Courant Institute New York University FAS CAS GSAS

Course Descriptions: AY 2015-16

Course Schedule

Undergraduate

Graduate


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

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 (fall); Monday, 5:10-7:00, Instructor TBA (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 (4th 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 (5th 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 (4th ed.). Upper Saddle River, NJ: Prentice-Hall/ Pearson Education.

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

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 (2nd 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, A. Rangan

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 (2nd ed.). Hoboken, NJ: John Wiley & Sons/ Wiley-Interscience.

Recommended Text: Strang, G. (2005). Linear Algebra and Its Applications (4th 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 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 (3rd 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 (2nd 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 2150.001 Advanced Topics in Algebra (Introduction to Algebraic Geometry and Elliptic Curves)
3 points. Fall term.
Wednesday, 5:10-7:00, A. Pirutka.

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

The objects of study in algebraic geometry are systems defined by polynomial equations. Here are some examples:
1. x^2+y^2+z^2-w^2=0;
2. y^2=x^3-2x
3. x^2+y^2=u^2, x^2+z^2=v^2, y^2+z^2=w^2, x^2+y^2+z^2=t^2.

The first example gives a projective quadric; the second one defines an elliptic curve. The rational solutions of the third one provide a rectangular box such that the lengths of the edges, face diagonals, and long diagonals are rational numbers. The existence of such a solution is still not known.

We will start this introductory course with some topics from the commutative algebra, such as ideals in polynomial rings and the famous Nullstellensatz theorem. We will also discuss some projective geometry in dimension two. A large part of the course will be devoted to the study of elliptic curves over various fields: finite fields, fields of rational or complex numbers. For elliptic curves defined over finite fields we will discuss as well applications in cryptography.
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, Instructor TBA.
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

MATH-GA 2310.001, 2320.001 TOPOLOGY I, II

3 points per term. Fall and spring terms.
Tuesday, 7:10-9:00, R. Young (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 (2nd 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 2333.001 ADVANCED TOPICS IN TOPOLOGY (Topic TBA)

3 points. Fall term.
Thursday, 9:00-10:50, S. Cappell.

Course description not yet available.
MATH-GA 2350.001, 2360.001 DIFFERENTIAL GEOMETRY I, II

3 points per term. Fall and spring terms.
Wednesday, 1:25-3:15, J. Cheeger (fall); Wednesday, 1:25-3:15, Instructor TBA (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.
MATH-GA 2410.001 ADVANCED TOPICS IN GEOMETRY (Topic TBA)
3 points.  Spring term.
Tuesday, 1:25-3:15, J. Cheeger.

Course description not yet available.


ANALYSIS

MATH-GA 1002.001 MULTIVARIABLE ANALYSIS

3 points. Spring term.
Monday, 7:10-9:00, Instructor TBA.
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, P. Germain (fall); Thursday, 5:10-7:00, E. Hameiri (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, 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: Royden, H.L. (1988). Real Analysis (3rd 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 (3rd 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 (2nd 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, F. Hang (fall); Wednesday, 5:10-7:00 (spring), J. Shatah.

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 (8th 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 (3rd 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, R. Varadhan.

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 (3rd ed.). New York, NY: McGraw-Hill.

MATH-GA 2470.001 ORDINARY DIFFERENTIAL EQUATIONS

3 points. Spring term.
Tuesday, 9:00-10:50, F. Hang.
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, 9:00-10:50, N. Masmoudi (fall and 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 (2nd 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 (2nd Rev. ed.). Providence, RI: AMS Chelsea Publishing/ American Mathematical Society.

Evans, L.C. (2010). Graduate Studies in Mathematics [Series, Bk. 19]. Partial Differential Equations (2nd 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, P. Deift.

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 (1st ed.). New York, NY: John Wiley & Sons/ Wiley-Interscience.

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

MATH-GA 2563.001 HARMONIC ANALYSIS

3 points. Spring term.
Monday, 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 (Topic TBA)
3 points. Fall term.
Wednesday, 1:25-3:15, F. Lin.

Course description not yet available.
MATH-GA 2650.001 ADVANCED TOPICS IN ANALYSIS (Ergodic Theory)
3 points. Fall term.
Monday, 5:10-7:00, T. Austin

The study of dynamical systems for which the state space is a probability space with a probability measure that the dynamics leaves invariant. This material should be of interest to students of probability, functional analysis or harmonic analysis, and also has deep connections to combinatorics, analytic number theory and parts of group theory.

After the definition of dynamical systems and a number of examples, the first part of the course will include: the basic ergodic theorems; equidistribution phenomena for homeomorphisms of compact metric spaces, with some applications to number theory; an introduction to the abstract study of the structure of probability-preserving dynamical systems, covering constructions such as factors and joinings and phenomena such as ergodicity and weak and strong mixing.

After these we will move on to more advanced topics. Depending on time and the interests of the class, two likely choices are: multiple recurrence phenomena, which relate to a large area of combinatorial number theory centered around the Multidimensional Szemeredi Theorem; some facets of the Kolmogorov-Sinai entropy theory for measure-preserving dynamical systems.

MATH-GA 2660.001 ADVANCED TOPICS IN ANALYSIS (Functional Analysis)

3 points. Spring term.
Thursday, 1:25-3:15, P. Deift.
Course description not yet available.
MATH-GA 2660.002 ADVANCED TOPICS IN ANALYSIS (Topic TBA)
3 points. Spring term.
Monday, 1:25-3:15, F. Lin.
Course description not yet available.
MATH-GA 2660.003 ADVANCED TOPICS IN ANALYSIS (Complex Analysis)

3 points. Spring term.
Wednesday, 9:00-10:50, P. Germain.

Course description not yet available.
MATH-GA 2660.004 ADVANCED TOPICS IN ANALYSIS (Ergodic Theory)

3 points. Spring term.
Tuesday, 1:25-3:15, T. Austin.

The study of dynamical systems for which the state space is a probability space with a probability measure that the dynamics leaves invariant. This material should be of interest to students of probability, functional analysis or harmonic analysis, and also has deep connections to combinatorics, analytic number theory and parts of group theory.

After the definition of dynamical systems and a number of examples, the first part of the course will include: the basic ergodic theorems; equidistribution phenomena for homeomorphisms of compact metric spaces, with some applications to number theory; an introduction to the abstract study of the structure of probability-preserving dynamical systems, covering constructions such as factors and joinings and phenomena such as ergodicity and weak and strong mixing.

After these we will move on to more advanced topics. Depending on time and the interests of the class, two likely choices are: multiple recurrence phenomena, which relate to a large area of combinatorial number theory centered around the Multidimensional Szemeredi Theorem; some facets of the Kolmogorov-Sinai entropy theory for measure-preserving dynamical systems.


NUMERICAL ANALYSIS

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); Thursday, 5:10-7:00, J. Goodman (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 (Approximation Theory and Practice)

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

This course will be closely based on Prof. Trefethen's textbook Approximation Theory and Approximation Practice. A great deal of information about the book and the subject can be found at www.maths.ox.ac.uk/chebfun/ATAP/, including a PDF of the first six chapters. The subject matter is material that everyone who is serious about numerical computation needs to know, presented in an intensive mix of theory and Chebfun-based numerical practice.

Text: Trefethen, L.N. (2012). Approximation Theory and Approximation Practice. Philadelphia, PA: Society for Industrial & Applied Mathematics.

Cross-listing: CSCI-GA 2945.002.

The course will given twice a week, beginning September 8 and ending October 15.

MATH-GA 2012.001 ADVANCED TOPICS IN NUMERICAL ANALYSIS (Variational Inverse Problems)
3 points. Spring term.
Thursday, 9:00-10:50, G. Stadler.

Course description not yet available.

Cross-listing: CSCI-GA 2945.001.
MATH-GA 2012.002 ADVANCED TOPICS IN NUMERICAL ANALYSIS (Convex and Non-Smooth Optimization)
3 points. Spring term.
Tuesday, 5:10-7:00, M. Overton.

Course description not yet available.

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, Instructor TBA.

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. Donev.

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 (2nd 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 (2nd 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.

MATH-GA 2046.001 ADVANCED ECONOMETRIC MODELING AND BIG DATA

3 points. Fall term.
Thursday, 7:10-9:00, Instructor TBA.

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, Instructor 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

MATH-GA 2701.001 METHODS OF APPLIED MATHEMATICS

3 points. Fall term.
Monday, 1:25-3:15, O. Buhler.

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, E. Hameiri.

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, E. Vanden Eijnden.

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 2706.001 PARTIAL DIFFERENTIAL EQUATIONS FOR FINANCE

3 points. Spring term.
Monday, 5:10-7:00, Instructor TBA.

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, Instructor TBA.

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.

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, Instructor TBA (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, 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, Instructor 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, Instructor 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, Instructor 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, Instructor TBA (fall); Wednesday, 7:10-9:00, Instructor 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, 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 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, 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, Instructor 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 MATHEMATICS (Topic TBA)
3 points. Fall term.
Thursday, 3:20-5:05, A. Majda.

Course description not yet available.
MATH-GA 2830.002 ADVANCED TOPICS IN APPLIED MATHEMATICS (Fast Analysis-Based Algorithms)
3 points. Fall term.
Monday, 1:25-3:15, M. O'Neil.

Prerequisites: Knowledge of PDE, complex analysis, numerical methods, and experience in programming are strongly recommended.

This course will be an introduction to several numerical methods known as "fast analysis-based algorithms," including fast multiple methods, butterfly algorithms, hierarchical matrix compression and fast direct solvers.  These algorithms can be used to solve many of the PDE governing classical mathematical physics, namely electromagnetics, acoustics, heat diffusion, and fluid dynamics. There are additional applications in signal processing and data analysis.  Methods from potential theory, applied analysis, functional analysis, numerical linear algebra, complex analysis, and asymptotic analysis are central to the construction of almost all such algorithms.

Grading will be based on a course project.
MATH-GA 2830.003 ADVANCED TOPICS IN APPLIED MATHEMATICS (Convex Duality in Math Finance)
3 points. Fall term.
Wednesday, 5:10-7:00, P. Carr & Q. Zhu.

Course description not yet available.

MATH-GA 2840.001 ADVANCED TOPICS IN APPLIED MATHEMATICS (Data Analysis Methods for High Dimensional Time Series)

3 points. Spring term.
Wednesday, 1:25-3:15, D. Giannakis.

Course description not yet available.

MATH-GA 2840.002 ADVANCED TOPICS IN APPLIED MATHEMATICS (Optimization-Based Data Analysis)

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

Prerequisites: Basic knowledge of probability and linear algebra.

In recent years, optimization-based techniques have had a major impact in two areas of data analysis: inverse problems and statistical estimation in high-dimensional spaces. The main principle underlying these techniques is to exploit nonparametric prior knowledge about the data by solving tractable optimization problems. Some representative examples include sparse regression analysis of genomic data, compressed sensing in medical imaging and signal processing, and low-rank models in recommender systems and computer vision. This course provides an introduction to optimization-based methods by describing the theoretical tools used to analyze them, the computational algorithms developed to implement them and some of their main applications. The course will start with a brief introduction to basic concepts in convex optimization. Class materials will be posted online.

Recommended Text
: Boyd, S., & Vandenberge, L. (2004). Convex Optimization. New York: Cambridge University Press.

MATH-GA 2851.001 ADVANCED TOPICS IN MATH BIOLOGY (Math Neuroscience)

3 points. Fall term.
Thursday, 1:25-3:15, D. Cai & D. McLaughlin.

Course description not yet available.
MATH-GA 2851.002 ADVANCED TOPICS IN MATH BIOLOGY (Topic TBA)
3 points. Fall term.
Monday, 1:25-3:15, C. Peskin.

Course description not yet available.
MATH-GA 2852.001 ADVANCED TOPICS IN MATH BIOLOGY (Synaptic Transmission)
3 points. Spring term.
Wednesday, 1:25-3:15, C. Peskin.

Course description not yet available.

Cross-listing: BIOL-GA 2852.001.
MATH-GA 2852.002 ADVANCED TOPICS IN MATH BIOLOGY (Topic TBA)
3 points. Spring term.
Thursday, 1:25-3:15, A. Mogilner.

Course description not yet available.

Cross-listing: BIOL-GA 2852.002.
MATH-GA 2855.001 ADVANCED TOPICS IN MATH PHYSIOLOGY (Neuronal Networks)
3 points. Fall term.
Wednesday, 2:30-4:20, J. Rinzel.

Course description not yet available.

Cross-listing: BIOL-GA 2855.001.
MATH-GA 2861.001 ADVANCED TOPICS IN FLUID DYNAMICS (Plasma Physics)
3 points. Fall term.
Tuesday, 9:00-10:50, J. Freidberg.

Course description not yet available.
MATH-GA 2862.001 ADVANCED TOPICS IN FLUID DYNAMICS (Computational Methods for Fluid-Structure Interactions)
3 points. Spring term.
Tuesday, 1:25-3:15, M. Shelley.

Course description not yet available.
MATH-GA 3001.001 GEOPHYSICAL FLUID DYNAMICS
3 points. Fall term.
Tuesday, 9:00-10:50, D. Giannakis.

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 3004.001 ATMOSPHERIC DYNAMICS
3 points. Spring term.
Tuesday, 1:25-3:15, O. Pauluis.

What effects the large scale circulation of the atmosphere?   Like the antiquated heating system of a New York apartment, solar radiation unevenly warms the Earth, leading to gradients in energy in both altitude and latitude.  But unlike the simple convection of air in your drafty home, the effects of rotation, stratification, and moisture lead to exotic variations in weather and climate, giving us something to chat about over morning coffee... and occasionally bringing modern life to a standstill.

The goals of this course are to describe and understand the processes that govern atmospheric fluid flow, from the Hadley cells of the tropical troposphere to the polar night jet of the extratropical stratosphere, and to prepare you for research in the climate sciences.   Building on your foundation in Geophysical Fluid Dynamics, we will explore how stratification and rotation regulate the atmosphere's response to gradients in heat and moisture.  Much of our work will be to explain the zonal mean circulation of the atmosphere, but in order to accomplish this we’ll need to learn a great deal about deviations from the zonal mean: eddies and waves.  It turns out that eddies and waves, planetary, synoptic (weather system size) and smaller in scale, are the primary drivers of the zonal mean circulation throughout much, if not all, of the atmosphere.

There will also be a significant numerical modeling component to the course.  You will learn how to run an atmospheric model on NYU's High Performance Computing facility, and then design and conduct experiments to test the theory developed in class for a final course project.

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

Lorenz, E.N. (1967). The Nature and Theory of the General Circulation of the Atmosphere. World Meteorological Organization.

Walker, G. (2007). An Ocean of Air: Why the Wind Blows and Other Mysteries of the Atmosphere. Orlando, FL: Houghton Mifflin Harcourt.
MATH-GA 3010.001 ADVANCED TOPICS IN ATMOSPHERE-OCEAN SCIENCE (Environmental Fluid Dynamics Lab)
3 points. Fall term.
Wednesday, 9:00-10:50, D. Holland.

Registration subject to approval by course instructor.

Description available from course instructor.
MATH-GA 3010.002 ADVANCED TOPICS IN ATMOSPHERE-OCEAN SCIENCE (Geophysical Turbulence)
3 points. Fall term.
Tuesday, 1:25-3:15, S. Smith.

Course description not yet available.
MATH-GA 3011.001 ADVANCED TOPICS IN ATMOSPHERE-OCEAN SCIENCE (Climate Modeling)
3 points. Spring term.
Thursday, 1:25-3:15, E. Gerber.

Prerequisites: Basic knowledge of fluid dynamics and physics.

Amidst the shouting over climate change and what action should be taken about it, this course seeks to focus in on the science of climate prediction. We will work our way through the components of state-of-the-art climate models, endeavoring to understand, or at least appreciate, the science and approximations that lie behind predictions of future climate change. The course will proceed in seminar format. Participants will be expected to read background material on climate models and complete a research oriented project, running and/or analyzing the output from an IPCC class climate model. No experience in climate modeling is required, but a basic knowledge of fluid dynamics and physics will help.

Our goal is to work through the key components of a climate model. The Earth’s climate is determined by interactions between the atmosphere, oceans, cryosphere (ice sheets, glaciers, sea ice) and land surfaces (terrestrial hydrology, biology, etc.). This course will be biased towards the atmospheric component of a climate model, but we will seek to understand how interactions between the atmosphere and the other elements of the climate system are represented in models.

Recommended Texts: McGufie, K. & Henderson-Sellers, A. (2014). A Climate Modelling Primer (4th ed.). Hoboken, NJ: John Wiley & Sons.

Jacobson, M.Z. (2005). Fundamentals of Atmospheric Modeling (2nd ed.). New York, NY: Cambridge University Press.

Washington, W.M., & Parkinson, C.L. (2005). An Introduction to Three-Dimensional Climate Modeling (2nd ed.). Sausalito, CA: University Science Books.

Walker, G. (2007). An Ocean of Air: Why the Wind Blows and Other Mysteries of the Atmosphere. Orlando, FL: Houghton Mifflin Harcourt.

Weart, S.R. (2008). New Histories of Science, Technology, and Medicine [Series, Bk. 13]. The Discovery of Global Warming (Rev. ed.). Cambridge, MA: Harvard University Press.

Other materials and links will be posted on the course website.



PROBABILITY AND STATISTICS


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, Instructor 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); 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, 9:00-10:50, 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

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 (Topic TBA)
3 points. Fall term.
Tuesday, Thursday, 3:20-5:05, O. Zeitouni.

Course description not yet available.
MATH-GA 2931.002 ADVANCED TOPICS IN PROBABILITY (Statistical Mechanics and the Riemann Hypothesis)
3 points. Fall term.
Wednesday, Friday, 1:25-3:15, C. Newman.

Course description not yet available.
MATH-GA 2931.003 ADVANCED TOPICS IN PROBABILITY (Ergodic Theory of Markov Processes)
3 points. Fall term.
Monday, 9:00-10:50, Y. Bakhtin.

To understand the behavior of a deterministic or random dynamical system, it is often useful to find and study stationary regimes or invariant distributions, since they govern long-term statisticalproperties of the system. From this point of view, existence and uniqueness of invariant distributions and, more generally, description of all invariant distributions along with convergence to equilibrium become important questions that this course will be centered around.

The course will begin with a discussion of deterministic dynamics, then proceed to Markov processes and random dynamics. The state space
for Markov processes will be consecutively assumed to be finite, countable, finite dimensional, infinite dimensional, and appropriate methods and notions will be discussed such as compactness, Krylov--Bogolyubov method, coupling, regularity conditions of minorization type, Doeblin and Harris conditions, strong Feller property, Lyapunov functions, asymptotic strong Feller property, asymptotic coupling, one force -- one solution principle. In the end of this course, these notions will be applied to stochastic PDEs such as stochastic Navier--Stokes system and stochastic Burgers equation.

Good knowledge of probability and measure theory is required (at the level of Probability: Limit Theorems 1). No knowledge of ergodic theory, dynamical systems, or Markov processes will be assumed (although acquaintance with those subjects will make the course more meaningful). A significant amount of time in this course will be devoted to material that may be viewed as preparatory.

Revised April 2015