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All mathematics courses carry 3 points per term (except
Master’s Thesis Research [G63.3881], which carries 2 points, and Independent
Study courses, which range from 1 to 3 points). A majority of courses,
including essentially all those taken by part-time students, meet once a week
for a two-hour period beginning at 5:10 p.m. or at 7:10 p.m. A number of
courses are offered earlier in the day.
The course
listings below are representative of the mathematics program as a whole but do
not refer specifically to this academic year. Not every course is given every
year. Information on current offerings and course descriptions are available in
the office of the department and on the Web at www.math.nyu.edu/courses.
ALGEBRA AND NUMBER THEORY
Linear Algebra
G63.2110, 2120
Linear spaces and mappings. Matrices and linear equations.
Eigenvalues and eigenvectors. Jordan form. Special classes of matrices,
spectral theory.
Linear Algebra
G63.2111 Prerequisite: undergraduate linear algebra. This one-term format course
is intended primarily for doctoral students.
Linear operators. Spectral theory. Duality theorems.
Euclidean and symplectic structure. Matrix valued functions. Matrix
inequalities. Convexity.
Algebra
G63.2130, 2140 Prerequisite: elements of linear algebra.
Basic concepts including groups,
rings, modules, polynomial rings, field theory, and Galois
theory.
Advanced Topics in Algebra
G63.2160
Recent topics: algebraic curves and Abelian varieties; Lie
algebras and Lie groups; representation of finite groups and Lie groups;
orthogonal polynomials.
Number Theory
G63.2210
Introduction to the elementary methods of number theory.
Topics: arithmetic functions, congruences, the prime number theorem, primes in
arithmetic progression, quadratic reciprocity, the arithmetic of quadratic
fields.
Advanced Topics in Number Theory
G63.2250, 2260
Recent topics: ergodic theory and number theory; analytic
theory of automorphic forms; computational number theory and algebra.
GEOMETRY AND TOPOLOGY
Topology
G63.2310, 2320 Prerequisites: elements of point-set topology and algebra.
Survey of point-set topology. Funda-mental groups, homotopy,
covering spaces. Singular homology, calculation of homology groups,
applications. Homology and cohomology of manifolds. Poincaré duality. Vector
bundles. De Rham cohomology and differential forms.
Advanced Topics in Topology
G63.2333, 2334
Recent topics: toric varieties and their applications;
characteristic classes of invariants of manifolds; vector bundles and singular
varieties.
Differential Geometry
G63.2350, 2360
Theory of curves and surfaces. Riemannian geometry:
manifolds, differential forms, and integration. Covariant derivatives and
curvature. Differential geometry in the large. Curvature, geodesics, Jacobi
fields, comparison theorems, and Gauss-Bonnet theorem.
Advanced Topics in Geometry
G63.2400, 2410
Recent topics: geometry of physics; local index theory;
computational topology and geometry; analysis on metric measure spaces.
ANALYSIS
Multivariable Calculus
G63.1002 Intended for master’s students. Does not carry credit toward the Ph.D. degree.
Calculus of several variables: partial differentiation,
vector calculus, Stokes’ theorem, divergence theorem, infinite series, Taylor’s
theorem.
Introduction to Mathematical Analysis
G63.1410, 1420
Rigorous treatment of limits and continuity. Riemann
integral. Taylor series. Absolute and uniform convergence. Elements of ordinary
and partial differential equations. Functions of several variables and their
derivatives. The implicit function theorem, optimization, and Lagrange
multipliers. Theorems of Gauss, Stokes, and Green. Fourier series and
integrals.
Real Variables
G63.2430
Basics of the theory of measure and integration, elements of
Banach spaces. Metric spaces, Ascoli-Arzela theorem, Radon-Nikodym theorem,
Fourier transform, distributions. Sobolev spaces and imbedding theorems.
Geometric measure theory, harmonic analysis, functional analysis. Measure
theory and convergence theorems.
Complex Variables
G63.2450, 2460
Analytic functions. Cauchy’s theorem and its many
consequences. Fractional linear transformations and conformal mappings.
Introduction to Riemann surfaces. The Riemann mapping theorems. Entire
functions. Special functions.
Complex Variables
G63.2451 Prerequisite: advanced calculus or G63.1410. This one-term format course is intended primarily for doctoral students.
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.
Ordinary Differential Equations
G63.2470 Prerequisites: linear algebra and elements of complex variables.
Existence, uniqueness, and continuous dependence. Linear
ODE. Stability of equilibria. Floquet theory. Poincaré-Bendixson theorem.
Additional topics may include bifurcation theory, Hamiltonian mechanics, and
singular ODE in the complex plane.
Partial Differential Equations
G63.2490 Prerequisites: linear algebra, complex variables, and elements of ordinary differential equations.
First-order equations. Cauchy-Kowalewsky theorem. Constant-
coefficient, second-order equations: Laplace’s, heat, and
wave equations. Explicit representation formulas and qualitative methods, such
as the maximum principle. Nonlinear equations, e.g., Burger’s and minimal
surface equations.
Functional Analysis
G63.2550 Prerequisites: linear algebra, complex variables, and real variables.
Banach spaces. Functionals and operators. Principle of
uniform boundedness and closed graph theorem. Completely continuous mappings.
Invariant subspaces. Linear operators, spectral theorem for self-adjoint
operators. Hilbert- Schmidt operators. Semigroups. Fixed-point theorem.
Applications.
Advanced Topics in Functional Analysis
G63.2561, 2562
Recent topic: spectral theory.
Harmonic Analysis
G63.2563 Prerequisites: linear algebra, complex variables, and real variables.
Hardy-Littlewood maximal functions and Marcinkiewicz
integrals, singular integrals. Fourier series and Fourier integrals.
Interpolation theorems. Applications in partial differential equations.
Advanced Topics in Partial Differential Equations
G63.2610,
2620
Recent topics: semiclassical pseudodifferential operators
and applications; free boundary value problems; harmonic maps and their heat
flow; Fourier analysis and incompressible Navier-Stokes equations.
Advanced Topics in Ordinary Differential Equations
G63.2615, 2616
Recent topics: Hamiltonian mechanics; bifurcation theory;
nonlinear dynamics and chaos.
Advanced Topics in Analysis
G63.2650, 2660
Recent topics: coding, quantization, and compression;
dynamical systems; wavelets and time-frequency analysis; random matrices.
NUMERICAL ANALYSIS
Numerical Methods
G63.2010, 2020 Identical to G22.2420, 2421. Corequisite: linear algebra.
Numerical linear algebra. Approxima-tion theory. Quadrature
rules and numerical integration. Nonlinear equations and optimization. Ordinary
differential equations. Elliptic equations. Iterative methods for large, sparse
systems. Parabolic and hyperbolic equations.
Advanced Topics in Numerical Analysis
G63.2011, 2012
Recent topics: convex and nonsmooth optimization;
computational techniques for problems with evolving interfaces; numerical
methods for time-dependent partial differential equations.
Advanced Numerical Analysis: Computational Fluid Dynamics
G63.2030 Identical to G22.2945. Prerequisites: familiarity with numerical methods and linear algebra.
Problems from applications such as gas dynamics, combustion,
and oil reservoir simulation. Flows with shocks and discontinuities. Adaptive
methods. Issues of algorithm design and computer implementation. Parallel
computation.
Advanced Numerical Analysis: Nonlinear Optimization
G63.2031 Identical to G22.2945. Prerequisites: knowledge of linear algebra and computer programming.
Constrained and unconstrained optimization. Topics: Newton’s
method and modifications, conjugate gradient and other methods suited to large,
sparse systems, conditions of optimality; linear and quadratic programming.
Advanced Numerical Analysis: Finite Element Methods
G63.2040 Identical to G22.2945. Prerequisites: elements of Hilbert space and theory of elliptic equations.
Basic theory of elliptic equations and calculus of
variations. Conforming finite elements. Approximation and interpolation by
piecewise polynomial functions. Error bounds. Numerical integration.
Nonconforming and isoparametric elements. Mixed methods. Problems of parabolic
type.
Computing in Finance
G63.2041 Prerequisite: basic C/C++ and Java programming.
An integrated introduction to software skills and their
applications in finance including trading, research, hedging, and portfolio
management. Students develop object-oriented software, gaining skill in
effective problem solving and the proper use of data structures and algorithms
while working with real financial models using historical and market data.
Scientific Computing
G63.2043 Prerequisites: multivariate calculus and
linear algebra. Some programming experience recommended.
Methods for numerical applications in the physical and
biological sciences, engineering, and finance. Basic principles and algorithms;
specific problems from various application areas; use of standard software
packages.
Monte Carlo Methods and Simulation of Physical Systems
G63.2044 Identical to G22.2960. Prerequisite: basic probability.
Principles of Monte Carlo: sampling methods and statistics,
importance sampling and variance reduction, Markov chains and the Metropolis
algorithm. Advanced topics such as acceleration strategies, data analysis, and
quantum Monte Carlo and the fermion problem.
Computational Methods for Finance
G63.2045 Prerequisites: G63.2043 or G63.2020, and G63.2792.
Computational methods for calibrating models; valuing,
hedging, and optimizing portfolios; and assessing risk. Approaches include
finite difference methods, Monte Carlo simulation, and
fast-Fourier-transform-based methods.
APPLIED MATHEMATICS AND MATHEMATICAL PHYSICS
Methods of Applied Mathematics
G63.2701 Prerequisites: undergraduate advanced calculus, ordinary differential equations, and complex variables.
Convergent and divergent asymptotic series. Asymptotic
expansion of integrals: steepest descents, Laplace principle, Watson’s lemma,
and methods of stationary phase. Regular and singular perturbations of
differential equations, the WKB method, boundary-layer theory, matched
asymptotic expansions, and multiple-scale analysis. Rayleigh-Schrödinger
perturbation theory for linear eigenvalue problems, summation of series, Pade
approximation, averaging methods, renormalization groups, weakly nonlinear
waves, and geometric optics.
Fluid Dynamics
G63.2702 Prerequisites: introductory complex variables and partial differential
equations.
Conservation of mass, momentum, and energy. Eulerian and
Lagrangian formulations. Basic theory of inviscid incompressible and barotropic
fluids. Kinematics and dynamics of vorticity and circulation. Special solutions
to the Euler equations: potential flows, rotational flows, conformal mapping
methods. The Navier-Stokes equations and special solutions thereof. Boundary
layer theory. Boundary conditions. The Stokes equations.
Applied Functional Analysis
G63.2703 Prerequisites: undergraduate advanced calculus, complex variables, ordinary differential equations, some experience with partial differential equations.
Green’s functions, theory of distributions, generalized
Fourier Series, Hilbert and Banach spaces, Riesz representation theorem,
integral equations, Fredholm alternative, potential theory, Hilbert-Schmidt
kernels, Rayleigh-Ritz method, spectral theory and Sturm-Liouville problems,
boundary value problems, elasticity and finite elements, optimization, quadratic
variational problems and duality, calculus of variations.
Partial Differential Equations for Finance
G63.2706 Prerequisites: basic probability and linear algebra.
Partial differential equations and advanced probability for
financial applications. Dynamic programming, Hamilton-Jacobi equations, and
viscosity solutions. Parabolic equations, diffusions, and Feynman-Kac.
Stochastic games, stopping times, and free boundary problems.
Financial Econometrics and Statistical Arbitrage
G63.2707 Prerequisites: G63.2043, G63.2791, and familiarity with basic probability.
An introduction to econometric aspects of financial markets,
focusing on the observation and quantification of volatility and on practical
strategies for statistical arbitrage.
Financial Engineering Models for Corporate Finance
G63.2709 Prerequisites: G63.2751 and G63.2791.
Advanced stochastic modeling applications. This course uses
simulation as a unifying tool to model all major types of market, credit, and
actuarial risks. Application of financial theory to the conceptualization and
solution of multifaceted real-world problems.
Mechanics
G63.2710
Newtonian mechanics. Lagrangian and Hamiltonian mechanics.
Integrable systems. Billiards. Method of averaging. KAM theory. Melnikov
method.
Capital Markets and Portfolio Theory
G63.2751
A mathematically sophisticated introduction to the analysis
of investments. Core topics include expected utility, risk and return,
mean-variance analysis, equilibrium asset pricing models, and arbitrage pricing
theory.
Case Studies in Financial Modeling
G63.2752 Prerequisites: G63.2041 and G63.2792.
Advanced topics in quantitative finance, such as dynamic
hedging; the volatility surface; local volatility and stochastic volatility
models; jump-diffusions; volatility-dependent options; power-law tails
and their consequences; behavioral finance.
Risk Management
G63.2753 Prerequisites: G63.2791 and G63.2041 or equivalent programming.
Measuring and managing the risk of trading and investment
positions: interest rate positions, vanilla options positions, and exotic
options positions. The portfolio risk management technique of Value-at-Risk,
stress testing, and credit risk modeling.
Derivative Securities
G63.2791 Prerequisite: G63.2901.
A first course in derivatives valuation. Arbitrage, risk
neutral pricing, binomial trees. Black-Scholes theory, early exercise,
barriers, interest rate models, floors, caps, swaptions. Introduction to
credit-based instruments.
Continuous Time Finance
G63.2792 Prerequisites: G63.2791 and G63.2901.
Advanced option pricing and hedging using continuous time
models: the martingale approach to arbitrage pricing; interests rate models
including the
Heath-Jarrow-Morton approach and short rate models; the
volatility smile/ skew and approaches to accounting for it.
Interest Rate and Credit Models
G63.2794
An introduction to widely used fixed income models,
emphasizing their implementation and applications to pricing, hedging, and
trading strategies. Topics include extraction of the yield curve from market
data; pricing and hedging of interest-based instruments using binomial and
trinomial tree models calibrated to market data; and credit risk models
including applications to the pricing of collateralized debt obligations and
the evaluation of credit risk in loan portfolios.
Advanced Topics in Applied Mathematics
G63.2830, 2840
Recent topics: mathematical models of crystal growth; waves
and mean flows; theory and modeling of rare events; atmosphere-ocean data
analysis; models of primitive organisms; vorticity and incompressible flow;
oceanic processes.
Advanced Topics in Biology
G63.2851, 2852 Identical to G23.2851, 2852.
Recent topics: computational biology; mathematical
neuroscience; statistical analysis of genomic data; cardiac mechanics and
electrophysiology.
Advanced Topics in Mathematical Physiology
G63.2855, 2856 Identical to G23.2855, 2856.
Recent topics: physiological control mechanisms;
mathematical aspects of neurophysiology; mathematical aspects of visual
physiology; mathematical models in cell physiology; mathematical models of
neuronal networks.
Advanced Topics in Fluid Dynamics
G63.2862
Recent topics: fluid dynamics of animal locomotion; complex
fluids; asymptotic problems in fluid mechanics; introduction to molecular simulations.
Advanced Topics in Mathematical Physics
G63.2863, 2864
Recent topics: quantum computation; supersymmetry; quantum
dynamics; hydrodynamical limit of nonreversible particle systems.
PROBABILITY AND STATISTICS
Basic Probability
G63.2901
Probability as a tool in computer science, finance,
statistics, and the natural and social sciences. Independence. Random variables
and their distributions. Conditional probability. Laws of large numbers.
Central limit theorem. Random walk, Markov chains, and Brownian motion.
Selected applications.
Stochastic Calculus
G63.2902 Prerequisite: G63.2901 or equivalent.
An application-oriented introduction to those aspects of
diffusion processes most relevant to finance. Topics include Markov chains;
Brownian motion; stochastic differential equations; the Ito calculus; the
forward and backward Kolmogorov equations; and Girsanov’s theorem.
Probability: Limit Theorems
G63.2911, 2912 Prerequisite: familiarity with the Lebesgue integral or real variables.
The classical limit theorems: laws of large numbers, central
limit theorem, iterated logarithm, arcsine law. Further topics: large deviation
theory, martingales, Birkhoff’s ergodic theorem, Markov chains, Shannon’s
theory of information, infinitely divisible and stable laws, Poisson processes,
and Brownian motion. Applications.
Advanced Topics in Probability
G63.2931, 2932
Recent topics: stochastic analysis; random walks on
disordered systems; percolation and disordered Ising models; stochastic
differential equations and diffusion processes.
Advanced Topics in Applied Probability
G63.2936
Recent topics: stochastic control and optimal trading in
incomplete and inefficient markets; information theory and financial modeling;
stochastic differential equations and Markov processes.
Mathematical Statistics
G63.2962 Prerequisite: a working knowledge of probability at the undergraduate level.
Principles and methods of statistical inference. Topics:
large sample theory, minimum variance unbiased estimates, method of maximum
likelihood, sufficient statistics, Neyman-Pearson theory of hypothesis testing,
confidence intervals, regression, nonparametric methods.
DISCRETE MATHEMATICS AND LOGIC
Elements of Discrete Mathematics
G63.2050 Identical to G22.2340.
Sets, relations, and functions. Algebraic structures.
Recursion and induction. Combinatorial mathematics. Graph theory. Probability.
Discrete and Computational Geometry
G63.2063
Algorithms for geometric problems involving points, lines,
and convex sets. Topics: convex hull formation, planarity testing, and sorting.
Applications to robotics.
Advanced Topics in Discrete and Computational Geometry
G63.2163, 2164
Recent topics: algorithms in real algebraic geometry; random
graphs; combinatorial geometry.
RESEARCH
Independent Study
G63.3771, 3772, 3773, 3774 Prerequisite: permission of the department. 1-3 points.
Master’s Thesis Research
G63.3881 Prerequisite: permission of the thesis adviser. May not be repeated for credit. 2 points.
Ph.D. Research
G63.3991, 3992, 3993, 3994, 3995, 3996, 3997, 3998 Open only to students who have passed the oral preliminary examination for the Ph.D. degree. Prerequisite: permission of the dissertation adviser.
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