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
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MATH-GA 2110.001, 2120.001 LINEAR ALGEBRA I, II

3 points per term. Fall and spring terms.
Tuesday, 5:10-7:00 R. Kleeman (fall); Monday, 5:10-7:00, S. Marques (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.

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MATH-GA 2110.001 LINEAR ALGEBRA I

3 points. Spring term.
Tuesday, 5:10-7:00, Y. Deng.

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.

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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.

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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.

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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.

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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.

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MATH-GA 2210.001 INTRODUCTION TO NUMBER THEORY I

3 points. Spring term.
Wednesday, 7:10-9:00, A. Pirutka.

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

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

For additional information, see the course website.

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

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GEOMETRY AND TOPOLOGY
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MATH-GA 2310.001, 2320.001 TOPOLOGY I, II

3 points per term. Fall and spring terms.
Thursday, 5:10-7:00, 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.

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MATH-GA 2333.001 ADVANCED TOPICS IN TOPOLOGY (Characteristic Classes and Applications to Manifolds and Varieties)

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

Prerequisites: Familiarity with homology and cohomology. Some sessions will be run concurrently with this course to provide further algebraic topology background.

Geometrical introduction first to numerical invariants (e.g., signature and index, Euler characteristic, arithmetic genus, etc.) and then their generalizations to characterstic classes (e.g., Stiefel-Whitney, Chern, Pontryjagin, genera) of manifolds, of vector bundles and of singular varieties.  Sample applications from topology, geometry, algebraic geometry, analysis, combinatorics.  The course will not have exams but students will do work to demonstrate or apply some of the methods.

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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, R. Young (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.

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MATH-GA 2400.001 ADVANCED TOPICS IN GEOMETRY (Isometric Immersions Before and After Nash)

3 points.  Fall term.
Wednesday, 5:10-7:00, M. Gromov.

The course will cover most classical results on isometric immersion, including local theorem by Janet, and global 2d by Alexandrov, Nirenberg and Nash.

The course will continue with developments, following Nash, on the general, solved and unsolved, problems of inducing tensorial structures by maps between manifolds.  An improvement of Nash’s implicit theorem by Gunther will be included.

Text: Much of the course material is contained in the corresponding chapters of Partial Differential Equations, M. Gromov, Springer, 1986

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MATH-GA 2410.001 ADVANCED TOPICS IN GEOMETRY (Ricci Curvature)

3 points. Spring term.
Tuesday, 1:25-3:15, J. Cheeger.

The course will survey some of the regularity theory for manifolds with Ricci curvature bounds and the partial regularity theory for Gromov-Hausdorff limits of sequences of such manifolds.

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ANALYSIS
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MATH-GA 1002.001 MULTIVARIABLE ANALYSIS

3 points. Spring term.
Monday, 7:10-9:00, C. Borges.

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.

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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, J. Shatah (spring).

Fall Term

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

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

Spring Term

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

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

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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 (4th 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.

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MATH-GA 2450.001, 2460.001 COMPLEX VARIABLES I, II

3 points per term. Fall and spring terms.
Tuesday, 7:10-9:00, F. Hang (fall); Wednesday, 5:10-7:00 (spring), E. Hameiri.

Fall Term

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

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

Text: Brown, J., & Churchill, R. (2008). Complex Variables and Applications (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.

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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.

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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.

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

3 points per term. Fall and spring terms.
Tuesday, 11:00-12:50, 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.

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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.

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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.)

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MATH-GA 2610.001 ADVANCED TOPICS IN PDE (Analytic Aspects of Harmonic Maps)

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

Prerequisites: basic elliptic PDEs including Sobolev spaces.

The course would be aimed at analytic aspects of harmonic maps. The topics to be discussed would include (some may be brief) followings: a) Basics: Dirichlet principle for harmonic maps and conformal maps, elementary facts about harmonic maps, Bochner identity and second variations.  b) Energy minimizing maps: The case of dimension two, minimizing maps in higher dimensions and Schoen-Uhlenbeck theory, tangent maps and its uniqueness.  c) Weakly and stationary harmonic maps: the case of dimension two, stationary maps in higher dimensions, stable and stationary maps, a brief look at blow-up analysis.  d) Heat flows: Topics to be chosen as time permits.

References:  F.H.Lin & C.Y.Wang, The Analysis of Harmonic Maps and Their Heat Flows, World Scientific Publishing Co. Pte. Ltd (2008); R.Schoen, Analytic aspects of harmonic maps, MSRI Publ. 2, Springer, New York (1984);  L.Simon, Theorems on regularity and singularity of energy minimizing maps, Lectures in Math. ETH Zurich,  Birkhauser (1996).

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

3 points. Spring term.
Monday, 1:25-3:15, T. Buckmaster.

Prerequisites: A graduate level partial differential equations course and a course in differential geometry.

In 1954, John Nash introduced a remarkable new technique for constructing 'wild' and in some cases paradoxical solutions to partial differential equations. At the time, Nash was interested in the isometric embedding problem, namely the construction of isometric embedding of Riemannian manifolds into Euclidean space. Using a refinement of Nash's work due to Nicolaas Kuiper, one can for example construct an C^1 isometric embedding of the sphere S^2 into an arbitrary small ball in R^3. The regularity of the embedding prohibits embeddings that involve folds that one would expect from an approach mimicking the scrunching of paper. The result is all the more remarkable when considers the fact that the only C^2 isometric embedding of the sphere into R^3 is the standard embedding modulo rigid transformations.

See the following webpage for a rendering of what these type of embeddings look like: http://www.gipsa-lab.fr/~francis.lazarus/Hevea/Presse/index-en.html

Recently, the ideas used in the isometric embedding problem have been adapted in order to tackle a famously unresolved conjecture of Lars Onsager related to turbulent fluid flow. The conjecture states that weak solutions to the Euler equation belonging to the Holder space with Holder exponent greater than 1/3 conserve kinetic energy; whereas for any Holder space with exponent less than 1/3, there exists solutions which dissipate kinetic energy. The first part of this conjecture had been previously confirmed by Constantin, E and Titi using simple scaling arguments. Following earlier work of Vladimir Scheffer and Alexander Shnirelman, in a sequence of two papers, Camillo De Lellis and Laszlo Szekelyhidi Jr. demonstrated the construction of Holder continuous dissipative solutions to Euler equations for any exponent less than 1/10. In later work by Camillo De Lellis, Philip Isett, Lazlo Szekelyhidi Jr. and myself the techniques have improved to a point where the conjecture is very close to being resolved.

The aim of this course is to detail the results mentioned above.

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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.

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

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

This course is a continuation of the Functional Analysis course given in fall 2015. The spring course will cover the spectral theory of self-adjoint operators, bounded and unbounded. The lectures will develop the general theory and also consider concrete examples of self-adjoint operators, such as Schroedinger operators.

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MATH-GA 2660.002 ADVANCED TOPICS IN ANALYSIS (Classical and Complex Fluid Equations)

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

Prerequisites: Basic PDEs including Sobolev spaces.

The course would consist of two parts. For the first part, we shall discuss some questions and results for Euler and Navier Stokes equations. As the literature is so vast, one has to be highly selective on this part. We would emphasize on various approaches to existence and to regularity of solutions. In the second part of the course, we shall discuss a couple examples of complex fluids including viscoelastic fluids and liquid crystals. And discuss some recent progress on existence and regularity issues.

References:

A.Majda and A.Bettozzi: Vorticity and incompressible flow, Cambridge Texts in Applied Mathematics, Cambridge Univ. Press, 2002.

There are other books on the Euler & Navier-Stokes Equations, for examples, books by R.Temam, by C.Marchioro and M. Pulvirenti, by G.Galdi and by G. Seregin.

L.Caffarelli, R.Kohn and L.Nirenberg, Partial regularity of weak solutions of the Navier-Stokes equations, Comm. Pure and Appl. Math. 35, (1982), PP 771--832.

F. Lin,  Some analytical issues for elastic complex fluids. Comm. Pure Appl. Math. 65 (2012), no. 7, 893–919.

F. Lin and C.Y.Wang, Recent developments of analysis for hydrodynamic flow of nematic liquid crystals. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372 (2014), no. 2029, 20130361, 18 pp.

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MATH-GA 2660.003 ADVANCED TOPICS IN ANALYSIS (Dynamics of the Nonlinear Schroedinger Equation)

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

I will present the mathematical theory of the Nonlinear Schrodinger Equation. This equation is very interesting in its own right, as well as a simple model for a large class of evolution problems.  A broad range of tools from Harmonic Analysis, Nonlinear Analysis, and Hamiltonian Dynamics will come into play. Topics will include Strichartz estimates, scattering, stationary states and their stability, Morawetz estimates, Schrodinger maps. A basic knowledge of PDE is required, but I will try to essentially start from scratch.

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MATH-GA 2660.004 ADVANCED TOPICS IN ANALYSIS (Calculus of Variations)

3 points. Spring term.
Thursday, 5:10-7:00, G. Francfort.

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

This course focuses on that part of the Calculus of Variations which is concerned with the search for minimizers for an integral functional. We do not discuss critical points and barely touch regularity. Also, we do not strive for the most general context, but rather illustrate the various concepts through examples coming mostly from the mechanics of materials.After a short review of the basic tenet of the direct method, we look at convexity and duality as a paradigm for the existence theory of minimizers, and convexification as a paradigm for the lack of existence of such minimizers.

We then introduce quasi-convexity and quasi-convexification/relaxation in a vectorial context. We prove higher integrability of minimizers in a standard setting. For functionals that are parameter dependent, we discuss Gamma-convergence and introduce homogenization of periodic structures within that context.

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

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

Other sources will be indicated during class.

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

3 points per term. Fall and spring terms.
Thursday, 5:10-7:00, G. Stadler (fall); Tuesday, 5:10-7:00, J. Goodman (spring).

Fall Term

Prerequisites: A good background in linear algebra, and some 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. Alternatively, you can also use Python for the homework assignments. You are encouraged but not required to learn and use a compiled language.

This course is part of a two-course series meant to introduce graduate students in mathematics to the fundamentals of numerical mathematics (but any Ph.D. student seriously interested in applied mathematics should take it). It will be a demanding course covering a broad range of topics, 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, and Fourier transforms. This course will not cover differential equations, which form the core of the second part of this series, Numerical Methods II.

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

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

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

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

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

If you want to brush up your MATLAB:

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

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

Cross-listing: CSCI-GA 2420.001.

Spring Term

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

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

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

Cross-listing: CSCI-GA 2421.001.

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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.

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MATH-GA 2012.001 ADVANCED TOPICS IN NUMERICAL ANALYSIS (Computational and Variational Methods for Inverse Problems)

3 points. Spring term.
Thursday, 9:00-1050, G. Stadler.

This course provides an introduction to inverse problems that are governed by systems of partial differential equations (PDEs), and to their numerical solution.  The focus of the course is on variational formulations, ill-posedness, regularization, variational discretization, large-scale solution algorithms for inverse problems and the computation of derivatives using adjoint methods.  Depending on the interest of the participants, the course will provide an introduction to the Bayesian framework for inverse problems additionally to the deterministic approach, and will draw connections between the two.  Examples will be drawn from different areas of science and engineering, including image processing, continuum mechanics, and geophysics.

Recommended texts:

C. Vogel: Computational Methods for Inverse Problems, SIAM 2002, http://dx.doi.org/10.1137/1.9780898717570

H. Engl, M. Hanke, A. Neubauer: Regularization of Inverse Problems, Dordrecht, 2nd edition, 1996.

F. Troeltzsch: Optimal Control of Partial Differential Equations: Theory, Methods and Applications, Graduate Studies in Mathematics, Vol. 112, AMS, 2010.  http://dx.doi.org/10.1090/gsm/112

A. Tarantola: Inverse Problem Theory and Methods for Model Parameter, Estimation, SIAM 2005.  http://dx.doi.org/10.1137/1.9780898717921

I also have unfinished lecture notes, which I plan to complement/extend while teaching this class.

Cross-listing: CSCI-GA 2945.001.

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MATH-GA 2012.002 ADVANCED TOPICS IN NUMERICAL ANALYSIS (Convex and Nonsmooth Optimization)

3 points. Spring term.
Monday, 5:10-7:00, M. Overton.

Convex optimization problems have many important properties, including a powerful duality theory and the property that any local minimum is also a global minimum. Nonsmooth optimization refers to minimization of functions that are generally not convex, usually locally Lipschitz, and typically not differentiable at their minimizers. Topics in convex optimization that will be covered include duality, self-concordance and global Newton methods, primal-dual interior-point methods for conic programs, including linear programs, quadratic cone programs and semidefinite programs, and using CVX to solve convex programs in practice. Topics in nonsmooth optimization that will be covered include variational analysis, subgradients and subdifferentials, Clarke regularity, and algorithms, including gradient sampling and BFGS, for nonsmooth, nonconvex optimization. Homework will be assigned, both mathematical and computational. Students may submit a final project or take an oral final exam.

Cross-listing: CSCI-GA 2945.002.

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MATH-GA 2041.001 COMPUTING IN FINANCE

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

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

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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 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.

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MATH-GA 2045.001 COMPUTATIONAL METHODS FOR FINANCE

3 points. Fall term.
Tuesday, 7:10-9:00, J. Guyon & S. Corlay.

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

The classical curriculum of mathematical finance programs generally covers the link between linear parabolic partial differential equations (PDEs) and stochastic differential equations (SDEs), resulting from Feynmam-Kac's formula. However, the challenges faced by today's practitioners mostly involve nonlinear PDEs. The aim of this course is to provide the students with the mathematical tools and computational methods required to tackle these issues, and illustrate the methods with practical case studies such as American option pricing, uncertain volatility, uncertain mortality, different rates for borrowing and lending, calibration of models to market smiles, credit valuation adjustment (CVA), portfolio optimization, transaction costs, illiquid markets, super-replication under delta and gamma constraints, etc.

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

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

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MATH-GA 2046.001 ADVANCED ECONOMETRIC MODELING AND BIG DATA

3 points. Fall term.
Thursday, 7:10-9:00, G. Ritter.

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

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

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MATH-GA 2048.001 SCIENTIFIC COMPUTING IN FINANCE

3 points. Spring term.
Wednesday, 5:10-7:00, Y Li & H. Cheng.

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.

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APPLIED MATHEMATICS
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MATH-GA 2701.001 METHODS OF APPLIED MATHEMATICS

3 points. Fall term.
Monday, 1:25-3:15, D. Giannakis.

Prerequisites: Elementary linear algebra and differential equations.

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

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

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

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

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

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

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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.

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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.

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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.

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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.

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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.

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MATH-GA 2710.001MECHANICS

3 points. Spring term.
Wednesday, 1:25-3:15, A. Cerfon.

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

Recommended texts: Structure and Interpretation of Classical Mechanics, G.J. Sussman and J. Wisdom, with Meinhard E. Mayer, MIT Press (2007)

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.

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MATH-GA 2751.001 RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS

3 points. Fall and spring terms.
Tuesday, 7:10-9:00, P. Kolm (fall); Wednesday, 7:10-9:00, M. Avellaneda (spring).

Fall Term

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

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

Spring Term

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

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

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

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

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MATH-GA 2752.001 ACTIVE PORTFOLIO MANAGEMENT

3 points. Spring term.
Monday, 5:10-7:00, J. Benveniste.

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.

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MATH-GA 2753.001 ADVANCED RISK MANAGEMENT

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

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

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

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

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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.

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MATH-GA 2757.001 REGULATION AND REGULATORY RISK MODELS

3 points. Fall term.
Wednesday, 7:10-9:00, K. Abbott & L. Andersen.

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

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

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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, A. Javaheri (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.

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MATH-GA 2792.001 CONTINUOUS TIME FINANCE

3 points. Fall and spring terms.
Monday, 7:10-9:00, A. Javaheri & S. Ghaman (fall); Wednesday, 7:10-9:00, P. Carr & B. Dupire (spring).

Prerequisites: Derivative Securities and Stochastic Calculus, or equivalent.

Fall Term

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

Spring Term

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

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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.

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MATH-GA 2797.001 CREDIT MARKETS AND MODELS

3 points. Fall term.
Wednesday, 7:10-9:00, B. Flesaker.

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

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

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

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MATH-GA 2798.001 INTEREST RATE AND FX MODELS

3 points. Spring term.
Thursday, 7:00-8:50, F. Mercurio & T. Fisher.

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.

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MATH-GA 2830.001 ADVANCED TOPICS IN APPLIED MATHEMATICS (Quantifying Uncertainty in Complex Turbulent Systems)

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

Prerequisite: some knowledge of elementary stochastic diff. equations and nonlinear dynamics.

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, coarse-graining, long-range forecasting, model error 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 and his PhD students and postdocs will jointly give the lectures. All students that want credit for class need to participate in at least one team lecture; the initial team for various lectures consists of postdocs and graduate students working with Professor Majda and others can join.

Background text: there is no formal text but online papers, notes and lectures will be available for the class.

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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.

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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.

Prerequisites: Advanced calculus and probability and statistics.

Financial models often involve maximizing a concave utility, or assessing uncertainty using a convex risk measure. As a result, tools from convex analysis have become indispensable in financial math research and practice. This course systematically explores the use of convex analysis methods -- in particular, duality theory -- in financial math. We start with a crash course on convex duality, emphasizing its relationship with Lagrange multipliers. Then we show how many important topics in financial math can be treated in a uniform framework using convex duality. Topics to be discussed include Markowitz portfolio theory and the closely related CAPM and Sharp ratio; the fundamental theorem of asset pricing; coherent risk measures and the closely related "good deal bounds" for pricing financial assets; the duality between delta hedging and option pricing; and conic finance, which explicitly models the disparity of bid and ask prices and explores the consequences.

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MATH-GA 2830.004 ADVANCED TOPICS IN APPLIED MATHEMATICS (Proximal Methods in Optimization and Data Science)

3 points. Fall term.
Tuesday, Thursday 1:25-3:15, P. Combettes.

Prerequisite: calculus, basic linear algebra, elementary functional analysis.

This course is intended to provide an account of proximal tools in convex optimization with a view towards their applications in certain areas of data science (signal and image processing, inverse problems, statistical data analysis, machine learning, etc.). The basic theory willbe provided and a strong emphasis will be placed on algorithm design and concrete applications.

Course overview and motivations: iteration principles; fixed point algorithms; convex sets and convex cones; best approximation paradigms; projection methods in convex feasibility problems – applications to data fusion and image recovery; convex functions; conjugation of convex functions; duality in convex optimization; subdifferential calculus; subgradient algorithms for convex feasibility and best approximation – applications in inverse problems; monotone operators; proximity operators; proximal calculus; forward-backward splitting and variants (Dykstralike methods, Chambolle-Pock algorithm, dual ascent method, etc.); Douglas-Rachford splitting and variants (parallel proximal algorithm, alternating direction method of multipliers, composite primal-dual method, etc.); the monotone+skew decomposition principle – primal-dual algorithms; proximal modeling of statistical information; proximal information extraction; proximal sparsity enforcement; proximal data classification; proximal image reconstruction; proximal learning; scalability: proximal methods in big data problems; special topics.

Grades will be determined by individual projects.

References:

H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York, 2011;

P. L. Combettes, The convex feasibility problem in image recovery, in: Advances in Imaging and Electron Physics (P. Hawkes, Ed.), vol. 95, pp. 155–270. Academic Press, New York, 1996;

P. L. Combettes & J.-C. Pesquet, Proximal splitting methods in signal processing, in Fixed-Point Algorithms for Inverse Problems in Science and Engineering (H. H. Bauschke et al., eds), pp. 185–212. Springer, New York, 2011;

S. Sra, S. Nowozin, and S. J. Wright, Optimization for Machine Learning. MIT Press, Cambridge, MA, 2012.

The course will run twice a week from October 27 to December 15, 2015.

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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.

The main theme of this seminar-style course is methods for extracting temporal and spatial patterns of variability from high dimensional time series. Such problems arise in many data rich areas, including geosciences, fluid dynamics, and molecular dynamics. Following a review of classical techniques, such as principal components analysis and singular spectrum analysis, we will focus on methods based on spectral graph theory, which attempt to recover from the data patterns of high dynamical significance (rather than high explained variance). Examples will be drawn mainly from atmosphere-ocean science and low-dimensional dynamical systems. A background in ergodic theory and differential geometry would be helpful, but no previous knowledge of these topics will be required. Graduate students at any level are welcome to attend and contribute. This course will be graded based on attendance and a presentation at the end of the semester.

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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.

The course's website is http://www.cims.nyu.edu/~cfgranda/pages/OBDA_spring16/.

Recommended Text (revised 1/11/16):  https://web.stanford.edu/~hastie/StatLearnSparsity_files/SLS_corrected_1.4.16.pdf

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

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

The course begins by covering fundamentals of physiological properties of neurons, from neuronal and synaptic dynamics, to rate vs. spike codings, such as compressive sensing. Then it delves into various mathematical aspects of neuronal network modeling, addressing issues of neuronal model reductions (for example, reduction from Hodgkin-Huxley models to exponential integrate-and-fire models), dynamical systems approach, stochastic processes and linear/nonlinear system analysis in neuronal network dynamics. It covers, in detail, population dynamics of networks, functional connectivity and structural connectivity of networks. The course strives to bring students with applied mathematics, physical science, or neuroscience background, quickly to research topics in theoretical modeling in neuroscience.

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MATH-GA 2851.002 ADVANCED TOPICS IN MATH BIOLOGY (Biomolecular Motors)

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

Prerequisites: probability and partial differential equations as used in applications.

Biological cells contain molecular machinery that is used for cell motility, for intracellular transport of vesicles and organelles, to move proteins across internal membranes, to partition chromosomes at cell division, to store metabolic energy in the form of ATP, and to copy genetic information. Unlike the macroscopic machinery of our everyday experience, these biomolecular motors function in a regime dominated by thermal fluctuations. Throughout the course, mathematical modeling and computer simulation will be used to elucidate the diverse mechanisms of biomolecular motors, with particular emphasis on the probabilistic aspect of their function. Topics to be studied include: cross-bridge dynamics in muscle, kinesin as a molecular walker, optimal "dynamic instability" of microtubules for chromosome capture, depolymerization-driven transport of chromosomes during mitosis, the role of elasticity (including chromosome flexibility) in the function of biomolecular motors, a look-ahead mechanism for RNA polymerase and the influence of look-ahead on the speed and error rate of transcription, and rotary molecular motors that are driven by ion gradients such as ATP synthase and the bacterial flagellar motor. Course requirements include homework and a computing project, both of which may be done by students working together in teams. Presentation of the computing project to the class is encouraged.

Text: A reprint collection will be available online.

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MATH-GA 2852.001 ADVANCED TOPICS IN MATH BIOLOGY (Synaptic Transmission)

3 points. Spring term.
Wednesday, 1:25-3:15, C. Peskin.

Prerequisites: familiarity with differential equations and probability as used in applications.

Synaptic transmission is the process by which neurons communicate with each other, and also with the outside world via sensory and motor systems.  The synapse is thought to be the site at which learning occurs and at which memories are encoded.  The synapse is also an important source of noise in the nervous system, via the stochastic release of discrete packets of neurotransmitter from synaptic vesicles.  Finally, the synapse is a major site of adaptivity, in which sensitivity is continually  being adjusted so that outputs depend more upon relative changes in the input than upon absolute input levels.

In this course, we study mathematical models of synaptic transmission under the headings of presynaptic processes, synaptic cleft processes, and postsynaptic processes. Among presynaptic processes, we focus on stochastic vesicle release and its consequences for interneuronal communication.  Here we study the role of calcium channels and local calcium diffusion in regulating vesicle release, and find the biophysical basis of short-term depression and facilitation.  In the synaptic cleft, we study the dynamics of decay of the neurotransmitter signal as influenced by binding to postsynaptic receptors, uptake mechanisms, enzymatic degradation, and diffusion of neurotransmitter out of the cleft.  On the postsynaptic side, we study the mechanisms underlying longterm adaptations responsible for learning and memory based on regulation of the numbers and kinds of postsynaptic receptors that are present in the postsynaptic membrane.  In particular we study the biophysical basis of the Hebb learning rule, in which a synapse  is strengthened not merely by use, but by effective use, i.e., when synaptic transmission is soon followed by the occurrence of an action potential in the postsynaptic neuron.  Also, we study the significance of dendritic spines for postsynaptic signal processing, in particular the possible role of spines as synaptic amplifiers, and the regulation of spine neck resistance as a possible learning mechanism.
Cross-listing: BIOL-GA 2852.001.

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MATH-GA 2852.002 ADVANCED TOPICS IN MATH BIOLOGY (The Art of Modeling in Cell and Developmental Biology)

3 points. Spring term.
Thursday, 1:25-3:15,
A. Mogilner.

Prerequisites: calculus, elementary background in ODE and probability

We will go through case studies of seminal mathematical models relevant to cell biology. The biological topics we will discuss will include: bacterial and eukaryotic chemotaxis, reaction-diffusion pattern formation in cell and developmental biology, mechanics in cell and developmental biology, cell cycle and biochemical oscillations, motifs in genetic and biochemical networks, cytoskeletal dynamics, mitosis, cytokinesis, cell movements, tissue morhogenesis. The mathematical topics we will discuss will include: scaling and non-dimensionalization; perturbation theory; nonlinear ODEs and PDEs; integro-differential equations; Langevin and Fokker-Planck equations; Monte-Carlo simulations; Gillespie algorithm; Boulean networks.

The course will include primers in cell biology, biophysics and applied mathematics.  We will use matlab to solve simple model equations, and some user-friendly software to solve more complex ones. We will read and analyze a few research papers combining experiment and modeling. The focus of the course will be not on using biology as an inspiration for sophisticated math, but on learning how to develop and use mathematical or computational models together with experimental biologists, how to come up with the right questions, methods, numbers, how to translate real data into modeling assumptions, how to analyze the model fast using scaling, intuition and rough estimates, how to test the model by thinking up new experiments, etc.

No single book will be used extensively.  Rather, I will provide a list of books excerpts from which we will be using; those will all be provided online.  We will also use review papers published in major journals (all available online).

Cross-listing: BIOL-GA 2852.002.

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MATH-GA 2855.001 ADVANCED TOPICS IN MATH PHYSIOLOGY (Neuronal Networks)

3 points. Fall term.
Wednesday, 2:30-4:20, J. Rinzel.

Prerequisite: familiarity with applied differential equations; most neurobiological background will be provided.

This course will involve the formulation and analysis of differential equation models for neuronal ensembles and neuronal computations. Spiking and firing rate mechanistic treatments of network dynamics as well as probabilistic behavioral descriptions will be covered. We will consider mechanisms of coupling, synaptic dynamics, rhythmogenesis, synchronization, bistability, adaptation. Applications will likely include: central pattern generators and frequency control, perceptual bistability, working memory, decision-making and neuro-economics, feature detection in sensory systems, cortical oscillations (gamma, up-down states).

Students will undertake computing projects related to the course material: some in homework format and a term project with report and oral presentation.

Cross-listing: BIOL-GA 2855.001

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MATH-GA 2861.001 ADVANCED TOPICS IN FLUID DYNAMICS (Introduction to Plasma Physics)

3 points. Fall term.
Tuesday, 9:00-10:50, J. Freidberg.

The course will provide a basic description of plasma physics. The focus will be primarily on plasmas with low collisionality which are dominated by long range electromagnetic forces. The material will cover basic physical concepts plus some applications to fusion and astrophysics. Background in electromagnetics and partial differential equations will be very useful. Topics to be covered include (1) definition of a plasma, (2) single particle motion in given magnetic and electric fields, (3) guiding center theory including cross field drifts and the mirror effect, (4) Coulomb collisions, (5) runaway electrons, (6) transport in velocity and physical space, (7) self-consistent plasma fluid models, MHD equilibrium and stability, (8) cold plasma waves, (9) kinetic theory, and (10) collisionless Landau damping.

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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.

Fluid-Structure Interactions describes a class of dynamics problems wherein flexible or moveable structures interact with and through a surrounding fluid. This class of problems is also fundamental and lies at the heart of swimming and flight, the dynamics of complex fluids, and biological processes within the cell. These problems can be challenging to model as the presence of immersed structures can make the fluid domain geometrically complicated, and the dynamics of the both flow and structure are intertwined and must be determined simultaneously. Further, fluid-structure interactions are often multi-scale, with disparate space and/or time-scales arising from response forces like elasticity, or from geometric anisotropy. As such, they can require special purpose approaches to simulate well and I will  discuss several of them (such as boundary integral, immersed boundary, and penalty methods).

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MATH-GA 3001.001 GEOPHYSICAL FLUID DYNAMICS

3 points. Fall term.
Tuesday, 9:00-10:50, O. Buhler.

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.

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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.

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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.

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MATH-GA 3010.002 ADVANCED TOPICS IN ATMOSPHERE-OCEAN SCIENCE (Geophysical Turbulence)

3 points. Fall term.
Tuesday, 1:25-3:15, S. Smith.

Prerequisites: fluid dynamics, geophysical fluid dynamics.

The dynamical structure and circulation of planetary atmospheres and oceans are shaped by turbulent motions occurring at scales ranging from that of the planet itself, to the scale where viscosity absorbs its energy (millimeters in Earth’s atmosphere). These motions are largely characterized by high rotation and stratification, but also by their ubiquity near boundaries. This course will survey a wide range of observed turbulent processes, with a focus on their phenomenology, interpretation through simplified models, and the tools necessary to analyze them. Topics to be covered may include: scaling, power spectra and structure functions, Reynolds averaging and eddy diffusivity, generating instabilities, inertial-range theory, rotating and stratified limits, geostrophic turbulence, coherent structure formation, boundary layer turbulence, convection, passive scalar advection by turbulent flows, closure theories and parameterization.

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MATH-GA 3011.001 ADVANCED TOPICS IN ATMOSPHERE-OCEAN SCIENCE (Climate Modeling)

3 points. Spring term.
Thursday, 9:00-10:50, 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.

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PROBABILITY AND STATISTICS
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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.

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MATH-GA 2902.001 STOCHASTIC CALCULUS

3 points. Fall and spring terms.
Monday, 7:10-9:00, J. Goodman(fall); Thursday, 7:10-9:00, A. Kuptsov (spring).

Prerequisites: MATH-GA 2901 Basic Probability or equivalent.

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

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

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

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

3 points per term. Fall and spring terms.
Wednesday, 11:00-12:50, E. Lubetzky (fall); Wednesday, 9:00-10:50, H. McKean (spring).

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

Fall Term

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.

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MATH-GA 2931.001 ADVANCED TOPICS IN PROBABILITY (Motion in Random Media)

3 points. Fall term.
Tuesday, Thursday, 3:20-5:05, O. Zeitouni.
The course will begin on September 15 and end on October 29.

The course will discuss limit theorems for motion in random media (mostly random walks in random environments), including some topics in homogenization theory. It will emphasize some of the outstanding open problems in this area.

Topics to be covered:  (1) The model of RWRE: hitting times, law of large numbers and fluctuations in dimension 1; (2) Introduction to homogenization: a) the reversible case: CLT for additive functionals of Markov chains, the conductance model, RWRE in dimension 1, b) an non-reversible example: balanced RWRE; (3)  Introduction to regeneration times and limit theorems for RWRE in the ballistic situation.  Space-time environments; (4) Perturbative methods.

The course will be partially based on Lecture Notes Math (vol. 1837), and on the book Fluctuations in Markov Processes, Komorowski, Landim & Olla.

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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.

We review a number of old results concerning certain statistical mechanics models and their possible connections to the Riemann Hypothesis.

A standard reformulation of the Riemann Hypothesis (RH) is: The (two-sided) Laplace transform of a certain specific function Ψ on the real line is automatically an entire function on the complex plane; the RH is equivalent to this transform having only pure imaginary zeros. Also Ψ is a positive integrable function, so (modulo a multiplicative constant C) is a probability density function.

A (finite) Ising model is a specific type of probability measure P on the points S=(S1,...,SN) with each Sj = +1 or -1. The Lee-Yang theorem (of T. D. Lee and C. N. Yang) implies that for non-negative a1, ..., aN, the Laplace transform of the induced probability distribution of a1 S1 + ... + aN SN has only pure imaginary zeros. There are also other models, where the variables are real-valued or vector-valued which have moment generating functions with only pure imaginary zeros.

An intriguing question is whether it's possible to find a sequence in N of models and generating functions so that the limit as N tends to infinity of such distributions has density exactly C Ψ. We'll discuss some of the cases where one can study the limiting distribution and some hints as to how one might try to find the "right" choice.

Some background references: For Lee-Yang type theorems: C. M. Newman, CPAM 27 (1974) 143--159; E. Lieb and A.D. Sokal, CMP 80 (1981) 153--179; J. Froehlich and P-F Rodriguez, JMP 53 (2012) 095218.  For the connection to the Riemann Hypothesis: C. M. Newman, Z. Wahr. (PTRF) 33 (1975) 75--93 (especially p. 90).  For some limiting distributions: B. Simon and R.B. Griffiths, CMP 33 (1973) 145--164; F. Camia, C. Garban and C. M. Newman, arXiv 1307.3926, AIHP to appear.

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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.

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MATH-GA 2932.001 ADVANCED TOPICS IN PROBABILITY (Random Graphs)

3 points.  Spring term.
Tuesday, 5:10-7:00, J. Spencer.

Prerequisites: "Mathematical Maturity." This topic takes from several areas but the material will be developed in the course. An acquaintance with, say, variance (in probability) and/or chromatic number (in graph theory) will be helpful but not mandatory.

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.

Today we are very interested in algorithmic implementation, both deterministically and with random algorithms. There is further 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. Probability results include Chernoff Bounds, Martingales, the Lovasz Local Lemma (including the algorithmic implementation) and the Janson Inequalities and will be derived from scratch. Topics include: Ramsey umbers, Continuous Time Greedy Algorithms, Graph Coloring, Discrepancy, the Liar Game and the Tenure Game.  Of particular pragmatic interest:  asymptotic calculations permeate the course and approaches to finding symptotics of various sums and products will be emphasized throughout.

Texts: Noga Alon, Joel Spencer, The Probabilistic Method, 3rd edition, John Wiley, 2009 (Note: the fourth edition, if it has appeared, is also OK. Earlier editions are acceptable but not optimal); Joel Spencer, Asymptopia, American Math Society, 2014.

Cross-listed as CSCI-GA 3230.001

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Revised September 2015