPICS IN APPLIED MATHEMATICS
(Data Analysis Opt Transp)**

3 points. Spring term.

Monday, 1:25-3:15, E. Tabak..

**Prerequisite**: TBA.

Texts: TBA.

**MATH-GA 2851.001 ADVANCED
TOPICS IN MATH BIOLOGY (Mathematical Neural Science)**

3 points. Fall term.

Wednesday, 1:25-3:15, D. McLaughlin
& D. Cai.

This course covers the mathematical representations of neuronal systems, and the mathematical methods used to analyze these representations. Many of these methods are the same as those used to analyze experimental data in neural science.

The course begins with the fundamental physiological properties of neurons, including neuronal and synaptic dynamics. The Hodgkin-Huxley equations for single neuronal dynamics, with synaptic couplings, will be briefly summarized -- as well as their reductions to simpler representations such as integrate and fire neurons and “exponential” integrate and fire neurons. The course then turns to its focus – representations of systems of excitatory and inhibitory neurons. It will cover, in detail, population dynamics of such neuronal networks, including coarse-grained representations (mean-field firing rate models, models which incorporate fluctuations, and models which incorporate multi-neuron effects), and functional and structural connectivity. The course strives to bring students from a background in either the physical/mathematical sciences or neural science quickly to research topics in theoretical neural science.

Topics to be covered will be taken from the following longer list:

- Basic Neurophysiology
- Anatomy of a Neuron
- Synapses and synaptic coupling
- Hodgkin-Huxley Equations
- Integrate and Fire Representations, including “exponential integrate and fire”

- Neuronal Coding – rate vs spike coding
- Signal Detection Theory
- Discrimination
- ROC
- Nieman-Pearsen Lemma
- Spike train analysis
- Renewal processes
- Wiener-Khinchin Theorm
- Power spectrum
- Coarse-grained representations for populations of neurons
- Mean-field firing rate models
- Mean-field representations with fluctuations
- Other methods to model fluctuations
- Kinetic Theory
- Multi-neuron effects
- Fokker-Planck equation for a single neuron
- Kramer-Moyal Expansion
- Pawula Theorem
- Compressed Sensing
- System analysis
- Weiner Expansion
- Bussgang Theorem
- Receptive Fields
- Functional vs anatomical connectivity
- Granger causality
- Early visual system (Retina → LGN → V1)
- Receptive Fields
- Simple and complex cells

Texts: TBA.

**MATH-GA 2855.001
ADVANCED TOPICS IN MATH PHYSIOLOGY (TBA)**

3 points. Fall term.

Wednesday, 2:30-4:20, J.
Rinzel.

**Prerequisite**: TBA.

Texts: TBA.

**MATH-GA 3001.001
GEOPHYSICAL FLUID DYNAMICS**

3 points. Fall term.

Tuesday, 9:00-10:50, E. Gerber.

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
ADVANCED TOPICS IN ATMOSPHERE-OCEAN SCIENCE (Climate Change)**

3 points. Spring term.

Tuesday, 1:25-3:15, S. Smith.

**Prerequisites**: TBA.

Texts: TBA.

**MATH-GA 3010.001
ADVANCED TOPICS IN ATMOSPHERE-OCEAN SCIENCE
(TBA)**

3 points. Fall term.

Wednesday, 9:00-10:50, D. Holland.

**Prerequisites**: TBA.

Texts: TBA.

**MATH-GA 3010.002
ADVANCED TOPICS IN ATMOSPHERE-OCEAN SCIENCE (Climate Change)**

3 points. Fall term.

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

**Prerequisites**: TBA.

Texts: TBA.

** PROBABILITY AND
STATISTICS**

(Show Detailed
Descriptions for All Courses)

**MATH-GA 2901.001 BASIC
PROBABILITY**

3 points. Fall and spring terms.

Wednesday, 5:10-7:00, Y. Bakhtin (fall); Wednesday, 7:10-9:00, E. Vanden Eijnden
(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, P. Bourgade(fall); Thursday, 7:10-9:00, TBA (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); TBA (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, 11:00-12:50, E. Lubetzky
(fall); Wednesday, 9:00-10:50, H.
McKean (spring).

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

**Fall Term**

First semester in an annual sequence of Probability Theory, aimed primarily for Ph.D. students. Topics include laws of large numbers, weak convergence, central limit theorems, conditional expectation, martingales and Markov chains.

**Text**: Varadhan, S.R.S. (2001). *Courant
Lecture Series in Mathematics* [Series, Bk. 7].

**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 (Extremes via second moments for logarithmic correlated fields: Gaussian and beyond)**

3 points. Fall term.

Tuesday, Thursday, 3:20-5:05, O. Zeitouni.

**Prerequisites**: A graduate course in probability. Prerequisites on Gaussian processes will be introduced as needed.

We will discuss the extremes of logarithmically correlated fields, starting with the classical case of branching random walks and moving to the general Gaussian case and beyond.

Texts: Notes will be made available, there is no textbook.

**MATH-GA 2931.002
ADVANCED TOPICS IN PROBABILITY (Coarsening and Related Interacting Particle Systems)**

3 points. Fall term.

Wednesday, Friday, 1:25-3:15, C. Newman.

**Prerequisites**: Recommended: Probability Theory at the level of Probability Limit Theorems I and II

An introduction to some of the stochastic processes where the states, evolving in time, are assignments of, say, +1 or -1 to points in d-dimensional lattices or other graphs, which are motivated by nonequilibrium statistical physics at zero (or low) temperature. A typical transition/update rule is to agree with a strict majority of nearest neighbors or else, in the case of a tie, to toss a fair coin. We will discuss both known results and many open problems.

One old and one recent reference:

Nanda-Newman-Stein: Dynamics of Ising spin systems at zero temperature, A. M. S. Transl. (2) 198 (2000) 183-194.

Damron-Kogan-Newman-Sidoravicius: Coarsening with a frozen vertex, Elec. Commun. Prob. 21 (2016) 1-4.

**MATH-GA 2931.003
ADVANCED TOPICS IN PROBABILITY (Entropy and Ergodic Theory)**

3 points. Fall term.

Thursday, 1:25-3:15, T. Austin.

**Prerequisites**: TBA.

This will be an introduction to entropy and its many roles in different branches of mathematics, especially information theory, probability, ergodic theory and statistical mechanics. The aim is to give a quick overview of many topics, emphasizing a few basic combinatorial problems that they have in common and which are responsible for the ubiquity of entropy.

The course divides roughly into five parts:

- Shannon Entropy and Information Theory;
- Large Deviations and Measure Concentration;
- Statistical Mechanics and Thermodynamics;
- Ergodic Theory;
- The Thermodynamic Formalism.

Parts 1, 2 and 3 will introduce entropy in settings that can be described in terms of independent random variables. Parts 4 and 5 will generalize much of that story to stochastic processes having an invariance under the passage of time or spatial translation. The natural setting for those generalizations is provided by ergodic theory. In order to cover a wide range of subjects, I will have to sacrifice a lot of generality, and sometimes omit technical details. To make up for this, students will be expected to support the lectures with reading from a variety of sources.

Texts: TBA.

*Revised September 2015*