Course Descriptions: AY 2011-12

MATH-GA 2110.001, 2120.001  LINEAR ALGEBRA I, II

3 points per term.  Fall and spring terms.
Tuesday, 5:10-7:00, W. Widlund (fall); Monday, 5:10-7:00, F. Greenleaf (spring).

Fall Term

Prerequisite: 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: Linear Algebra, Friedberg, Insel & Spence, Prentice-Hall, 4th Ed.

Strongly recommended text: Schaum’s Outline Series: Linear Algebra, S. Lipschuts

Spring Term

Prerequisite: Linear Algebra I  or permission of the instructor.

Trace, determinant, characteristic and minimal polynomial. Eigenvalues, diagonalization, spectral theorem and spectral mapping theorem.  Rayleigh quotient and minimax theorem.  When diagonalization fails: nilpotent operators and their structure, generalized eigenspace decomposition, Jordan canonical form. 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: Linear Algebra, Friedberg, Insel & Spence, Prentice-Hall, 4th Ed.

Cross-listed as MATH-UA 0141, 0142.

MATH-GA 2110.001  LINEAR ALGEBRA I

3 points.  Spring term.
Tuesday, 5:10-7:00, B. De Oliveira.

Prerequisite: 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:  Linear Algebra, P. Lax, Wiley - Interscience

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

3 points.  Fall term.
Thursday, 9:00-10:50, E. Vanden Eijnden.

Prerequisite: 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: Linear Algebra, P. Lax, Wiley-Interscience Publications

Optional text:  Linear Algebra and Its Applications, G. Strang

MATH-GA 2130.001, 2140.001  ALGEBRA I, II

3 points per term.  Fall and spring terms.
Wednesday, 7:10-9:00, B. Bakker (fall); Monday, 7:10-9:00, F. Bogomolov (spring)

Prerequisite:  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.

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:  Algebra, M. Artin, Prentice Hall

Supplementary texts:  Algebra, S. Lang; Linear Representations of Finite Groups, J. P. Serre; Undergraduate Algebraic Geometry, M. Reid; Representations and Characters of Groups, G. James and M. Liebeck; Cambridge Math Textbooks, 1993; Representation Theory, W. Fulton and J. Harris, Springer-Verlag; The Symmetric Group, B. E. Sagan, Wadsworth & Brooks/Cole Math. Series; Representations of Compact Lie Groups, T. Brocker and T. tom Dieck

MATH-GA 2170.001 INTRODUCTION TO CRYPTOGRAPHY

3 points.  Spring term.
Wednesday, 5:00-6:50, Y. Dodis.

Prerequisites: a 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-listed with CSCI-GA 3210.001

MATH-GA 2210.001  NUMBER THEORY

3 points.  Spring term.
Wednesday, 7:10-9:00, I. Karzhemanov.

Prerequisites: basic complex analysis and algebra helpful.

Introduction to elementary methods of number theory. Topics: arithmetic functions, congruences, the prime number theorem, primes in arithmetic progression, quadratic reciprocity, the arithmetic of number fields,  approximations and transcendence theory, p-adic numbers, diophantine equations of degree 2 & 3.

Text:  A course in Arithmetic, J. P. Serre, Springer GTM, #7




MATH-GA 2310.001, 2320.001  TOPOLOGY I, II

3 points per term.  Fall and spring terms.
Monday,  7:10-9:00, (fall); Tuesday, 7:10-9:00, (spring), S. Cappell.

Fall term

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

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

Useful texts: Algebraic Topology, A. Hatcher (on-line at http://www.math.cornell.edu/~hatcher/#ATI);  Topology,  J. Munkres, Prentice Hall 2000, 2nd Ed.; Differential Topology, Guillemin & Pollack, Prentice Hall; Topology from a Differential Viewpoint, J. Milnor,  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 Poincare duality.  Products and ring structures.  Vector bundles, tangent bundles, De Rham cohomology and differential forms.

MATH-GA 2350.001, 2360.001  DIFFERENTIAL GEOMETRY I, II

3 points per term.  Fall and spring terms.
Monday, 1:25-3:15, L. Guth (fall); Thursday, 9:30-11:20, J. Cheeger (spring).

Prerequisites: multivariable calculus and linear algebra.

Fall Term

Differentiable manifolds, tangent bundle, embedding theorems, vector fields and differential forms.  Riemannian metrics and connections, geodesics, exponential map, and Jacobi fields. Generalizations of differential geometric concepts and applications.

Text: Differential Topology, Victor Guillemin & Alan Pollack, AMS Chelsea publishing, American Mathematical Society, Providence RI, republished in 2010

Spring Term

Differential forms. Integration on manifolds.  Sard's Theorem. DeRham cohomology. Morse theory.  Submanifolds and second fundamental form.  Applications to geometric problems.

MATH-GA 2400.001  ADVANCED TOPICS IN GEOMETRY (Quantitative Differentiation)

3 points.  Fall term.
Wednesday, 1:25-3:15, J. Cheeger.

In the simplest instance quantitative differentiation can be explained as follows. Let f be a real valued function on the unit interval with |f'|<=1. One can give a precise quantitative interpretation to the statement: “f looks as linear as one likes at most locations and scales". This is not implied by Taylor's theorem because there is not enough information to estimate the remainder term. Recently it has been recognized that the above model case is a particular instance of a more generally applicable principle, which applies in contexts such as Einstein manifolds, harmonic maps, minimal hypersurfaces and Lipschitz  maps to Banach spaces including the space L_1.  We will start by explaining the model case from a general perspective.  This requires only calculus.  Then we will survey some of the recent applications.  For some of these some knowledge of Riemannian geometry would be helpful.

Grading: this course will be graded as a seminar course; students may be expected to give several of the lectures.

MATH-GA 2400.002  ADVANCED TOPICS IN GEOMETRY (Group Actions on Trees)

3 points.  Fall term.
Thursday, 9:30-11:120, B. Kleiner.

Prerequisites: covering space theory, basic topology, basic Riemannian geometry.

A tree is a connected graph without cycles.  The theory of groups acting on trees (Bass-Serre theory) was developed as a geometric framework for working with splittings of groups (generalizations of free products with amalgamation).  Groups actions on trees soon became a standard tool in geometric group theory, in geometric topology (especially 3-manifolds), and hyperbolic geometry (aka Kleinian groups).  Starting in the 1970's with the work of Thurston on hyperbolic structures, group actions on R-trees -- more general tree-like objects -- entered the story.  Initially they were used  study the degeneration of different types of geometric structures (e.g. to compactify Teichmuller space), and then found a variety of applications in asymptotic geometry.

After introducing the basic theory, the course will cover different topics related to group actions on trees: (1) Stallings' theorem on ends of groups, Serre's conjecture about groups of cohomological dimension 1;
(2) Incompressible surfaces in 3-manifolds, and the Jaco-Shalen-Johansson decomposition; Canonical decompositions of finitely presented groups (after Rips, Sela, Bowditch, Dunwoody-Swenson, Dunwoody-Sageev, Scott-Swarup); (3) Degeneration of hyperbolic structures and actions on R-trees; (4)Measured laminations and R-trees; (5) Thurston's compactification of Teichmuller space using measured laminations; (6) Rips' machine and applications; (7) Asympototic geometry and splittings.

Grading: this course will be graded as a seminar course.

MATH-GA 2410.001  ADVANCED TOPICS IN GEOMETRY (Fundamental Group in Topology, Geometry and Dynamics)

3 points.  Spring term.
Thursday, 1:25-3:15, M. Gromov.

The course will include some background on these three subjects and on some group theory.  It will also include the following topics, which I will try to present in a unified way: (1) topological and homotopy invariance results on Pontryagin classes, starting from S. Novikov’s theorems; (2) global structural stability theorems for hyperbolic, including basic “local” results by Smale and Anosov; (3) geometry of the Abel-Kacobi-Albanese construction, harmonic maps and rigidity theorem of Siu for Kahler manifolds.

Grading: this course will be graded as a seminar course.

MATH-GA 2410.002  ADVANCED TOPICS IN GEOMETRY (Embeddings of Discrete Spaces)

3 points.  Spring term.
Tuesday, 1:25-3:15, A. Naor.

Prerequisites: basic analysis, probability theory, and measure theory.

The structure of certain discrete spaces such as graphs and groups can often be better understood when one represents them as subsets of certain well-understood geometries, such as Banach spaces or trees. This course will cover the basics of embedding theory, starting with the fundamental theorems of Assouad and Bourgain, and will proceed to more advanced topics (including coarse embeddings, nonlinear Dvoretzky problems, and representations arising from semidefinite relaxations).

Recommended text: Chapter 15 of Matousek's Lectures on Discrete e Geometry is a good introductory text (not mandatory); the course will mainlyh cover topics that appear only in research papers.

Grading: this course will be graded as a seminar course requiring a presentation.

MATH-GA 2410.003  ADVANCED TOPICS IN GEOMETRY (Metric Geometry)

3 points.  Spring term.
Monday, 1:25-3:15, L. Guth.

Prerequisites: a first course in algebraic topology (including an introduction to homology).  A first course in Riemannian geometry is a good idea (it can be taken simultaneously).

We will study geometric inequalities about lengths, areas, volumes, and diameters.  One fundamental inequality is the isoperimetric inequality in Euclidean space.  We will focus on several inequalities in a similar spirit, which are not as well known.  They include the (Federer-Fleming) isoperimetric inequality for higher-codimension surfaces in Euclidean space, the waist inequality, and the systolic inequality.

One theme of the course is different ways to describe the size of a Riemannian manifold.  Two classical measures of the size are the volume and the diameter.  But two manifolds with equal volume and equal diameter may have very different shapes.  What other quantitative measures of the size may we define?  We will explore some variations of the idea of `width' of a Riemannian manifold and prove quantitative estimates about them.

A second theme of the course is to connect quantitative estimates with topology.  I considered calling the course quantitative topology.  For example, we know that Euclidean space R^n is contractible, and so any map f from S^k to R^n extends to a map from B^{k+1} to R^n.  Now we inject quantitative geometric information about f into the hypothesis.  Suppose we know that the (k-dimensional) volume of f(S^k) is < 1.  Can we find an extension F with a bound on the (k+1)-dimensional volume of F(B^{k+1})?  This gives another point of view about some variations on the isoperimetric inequality.

Grading:  students will turn in a few problem sets over the semester.




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

Fall term

Functions of one variable: rigorous treatment of limits and continuity.  Derivatives.  Riemann integral.  Taylor series.  Convergence of infinite series and integrals.  Absolute and uniform convergence.  Infinite series of functions.  Fourier series.

Spring term

Functions of several variables and their derivatives.  Topology of Euclidean spaces.  The implicit function theorem, optimization and Lagrange multipliers.  Line integrals, multiple integrals, theorems of Gauss, Stokes, and Green.

Required text: Introduction to Analysis, W. R. Wade, Prentice Hall

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, 5:10-6:25, L.S. Young.

Note: Master's students should consult 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.

Required text: Real Analysis, Halsey Royden, Prentice Hall, 3rd Edition, 1988. (NOT the new, 4th edition).

Supplementary texts: Introductory Real Analysis,  Andrei N. Kolmogorov and Sergei V. Fomin; Real and Complex Analysis, Walter Rudin; Real Analysis: Modern Techniques and Their  Applications, Gerald B. Folland.

MATH-GA 2450.001,2460.001 COMPLEX VARIABLES I, II

3 points per term.  Fall and spring terms.
Wednesday, 5:10-7:00, E. Hameiri (fall); Tuesday, 5:10-7:00, E. Belbruno (spring).

Fall Term

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:  Introduction to Complex Variables and Applications, Brown & Churchill

Spring Term

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.}

Cross-listed as MATH-UA 0393.001, 0394.001

Text: Complex Analysis, Alfors

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

3 points.  Fall term.
Mondays, Wednesdays, 10:45-12:00, H. Weitzner.

Note: Master's students should consult course instructor before registering for this course.

Prerequisites: advanced calculus, or MATH-GA 1410 Introduction to Math Analysis I.  Concurrent registration is not permitted.

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:  Complex Analysis: An Introduction, Lars V. Ahlfors, McGraw-Hill, 3rd Ed.

MATH-GA 2470.001  ORDINARY DIFFERENTIAL EQUATIONS

3 points.  Spring term.
Wednesday, 5:10-7:00, F. Hang.

Prerequisites:  linear algebra, real variables.

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 Poincare-Bendixson theorems.

Recommended text: Ordinary Differential Equations and Dynamical Systems, G. Teschl; Theory of Ordinary Differential Eqations, Coddington & Levinson

MATH-GA 2490, 2500.001  PARTIAL DIFFERENTIAL EQUATIONS I, II
                               
3 points per term.  Fall and spring terms.
Tuesday, 5:10-7:00,(fall); 9:30-11:20 (spring), P. Germain.

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

Fall Term

Prerequisites:  a good knowledge of undergraduate level linear algebra and ODE.

This course is a basic introduction to PDEs and is designed for students who are interested in applied mathematics or analysis and PDEs.  The concentration is on concrete examples of PDEs that arise in various physical systems, and methods of solving these problems will be introduced.  The class will cover the following topics:  first-order equations, methods of characteristics, conservation laws, shocks, weak solutions, Hamilton-Jacobi theory and caustics; wave equations, the method of spherical means, Duhamel's principle; the heat equation, the fundamental solution, diffusion and Brownian motion; Laplace's equation, maximum principle, fundamental solutions, Dirichlet and Neumann problems, boundary layer potential; Fourier methods and dispersive equations.

Recommended text: Partial Differential Equations, Lawrence C. Evans; class notes.
               
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.

Suggested texts: Partial Differential Equations, Paul R. Garabedian; Partial Differential Equations, L. C. Evans; Partial Differential Equations, Fritz John

MATH-GA 2550.001  FUNCTIONAL ANALYSIS

3 points.  Spring term.
Monday, 9:30-11:20, N. Masmoudi.

Prerequisite: Linear algebra. Real variables or the equivalent. 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: Functional Analysis, P. Lax, (Pure & Applied Mathematics, New York), Wiley-Interscience, John Wiley & Sons, 2002; Recommended  texts: Methods of Modern Mathematical physics Vol. I: Functional Analysis, M. Reed & B. Simon, Academic Press, New York-London, 1972

MATH-GA 2563.001  HARMONIC ANALYSIS

3 points. Fall term.
Wednesday, 9:30-11:20, 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, stationary phase. Topics in real variable methods: maximal functions, Hilbert and Riesz transforms and singular integral operators, BMO and Hardy spaces, Carleson measures and boundary behavior of harmonic functions.

References: Singular Integrals and Differentiability Properties of Functions, E.Stein; Introduction to Fourier Analysis on Eucledian Spaces, E.Stein & G.Weiss; Fourier Integrals in Classical Analysis, C.Sogge

MATH-GA 2610.001  ADVANCED TOPICS IN PDE (Homogenization of Elliptic Operators)

3 points.  Fall term.
Wednesday, 1:25-3:15, F.H. Lin.

Prerequisites: real variables and Hilbert spaces, basic elliptic PDE.

With a brief preparation, we shall started with general discussions on G- and H- convergences of elliptic operators. Then we proceed further for the homogenization of elliptic operators, compensate compactness, periodic homogenization and uniform estimates, and nonlinear homogenizations.  If time permit, we may also discuss H-measures and semiclaasical measures.

References:  Weak Convergence Methods, C.Evans, CBMS Lecture Notes 74, AMS 1989;  Asymptotic Analysis for Periodic Structures, A. Bensoussan, J. L. Lions, & G. C. Papanicolaou, North Holland, 1978; Homogenization of Differential Operators and Integral Functionals, V. V. Jikov, S. M. Kozlov, & O. A. Oleinik, Springer-Verlag, Berlin, 1994

Grading:  this course will be graded as a seminar course.

MATH-GA 2610.002  ADVANCED TOPICS IN PDE (Variational Methods and Gamma-convergence)

3 points.  Fall term.
Tuesday, Thursday, 3:20-5:05, S. Serfaty.

This course will be given twice a week and will run from October 13 to November 29, 2011.

Prerequisite: some knowledge of RV II, Functional Analysis and PDE.

Variational problems, which consist in minimizing a certain ”energy" functional among a suitable class of functions, arise in the study of PDEs as well as in optimization problems related to many branches of science (e.g. physics, materials, chemistry, economics, operation research). This course will review the basics of solving variational problems and their associated PDEs, and present the notion of Gamma-convergence used for analyzing asymptotics of variational problems. It will emphasize general methods as well as a diversity of examples.

A rough outline includes: (1) from energy to PDEs: computing variations; (2) the direct method of the calculus of variations; (3) quasiconvexity and applications; (4) constrained problems, the example of the obstacle problem; (5) relaxation, the example of optimal transportation; Gamma-convergence: definition and properties; (6) application to homogenization; (7) application to interface problems: the examples of Allen-Cahn and Aviles-Giga; (8) nonminimizing critical points: minmax principles, Palais-Smale condition, Noether's theorem; (9) harmonic maps and monotonicity formula.

Suggested texts: Variational Methods, M. Struwe; Direct methods in the calculus of variations, B. DaCorogna, Modern Methods in the calculus of variations, I. Fonseca, G. Leoni, Gamma-convergence for Beginners,
A. Braides, Conservation laws and moving frames, F. Helein, Harmonic maps

Grading: this course will be graded as a seminar course.

MATH-GA 2620.001  ADVANCED TOPICS IN PDE (Topic TBA)

*** THIS COURSE HAS BEEN CANCELLED ***

MATH-GA 2620.002  ADVANCED TOPICS IN PDE (Elliptic and Parabolic Methods)

3 points.  Spring term.
Wednesday, 1:25-3:15, F.H. Lin.

Prerequisites: basic PDE (Evans Notes) and real variables.

The purpose of this course is to explain various elliptic and parabolic methods that are used in the studies several variational problems of geometric nature. In particular, we shall discuss some analytic aspects of theory of harmonic mappings, the regularity theory of solutions in free boundary problems.

References: The Analysis of Aarmonic Maps and Their Heat Flows, F. H. Lin & C. Y. Wang, World Scientific Publishing Co. Pte. Ltd (2008); A Geometric Approach to Free Boundary Problems, Graduate Studies in Math, L. Caffarelli & S. Salsa, Vol.68, Amer. Math. Soc. (2005); Elliptic Partial Differential Equations, Q. Han & F. H. Lin, Courant Lecture Notes, Vol. #1, AMS (2010), Second Ed.
 
Grading: this course will be graded as a seminar course.

MATH-GA 2650.001  ADVANCED TOPICS IN ANALYSIS (Integrable systems)

3 points. Fall term.
Monday, 1:25-3:15, P. Deift.

Prerequisites: real analysis, complex analysis and linear algebra.

Introduction to Hamiltonian mechanics.  Integrable systems: Liouville integrable systems.  Lax pairs.  Finite dimensional integrable systems: examples.  Infinite dimensional integrable systems: examples. Introduction to perturbation theory of integrable systems.

Grading: this course will be graded as a seminar course.

MATH-GA 2650.002  ADVANCED TOPICS IN ANALYSIS (Free Boundries and Boundary Layers)

3 points.  Fall term.
Monday, 9:30-11:20, N. Masmoudi.

Prerequisite: some background in PDE.

The goal of the course is to study the inviscid limit of the free-boundary Navier-Stokes system and recover at the limit the free-boundary Euler  system. This will  allow us to link together two important problems in Ffuid Mmchanics, namely free boundary problems and boundary layers.

Grading: this course will be graded as a seminar course.

MATH-GA 2660.001 ADVANCED TOPICS IN ANALYSIS (Spectral Theory)

3 points.  Spring term.
Monday, 1:25-3:15, P. Deift.

Prerequisites: functional analysis, complex analysis.

In the course we will develop the spectral theory of specific classes of self-adoint operators. In particular we will consider second order ordinary differential operators and discuss Sturm-Liouville theory and Weyl limit point-limit circle theory. We will also analyze the spectral theory of common atomic Hamiltonians such as the hydrogen atom.

Grading: this course will be graded as a seminar course.

MATH-GA 2660.002 ADVANCED TOPICS IN ANALYSIS (Introduction to Ergodic Theory)

3 points.  Spring term.
Tuesday, 1:25-3:15, L.S. Young.

Prerequisite: real variables.

I will begin with basic concepts in abstract ergodic theory, i.e. the study of measure-preserving transformations of abstract measure spaces.  Recurrence properties, ergodicity, mixing properties and entropy will be discussed. This is followed by the ergodic theory of continuous maps, where we study topological analogs of ergodic theory ideas. The last part of the course is on differentiable systems; the main topic here is Lyapunov exponents.

Recommended text: An Introduction to Ergodic Theory, Peter Walters, (Graduate Texts in Mathematics, Springer)

Grading: this course will be graded as a seminar course (homework will be assigned and discussed).




MATH-GA 2010.001, 2020.001  NUMERICAL METHODS I, II

3 points per term.  Fall and spring terms.
Thursday, 5:10-7:00 O. Widlund (fall); Thursday, 7:10-9:00, 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 homework assignments.

Floating point arithmetic; conditioning and stability; numerical linear algebra, including direct methods for systems of linear equations, eigenvalue problems, LU, Cholesky, QR and SVD factorizations; interpolation by polynomials and cubic splines; Numerical quadrature; nonlinear systems of equations and unconstrained optimization; Fourier transforms; Monte Carlo methods.

Computer programming assignments and individual research projects will form an essential part of the course.

Required text:  Numerical Linear Algebra, David Bau III & Lloyd N. Trefethen, SIAM, 2000

Optional reading: Scientific Computing with MATLAB and Octave, Alfio M. Quarteroni & Fausto Saleri, Springer, 2006, available electronically through the library; An Introduction to Programming and Numerical Methods in MATLAB, Stephen R. Otto & James P. Denier, Springer, 2005, available electronically through the library; An Introduction to Scientific Computing: Twelve Computational Projects Solved with MATLAB, Ionut Danaila, Pascal Joly, Sidi Mahmoud Kaber, & Marie Postel, Springer, 2009.

Cross-listed as CSCI-GA 2420.001

Spring term                                        

Prerequisite: 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; (5) parabolic and hyperbolic partial differential equations.

Text:  A First Course in the Numerical Analysis of Differential Equations, A. Iserles, Cambridge University Press, 1st Ed.

Cross-listed as CSCI-GA 2421.001

MATH-GA 2011.001 ADVANCED TOPICS IN NUMERICAL ANALYSIS (Fast Multipole Methods)

3 points.  Fall term.
Tuesday, 1:25-3:15, M. Tygert.

Prerequisites: linear algebra, Calculus, complex-analytic contour integration, and some familiarity with convolution, the Fourier transform, sampling, and the fast Fourier transform.

An introduction to the numerical treatment of elliptic partial differential equations such as the Laplace, Helmholtz, and Maxwell equations, via fast multipole methods (FMMs) and related techniques of
matrix compression.

Texts: lecture notes and recommended references are available online at http://cims.nyu.edu/~tygert/gradcourse

Grading: this course will be graded as a seminar course.

Cross-listed as CSCI-GA 2945.002

MATH-GA 2011.002 ADVANCED TOPICS IN NUMERICAL ANALYSIS (Coarse-Grained Models of Materials)

3 points.  Fall term.
Wednesday, 1:25-3:15, A. Donev.

Describing the full atomistic structure of materials at relevant time and length scales is typically not computationally feasible.  A well-known alternative is to use classical continuum models such as the Navier-Stokes equations or linear elasticity.  At time and length scales in-between the atomistic and macroscopic, however, lies a continuum of mesoscopic scales at which is is necessary to use coarse-grained models.  These mesoscopic models try to capture some of the microscopic details but at a fraction of the cost of a full molecular-dynamics simulation.  These include particle models such as the Direct Simulation Monte Carlo method for gas flows or Dissipative Particle Dynamics for fluid flows, stochastic continuum models such as the fluctuating Navier-Stokes equations for small-scale fluid flows, hybrid models such as the quasi-continuum method, geometric models such as hard-particle packings for granular materials or graph models for magnetic materials, and others. 

In this course, we will discuss the fundamental ideas behind coarse-grained models, as well as computational algorithms for mesoscopic modeling of gases, liquids, solids, and granular materials.  Students will study a review or seminal paper and discuss what they have learned in class.

Cross-listed as CSCI-GA 2945.003

Grading: this course will be graded as a seminar course requiring a presentation.

MATH-GA 2012.001 ADVANCED TOPICS IN NUMERICAL ANALYSIS (Computational Electromagnetics)
  
3 points.  Spring term.
Tuesday, 9:30-11:20, L. Greengard.

Prerequisites: Complex analysis, partial differential equations, numerical analysis, basic functional analysis.

The field of computational electromagnetics is devoted to the solution of Maxwell's equations, with application (for example) to antenna and chip design, electromagnetic compatibility, wave scattering, and optics. We will review the basic theory, followed by an overview of  finite difference, finite element, and integral equation methods. The course will emphasize scattering theory from the integral equation perspective.

Recommended text: Theory and Computation of Electromagnetic Fields, Jian-Ming Jin, Wiley-IEEE Press, 2010.

Grading: this course will be graded as a seminar course; student presentations are encouraged but not required.

MATH-GA 2012.002 ADVANCED TOPICS IN NUMERICAL ANALYSIS (Computational Neuroscience)
  
3 points.  Spring term.
Wednesday, 1:25-3:15, A. Rangan.

Prerequisites: familiarity with ordinary and partial differential equations, as used in applications.  Some familiarity with a programming language (such as Matlab, Java, C, etc.) is expected.  The necessary background in biology will be explained within the course, and so there is no biology prerequisite.

This course focuses on the computational techniques used to model neuronal networks. Along the way, we will discuss and comment on the biological phenomena which have been successfully modeled using these techniques. We will also try and highlight the features of these models which enable them to capture different types of network dynamics.

Topics include: (1) the analytical and numerical consequences of various approximate neuron-models (e.g., Hodgkin-Huxley and Integrate-and-fire models, and reductions from deterministic point-neuron models to stochastic processes). (2) large-scale models of the early olfactory system (e.g., the antennal-lobe models of Bazhenov, Laurent, Kopell and others), and some numerical details necessary for the fast simulation of stiff neuron-neuron interactions.  (3) large-scale models of the early visual system (e.g., the "NYU" model of McLaughlin, Shelley and Shapley, as well as models of primary visual cortex developed by Miller, Roque and others) and numerical details necessary for fast simulation of spatially extended systems. (4) numerical methods for the coarse-grained population-dynamics models introduced by Knight, Abbott, van Vreeswijk, Sirovich and others – focusing on the numerical details required to resolve the discontinuous solutions which arise from these population-dynamics equations.

Grading: this course will be based on homeowrk assignments, some of which will involve computation, as well as a computing project.  There will not be an exam/  Students are encouraged to collaborate and work in teams, both for the homework and the project. 

MATH-GA 2041.001  COMPUTING IN FINANCE

3 points.  Fall term.
Thursday, 7:10-9:00,  K. Laud & E. Fishler.

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

MATH-GA 2043.001  SCIENTIFIC COMPUTING

3 points. Fall term.
Thursday, 5:10-7:00, M. Shelley.

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

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.

No single textbook is required for this class, but there are several optional texts which are recommended: Numerical Linear Algebra, David Bau III & Lloyd N. Trefethen, SIAM, 2000; Scientific Computing with MATLAB and Octave, Alfio M. Quarteroni & Fausto Saleri, Springer, 2006 (available electronically through the library);  An Introduction to Programming and Numerical Methods in MATLAB, Stephen R. Otto & James P. Denier, Springer, 2005 (available electronically through the library)

Cross-listed as CSCI-GA 2112.001

MATH-GA 2043.001  SCIENTIFIC COMPUTING

3 points. Spring term.
Thursday, 5:10-7:00, A. Donev.

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

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.

Required text: Scientific Computing with MATLAB and Octave, Alfio M. Quarteroni & Fausto Saleri, Springer, 2006, available electronically through the library.

Optional reading: An Introduction to Programming and Numerical Methods in MATLAB, Stephen R. Otto & James P. Denier, Springer, 2005, available electronically through the library; Scientific Computing with Case Studies,  Dianne P. O'Leary, SIAM, 2008

Cross-listed as CSCI-GA 2112.001

MATH-GA 2045.001  COMPUTATIONAL METHODS FOR  FINANCE

3 points.  Fall term.
Tuesday, 7:10-9:00, A. Hirsa.

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

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

Cross-listed as B40.7311.010
 


MATH-GA 2701.001 METHODS OF APPLIED MATHEMATICS

3 points.  Fall term.
Tuesday, 125-3:15, O. Bühler.

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

Supplementary reading: Scaling, self-similarity, and intermediate asymptotics, G.I. Barenblatt; Perturbation Methods, E.J. Hinch, Advanced Mathematical Methods for Scientists and Engineers, C.M. Bender & S.A. Orszag; Linear and Nonlinear Waves, G.B. Whitham; Calculus of Variations, I.M. Gelfand & S.V. Fomin

MATH-GA 2702.001 FLUID DYNAMICS

3 points.  Fall term.
Monday, 9:30-11:20, O. Pauluis.

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: An Introduction to Theoretical Fluid Mechanics, Courant Lecture Notes, S. Childress.
Recommended text: Elementary Fluid Dynamics, D. Acheson, Oxford Applied Mathematics & Computing Science Series.

MATH-GA 2704.001 APPLIED STOCHASTIC ANALYSIS

3 points. Spring term.
Thursday, 1:25-3:15, E. Vanden Eijnden.

The class will provide an introduction to probability and stochastic processes theory from an applied perspective. Topics will include definition of random variables, limit theorems, Markov chain and Markov processes, Wiener processes, stochastic differential equations, Fokker-Planck equations, large deviations and rare events, Monte Carlo methods and introduction to statistical mechanics.

MATH-GA 2706.001  PARTIAL DIFFERENTIAL EQUATIONS FOR FINANCE

3 points.  Spring term.
Monday, 5:10-7:00, O. Bühler.

Prerequisite: Stochastic Calculus or equivalent.

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

MATH-GA 2707.001  TIME SERIES ANALYSIS AND STATISTICAL ARBITRAGE

3 points.  Fall term.      
Monday, 7:10-9:00, F. Asl & R. Reider.

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

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

MATH-GA 2708.001 ALGORITHMIC TRADING AND QUANTITATIVE STRATEGIES

3 points.  Spring term.
Tuesday, 7:10-9:00, P. Kolm. & L. Maclin.

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

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

MATH-GA 2709.001  FINANCIAL ENGINEERING MODELS FOR CORPORATE FINANCE

3 points.  Fall term.
Thursday, 7:10-9:00, D. Shimko & B. Humphreys.

Prerequisites: MATH-GA 2751 Capital Markets & Portfolio Theory and MATH-GA 2791 Derivative Securities.

This course covers advanced stochastic modeling applications in finance.  Combining capital markets, corporate finance and statistical knowledge, this course uses simulation as a unifying tool to model all major types of market, credit and actuarial risks.  Emphasis is placed on rigorous application of financial theory to the conceptualization and solution of multifaceted real-world problems.  These problems arise in security design and risk management strategy.

MATH-GA 2710.001  MECHANICS                  

3 points.  Spring term.
Thursday, 9:30-11:20, R. Kohn.

This course provides a mathematical introduction to Newtonian dynamics, and continuum mechanics.
For students preparing to do research on physical applications, the class provides an introduction to core concepts. No prior exposure to physics is expected.  Problems of physical interest will be treated and methods of calculus of variation will be employed.

MATH-GA 2751.001  RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS
      
3 points.  Fall term.
Tuesday, 7:10-9:00, P. Kolm.

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.

MATH-GA 2751.001   RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS

3 points.  Spring term.
Wednesdays, 7:10-9:00, M. Avellaneda.

Details to be determined (generally consistent with the fall term description).

MATH-GA 2752.001 ACTIVE PORTFOLIO MANAGEMENT

3 points.  Spring term.
Monday, 5:10-7:00, R. Lindsey.

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

MATH-GA 2753.001  ADVANCED RISK MANAGEMENT

3 points.  Spring term.
Monday, 7:10-9:00, K. Abbott.

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

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

Text: Financial Risk Management, S. Allen, John Wiley & Sons, 2003

MATH-GA 2754.001  CASE STUDIES IN FINANCIAL MODELING

3 points. Fall term.
Wednesday, 7:10-9:00, J. Dash.

Prerequisites: Stochastic Calculus, Computing in Finance, and Continuous Time Finance.

This course will consist of five parts: (1) Securities Modeling (6 classes):  Barrier Options, Exotic Options, CVR and other Deals, Path Integrals (Introduction).   (2) Risk Modeling (4 classes): Fat Tail Volatility, Stressed Correlations, Stressed VAR and Economic Capital, 1001 Risks (Systems, Models, Data, Systemic).   (3) Underlying Variables Modeling at both Long and Short Time Scales (1 class): The Macro Micro Model. (4) Student Presentations (1 class). (5) Guest Lecture (1 class).

Required text: Quantitative Finance and Risk Management, A Physicist’s Approach, J.W. Dash, World Scientific, 3rd printing, 2008

MATH-GA 2755.001  PROJECT AND PRESENTATION (MATH FINANCE)

3 points.  Fall Term.
Monday, 5:10-7:00, 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 2755.001  PROJECT AND PRESENTATION (MATH FINANCE)

3 points.  Spring Term.
Wednesday, 5:10-7:00, P. Kolm.

Description as above.

MATH-GA 2791.001  DERIVATIVE SECURITIES  
      
3 points.  Fall term.
Wednesday, 7:00-9:00, M. Avellaneda.

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

Cross-listed as FINC-GB 7312.010

MATH-GA 2791.001  DERIVATIVE SECURITIES  
      
3 points.  Spring term.
Monday, 7:10-9:00, B. Flesaker.

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 term.                      
Monday, 7:10-9:00, P. Carr & A. Jahaveri.

Prerequisites: Derivative Securities and Stochastic Calculus, or equivalent.

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.

Cross-listed as FINC-GB 7310.010

MATH-GA 2792.001  CONTINUOUS TIME FINANCE
      
3 points.  Spring term.                      
Wednesday, 7:10-9:00, B. Dupire & F. Mercurio.

Prerequisites: Derivative Securities and Stochastic Calculus, or equivalent

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 MORTGAGE-BACKED SECURITIES AND ENERGY DERIVATIVES

3 points.  Spring term.
Tuesday, 7:10-9:00, G. Swindle & L. Tatevossian.

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

The first part of the course will cover the fundamentals and building blocks of understanding how mortgage-backed securities are priced and analyzed. The focus will be on prepayment and interest rate risks, benefits and risks associated with mortgage-backed structured bonds and mortgage derivatives. Credit risks of various types of mortgages will also be discussed. The second part of the course will focus on energy commodities and derivatives, from their basic fundamentals and valuation, to practical issues in managing structured energy portfolios. We develop a risk neutral valuation framework starting from basic GBM and extend this to more sophisticated multifactor models. These approaches are then used for the valuation of common, yet challenging, structures. Particular emphasis is placed on the potential pitfalls of modeling methods and the practical aspects of implementation in production trading platforms. We survey market mechanics and valuation of inventory options and delivery risk in the emissions markets.
                                           
Suggested texts:  Salomon Smith Barney Guide to Mortgage-Backed and Asset-Backed Securities, Lakhbir Hayre; The Handbook of Mortgage-Backed Securities, Frank Fabozzi; Energy and Power Risk Management, Eydeland & Wolyniec; Electricity Markets: Pricing, Structures and Economics, Chris Harris

MATH-GA 2797.001 CREDIT MARKETS AND MODELS

3 points.  Fall term.
Wednesday, 7:10-9:00, V. Finkelstein.

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.

Suggested texts: Modeling Single-Name and Multi-Name Derivatives, Dominic O’Kane, 2008; Options, Futures, & Other Derivative Securities, John Hull, 7th Edition.

MATH-GA 2798.001 INTEREST RATE AND FX MODELS

3 points.  Spring term.
Thursday, 7:10-9:00, L. Andersen & A. Lesniewski.

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  (Quantifying Uncertainty in Complex Systems)

3 points. Fall term.
Thursday, 3:20-5:05, A. Majda.

Prerequisites: some knowledge of elementary stochastic diff. equations and nonlinear dynamics although the first two lectures will provide a review.

In many situations in science and engineering, the analysis and prediction of phenomena often occur through complex dynamical equations which have significant model errors compared with the true signal in nature. Clearly, it is important both to improve the imperfect model’s capabilities to recover crucial features of the natural system and also to model the sensitivities of the natural system to changes in external or internal parameters. These efforts are hampered by the fact that the actual dynamics of the natural system are unknown. An important example with major societal impact is the Earth’s climate. This course is a discussion of cutting edge mathematical developments in quantifying uncertainty. Mathematical ideas involving fluctuation dissipation theorems, empirical information theory, and uncertainty propagation will be developed in a suite of instructive elementary and more complex examples. This is a seminar style course where Prof. Majda will give about half the lectures; his post docs, Branicki, Giannakis, and Sapsis will give the other lectures and the graduate students attending will be invited (this is not require) to participate in these lectures and small class projects.

Grading: this course will be graded as a seminar course.

Background text: Information Theory and Stochastics for Multi scale Nonlinear Systems, Majda, Abramov, Grote, CRM Series, Vol. 25, American Mathematical Society, 2005

MATH-GA 2840.001  ADVANCED TOPICS IN APPLIED MATHEMATICS (Math Strategies for Real-Time Filtering of Turbulent Signals in Complex Systems)

3 points.  Spring term.
Thursday, 3:20-5:05, A. Majda.

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

Lecture notes and material from upcoming book will be handed out as the course proceeds.

Grading: this course will be as a seminar course; interested students can participate optionally in some projects based on the course material.

MATH-GA 2840.002  ADVANCED TOPICS IN APPLIED MATHEMATICS (Mathematics and Signal Processing of Analog/Digital Conversion)

3 points.  Spring term.
Monday, 9:30-11:20, S. Güntürk.

Prerequisites: working knowledge of Real Analysis; some exposure to harmonic analysis and dynamical systems will be useful, but not strictly required.

This course will introduce the mathematical theory of conventional as well as abstract analog-to-digital conversion systems. Topics will include: scalar and vector quantization, number systems and associated dynamics, sampling theory and quantization, sigma-delta modulation. Time permitting, emerging systems such as compressive sampling will also be covered.

Text: relevant book chapters and papers will be discussed throughout the course.

Grading: this course will be graded as a seminar course.

MATH-GA 2840.003 ADVANCED TOPICS IN APPLIED MATHEMATICS (Dynamic Computational Statistical  models  for Socio-economic and Geo-political Systems)

3 points. Spring term.
Tuesday, 9:30-11:20 a.m., D. Mordecai

Prerequisites: A genuine interest in computational statistical modeling of social phenomena is expected. Some foundation or background in theory or applications of mathematical statistics, matrices, PDEs/finite difference methods, Markov Chain numerical methods, and/or network theory is suggested, but not required. The class will be suitable for students with appropriate backgrounds from areas such as Economics, Epidemiology, Finance, Marketing, Management, Politics, Public Policy, or Sociology, as well as for students of Mathematics or Computer Science.

A comparative multi-disciplinary survey of the developments and application of computational statistics models to the simulation, measurement, and analysis of evolutionary dynamics underlying social and institutional structures, as applied to the following phenomena: cultural and technological propagation and adoption; information dissemination and aggregation; search, matching, intermediation; and network effects across populations within geo-political, socioeconomic, financial systems. Areas of application to be considered will include: financial, labor, housing, consumption, and trade effects of consumer behavior, as well as global population, immigration, socio-political conflict, environmental, epidemiological, aging, and health trends. A extensive list of suggested texts and articles will be provided to participants. Additional information (including references for background reading) will be available well before the start of the Spring 2012 semester.

Grading: to be based upon in-class discourse and small group activities (paper, project, or problem).

MATH-GA 2852.001 ADVANCED TOPICS IN MATH BIOLOGY (Biophysical Modeling of Cells and Populations)

3 points.  Spring term.
Thursday, 9:30-12:15, E. Kussell.

Prerequisites: previous exposure to differential equations; familiarity with programming or computational software such as Matlab or Mathematica.

This course develops the biophysical approach to modeling biological systems, applied to classic problems of molecular biology, as well as to systems of recent interest.  The course is organized in a bottom-up way, beginning with models of cooperativity in binding, of promoter recognition and activation, continuing with models of networks, diffusion and pattern formation, and working towards a population-level description of biological systems.  Diverse biological examples will be presented to illustrate key concepts in biophysical modeling.

Grading: 20% Problem Sets (due dates on syllabus); 40% Discussions (almost every week, student-led); 40% Final Project (presentation + paper)

Texts: Physical Biology of the Cell, R. Phillips, J. Kondev, J. Theriot, Garland Science, 2008; Random Walks in Biology, H. C. Berg, Princeton University Press, 1993; A Genetic Switch: Phage Lambda Revisited, M. Ptashne, 3rd edition, Cold Spring Harbor Laboratory Press, 2004.

Cross-listed with BIOL-GA 1131.001

MATH-GA 2855.001 ADVANCED TOPICS IN MATH BIOLOGY (Cardiac Mechanics and Electrophysiology)

3 points.  Fall term.
Thursday, 1:25-3:15, C. Peskin.

Prerequisite: familiarity with partial differential equations as used in applications. There is no biology prerequisite, as the necessary biological background will be explained within the course.

This course is about the equations of a heartbeat, which are partial differential equations.  The Navier-Stokes equations of a viscous incompressible fluid, suitably modified to include fluid-structure interaction with the muscular heart walls and the flexible heart valve leaflets, describe the mechanical function of the heart. Cardiac mechanics is coordinated and controlled by three-dimensional electrical activity described by Hodgkin-Huxley equations in their bidomain form, which models both intracellular and extracellular current and voltage, coupled by transmembrane ionic and capacitive currents.  Both mechanical and electrical activity are strongly influenced by the fiber architecture of the heart, the differential geometry of which is governed by partial differential equations derived primarily from considerations of mechanical equilibrium. Meanwhile, at the molecular level, the contractile machinery of the heart is described by population dynamics equations that govern the attachment, motion, and detachment of myosin cross bridges interacting with the actin  filaments of the muscle. Excitation-contraction coupling is mediated by calcium ions, which are released from intracellular stores by a positive-feedback mechanism known as calcium-induced calcium release (CICR).  The dynamics of CICR are responsible for intracellular calcium oscillations and for intracellular calcium waves governed by reaction diffusion equations. The emphasis of the course will be on the formulation of detailed realistic models for the phenomena described above, and on the numerical solution of the model equations.  Simplified models that allow for analytic or asymptotic solution will also be introduced for
comparison.

Requirements of the course will be homework assignments, some of which involve computing, and a computing project, but no exam.  Both the homework and the computing project may be done collaboratively by students working together in teams.

Grading:  This course will be graded as a regular course.

Cross-listed as BIOL-GA 2855.001.

MATH-GA 3001.001  GEOPHYSICAL FLUID DYNAMICS

3 points.  Fall term.
Tuesday, 9:30-11:20, S. Smith.

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: Atmospheric and Oceanic Fluid Dynamics, G. Vallis, Cambridge 2006; Lectures on Geophysical Fluid Dynamics, R. Salmon, Oxford 1998; Geophysical Fluid Dynamics, J. Pedlosky, Springer-Verlag 1987

MATH-GA 3004.001  ATMOSPHERIC DYNAMICS

3 points.  Spring term.
Monday, 1:25-3:15, E. Gerber.

Prerequisite: Geophysical Fluid Dynamics or instructor’s permission.

This lecture course offers a general overview of the physical processes that determine the state of the Earth atmosphere. The focus here is to describe the main features of the planetary circulation, and to explain how they arise as a dynamical response of the atmosphere to different external forcings such as solar radiation or topography. Students should have some knowledge in geophysical fluid dynamics before taking this course. Topics to be covered include: solar forcing, the mean-state of the atmosphere, Hadley and monsoonal circulations, dynamics of the midlatitudes stormtracks, energetics, zonally asymmetric circulations, equatorial dynamics, and the interaction between moist convection and large-scale flow. Students will be assigned bi-weekly homework assignments and some computer exercises, and will be expected to complete a final project or exam, as per instructor's decision.  This course will be supplemented with out-of-class instruction.

Required text: Atmospheric and Oceanic Fluid Dynamics , G. Vallis, Cambridge 2006
Recommended Texts:   An Introduction to Dynamic Meteorology, J. R. Holton, Academic Press;  Atmospheric Dynamics, C. F  Bohren, & B.A. Albrecht, Oxford University Press; additional material in the form of published articles will be provided to the students to complement the textbook

MATH-GA 3011.001  ADVANCED TOPICS IN ATMOSPHERE-OCEAN SCIENCE (Intro to Information Theory with Application Dynamical Systems Predictability)

3 points.  Spring term.
Tuesday, 1:25-3:15, R. Kleeman.

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

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

Recommended texts: Elements of Information Theory, Cover & Thomas; review paper by instructor on information theory and predictability.

Grading: this course will be graded as a seminar course.



MATH-GA 2901.001  BASIC PROBABILITY

3 points.  Fall term.
Thursday, 5:10-7:00, E. Vanden Eijnden.

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

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.

Optional text:  Probability and Random Processes, G. Grimmett & D. Stirzaker, 3rd Ed.

MATH-GA 2901.001  BASIC PROBABILITY

3 points.  Spring term.
Wednesday, 7:10-9:00, S. Chatterjee.

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

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.

Optional texts:  Probability and Random Processes, G. Grimmett & D. Stirzaker, 3rd Ed.;  One Thousand Exercises in Probability, G. Grimmett & D. Stirzaker, Oxford University Press

MATH-GA 2902.001  STOCHASTIC CALCULUS

3 points.  Fall term.
Monday, 7:10-9:00, J. Goodman.

Prerequisite: 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.

Problem session: Thursday, 5:30-6:30 (optional).

MATH-GA 2902.001  STOCHASTIC CALCULUS

3 points.  Spring term.
Thursday, 7:10-9:00, A.  Kuptsov.

Prerequisite: 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.

Problem session: Monday, 5:30-6:30 (optional).

Text: Stochastic Calculus, A Practical Introduction, Richard Durrett, CRC Press, Probability & Stochastics Series

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

3 points per term.  Fall and spring terms.
Tuesday, 9:30-11:20, C. Newman, (fall); Wednesday, 9:30-11:20, H. McKean (spring).

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

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.

Spring term

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

Recommended text:  Stochastic Processes,  S. R. S. Varadhan, CIMS - AMS, 2007

MATH-GA 2931.001 ADVANCED TOPICS IN PROBABILITY (Topics in Percolation Theory)

3 points.  Fall term.
Monday, 1:25-3:15, S. Chatterjee.

Prerequisite: graduate probability.

Description: In physics, chemistry and materials science, percolation concerns the movement and filtering of fluids through porous materials. During the last five decades, percolation theory, an extensive mathematical model of percolation, has brought new understanding and techniques to a broad range of topics in physics, materials science, complex networks, epidemiology, and other areas. This course will aim to give an introduction to the deep mathematical theory that has developed around this topic through the contributions of many authors.

Grading: As a seminar course requiring a presentation.

MATH-GA 2932.001 ADVANCED TOPICS IN PROBABILITY (Large Deviations and Applications)

3 points.  Spring term.
Wednesday, 1:25-3:15, R. Varadhan.

We will  study Large Deviations through several examples. Sums of independent variables, empirical distribution of Markov chains, small random perturbations of dynamical systems, and interacting particle systems are some of the contexts through which we will  develop the general theory.

Grading: this course will be graded as a seminar course.

MATH-GA 2932.002 ADVANCED TOPICSIN PROBABILITY (Random Graphs)

3 points.  Spring term.
Tuesday, 5:00-6:50, J. Spencer.

Equally appropriate titles would have been "Probabilistic Combinatorics" or "The Probabilistic Method" or (personal favorite) "Erdos Magic." The Probabilistic Method is a lasting legacy of the late Paul Erdos. For "Uncle Paul" the purpose was to prove the existence of a graph, coloring, tournament, or other combinatorial object. A random object would be described, and then one would show that that object had the desired properties with positive probability.

Probability results include Chernoff Bounds (Large Deviation Bounds), Martingales, the Lovasz Local Lemma and the Janson Inequalities. Random walks and excursions with specific distributions are examined.  We generally restrict to discrete distributions, considering their asymptotics.  We study discrete percolation events, notably the "sudden" appearance of the giant component in random graphs, using Galton-Watson branching processes.  The probability is done "from scratch,"  an acquaintance with graduate probability courses is NOT a prerequisite.  We are interested in algorithmic implementation, both deterministically and with random algorithms. There is great interest (the official title) in the study of random discrete structures (not just graphs, though that is the main one) for their own sake. The course involves probability, Discrete Math, and algorithms.

Topics include: Ramsey Numbers, Continuous Time Greedy Algorithms, Graph Coloring, Discrepancy, the Liar Game and the Tenure Game. Methods of asymptotic analysis permeate the course.  In applications exact results are often impossible and practical methods are given to find main asymptotic terms.

Text: The Probabilistic Method, Noga Alon & Joel Spencer. third  edition,  Publisher: John Wiley, 2009

Grading:  There will be an assignment every week.  At the end of the course there will be a regularly scheduled final, or, if the class size allows, individual oral exams.

MATH-GA 2936.001  ADVANCED TOPICS IN APPLIED PROBABILITY (Quantitative Investment Strageties and Hedge Funds)

3 points.  Spring term.
Tuesday, 5:10-7:00, M. Avellaneda.

Prerequisites: Finance: a course on investments including equities, futures, fixed-income securities and options Mathematics: Calculus, Probability and Statistics, programming in Matlab or VBA and working knowledge of one statistical package (R, SAS, NAG, etc.).

The course will introduce the class to quantitative investments from the hedge-fund (buy-side) perspective. We will cover a broad spectrum of quantitative investments from the point of view of hedge funds and the ”buy-side.’’

Topics: (1) Hedge fund economics and legal frameworks in the US, Euroland and Brazil: rules deriving from the US Investment Act of 1940, UCITS & Instrucao CVM 359; market access and regulatory constraints for onshore/offshore entities; hedge-fund replication platforms and managed accounts; introduction to management and incentive fees.  (2) Overview of main quantitative strategies in use today: index and ETF arbitrage; statistical arbitrage; high-frequency trading; the limit order book and the opportunity that it presents; statistical Arbitrage; CTA strategies. (3) Statistical Arbitrage: when, where and how: relation between statistical arbitrage and cross-sectional market volatility; universe and signal-generation techniques; dynamic risk-management; backtesting strategies in Matlab/Excel/VBA/C.  (4) ETF and Index Arbitrage: the ETF revolution; physical versus synthetic structures; index ETFs; commodity ETFs; leveraged ETFs; high-frequency creation/redemption strategies; funding costs and EOD/Multi-period strategies.  (4) Volatility arbitrage: VIX and its term structure; contango and backwardation in VIX futures and ETFs; dispersion trading: index options versus options on the components; using Weighted Monte-Carlo to generate dispersion signals; using Steepest-Descent Approximations; hedging and managing dispersion trades across the lifetime of the trade; single-name versus industry-ETF volatility trading; Variance, volatility and correlation swaps.

Grading: this course will be graded as a regular course.

MATH-GA 2962.001  MATHEMATICAL STATISTICS
       
3 points.  Spring term.
Wednesday, 5:10-7:00, M. Tygert.

Prerequisite: a working knowledge of probability at the undergraduate or introductory graduate level.

descriptive statistics, the binomial, Poisson, exponential, normal, chi-square, t, and F distributions, hypothesis testing, confidence intervals, point estimation (maximum-likelihood methods, consistency, efficiency, sufficiency, etc.), regression, correlation, analysis of variance, Bayesian inference, nonparametric methods, sequential tests, loss, risk, and decision theory.

Required text: Principles of Statistics, M.G. Bulmer, Dover Publications, 1979





Revised January 2012