Course Descriptions: AY 2013-14
ALGEBRA AND NUMBER THEORY
MATH-GA 2110.001, 2120.001 LINEAR ALGEBRA I, II
3 points per term. Fall and spring terms.
Tuesday, 5:10-7:00, F.
Greenleaf (fall); Monday, 5:10-7:00 F.
Greenleaf (spring).
Fall Term
Prerequisite: undergraduate linear algebra or permission of the instructor.
Linear spaces, subspaces. Linear dependence, linear independence; span, basis, dimension, isomorphism. Quotient spaces. Linear functionals, Dual spaces. Linear mappings, null space, range, fundamental theorem of linear algebra. Underdetermined systems of linear equations. Composition, inverse, transpose of linear maps, algebra of linear maps. Similarity transformations. Matrices, matrix Multiplication, Matrix Inverse, Matrix Representation of Linear Maps, Determinant, Laplace expansion, Cramer's rule. Eigenvalue problem, eigenvalues and eigenvectors, characteristic polynomial, Cayley-Hamilton theorem. Diagonalization.
Text: Linear Algebra, Friedberg, Insel & Spence, Prentice-Hall, 4th Ed.
Strongly recommended text: Schaum’s Outline Series: Linear Algebra, S. Lipschuts
Spring Term
Prerequisite: Linear Algebra I or permission of the instructor.
Trace, determinant, characteristic and minimal polynomial.
Eigenvalues, diagonalization, spectral theorem and spectral mapping
theorem. Rayleigh quotient and minimax theorem. When
diagonalization fails: nilpotent operators and their structure,
generalized eigenspace decomposition, Jordan canonical form.
Complexification and diagonalization over R. Matrix norms, series and
the matrix exponential map,
applications to ODE. Bilinear and quadratic forms and their normal
forms. The classical matrix groups: unitary, orthogonal,
symplectic. Implicit function theorem, smooth surfaces in R^n and
their tangent spaces. Intro to matrix Lie algebras and Lie groups.
Text: Linear Algebra, Friedberg, Insel & Spence, Prentice-Hall, 4th Ed.
Cross-listed as MATH-UA 0141, 0142.
MATH-GA 2110.001 LINEAR ALGEBRA I
3 points. Spring term.
Tuesday, 5:10-7:00, H. Weitzner.
Prerequisite: undergraduate Linear Algebra or permission of the instructor.
Linear spaces, subspaces. Linear dependence, linear independence; span, basis, dimension, isomorphism. Quotient spaces. Linear functionals, dual spaces. Linear mappings, null space, range, fundamental theorem of linear algebra. Underdetermined systems of linear equations. Composition, inverse, transpose of linear maps, algebra of linear maps. Similarity transformations. Matrices, matrix Multiplication, Matrix Inverse, Matrix Representation of Linear Maps determinant, Laplace expansion, Cramer's rule. Eigenvalue problem, eigenvalues and eigenvectors, characteristic polynomial, Cayley-Hamilton theorem. Diagonalization.
Text: Linear Algebra, P. Lax, Wiley - Interscience
MATH-GA 2111.001 LINEAR ALGEBRA (one-term format)
3 points. Fall term.
Thursday, 9:00-10:50, A.
Rangan.
Prerequisite: undergraduate linear algebra.
Linear algebra is two things in one: a general methodology for solving linear systems, and a beautiful abstract structure underlying much of mathematics and the sciences. This course will try to strike a balance between both. We will follow the book of our own Peter Lax, which does a superb job in describing the mathematical structure of linear algebra, and complement it with applications and computing. The most advanced topics include spectral theory, convexity, duality, and various matrix decompositions.
Text: Linear Algebra, P. Lax, Wiley-Interscience Publications
Optional text: Linear Algebra and Its Applications, G. Strang
MATH-GA 2130.001, 2140.001 ALGEBRA I, II
3 points per term. Fall and spring terms.
Tuesday, 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.
Spring term
Representations of finite groups. Characters, orthogonality of characters of irreducible representations, a ring of representations. Induced representations, Artin’s theorem, Brauer’s theorem. Representations of compact groups and the Peter-Weyl theorem. Lie groups, examples of Lie groups, representations and characters of Lie group. Lie algebras associated with Lie groups. Applications of the group representations in algebra and physics. Elements of algebraic geometry.
Text: Algebra, M. Artin, Prentice Hall
Supplementary texts: Algebra, S. Lang; Linear Representations of Finite Groups, J. P. Serre; Undergraduate Algebraic Geometry, M. Reid; Representations and Characters of Groups, G. James and M. Liebeck; Cambridge Math Textbooks, 1993; Representation Theory, W. Fulton and J. Harris, Springer-Verlag; The Symmetric Group, B. E. Sagan, Wadsworth & Brooks/Cole Math. Series; Representations of Compact Lie Groups, T. Brocker and T. tom Dieck
MATH-GA 2160.001 ADVANCED TOPICS IN ALGEBRA (Algebraic Transformation Groups and Invariant Theory)
3 points. Spring term.
Tuesday, 1:25-3:15, C. Boehning.
Prerequisites: a working knowledge of the basics in algebraic geometry and the representation theory of linear algebraic groups. See Basic Algebraic Geometry I, I. R. Shafarevich, Springer (for algebraic geometry); and Linear Algebraic Groups, Graduate Texts in Mathematics, Vol. 21, Springer (for linear algebraic groups and their representations).
The subject of transformation groups deals with symmetries of
mathematical objects. The idea of invariants is ubiquitous in
mathematics and is indispensable in any type of classification result.
Moreover, the richest
geometric structures (and those which are most amenable to detailed
investigation) can usually be found in the presence of a group action:
the beauty of the classical geometries (affine, projective, Euclidean,
spherical, hyperbolic), the omnipresence of Grassmannians, flag
varieties, Schubert varieties, toric varieties in algebraic geometry
and the (abelian) anabelian approaches to birational geometry through
Galois groups of function
fields of varieties all provide illustrations of this yoga.
It should therefore be not too surprising that in 1864 the important English mathematician (and later founder of the American Journal of Mathematics) Sylvester wrote (with enthusiasm and flowery language) "As all the roads lead to Rome so I find in my own case at least that all algebraic inquiries, sooner or later, end at the Capitol of modern algebra over whose shining portal is inscribed the Theory of Invariants"; and, more than a hundred years later, in an 1984 paper, Kung and Rota compared the theory of invariants, pronounced dead many times in the meantime, to "the Arabian phoenix rising out of its ashes".
The purpose of the course is to describe some salient features of the modern phenotype of the phoenix. Here is a rough outline of the topics we propose to cover, in the form of some buzzwords (and possibly subject to changes): (1) basic notions in the geometry of group actions: complexity, rank and modality of an action, Borel-Weil theorem, classes of homogeneous spaces, local structure theorems for actions, etale slices, Luna stratification, stabilizers in general position, quotient morphisms; (2) Hilbert null cones and their stratification, stability, linear group quotients and their rationality/stable rationality properties; (3) some aspects of equivariant cohomology in algebraic geometry, Borel construction, group cohomology, stable and unramified cohomology and their relation to rationality properties of linear group quotients
Grading: this course will be graded as a seminar course.
Reference texts: Algebraic Geometry IV, Linear Algebraic Groups, Invariant Theory, Encyclopedia of Mathematical Sciences Vol. 55 V.L. Popov & E.B. Vinberg, Springer, 1994; Théorie des Invariants & Géométrie des Variétés Quotient, Travaux en Cours, G. W. Schwarz & M. Brion, 2000; Homogeneous Spaces and Equivariant Embeddings, Encyclopedia of Mathematical Sciences 138, D.A. Timashev, Springer-Verlag, 2011; Transformation Groups, De Gruyter Studies in Mathematics 8, T. tom Dieck, de Gruyter, 1987
MATH-GA 2210.001 NUMBER THEORY
3 points. Spring term.
Tuesday, 7:10-9:00, B. Bakker.
Prerequisites: basic complex analysis and algebra helpful.
Introduction to elementary methods of number theory. Topics: arithmetic functions, congruences, the prime number theorem, primes in arithmetic progression, quadratic reciprocity, the arithmetic of number fields, approximations and transcendence theory, p-adic numbers, diophantine equations of degree 2 & 3.
Text: A course in Arithmetic, J. P. Serre, Springer GTM, #7
GEOMETRY AND TOPOLOGY
MATH-GA 2310.001 TOPOLOGY I, II
3 points per term. Fall and spring terms.
Thursday, 7:10-9:00, S. Cappell
(fall);
Wednesday,
7:10-9:00,
Instructor
TBA (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.
Useful texts: Algebraic Topology, A. Hatcher (on-line at http://www.math.cornell.edu/~hatcher/#ATI); Topology, J. Munkres, Prentice Hall 2000, 2nd Ed.; Differential Topology, Guillemin & Pollack, Prentice Hall; Topology from a Differential Viewpoint, J. Milnor, Princeton University Press.
Spring term
Homology and cohomology from simplicial, singular, cellular, axiomatic and differential form viewpoints. Axiomatic characterizations and applications to geometrical problems of embedding and fixed points. Manifolds and Poincaré duality. Products and ring structures. Vector bundles, tangent bundles, De Rham cohomology and differential forms.
MATH-GA 2334.001, 2360.001 ADVANCED TOPICS IN TOPOLOGY (Vectors, Bundles & Characteristic Classes)
3 points. Spring term.
Thursday, 5:10-7:00, S. Cappell.
Course description available from the instructor.
MATH-GA 2350.001, 2360.001 DIFFERENTIAL GEOMETRY I, II
3 points per term. Fall and spring terms.
Monday, 1:25-3:15, B.
Kleiner
(fall); Thursday, 9:00-10:50, J. Cheeger
(spring).
Prerequisites: multivariable calculus and linear algebra.
Fall Term
Differentiable manifolds, tangent bundle, embedding theorems, vector fields and differential forms. Riemannian metrics and connections, geodesics, exponential map, and Jacobi fields. Generalizations of differential geometric concepts and applications.
Text: Differential Topology, Victor Guillemin & Alan Pollack, AMS Chelsea publishing, American Mathematical Society, Providence RI, republished in 2010
Spring Term
Differential forms. Integration on manifolds. Sard's Theorem. DeRham cohomology. Morse theory. Submanifolds and second fundamental form. Applications to geometric problems.
MATH-GA 2410.001 ADVANCED TOPICS IN GEOMETRY (Entropy in Dynamics, Geometry and Algebra)
3 points. Spring term.
Thursday, 1:25-3:15, M. Gromov.
The course will start with explaining the functorial meaning on the
Boltzmann-Shannon entropy that allows its generalizations
and applications in many different contexts. The
following topics will be discussed: (1) entropy and homology; (2)
entropy and volume; (3) entropy and random walk in discrete groups; (4)
entropy and isoperimetric inequalities. If time permits, the
course will also discuss the von Neumann entropy in Hilbert spaces.
ANALYSIS
MATH-GA 1410.001, 1420.001 INTRODUCTION TO MATHEMATICAL ANALYSIS I, II
3 points per term. Fall and spring terms.
Monday, 5:10-7:00, S. Güntürk
(fall); Thursday, 5:10-7:00, (spring) X.
Hu.
Fall term
Functions of one variable: rigorous treatment of limits and
continuity. Derivatives. Riemann integral. Taylor
series. Convergence of infinite series and integrals.
Absolute and uniform convergence. Infinite series of
functions. Fourier series.
Main text: Advanced Calculus, Avner Friedman, Dover
publications.
Optional supplementary text: Introduction to Analysis,
W. R.
Wade, Prentice Hall
Spring term
Functions of several variables and their derivatives. Topology of Euclidean spaces. The implicit function theorem, optimization and Lagrange multipliers. Line integrals, multiple integrals, theorems of Gauss, Stokes, and Green.
Required text: Introduction to Analysis, W. R. Wade, Prentice Hall
Recitation/Problem Session: 7:15-8:30 (following the course in both terms).
MATH-GA 2430.001 REAL
VARIABLES
3 points. Fall term.
Monday, Wednesday, 9:35-10:50, N. Masmoudi.
Note: Master's students should consult course instructor before registering for this course.
Prerequisites: A familiarity with rigorous mathematics, proof writing, and the epsilon-delta approach to analysis, preferably at the level of MATH-GA 1410, 1420 Introduction to Mathematical Analysis I, II.
Measure theory and integration. Lebesgue measure on the line and abstract measure spaces. Absolute continuity, Lebesgue differentiation, and the Radon-Nikodym theorem. Product measures, the Fubini theorem, etc. L^p spaces, Hilbert spaces, and the Riesz representation theorem. Fourier series.
Required text: Real Analysis, Halsey Royden, Prentice Hall, 3rd Edition, 1988. (NOT the new, 4th edition).
Supplementary texts: Introductory Real Analysis, Andrei N. Kolmogorov and Sergei V. Fomin; Real and Complex Analysis, Walter Rudin; Real Analysis: Modern Techniques and Their Applications, Gerald B. Folland.
MATH-GA 2450.001, 2460.001 COMPLEX VARIABLES I, II
3 points per term. Fall and spring terms.
Thursday, 5:10-7:00, Instructor TBA (fall); Wednesday, 5:10-7:00, E.
Hameiri (spring).
Prerequisites: advanced calculus, or MATH-GA 1410 Introduction to Math Analysis I. Concurrent registration is not permitted.
Fall Term
Complex numbers; analytic functions, Cauchy-Riemann equations; linear fractional transformations; construction and geometry of the elementary functions; Green's theorem, Cauchy's theorem; Jordan curve theorem, Cauchy's formula; Taylor's theorem, Laurent expansion; analytic continuation; isolated singularities, Liouville's theorem; Abel's convergence theorem and the Poisson integral formula.
Text: Introduction to Complex Variables and Applications, Brown & Churchill
Spring Term
The fundamental theorem of algebra, the argument principle; calculus of residues, Fourier transform; the Gamma and Zeta functions, product expansions; Schwarz principle of reflection and Schwarz-Christoffel transformation; elliptic functions, Riemann surfaces; conformal mapping and univalent functions; maximum principle and Schwarz's lemma; the Riemann mapping theorem.}
Text: Complex Analysis, Alfors
Cross-listed as MATH-UA 0393.001, 0394.001
MATH-GA 2451.001 COMPLEX VARIABLES (one-term format)
3 points. Fall term.
Tuesday, Thursday, 1:25-2:40, Instructor TBA.
Note: Master's students should consult course instructor before registering for this course.
Prerequisites: advanced calculus, or MATH-GA 1410 Introduction to Math Analysis I. Concurrent registration is not permitted.
Complex numbers, the complex plane. Power series, differentiability of convergent power series. Cauchy-Riemann equations, harmonic functions. conformal mapping, linear fractional transformation. Integration, Cauchy integral theorem, Cauchy integral formula. Morera's theorem. Taylor series, residue calculus. Maximum modulus theorem. Poisson formula. Liouville theorem. Rouche's theorem. Weierstrass and Mittag-Leffler representation theorems. Singularities of analytic functions, poles, branch points, essential singularities, branch points. Analytic continuation, monodromy theorem, Schwarz reflection principle. Compactness of families of uniformly bounded analytic functions. Integral representations of special functions. Distribution of function values of entire functions.
Text: Complex Analysis: An Introduction, Lars V. Ahlfors, McGraw-Hill, 3rd Ed.
MATH-GA 2470.001 ORDINARY DIFFERENTIAL EQUATIONS
3 points. Spring term.
Wednesday, 5:10-7:00, F.
Hang.
Prerequisites: linear algebra, real variables.
Existence and uniqueness of initial value problems. Linear equations. Complex analytic equations. Boundary value problems and Sturm-Liouville theory. Comparison theorem for second order equations. Asymptotic behavior of nonlinear systems. Perturbation theory and Poincaré-Bendixson theorems.
Recommended text: Ordinary Differential Equations and Dynamical Systems, G. Teschl; Theory of ordinary differential equations, Coddington & Levinson
MATH-GA 2490.001, 2500.001 PARTIAL DIFFERENTIAL
EQUATIONS I, II
3 points per term. Fall and spring terms.
Tuesday, 5:10-7:00, R. Kohn
(fall); Tuesday, 9:00-10:50, N. Masmoudi
(spring).
Fall Term
Prerequisites: a good knowledge of undergraduate level linear algebra and ODE.
This course is a basic introduction to PDEs and is designed for students who are interested in applied mathematics or analysis and PDEs. The concentration is on concrete examples of PDEs that arise in various physical systems, and methods of solving these problems will be introduced. The class will cover the following topics: first-order equations, methods of characteristics, conservation laws, shocks, weak solutions, Hamilton-Jacobi theory and caustics; wave equations, the method of spherical means, Duhamel's principle; the heat equation, the fundamental solution, diffusion and Brownian motion; Laplace's equation, maximum principle, fundamental solutions, Dirichlet and Neumann problems, boundary layer potential; Fourier methods and dispersive equations.
Recommended text: Partial Differential Equations, Lawrence
C. Evans; class notes.
Spring Term
Prerequisites: MATH-GA 2490.001 PDE I and MATH-GA 2430.001 Real Variables, or the equivalent.
This course is a continuation of MATH-GA 2490 and is designed for students who are interested in analysis and PDEs. The course gives an introduction to Sobolev spaces, Holder spaces, and the theory of distributions; elliptic equations, harmonic functions, boundary value problems, regularity of solutions, Hilbert space methods, eigenvalue problems; general hyperbolic systems, initial value problems, energy methods; linear evaluation equations, Fourier methods for solving initial value problems, parabolic equations, dispersive equations; nonlinear elliptic problems, variational methods; Navier-Stokes and Euler equations.
Suggested texts: Partial Differential Equations, Paul R. Garabedian; Partial Differential Equations, L. C. Evans; Partial Differential Equations, Fritz John
MATH-GA 2550.001 FUNCTIONAL ANALYSIS
3 points. Spring term.
Monday, 9:00-10:50, S.
Güntürk .
Prerequisite: Linear algebra. Real variables or the equivalent. Some complex function-theory would be helpful.
The course will concentrate on concrete aspects of the subject and on the spaces most commonly used in practice such as Lp(1? p ? ?), C, C?, and their duals. Working knowledge of Lebesgue measure and integral is expected. Special attention to Hilbert space (L2, Hardy spaces, Sobolev spaces, etc.), to the general spectral theorem there, and to its application to ordinary and partial differential equations. Fourier series and integrals in that setting. Compact operators and Fredholm determinants with an application or two. Introduction to measure/volume in infinite-dimensional spaces (Brownian motion). Some indications about non-linear analysis in an infinite-dimensional setting. General theme: How does ordinary linear algebra and calculus extend to d=? dimensions?
Recommended texts: Functional Analysis, P. Lax, (Pure & Applied Mathematics, New York), Wiley-Interscience, John Wiley & Sons, 2002; Methods of Modern Mathematical physics Vol. I: Functional Analysis, M. Reed & B. Simon, Academic Press, New York-London, 1972
MATH-GA 2563.001 HARMONIC ANALYSIS
3 points. Fall term.
Wednesday, 9:00-10:50, F.
Hang.
Prerequisites: real analysis; basic knowledge of complex variables and functional analysis.
Classical Fourier Analysis: Fourier series on the circle, Fourier transform on the Euclidean space, stationary phase. Topics in real variable methods: maximal functions, Hilbert and Riesz transforms and singular integral operators, BMO and Hardy spaces, Carleson measures and boundary behavior of harmonic functions.
References: Singular Integrals and Differentiability Properties of Functions, E.Stein; Introduction to Fourier Analysis on Eucledian Spaces, E.Stein & G.Weiss; Fourier Integrals in Classical Analysis, C.Sogge
MATH-GA 2620.002 ADVANCED TOPICS IN PDE (Mathematical Fluid Mechanics)3 points. Spring term.
Wednesday, 9:00-10:50, P. Germain.
I will focus on incompressible fluid mechanics, the equations of interest being Navier-Stokes (NS: viscous case) and Euler (E: inviscid case). I will present the main known results on these two equations including:
existence of weak solutions of NS (Leray), local well-posedness for E and NS (Fujita-Kato), regularity of weak solutions of NS (Cafarelli-Kohn-Nirenberg), non-uniqueness for E (Shnirelman-De Lellis-Szekelehydi).
This class will present very recent results, and should provide a good introduction to research question of current interest.
Grading: this course will be graded as a seminar course.
MATH-GA 2650.001 ADVANCED TOPICS IN ANALYSIS (Random Matrices)
3 points. Fall term.
Monday, 9:00-10:50, P. Deift.
MATH-GA 2660.001 ADVANCED TOPICS IN ANALYSIS (Calculus of Variations)
Tuesday, 1:25-3:15, R. Kohn.
Prerequisites: Real Variables I and PDE I.
A modern introduction to the calculus of variations, with equal emphasis on theory and applications. Topics will include: existence of solutions and convergence of numerical schemes; convex duality; one-dimensional variational problems; multidimensional nonconvex problems; relaxation; Gamma convergence; homogenization; and pattern formation due to singular perturbation. Along the way, we'll discuss many applications including minimal surfaces, optimal control, nonlinear elasticity, composite materials, and the wrinkling of thin sheets.
Grading: this course will be graded as a seminar course.
NUMERICAL ANALYSIS
MATH-GA 2010.001, 2020.001 NUMERICAL METHODS I, II
3 points per term. Fall and spring terms.
Thursday, 5:10-7:00 O.
Widlund
(fall); Thursday, 7:10-9:00, O. Widlund
(spring).
Fall term
Prerequisites: Calculus, linear algebra, and programming experience in C, Fortran, Matlab, or something similar.
Numerical errors, conditioning, and stability; function
approximation; numerical linear algebra; root-finding and optimization;
numerical integration, differentiation, and interpolation; spectral
methods; Monte Carlo methods
Required text: Numerical Methods, Germund Dahlquist &
Ake Bjorck, Dover Publications, 2003 (reprinted from the 1974 edition);
Optional texts: Numerical Recipes, William Press, Saul Teukolsky, William Vetterling, & Brian Flannery; A Brief Introduction to Numerical Analysis, Eugene Tyrtyshnikov; Matrix Computations, Gene Golub & Charles Van Loan.
Cross-listed as CSCI-GA 2420.001
Spring term
Prerequisite: numerical linear algebra, elements of ODE and PDE.
This course will cover fundamental methods that are essential for numerical solution of differential equations. It is intended for students familiar with ODE and PDE and interested in numerical computing; computer programming assignments form an essential part of the course. The course will introduce students to numerical methods for (1) nonlinear equations, Newton’s method; (2) ordinary differential equations, Runge-Kutta and multistep methods, convergence and stability; (3) finite difference and ;finite element methods; (4) fast solvers, multigrid method; (5) parabolic and hyperbolic partial differential equations.
Text: A First Course in the Numerical Analysis of Differential Equations, A. Iserles, Cambridge University Press, 1st Ed.
Cross-listed as CSCI-GA 2421.001
MATH-GA 2011.001 ADVANCED TOPICS IN NUMERICAL ANALYSIS (Data Science Project)
3 points. Fall term.
Tuesday, 5:10-7:00, L. Greengard.
Cross-listed as CSCI-GA 2945.001
MATH-GA 2011.002 ADVANCED TOPICS IN NUMERICAL ANALYSIS (Coarse-Grained Models of Materials)
3 points. Fall term.
Wednesday, 1:25-3:15, A. Donev.
Course description not yet available.
Cross-listed as CSCI-GA 2945.002
MATH-GA 2011.003 ADVANCED TOPICS IN NUMERICAL ANALYSIS (Approximation Theory and Practice)
3 points. Fall term.
Tuesday, Thursday, 1:25-3:15, N. Trefethen.
Cross-listed as CSCI-GA 2945.003
MATH-GA 2012.001 ADVANCED
TOPICS IN NUMERICAL ANALYSIS (Monte Carlo Methods)
3 points. Spring term.
Tuesday, 5:10-7:00, J. Goodman.
Further information about the course is available at: http://www.math.nyu.edu/faculty/goodman/teaching/MonteCarlo12/ClassHomePage.php.html
Grading: this course will be graded as a regular course.
Crossed-listed as CSCI-GA 2945.001
MATH-GA 2012.002
ADVANCED
TOPICS IN NUMERICAL ANALYSIS (Computational Fluid Dynamics)
3 points. Spring term.
Monday, 1:25-3:15, A. Donev.
This course will cover advanced numerical techniques for solving PDEs, with a particular focus on fluid dynamics. This includes advection-diffusion-reaction equations, compressible and incompressible Navier-Stokes equations, and fluid-structure coupling. Basic familiarity with temporal integrators for ODEs (multistep, Runge-Kutta), methods for solving PDEs (finite difference, finite volume, finite elements for parabolic and elliptic problems), iterative solvers for linear systems, and the Navier-Stokes equations will be assumed. Topics covered will include higher-order spatio-temporal discretizations for advection-diffusion equations, artificial dissipation and dispersion, compressible flow (conservation laws, limiters, shock-capturing methods, boundary layers, turbulence), incompressible flow (projection methods, Stokes solvers, spectral methods), and fluid-structure coupling (boundary-integral formulations, immersed boundary methods). Each student will be required to do a computational project on a subject of choice and present it in class.
Grading: this course will be graded as a regular course.
Cross-listed as CSCI-GA 2945.002
MATH-GA 2041.001 COMPUTING IN FINANCE
3 points. Fall term.
Thursday, 7:10-9:00, L. Maclin & E. Fishler.
This course will introduce students to the software development
process, including applications in financial asset trading, research,
hedging, portfolio management, and risk management. Students will use
the Java
programming language to develop object-oriented software, and will
focus on the most broadly important elements of programming - superior
design, effective problem solving, and the proper use of data
structures and algorithms. Students will work with market and
historical data to run simulations and test strategies. The course is
designed to give students a feel for the practical considerations of
software development and
deployment. Several key technologies and recent innovations in
financial computing will be presented and discussed.
MATH-GA 2043.001 SCIENTIFIC COMPUTING
3 points. Fall term.
Thursday, 5:10-7:00, A.
Rangan.
Prerequisites: undergraduate multivariate calculus and
linear
algebra. Programming experience strongly recommended but not required.
This course is intended to provide a practical introduction to computational problem solving. Topics covered include: the notion of well conditioned and poorly conditioned problems, with examples drawn from linear algebra; the concepts of forward and backward stability of an algorithm, with examples drawn from floating point arithmetic and linear-algebra; basic techniques for the numerical solution of linear and nonlinear equations, and for numerical optimization, with examples taken from linear algebra and linear programming; principles of numerical interpolation, differentiation and integration, with examples such as splines and quadrature schemes; an introduction to numerical methods for solving ordinary differential equations, with examples such as multistep, Runge Kutta and collocation methods, along with a basic introduction of concepts such as convergence and linear stability; An introduction to basic matrix factorizations, such as the SVD; techniques for computing matrix factorizations, with examples such as the QR method for finding eigenvectors; Basic principles of the discrete/fast Fourier transform, with applications to signal processing, data compression and the solution of differential equations.
This is not a programming course but programming in homework projects with Matlab/Octave and/or C is an important part of the course work. As many of the class handouts are in the form of Matlab/Octave scripts, students are strongly encouraged to obtain access to and familiarize themselves with these programming environments.
The course textbook is Numerical Methods:
Design, Analysis, and Computer Implementation of Algorithms by
Anne Greenbaum & Timothy P. Chartier. A reserve copy will be
available on 2h reserve in the Courant Library. Some lectures will be
drawn in part from the draft of an upcoming book Principles of Scientific Computing
by Profs. Jonathan Goodman and David Bindel.
Cross-listed as CSCI-GA 2112.001
MATH-GA 2043.001 SCIENTIFIC COMPUTING
3 points. Spring term.
Thursday, 5:10-7:00, Y. Chen.
Prerequisites: Undergraduate multivariate calculus and linear algebra. Programing experience strongly recommended but not required.
A practical introduction to computational problem solving. Conditioning of problems and stability of algorithms; floating point arithmetic; principles of reliable and robust computational software; scientific visualization; applied approximation theory, including numerical interpolation, differentiation and integration; solution of linear and nonlinear systems of equations and optimization; Eigenvalue problems and SVD decomposition; ordinary differential equations; Fourier transforms; Introduction to Monte Carlo simulation.
This is not a programming course but programming in homework projects with Matlab (Python, Fortran, C/C++, or other language of your choice) is an important part of the course work.
Required text: Scientific Computing with MATLAB and Octave, Alfio M. Quarteroni & Fausto Saleri, Springer, 2006, available electronically through the library.
Optional reading: An Introduction to Programming and Numerical Methods in MATLAB, Stephen R. Otto & James P. Denier, Springer, 2005, available electronically through the library; Scientific Computing with Case Studies , Dianne P. O'Leary, SIAM, 2008
Cross-listed as CSCI-GA 2112.001
MATH-GA 2045.001 COMPUTATIONAL METHODS FOR FINANCE
3 points. Fall term.
Tuesday, 7:10-9:00, A. Hirsa.
Prerequisites: Scientific Computing or Numerical Methods II, Continuous Time Finance, or permission of instructor.
Computational techniques for solving mathematical problems arising in finance. Dynamic programming for decision problems involving Markov chains and stochastic games. Numerical solution of parabolic partial differential equations for option valuation and their relation to tree methods. Stochastic simulation, Monte Carlo, and path generation for stochastic differential equations, including variance reduction techniques, low discrepancy sequences, and sensitivity analysis.
Cross-listed as FINC-GB 7311
MATH-GA 2170.001 INTRODUCTION TO CRYPTOGRAPHY
3 points. Fall term.
Monday, 5:10-7:00, O.Regev.
Course description not yet available.
Cross-listed as CSCI-GA 3210.001
MATH-GA 2701.001 METHODS OF
APPLIED MATHEMATICS
3 points. Fall term.
Tuesday, 1:25-3:15, S. Smith.
Prerequisites: elementary linear algebra and differential equations.
This is a first-year course for all incoming PhD and Master students interested in pursuing research in applied mathematics. It provides a concise and self-contained introduction to advanced mathematical methods, especially in the asymptotic analysis of differential equations. Topics include scaling, perturbation methods, multi-scale asymptotics, transform methods, geometric wave theory, and calculus of variations
Supplementary reading: Scaling, self-similarity, and intermediate asymptotics, G.I. Barenblatt; Perturbation Methods, E.J. Hinch, Advanced Mathematical Methods for Scientists and Engineers, C.M. Bender & S.A. Orszag; Linear and Nonlinear Waves, G.B. Whitham; Calculus of Variations, I.M. Gelfand & S.V. Fomin
MATH-GA 2702.001 FLUID DYNAMICS
3 points. Fall term.
Thursday, 9:00-10:50, D.
Giannakis.
Prerequisites: introductory complex variable and partial differential equations.
The course will expose students to basic fluid dynamics from a mathematical and physical perspectives, covering both compressible and incompressible flows. Topics: conservation of mass, momentum, and Energy. Eulerian and Lagrangian formulations. Basic theory of inviscid incompressible and compressible fluids, including the formation of shock waves. Kinematics and dynamics of vorticity and circulation. Special solutions to the Euler equations: potential flows, rotational flows, irrotationall flows and conformal mapping methods. The Navier-Stokes equations, boundary conditions, boundary layer theory. The Stokes Equations.
Text: An Introduction to Theoretical Fluid Mechanics, S.
Childress, Courant Lecture Notes.
Recommended text: Elementary Fluid Dynamics, D. Acheson,
Oxford Applied Mathematics & Computing Science Series,
MATH-GA 2704.001 APPLIED STOCHASTIC ANALYSIS
3 points. Spring term.
Thursday, 1:25-3:15, E. Vanden
Eijnden.
The class will provide an introduction to probability and stochastic processes theory from an applied perspective. Topics will include definition of random variables, limit theorems, Markov chain and Markov processes, Wiener processes, stochastic differential equations, Fokker-Planck equations, large deviations and rare events, Monte Carlo methods and introduction to statistical mechanics.
MATH-GA 2706.001 PARTIAL DIFFERENTIAL EQUATIONS FOR FINANCE
3 points. Spring term.
Monday, 5:10-7:00, O. Bühler.
Prerequisite: Stochastic Calculus or equivalent.
An introduction to those aspects of partial differential equations and optimal control most relevant to finance. Linear parabolic PDE and their relations with stochastic differential equations: the forward and backward Kolmogorov equation, exit times, fundamental solutions, boundary value problems, maximum principle. Deterministic and stochastic optimal control: dynamic programming, Hamilton-Jacobi-Bellman equation, verification arguments, optimal stopping. Applications to finance, including portfolio optimization and option pricing -- are distributed throughout the course.
MATH-GA 2707.001 TIME SERIES ANALYSIS AND STATISTICAL ARBITRAGE
3 points. Fall term.
Monday, 7:10-9:00, F. Asl & R. Reider.
Prerequisites: Derivative Securities, Scientific Computing, Computing for Finance, and Stochastic Calculus.
The term "statistical arbitrage" covers any trading strategy that uses statistical tools and time series analysis to identify approximate arbitrage opportunities while evaluating the risks inherent in the trades (considering the transaction costs and other practical aspects). This course starts with a review of Time Series models and addresses econometric aspects of financial markets such as volatility and correlation models. We will review several stochastic volatility models and their estimation and calibration techniques as well as their applications in volatility based trading strategies. We will then focus on statistical arbitrage trading strategies based on cointegration, and review pairs trading strategies. We will present several key concepts of market microstructure, including models of market impact, which will be discussed in the context of developing strategies for optimal execution. We will also present practical constraints in trading strategies and further practical issues in simulation techniques. Finally, we will review several algorithmic trading strategies frequently used by practitioners.
MATH-GA 2708.001 ALGORITHMIC TRADING AND QUANTITATIVE STRATEGIES
3 points. Spring term.
Tuesday, 7:10-9:00, P. Kolm & L. Maclin.
Prerequisites: Computing in Finance, and Risk Portfolio Management with Econometrics, or equivalent.
In this course we develop a quantitative investment and trading framework. In the first part of the course, we study the mechanics of trading in the financial markets, some typical trading strategies, and how to work with and model high frequency data. Then we turn to transaction costs and market impact models, portfolio construction and robust optimization, and optimal betting and execution strategies. In the last part of the course, we focus on simulation techniques, back-testing strategies, and performance measurement. We use advanced econometric tools and model risk mitigation techniques throughout the course. Handouts and/or references will be provided on each topic.
MATH-GA 2710.001 MECHANICS
3 points. Spring term.
Thursday, 1:25-3:15, B.
Wirth.
This course provides a basic mathematical introduction to solid
continuum mechanics, classical Newtonian mechanics, and statistical
mechanics. Prior knowledge in physics is not required. Key topics
include: nonlinear elasticity; linear elasticity; dimension reduction
(plate theory); calculus of variations methods; Hamilton's equations;
action minimization; Liouville's theorem; microcanonical and canonical
ensemble; entropy.
A first impression can be obtained from the following texts: Mathematical Elasticity I:
Three-dimensional Elasticity, P.G. Ciarlet, North-Holland
(1988); A Brief Introduction to
Classical, Statistical, and Quantum Mechanics, O. Bühler,
Courant Lectures Notes in Mathematics vol. 13 (2006).
MATH-GA 2751.001 RISK
AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS
3 points. Fall term.
Tuesday, 7:10-9:00, P. Kolm.
Prerequisites: univariate statistics, multivariate calculus, linear algebra, and basic computing (e.g. familiarity with Matlab or co-registration in Computing in Finance).
A comprehensive introduction to the theory and practice of portfolio management, the central component of which is risk management. Econometric techniques are surveyed and applied to these disciplines. Topics covered include: factor and principal-component models, CAPM, dynamic asset pricing models, Black-Litterman, forecasting techniques and pitfalls, volatility modeling, regime-switching models, and many facets of risk management, both theory and practice.
MATH-GA 2751.001 RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS
3 points. Spring term.
Wednesdays, 7:10-9:00, M. Avellaneda.
Details to be determined (generally consistent with the fall term description).
MATH-GA 2752.001 ACTIVE PORTFOLIO MANAGEMENT
3 points. Spring term.
Monday, 5:10-7:00, R. Lindsey.
Prerequisites: Risk & Portfolio Management with Econometrics, Computing in Finance.
The first part of the course will cover the theoretical aspects of portfolio construction and optimization. The focus will b e on advanced techniques in portfolio construction, addressing the extensions to traditional mean-variance optimization including robust optimization, dynamical programming and Bayesian choice. The second part of the course will focus on the econometric issues associated with portfolio optimization. Issues such as estimation of returns, covariance structure, predictability, and the necessary econometric techniques to succeed in portfolio management will be covered. Readings will be drawn from the literature and extensive class notes.
MATH-GA 2753.001 ADVANCED RISK MANAGEMENT
3 points. Spring term.
Monday, 7:10-9:00, K. Abbott.
Prerequisites: Derivative Securities, Computing in Finance or equivalent programming.
The importance of financial risk management has been increasingly recognized over the last several years. This course gives a broad overview of the field, from the perspective of both a risk management department and of a trading desk manager, with an emphasis on the role of financial mathematics and modeling in quantifying risk. The course will discuss how key players such as regulators, risk managers, and senior managers interact with trading. Specific techniques for measuring and managing the risk of trading and investment positions will be discussed for positions in equities, credit, interest rates, foreign exchange, commodities, vanilla options, and exotic options. Students will be trained in developing risk sensitivity reports and using them to explain income, design static and dynamic hedges, and measure value-at-risk and stress tests. Students will create Monte Carlo simulations to determine hedge effectiveness. Extensive use will be made of examples drawn from real trading experience, with a particular emphasis on lessons to be learned from trading disasters.
Text: Financial Risk Management, S. Allen, John Wiley & Sons, 2003
MATH-GA 2755.001 PROJECT AND PRESENTATION (MATH FINANCE)
3 points. Fall term.
Monday, 5:10-7:00, P. Kolm.
Students in the Mathematics in Finance program conduct research
projects individually or in small groups under the supervision of
finance professionals. The course culminates in oral and written
presentations of the research results.
MATH-GA 2755.001 PROJECT AND PRESENTATION (MATH FINANCE)
3 points. Spring term.
Wednesday, 5:10-7:00, P.
Kolm.
See course description above.
MATH-GA 2791.001
DERIVATIVE SECURITIES
3 points. Fall term.
Wednesday, 7:10-9:00, M. Avellaneda.
An introduction to arbitrage-based pricing of derivative securities. Topics include: arbitrage; risk-neutral valuation; the log-normal hypothesis; binomial trees; the Black-Scholes formula and applications; the Black-Scholes partial differential equation; American options; one-factor interest rate models; swaps, caps, floors, swaptions, and other interest-based derivatives; credit risk and credit derivatives.
Cross-listed as FINC-GB 7312
MATH-GA 2791.001 DERIVATIVE SECURITIES
3 points. Spring term.
Monday, 7:10-9:00, B. Flesaker.
See course description above.
MATH-GA 2792.001
CONTINUOUS TIME FINANCE
3 points. Fall
term.
Monday, 7:10-9:00, P. Carr & A. Javaheri.
Prerequisites: Derivative Securities and Stochastic Calculus, or equivalent.
A second course in arbitrage-based pricing of derivative securities. The Black-Scholes model and its generalizations: equivalent martingale measures; the martingale representation theorem; the market price of risk; applications including change of numeraire and the analysis of quantos. Interest rate models: the Heath-Jarrow-Morton approach and its relation to short-rate models; applications including mortgage- backed securities. The volatility smile/skew and approaches to accounting for it: underlyings with jumps, local volatility models, and stochastic volatility models.
Cross-listed as FINC-GB 7310.010
MATH-GA 2792.001 CONTINUOUS TIME FINANCE
3 points. Spring
term.
Wednesday, 7:10-9:00, B. Dupire & F. Mercurio.
Prerequisites: Derivative Securities and Stochastic Calculus, or equivalent
A second course in arbitrage-based pricing of derivative securities. Concerning equity and FX models: we'll discuss numerous approaches that are used in practice in these markets, such as the local volatility model, Heston, SABR, and stochastic local volatility. The discussion will include calibration and hedging issues and the pricing of the most common structured products. Concerning interest rate models: we'll start with a thorough discussion of one-factor short-rate models (Vasicek, CIR, Hull-White) then proceed to more advanced topics such as two-factor Hull-White, forward rate models (HJM) and the LIBOR market model. Throughout, the pricing of specific payoffs will be considered and practical examples and insights will be provided. We'll conclude with a discussion of inflation models.
MATH-GA 2796.001 MORTGAGE-BACKED SECURITIES AND ENERGY DERIVATIVES
3 points. Spring term.
Tuesday, 7:10-9:00, G. Swindle & L. Tatevossian.
Prerequisites: basic bond mathematics and bond risk measures (duration and convexity); Derivative Securities, Stochastic Calculus.
The first part of the course will cover the fundamentals and
building blocks of understanding how mortgage-backed securities are
priced and analyzed. The focus will be on prepayment and interest rate
risks, benefits and risks associated with mortgage-backed structured
bonds and mortgage derivatives. Credit risks of various types of
mortgages will also be discussed. The second part of the course will
focus on energy commodities and derivatives, from their basic
fundamentals and valuation, to practical issues in managing structured
energy portfolios. We develop a risk neutral valuation framework
starting from basic GBM and extend this to more sophisticated
multifactor models. These approaches are then used for the valuation of
common, yet challenging, structures. Particular emphasis is placed on
the potential pitfalls of modeling methods and the practical aspects of
implementation in production trading platforms. We survey market
mechanics and valuation of inventory options and delivery risk in the
emissions markets.
Suggested texts: Salomon Smith Barney Guide to
Mortgage-Backed and Asset-Backed Securities, Lakhbir Hayre; The
Handbook
of
Mortgage-Backed
Securities, Frank Fabozzi; Energy
and Power Risk Management, Eydeland & Wolyniec; Electricity
Markets:
Pricing,
Structures
and
Economics, Chris Harris
MATH-GA 2797.001 CREDIT
MARKETS AND MODELS
3 points. Fall term.
Wednesday, 7:10-9:00, V. Finkelstein.
Prerequisites: Computing in Finance (or equivalent), Derivative Securities (or equivalent), familiarity with analytical methods applied to interest rate derivatives.
This course addresses a number of practical issues concerned with modeling, pricing and risk management of a range of fixed-income securities and structured products exposed to default risk. Emphasis is on developing intuition and practical skills in analyzing pricing and hedging problems. In particular, significant attention is devoted to credit derivatives.
We begin with discussing default mechanism and its mathematical representation. Then we proceed to building risky discount curves from market prices and applying this analytics to pricing corporate bonds, asset swaps, and credit default swaps. Risk management of credit books will be addressed as well. We will next examine pricing and hedging of options on assets exposed to default risk.
After that, we will discuss structural (Merton-style) models that connect corporate debt and equity through the firm’s total asset value. Applications of this approach include the estimation of default probability and credit spread from equity prices and effective hedging of credit curve exposures.
A final segment of the course will focus on credit structured products. We start with cross-currency swaps with a credit overlay. We will next analyze models for pricing portfolio transactions using Merton-style approach. We also will discuss portfolio loss model based on a transition matrix approach. These models will then be applied to the pricing of collateralized debt obligation tranches and pricing counterparty credit risk taking wrong-way exposure into account.
Suggested texts: Modeling Single-Name and Multi-Name Derivatives, Dominic O’Kane, 2008; Options, Futures, & Other Derivative Securities, John Hull, 7th Edition.
MATH-GA 2798.001 INTEREST RATE AND FX MODELS
3 points. Spring term.
Wednesday, 5:10-7:00, L. Andersen & A. Lesniewski.
Prerequisites: Derivative Securities, Stochastic Calculus, and Computing in Finance (or equivalent familiarity with financial models, stochastic methods, and computing skills).
The course is divided into two parts. The first addresses the fixed-income models most frequently used in the finance industry, and their applications to the pricing and hedging of interest-based derivatives. The second part covers the foreign exchange derivatives markets, with a focus on vanilla options and first-generation (flow) exotics. Throughout both parts, the emphasis is on practical aspects of modeling, and the significance of the models for the valuation and risk management of widely-used derivative instruments.
MATH-GA 2830.001 ADVANCED TOPICS IN APPLIED MATH (Multi-Scale Modelling in Climate Atmosphere Ocean Science)
3 points. Fall term.
Thursday, 3:15-5:00,
A.Majda.
Topics include: This semester’s grad course will have a seminar-style format revolving around cutting edge topics in multiscale modeling in climate science asymptotic modeling, superparameterization, and dynamic stochastic super-resolution which are exciting current research topics. Both complex models and applied math toy models will be used and emphasized throughout the course. CIMS graduate students at any level are welcomed to attend and contribute; to receive graduate credit for the course, a student needs to participate in at least one lecture.
Text References: Introduction to P.D.E.’s and Waves for the Atmosphere and Ocean: Courant Lecture Notes, A. Majda,Vol. 9, American Mathematical Society & Courant Institute of Mathematical Sciences, 2002; K-M-S ↔ Survey paper Climate Science in the Tropics: Waves, Vortices, and PDE, B. Khouider, A. Majda, S. Stechmann, submitted, Nonlinearity, March 17, 2012 (invited review paper) ; Challenges in Climate Science and Contemporary Applied Mathematics, Invited Paper for the Special Volume of Communications on Pure and Applied Mathematics for 75th Anniversary of the Courant Institute, A. Majda,CPAM, LXV 0920-0948 (2012)MATH-GA
2830.002
ADVANCED TOPICS IN APPLIED
MATH (Optimization and Data Analysis)
3 points. Fall term.
Tuesday, 9:00-10:50,
E. Tabak.
MATH-GA 2840.001 ADVANCED TOPICS IN APPLIED MATH (Plasma Physics: Theory and Computation)
3 points. Spring term.
Thursday, 9:00-10:50, A. Cerfon.
Prerequisites: elementary notions of classical mechanics and electromagnetics, working knowledge of undergraduate multivariable calculus and ODEs, introductory notions of PDE and complex variables
This seminar course offers a general overview of the mathematical
models and numerical methods for plasma physics, the field of physics
which describes large ensembles of particles interacting through
electromagnetic fields. Applications cover a wide range of disciplines:
astrophysics, planetary science, magnetic and inertial fusion, particle
accelerators, etc. In the first part of the course, we will derive and
study fluid models, including the very well-known MagnetoHydroDynamics
(MHD) model. We will look at the behavior of MHD plasmas, with a
particular focus on MHD waves and instabilities, MHD dynamos, and
magnetic reconnection.
In the second part of the course, we will investigate kinetic models,
which are crucial for the description of plasmas that are not in local
thermal equilibrium. With such models, we will be able to look into
purely kinetic effects such as wave particle interactions (including
Landau damping) and phase space instabilities.
Recommended introductory texts: Introduction to plasma physics,
R.J. Goldston & P.H. Rutherford, Taylor & Francis (1995); Plasma
Physics
and
Fusion
Energy, J.P. Freidberg, Cambridge University
Press (2007).
Recommended More advanced texts: The Framework of Plasma Physics,
R.D. Hazeltine & F.L Waelbroeck, Westview Press (Frontiers in
Physics) (2004); Plasma Confinement, R.D. Hazeltine & J.D.
Meiss, Dover Publications (2003); Principles of
Magnetohydrodynamics; with Applications to Laboratory and Astrophysical
Plasmas, J.P. Goedbloed & S. Poedts, Cambridge University Press
(2004).
Recommended general text on kinetic theory: Mathematical Methods in
Kinetic Theory, C.Cercignani, Springer (1990).
Grading: This course will be graded as a seminar course.
MATH-GA 2840.002
ADVANCED TOPICS IN APPLIED MATH (Data Analysis Methods for
High-dimensional Time Series)
3 points. Spring term.
Thursday, 3:20-5:00, D.
Giannakis.
Grading: this course will be graded based on attendance and a student presentation.
MATH-GA 2851.001 ADVANCED TOPICS IN MATH BIOLOGY (Topic TBA)
3 points. Fall term.
Wednesday, 1:25-3:15, D.
Cai.
Course description not yet available.
Cross-listed as BIOL-GA 2851.001.
MATH-GA 2852.001 ADVANCED TOPICS IN MATH BIOLOGY (Stochastic Problems in Cellular, Molecular and Neural Science)
3 points. Spring term.
Wednesday, 1:25-3:15, D.
Tranchina.
Prerequisites:
elementary background in ODEs, PDEs, probability theory, Fourier
transforms.
A variety of topics of current interest in biology and neural science
will be addressed. Topics include: (1) Stochastic gene expression:
analytical modeling of stochastic messenger RNA synthesis and
degradation; discrete and continuous models; master equation;
generating function; steady-state distributions; temporal evolution of
the distributions; stochastic protein product. (2) Stochastic cell
divisions and population growth:
mean growth rate; age distributions. (3) Single-photon responses of
retinal rods; statistical measures of variability; reproducibility of
the single-photon response; explicit biochemical kinetic models; model
testing with Monte Carlo simulations. (4) Optimal filtering of
photon noise in vision. (5) Stochastic behavior of neurons in the
central nervous system: models for synaptic noise; spike train
statistics and renewal theory. (6) Probability density methods for
large-scale modeling of neural networks: partial differential-integral
equations; Fokker-Plank approximation; applications to modeling
visual cortex.
Lecture notes will be distributed, and assigned readings will be
recommended.
Grading: this course will be graded as a seminar course.
Cross-listed as BIOL-GA 2852.001.
MATH-GA
2852.002 ADVANCED TOPICS IN MATH BIOLOGY (Biophysical Modeling of
Cells
and Populations)
3 points. Spring term.
Thursday, 9:30-12:15, E. Kussell.
Prerequisites: This course is geared towards a highly diverse group of
students, including biologists, mathematicians, and physicists. For
this reason, there are no hard prerequisites. Comfort with mathematics
is the only requirement (e.g. familiarity with differential equations).
Previous exposure to programming
or computational software such as Matlab or Mathematica will be useful
for the projects, though the ambitious student could learn these along
the way.
The course covers modeling of biological systems at multiple levels. In
the first part, we begin with some basic molecular biology, including
cooperative binding and simple induction of genes. We develop a general
approach to quantitative modeling of transcriptional regulation. We
apply this to study small genetic circuits with feedback loops. We
study pattern formation, and the mechanisms by which cells perceive
spatial information from chemical signals. We investigate how
biological systems can function robustly in the face of noise. In the
second part, we study cellular behaviors within heterogeneous
populations. We introduce population models, and relate these to the
molecular/cellular models of the first part. Diverse biological
examples (see Syllabus & Reading List) will be presented over the
course of the semester to illustrate key concepts in modeling.
Recommended texts: Physical Biology of the Cell, R. Phillips, J.
Kondev, J. Theriot, Garland Science, 2008;
Random Walks in Biology, Howard C. Berg, Princeton University Press,
1993; A Genetic Switch: Phage Lambda Revisited, M. Ptashne, 3rd
edition, Cold Spring Harbor Laboratory Press, Cold Spring Harbor, New
York, 2004.
Grading: The course will be graded as a regular
course.
Cross-listed as BIOL-GA 1131.001
MATH-GA 2855.001 ADVANCED TOPICS IN MATH PHYSIOLOGY (Physiological Control Mechanisms)
3 points. Fall term.
Monday, 1:25-3:15, C. Peskin.
Cross-listed as BIOL-GA 2855.001.
MATH-GA 2856.001 ADVANCED TOPICS IN MATH PHYSIOLOGY (Nonlinear ynamics of Neuronal Systems)
3 points. Spring term.
Friday, 3:30-5:20, J. Rinzel.
This is an upper-level undergraduate/graduate course in computational neuroscience, to be taught by Professors John Rinzel (NYU) and Horacio Rotstein (NJIT). We will develop and simulate mathematical (differential equation) models to understand the dynamical properties of neurons, synapses, and networks/systems. The foundations of dynamical systems theory and neurophysiology will be covered as needed. We will study neuro-mechanistic models for spiking, synaptic integration, coupling and coordination in networks, and mean-field firing-rate activity.
Case studies will include: network rhythms, sensory processing and perceptual/cognitive dynamics such as decision-making, perceptual grouping and competition. Simulations including visualization and animation, will be run using Matlab, The course will involve classroom lectures and interactive computing lab sessions, homework and a simulation project.
Text: Spikes, Decisions and Actions: The Dynamical Foundations of Neuroscience, H. R. Wilson, Oxford Univ, Press, 1999, available on line at no cost at: http://cvr.yorku.ca/webpages/wilson.htm#book .
This course will be graded as a regular course.
Cross-listed as BIOL-GA 2855.001, NEUR-GA 3042.010 and MATH-UA 0395.001
MATH-GA 2861.001 ADVANCED TOPICS IN FLUID DYNAMICS (Wave and Mean Flows)
3 points. Fall term.Monday, 1:25-3:15, O. Bühler.
Course description not yet available.
MATH-GA 2861.002 ADVANCED TOPICS IN FLUID DYNAMICS (Complex Fluids)
3 points. Fall term.Wednesday, 7:10-9:00, M. Shelley.
Course description not yet available.
MATH-GA 3001.001 GEOPHYSICAL FLUID DYNAMICS
3 points. Fall term.
Tuesday, 9:00-10:50, O. Pauluis.
This course serves as an introduction to the fundamentals of geophysical fluid dynamics. No prior knowledge of fluid dynamics will be assumed, but the course will move quickly into the subtopic of rapidly rotating, stratified flows. Topics to be covered include (but are not limited to): the advective derivative, momentum conservation and continuity, the rotating Navier-Stokes equations and non-dimensional parameters, equations of state and thermodynamics of Newtonian fluids, atmospheric and oceanic basic states, the fundamental balances (thermal wind, geostrophic and hydrostatic), the rotating shallow water model, vorticity and potential vorticity, inertia-gravity waves, geostrophic adjustment, the quasi-geostrophic approximation and other small-Rossby number limits, Rossby waves, baroclinic and barotropic instabilities, Rayleigh and Charney-Stern theorems, geostrophic turbulence. Students will be assigned bi-weekly homework assignments and some computer exercises, and will be expected to complete a final project. This course will be supplemented with out-of-class instruction.
Recommended texts: Atmospheric and Oceanic Fluid Dynamics, G. Vallis, Cambridge 2006; Lectures on Geophysical Fluid Dynamics, R. Salmon, Oxford 1998; Geophysical Fluid Dynamics, J. Pedlosky, Springer-Verlag 1987
MATH-GA 3003.001 OCEAN DYNAMICS
3 points. Spring term.
Tuesday, 1:25-3:15, R.
Kleeman.
Prerequisite: Geophysical Fluid Dynamics or instructor’s permission.
This lecture course offers a general overview of the physical processes that determine the state of the Earth atmosphere. The focus here is to describe the main features of the planetary circulation, and to explain how they arise as a dynamical response of the atmosphere to different external forcings such as solar radiation or topography. Students should have some knowledge in geophysical fluid dynamics before taking this course. Topics to be covered include: solar forcing, the mean-state of the atmosphere, Hadley and monsoonal circulations, dynamics of the midlatitudes stormtracks, energetics, zonally asymmetric circulations, equatorial dynamics, and the interaction between moist convection and large-scale flow. Students will be assigned bi-weekly homework assignments and some computer exercises, and will be expected to complete a final project or exam, as per instructor's decision. This course will be supplemented with out-of-class instruction.
Required text: Atmospheric and Oceanic Fluid Dynamics , G. Vallis, Cambridge 2006
Recommended Texts: An Introduction to Dynamic Meteorology, J. R. Holton, Academic Press; Atmospheric Dynamics, C. F Bohren, & B.A. Albrecht, Oxford University Press; additional material in the form of published articles will be provided to the students to complement the textbook.
MATH-GA 3010.001 ADVANCED TOPICS IN ATMOSPHERE-OCEAN SCIENCE (Environmental Fluid Dynamics Lab)
3 points. Fall term.
Thursday, 1:25-3:15, D. Holland.
Registration subject to approval by course instructor.
Prerequisites: undergraduate calculus, physics.This course introduces topics in environmental fluid dynamics from an observational of view. Laboratory instrumentation centers on a turntable equipped with PIV and LIF equipment, and a rooftop environmental monitoring station equipped with various meteorological sensors. Students are also introduced to field equipment for oceanographic (CTD, ADCP) and glaciological (GPS, seismic) deployments. Each student will be assigned a term project on a particular piece of equipment and will integrate geophysical fluid dynamics theory with a direct measurement from an instrument.
Grading: This course will be graded as a seminar course.
MATH-GA 3011.001 ADVANCED TOPICS IN ATMOSPHERE-OCEAN SCIENCE (Climate Dynamics)
3 points. Spring term.
Monday, 1:25-3:15, O. Pauluis.
This course will focus on how the various components of the climate systems interact with each other and on how such interactions affect the evolution of the Earth climate. Among the process we will discuss are: radiation, the hydrological cycle, the planetary boundary layer, and the atmospheric and oceanic circulations. We will look at the role played by these different processes play in the current climate, and investigate their behavior in the context of specific climate phenomena. As part of the course, we will use a state-of-the-art model to perform climate simulations and analyze its underlying dynamics.
Required text: Physics of the atmosphere and climate, M.L Salby.
Grading: this course will be graded as a seminar course requiring a presentation.
PROBABILITY
AND
STATISTICS
MATH-GA 2901.001 BASIC PROBABILITY
3 points. Fall term.
Wednesday, 5:10-7:00, M. Tao.
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: Probability and Random Processes, G. Grimmett & D. Stirzaker, 3rd Ed. (on reserve); Introduction to probability, Charles M. Grinstead & J. Laurie Snell, 2nd Ed. (free online)
MATH-GA 2901.001 BASIC PROBABILITY
3 points. Spring term.
Wednesday, 7:10-9:00, M.
Tao.
MATH-GA 2902.001 STOCHASTIC CALCULUS
3 points. Fall term.
Monday, 7:10-9:00, J. Goodman.
Prerequisite: MATH-GA 2901 Basic Probability or equivalent.
Review of basic probability and useful tools. Bernoulli trials and random walk. Law of large numbers and central limit theorem. Conditional expectation and martingales. Brownian motion and its simplest properties. Diffusion in general: forward and backward Kolmogorov equations, stochastic differential equations and the Ito calculus. Feynman-Kac and Cameron-Martin Formulas. Applications as time permits.
Problem session: Thursday, 5:30-6:30 (optional).
MATH-GA 2902.001 STOCHASTIC CALCULUS
3 points. Spring term.
Thursday, 7:10-9:00, A.
Kuptsov.
Prerequisite: MATH-GA 2901 Basic Probability or equivalent.
Review of basic probability and useful tools. Bernoulli trials and random walk. Law of large numbers and central limit theorem. Conditional expectation and martingales. Brownian motion and its simplest properties. Diffusion in general: forward and backward Kolmogorov equations, stochastic differential equations and the Ito calculus. Feynman-Kac and Cameron-Martin Formulas. Applications as time permits.
Text: Stochastic Calculus, A Practical Introduction, Richard Durrett, CRC Press, Probability & Stochastics Series
Problem session: Monday, 5:30-6:30 (optional).
MATH-GA 2911.001, 2912.001 PROBABILITY: LIMIT THEOREMS I, II
3 points per term. Fall and spring terms.
Wednesday, 1:25-3:15, H.
McKean (fall); Monday, 9:30-11:20, H.
McKean (spring).
Prerequisites: a first course in probability, familiarity with Lebesgue integral, or MATH-GA 2430 Real Variables as mandatory corequisite.
Fall term
Probability, independence, laws of large numbers, limit theorems including the central limit theorem. Markov chains (discrete time). Martingales, Doob inequality, and martingale convergence theorems. Ergodic theorem.
Spring term
Independent increment processes, including Poisson processes and
Brownian motion. Markov chains (continuous time).
Stochastic differential equations and diffusions, Markov processes,
semigroups,
generators and connection with partial differential equations.
Recommended text: Stochastic Processes, S. R. S. Varadhan, CIMS - AMS, 2007
MATH-GA 2931.001 ADVANCED TOPICS IN PROBABILITY (Ergodic Theory)
3 points. Fall term.
Monday, 1:25-3:15, T. Austin.
MATH-GA 2931.002 ADVANCED TOPICS IN PROBABILITY (Gaussian Fields & Extrema of the Gaussian Free Field)
3 points. Fall term.
Tuesday, 5:10-7:00, O. Zeitouni.
MATH-GA 2962.001 MATHEMATICAL STATISTICS
3 points. Spring term.
Wednesday, 5:10-7:00, M. Tygert.
Prerequisite: a working knowledge of probability at the undergraduate or introductory graduate level.
descriptive statistics, the binomial, Poisson, exponential, normal, chi-square, t, and F distributions, hypothesis testing, confidence intervals, point estimation (maximum-likelihood methods, consistency, efficiency, sufficiency, etc.), regression, correlation, analysis of variance, Bayesian inference, nonparametric methods, sequential tests, loss, risk, and decision theory.
Required text: Principles of Statistics, M.G. Bulmer, Dover Publications, 1979
Revised May 2013