Math courses for NYU students
 V63.0121 Calculus I
Derivatives, antiderivatives, and integrals of functions of one
real variable. Trigonometric, inverse trigonometric, logarithmic
and exponential functions. Applications, including graphing,
maximizing and minimizing functions. Areas and volumes.
 V63.0122 Calculus II
Techniques of integration. Further applications. Plane analytic
geometry. Polar coordinates and parametric equations. Infinite
series, including power series.
 V63.0123 Calculus III
Functions of several variables. Vectors in the plane and space.
Partial derivatives with applications, especially Lagrange
multipliers. Double and triple integrals. Spherical and
cylindrical coordinates. Surface and line integrals. Divergence,
gradient, and curl. Theorem of Gauss and Stokes.
 V63.0140 Linear Algebra
Systems of linear equations, Gaussian elimination, matrices,
determinants, Cramer’s rule. Vectors, vector spaces, basis and
dimension, linear transformations. Eigenvalues, eigenvectors,
and quadratic forms.
 V63.0141 Honors Linear Algebra I  identical to
G63.2110
Linear spaces, subspaces, and quotient spaces; linear dependence
and independence; basis and dimensions. Linear transformation
and matrices; dual spaces and transposition. Solving linear
equations. Determinants. Quadratic forms and their relation to
local extrema of multivariable functions.
 V63.0142 Honors Linear Algebra II  identical to
G63.2120
 V63.0233 Theory of Probability
An introduction to the mathematical treatment of random
phenomena occurring in the natural, physical, and social
sciences. Axioms of mathematical probability, combinatorial
analysis, binomial distribution, Poisson and normal
approximation, random variables and probability distributions,
generating functions, Markov chains applications.
 V63.0234 Mathematical Statistics
An introduction to the mathematical foundations and techniques
of modern statistical analysis for the interpretation of data in
the quantitative sciences. Mathematical theory of sampling;
normal populations and distributions; chisquare, t, and F
distributions; hypothesis testing; estimation; confidence
intervals; sequential analysis; correlation, regression;
analysis of variance. Applications to the sciences.
 V63.0250 Mathematics of Finance
Introduction to the mathematics of finance. Topics include:
Linear programming with application pricing and quadratic.
Interest rates and present value. Basic probability: random
walks, central limit theorem, Brownian motion, lognormal model
of stock prices. BlackScholes theory of options. Dynamic
programming with application to portfolio optimization.
 V63.0252 Numerical Analysis
In numerical analysis one explores how mathematical problems can
be analyzed and solved with a computer. As such, numerical
analysis has very broad applications in mathematics, physics,
engineering, finance, and the life sciences. This course gives
an introduction to this subject for mathematics majors. Theory
and practical examples using Matlab will be combined to study a
range of topics ranging from simple rootfinding procedures to
differential equations and the finite element method.
 V63.0262 Ordinary Differential Equations
First and second order equations. Series solutions. Laplace
transforms. Introduction to partial differential equations and
Fourier series.
 V63.0263 Partial Differential Equations
Many laws of physics are formulated as partial differential
equations. This course discusses the simplest examples, such as
waves, diffusion, gravity, and static electricity. Nonlinear
conservation laws and the theory of shock waves are discussed.
Further applications to physics, chemistry, biology, and
population dynamics.
 V63.0282 Functions of a Complex Variable
Complex numbers and complex functions. Differentiation and the
CauchyRiemann equations. Cauchy’s theorem and the Cauchy
integral formula. Singularities, residues, and Laurent series.
Fractional Linear transformations and conformal mapping.
Analytic continuation. Applications to fluid flow etc.
 V63.0325 Analysis I
The real number system. Convergence of sequences and series.
Rigorous study of functions of one real variable: continuity,
connectedness, compactness, metric spaces, power series, uniform
convergence and continuity.
 V63.0326 Analysis II
Functions of several variables. Limits and continuity. Partial
derivatives. The implicit function theorem. Transformation of
multiple integrals. The Riemann integral and its extensions.
 V63.0375 Topology (optional)
Settheoretic preliminaries. Metric spaces, topological spaces,
compactness, connectedness, covering spaces, and homotopy
groups.
 G63.1410.001, 1420.001 INTRODUCTION TO MATHEMATICAL
ANALYSIS I, II
Fall term
Functions of one variable: rigorous treatment of limits and
continuity. Derivatives. Riemann integral. Taylor series.
Convergence of infinite series and integrals. Absolute and
uniform convergence. Infinite series of functions. Fourier
series.
Spring term
Functions of several variables and their derivatives. Topology
of Euclidean spaces. The implicit function theorem, optimization
and Lagrange multipliers. Line integrals, multiple integrals,
theorems of Gauss, Stokes, and Green.
 G63.2450.001, 2460.001 COMPLEX VARIABLES I, II
Fall Term
Complex numbers; analytic functions, CauchyRiemann 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 SchwarzChristoffel transformation; elliptic functions,
Riemann surfaces; conformal mapping and univalent functions;
maximum principle and Schwarz's lemma; the Riemann mapping
theorem.}
Text: Complex Analysis, Alfors
 G63.2470.001 ORDINARY DIFFERENTIAL EQUATIONS
Existence theorem: finite differences; power series. Uniqueness.
Linear systems: stability, resonance. Linearized systems:
behavior in the neighborhood of fixed points. Linear systems
with periodic coefficients. Linear analytic equations in the
complex domain: Bessel and hypergeometric equations.
Recommended text: Ordinary Differential Equations, Coddington &
Levinson
 G63.2490.001 PARTIAL DIFFERENTIAL EQUATIONS
(oneterm format)
Basic constantcoefficient linear examples: Laplace's equation,
the heat equation, and the wave equation, analyzed from many
viewpoints including solution formulas, maximum principles, and
energy inequalities. Key nonlinear examples such as scalar
conservation laws, HamiltonJacobi equations, and semilinear
elliptic equations, analyzed using appropriate tools including
the method of characteristics, variational principles, and
viscosity solutions. Simple numerical schemes: finite
differences and finite elements. Important PDE from mathematical
physics, including the Euler and NavierStokes equations for
incompressible flow.
Suggested texts: Partial Differential Equations, Paul R.
Garabedian, AMS; Partial Differential Equations, L. C. Evans,
AMS; Partial Differential Equations, Fritz John, Springer
 G63.2550.001 FUNCTIONAL ANALYSIS
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
infinitedimensional spaces (Brownian motion). Some indications
about nonlinear analysis in an infinitedimensional setting.
General theme: How does ordinary linear algebra and calculus
extend to d=? dimensions?
Mandatory text: Functional Analysis, P. Lax, (Pure & Applied
Mathematics, New York), WileyInterscience, John Wiley & Sons,
2002
Rec. text: Methods of Modern Mathematical physics Vol. I:
Functional Analysis, M. Reed & B. Simon, Academic Press, New
YorkLondon, 1972
 G63.2012.002 ADVANCED TOPICS IN NUMERICAL ANALYSIS
(Numerical Methods with Probability)
A continuation of Numerical Methods I, introducing statistical
and scientific applications of numerical linear algebra
(including randomized algorithms), digital signal processing
(including stochastic processes), spectral and adaptive schemes
for numerical integration, MonteCarlo techniques (including
Metropolis' and Hastings'), the enhancement of accuracy via
postprocessing, and other fundamentals. The focus is on basic
methods for solving problems encountered frequently in modern
science and technology.
Crosslisted as G22.2945.002
 G63.2902.001 STOCHASTIC CALCULUS
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. FeynmanKac and CameronMartin Formulas.
Applications as time permits.
Text: Stochastic Calculus, A Practical Introduction, Richard
Durrett, CRC Press, Probability & Stochastics Series
 G63.2911.001, 2912.001 PROBABILITY: LIMIT THEOREMS
I, II
Newman, (fall); H. McKean (spring).
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.
Spring text: Stochastic Processes, S. R. S. Varadhan, CIMS 
AMS, 2007
 G63.2931.001 ADVANCED TOPICS IN PROBABILITY (Markov
Processes and Diffusions).
In the first part of the course, we will give an introduction to
the general theory of Markov processes for both discrete and
continuous time. Our main focus will be the study of their
longtime behavior (transience, recurrence, ergodicity, mixing)
in the classical context of Harris chains, but also for a larger
class of processes that doesn't fit into this context. The
second part of the course will be aimed at applying the abstract
results from the first part to the more concrete framework of
elliptic diffusion processes. Lyapunov function techniques will
play a prominent role in this part of the course. The final part
of the course will be an introduction to the theory of
hypoelliptic diffusion processes. We will give a short
introduction to Mallivain calculus and use it to give a
probabilistic proof of Hörmander's famous "sums of squares"
theorem.
Recommended texts: Markov Chains and Stochastic Stability, Meyn
and Tweedie (available online at http://www.probability.ca/MT/);
Introduction to the Theory of Diffusion Processes, Krylov; The
Malliavin Calculus and Related Topics, Nualart
 G63.2932.001 ADVANCED TOPICS IN PROBABILITY (Large
Deviations and Applications)
Varadhan
Prerequisites: Probability: Limit Theorems I and II; familiarity
with some Markov Processes, Brownian motion, SDE, diffusions.
Standard Cramer Theory for sums of iid random variables, Ventcel
Freidlin theory for ordinary differential equations with small
noise and the exit problem. DonskerVaradhan theory of large
time behavior of Markov Processes. Applications to interacting
particle systems.
Recommended Texts on Large Deviations: Dembo & Zeitouni,
Deuschel & Stroock, Weiss & Schwartz
 G36.2011 Advanced Topics in Numerical Analysis: High
Performance Scientific Computing.
Topics: Serial and parallel performance tuning, parallel programing in MPI and OpenMP.

G63.2044 Monte Carlo Methods and Simulation of Physical Systems
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.

G63.2830, 2840 Advanced Topics in Applied
Mathematics
Recent topics: mathematical models of crystal growth; math
adventures in data mining; ice dymamics; vortex dynamics;
applied stochastic analysis; developments in statistical
learning; fluctuation dissipation theorems and climate change;
theory and modeling of rare events.