Text: (Required) Gerald B. Foland, Real Analysis: Modern Techniques and Their Applications (2nd ed.), ISBN #0-471-31716-0, Wiley-Interscience/John Wiley Sons, Inc., 1999,
and Elliott Lieb and michael Loss, Analysis. Graduate Studies in Mathematics vol 14, (2nd ed.) ISBN# 978-0821827833, AMS, 2010.
Prerequisites: 640:501 or permission of Instructor
Description: This course is a continuation of 640:501 from Fall 2012. The goal is to give an introduction to core topics in real and functional analysis that every professional mathematician should know.
The course will cover material from Chapters 4-8 of Folland's book, as well as overlapping material and applications in the text of Lieb and Loss:
Topological Spaces Basic properties, compact spaces, Stone-Weierstrass theorem Introduction to Functional Analysis, Normed vector spaces, Hahn-Banach theorem, bounded linear transformations, Closed graph and Open mapping theorem, applications of Baire category theorem, Hilbert spaces, topological vector spaces, weak and weak* convergence Lp Spaces, Integral inequalities, duality, bounded integral operators Introduction to Fourier analysis, Schwartz space, convolutions, Fourier transform and Fourier series, Plancherel theorem, Poisson summation formula, Lp and pointwise convergence of Fourier series, Integration on Locally Compact Spaces Continuous functions and Radon measures on locally compact spaces, dual of C(X), vague convergence of measures. A number of application to problems in the calculus of variations and PDE will be studied.
Text: The text for the course will be Function Theory of One Complex Variable, by R. E. Greene and S. G. Krantz (Wiley-Interscience, New York; ISBN 0-471-80468-1), but excursions will be made to other sources.
Prerequisites: Math 503 or the equivalent.
Description: This course will be a continuation of Math 503. We shall cover a variety of more advanced topics in the theory of functions of one complex variable. Depending on where Math 503 ends, we shall begin with a discussion of normal families of analytic maps and the Riemann mapping theorem. We shall cover results about harmonic, subharmonic and superharmonic maps and their applications to the Dirichlet problem (Perron’s method). Other topics will include the Weierstrass factorization theorem, basic theory of entire functions (Jensen’s formula, genus and order of entire functions and the Hadamard factorization theorem) and results on the range of analytic functions (the big and little Picard theorems).
Subtitle: An introduction to Navier-Stokes equations
Text: Textbooks are not required. Good references are: O.A. Ladyzhenskaya, The mathematical theory of viscous incompressible Second English edition, 1969; Roger Temam, Navier-Stokes equations. Theory and numerical analysis. Third edition, 1984; Jean Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math. 63 (1934), 193-248.
Prerequisites: Instructor permission.
Description: In this course, we will introduce some results on Navier-Stokes equations. The course will consist of two parts, each for about half a semester. For the first part, I will present the classical results on the existence of weak solutions in three space dimension (Jean Leray) and the existence and smoothness of Navier-Stokes solutions in two space dimension (Ladyzhenskaya).
For the second part, we study some more recent papers which include works on partial regularity of weak solutions in three space dimension (Caffarelli-Kohn-Nirenberg, and a simplified proof by Fanghua Lin), on Liouville theorems (Koch-Nadirashvili-Seregin-Sverak), and nonhomogeneous steady incompressible
Navier-Stokes equations in a two dimensional domain (Amick).
Prerequisites: Graduate PDE course, some basics of differential geometry.
Description: We will start the course with some basics of geometric analysis. We will introduce different geometric flows, emphasize the importance of the maximum principles and monotonicity formulas in studying all of them. In the first part of the course, we will focus more on the mean curvature flow. We will mention Huisken's celebrated result that says that every convex hypersurface shrinks to a point under the mean curvature flow, define the notion of weak solutions to the mean curvature flow, the regularity theory for the mean curvature flow and the singularity analysis. In the second part of the course, we will be dealing with the Monge -Ampere equation together with a selection of geometric applications, such as the application to the isoperimetric inequality, solution to the Minkowski problem. We will also mention some Monge-Ampere geometric flows such as the Gauss curvature flow.
Text: Ordinary Differential Equations with Applications by Carmen Chicone (Springer
Texts in Applied Mathematics, vol. 34, second edition; Springer, 2006; ISBN-13: 978-0387-
Lecture notes on Computational ODEs by M. Gameiro, J.-P. Lessard, J. Mireles-James,
and K. Mischaikow.
Prerequisites: An undergraduate course on ordinary differential equations, linear algebra,advanced calculus and some basic results from analysis, e.g., the definition of a Banach space.
Description: This is an introduction to the theory of ordinary differential equations. Wewill cover the classical results: existence and uniqueness theorems; linear theory including Floquet theory and elementary bifurcations; stable and unstable manifolds; boundary value
problems; and a brief introduction to chaotic dynamics. The novelty of the course is thatthe proofs will be presented in a manner which allows for rigorous computer verifcation. Using Matlab we will apply these new techniques to rigorously extract specific solutions and explore the dynamics of explicit nonlinear systems such as Lorenz, Swift-Hohenberg, Gray-Scott, Kuramoto-Sivashinsky, and Ginzburg-Landau.
Prerequisites: This course is a continuation of 509. The assumption is that the students know the basic facts about Sobolev spaces and injections, degree theory and critical point theory. For these two latter subjects, adjustments can be made for new students. Knowledge of the basic facts about Sobolev spaces is a prerequisite.
Subtitle: Convergence and Collapsing Theory in Riemannian Geometry
Text: References:1.“Riemannian Geometry” by Peter Petersen, 2nd Edition, GTM 171
2. Lecture notes (by Xiaochun Rong)
Prerequisites: 1. Basic notions and properties related to differentiable manifolds and Riemannian
manifolds such as: tangent bundles, differential forms, integral of forms,
Levi-Civita connection, geodesics, exponential maps, curvature tensors, first and
second variation of arc length, basic curvature comparison.
2. Basic notions and properties in topology: covering spaces, fundamental
groups, homology groups.
Description: Convergence and collapsing theory were introduced in late 80’s. Since then, it
has been a powerful tool in Riemannian geometry and related fields.
1.1. The Gromov-Hausdorff distances
1.2. An Alternative formulation
1.3. Equivariant Gromov-Haudorff distances
1.4. Pointed Gromov-Haudorff distances
Chapter 2. Smooth limit – Fibrations
2.1. The fibration theorem
2.2. Embedding via distance function
2.3. Proof of fibration theorem
2.4. Equivariant fibration theorem
Chapter 3. Convergence Theorems
3.1. Cheeger-Gromov convergence theorem
3.2. Some elliptic estimates
3.3. Harmonic radius estimates
Chapter 4. Singular Limits – Singular Frbrations
4.1. Singular fibration theorem
4.2. Controlled homotopy structure by geometry
4.3. π_2-finiteness theorem
4.4. Collapsed manifolds with pinched positive sectional curvature
Chapter 5 (Optional). Almsot flat manifolds
5.1. Gromov’s almost flat manifold theorem
5.2. The Margulis Lemma
5.3. Flat connection with small torsion
5.4. Flat connection with parallel torsion
5.5. Proof of Almost flat manifold theorem
Prerequisites: Math 535. Familiarity with commutative algebra is an
advantage, but is not required.
Description: This course continues the study of algebraic geometry from the fall by
replacing algebraic varieties with the more general theory of schemes,
which makes it possible to assign geometric meaning to an arbitrary
commutative ring. One major advantage of schemes is the availability
of a well-behaved fiber product. Combined with Grothendieck's
philosophy that properties of schemes should be expressed as
properties of morphisms between schemes, fiber products make the
theory very flexible. The goal of the course is to cover the basic
definitions and properties of schemes and morphisms, and to introduce
and study the cohomology of sheaves, which provides a powerful tool
for settling geometric questions. For example, one can use
cohomological methods to give a simple proof of the classical
Riemann-Roch theorem for curves.
Description: We will start with chapter 3 of Hatcher, which introduces cohomology. Cohomology has a group structure roughly similar to homology, but it has an additional ring structure that allows us to distinguish spaces that can’t be distinguished using homology. For instance, using cohomology, we will show that there is no orientation-reversing homotopy equivalence on CPn, n even.Topics include cohomology, universal coefficients, Hom, Ext, axioms for cohomology, cup and cap products, the cohomology ring, Künneth formula, Poincaré duality, Alexander duality, Tor and universal coefficients for homology, H-spaces, James construction, infinite symmetric products, cohomology of SO(n), Stiefel manifolds, transfer homomorphism, local coefficients. If time permits, we will continue with basic homotopy theory from Davis and Kirk, which, like Hatcher’s book, is available online in pdf format.
Text: There will be no textbook for the course. Good references for surface and 3-manifold theory are:
 Simon Donaldson, Riemann surfaces, Oxford Graduate Texts in Mathematics,. Vol. 22, Oxford University Press, Oxford, 2011, xiv+286 pp.
 Otto Forster, Lectures on Riemann Surfaces. 4th corrected printing. Springer-Verlag 1999.
 Farkas H., Kra I. Riemann surfaces (Springer 1980).
 Benedetti, Riccardo; Petronio, Carlo Lectures on hyperbolic geometry. Universitext. Springer-Verlag, Berlin, 1992.
Prerequisites: Basic Algebraic topology (16:640:540) and differential geometry (16:640:532)
Description: Our goal is to introduce a wide range of interesting 2- and 3-dimensional geometric and topological phenomena.
For surfaces, we plan to cover basic Riemann surfaces, hyperbolic geometry and the associated moduli spaces. The topics include: the uniformization theorem, Teichmuller theory, the moduli space of surfaces, the mapping class group and some basic hyperbolic groups.
For 3-manifolds, we will be focusing on Thurston’s equation for producing hyperbolic structures and their relationship to simplicial Chern-Simons theory. The topics include: Haken’s normal surface theory, Thurston’s equation for SL(2,C) representation and Casson invariant .
If time permits, we will also introduce discrete Riemann surface theory, quantum Teichmuller spaces and the work of Kashaev on topological quantum field theory for 3-manifolds derived from the quantum dilogarithm.
This is intended to be a self-contained course suitable for students who know basic algebraic topology and differential geometry.
Text: Jacobson, "Basic Algebra", Volumes 1 and 2, second
edition. These volumes are currently available from Dover
Prerequisites: Any standard course in abstract algebra for
undergraduates and/or Math 551
Description: Topics: This is the continuation of Math 551, aimed at a discussion of many fundamental algebraic structures. The course will cover the following topics (and perhaps some others).
1) Basic module theory and introductory homological algebra - most of Chapter 3 and part of Chapter 6 of Basic Algebra II: hom and tensor, projective and injective modules, resolutions, completely reducible modules, the Wedderburn-Artin theorem
2) Commutative ideal theory and Noetherian rings - part of Chapter 7 of Basic Algebra II: rings of polynomials, localization, primary decomposition theorem, Dedekind domains, Noether normalization
3) Galois Theory - Chapter 4 of Basic Algebra I and part of Chapter 8 of Basic Algebra II: algebraic and transcendental extensions, separable and normal extensions, the Galois group, solvability of equations by radicals
Subtitle: Lie Algebras With Built-in Structure Constants
Text: Course notes will be provided.
Prerequisites: The basic structure theory of simple Lie algebras and their representations, Chevalley groups, infinite dimensional Kac-Moody algebras and their associated groups will be covered. Some familiarity with finite dimensional Lie groups and Lie algebras is preferable though not required.
Description: We explore a new construction of simply laced Lie algebras with structure constants built into the Lie bracket.
We will study how this impacts the definition of Chevalley groups, which give an alternate construction of simple Lie groups deﬁned in terms of generators using data from the root system and automorphisms of the underlying Lie algebra.
In particular we will consider the Chevalley group corresponding to a lattice between the root and weight lattice, the representation ring and Tanaka duality.
We explore how this extends to infinite dimensional Kac-Moody algebras and their associated Kac-Moody groups.
Subtitle: Nonlinear sigma-models and Riemannian geometry
Text: Some papers to be distributed in the classes, including, in particular, E. Witten, Elliptic genera and quantum field theory, Comm. Math. Phys. 109 (1987), 525-536. B. Greene, String Theory on Calabi-Yau Manifolds, hep-th/9702155,Yi-Zhi Huang, Meromorphic open-string vertex algebras, arXiv:1204.1774, Yi-Zhi Huang, Meromorphic open-string vertex algebras and Riemannian manifolds,arXiv:1205.2977 and some other papers.
Prerequisites: Algebra and analysis at the level of first year graduate courses. Exposure to the theory of vertex operator algebras and Riemnnian geometry will be helpful but is not required.
Description: Nonlinear sigma-models are a class of two-dimensional quantum field theories from which physicists have obtained many deep conjectures in topology and geometry. Unfortunately nonlinear sigma-models are still not constructed mathematically.
In this course, I will cover the following topics:
1. Nonlinear sigma-models from a mathematical point of view; conjectures obtained by physicists, including, in particular, those on Calabi-Yau manifolds and elliptic genera.
2. The mathematical construction of linear sigma-models, that is, sigma-models
with flat target spaces.
3. My recent results in a program of constructing nonlinear sigma-models;
Laplacians and sheaves of meromorphic open-string vertex algebras and modules on
4. My recent results on Witten's Dirac-like operator on
a loop space and their connection with Witten genus.
Text: (Recommended) Macdonald, I. G., Affine Hecke algebras and
Cambridge Tracts in Mathematics, 157. Cambridge University Press,
Cambridge, 2003. x+175 pp.
Prerequisites: First year graduate algebra.
Description: The subject of Cherednik algebras has seen an explosion
of activity in recent years, with more than 150 papers on the subject
in the last 5-6 years alone. The purpose of this course is to give
students an introduction to this beautiful subject and quickly bring
them to a state-of-the-art understanding of some of the research in
the area. To this end we will adopt an IMMERSE-style course strategy,
which consists of directly reading research papers in the subject,
developing the necessary material as needed. Thus we will read two
papers in this course, of which  develops the structure theory of
Cherednik algebras and  develops the representation theory. There
are no prerequisites beyond basic first year courses. Knowledge of Lie
algebras will be helpful but is not required.
 Sahi, Siddhartha, Nonsymmetric Koornwinder polynomials and
duality. Ann. of Math. (2) 150 (1999), no. 1, 267–282.
 Etingof, Pavel; Stoica, Emanuel, Unitary representations of
rational Cherednik algebras. With an appendix by Stephen Griffeth.
Represent. Theory 13 (2009), 349–370.
Prerequisites: General first year graduate education, set theory.
Description: Abstract: Axiom of Determinacy (AD), which says that all two player games of perfect information are determined, is one of the most fundamental and well studied axioms of modern set theory. Since it contradicts the Axiom of Choice, its introduction was not short of controversies and many thought that it is inconsistent. By now classical work of Martin, Steel and Woodin showed that while the consistency of Axiom of Determinacy cannot be proven in ZFC alone it can be proven in a stronger but very natural theory namely in ZFC+There are infinitely many Woodin cardinals (and a bit more). The proof of their theorem exploits many deep techniques and bridges two areas of set theory, the study of canonical models of set theory such as Gödel's constructible universe and the study of models of AD. The goal of this course will be to introduce some of these techniques while keeping in mind the current research topics, such as the inner model problem, HOD analysis and related problems.
Text: There is a vast literature on sieve methods. My recent book with J.Friedlander “Opera de Cribro” AMS Colloquium Publications, vol 57, is sufficient and recommended for casual reading. I will not follow the book completely or precisely, but rather select parts which the participants will find to be most attractive.
Prerequisites: Students are not required to know abstract mathematics; nevertheless in the later stage of the course some technical skill in classical Fourier analysis will be useful.
Description: This course is for graduate students interested in number theory in a broad sense. Conceptually the sieve methods are elementary, but some parts are quite advanced, particularly in applications to problems of Diophantine type for which the sieve was not primarily designed. Historically the sieve was a tool to solve problems about prime numbers, such as the Goldbach conjecture or the twin prime conjecture. Of course I shall cover these questions in great detail, but there will be much more about the theory itself. Selected topics are:
- The Eratosthenes sieve
- The Brun combinatorial sieve
- The Selberg sieve
- The Bombieri sieve
- The parity phenomena
- Producing primes by sieve
- Small gaps between primes
- Primes represented by polynomials
- Connections with harmonic analysis
- Zillions of applications
Description: Experimental Mathematics used to be considered an oxymoron, but the future of mathematics is in that direction. In addition to learning the philosophy and methodology of this budding field, students will become computer-algebra wizards, and that should be very helpful in whatever mathematical specialty they are doing (or will do) research in.
We will first learn Maple, and how to program in it. This semester we will focus on Games (of all kinds!), classical (the subject of so-called Game Theory for which John Nash (and quite a few other people) got a Nobel), Combinatorial (from Nim to Chess, via Backgammon), and Gambling (what Wall Street does).
In addition to the actual, very important content, students will master the methodology of computer-generated and computer-assisted research that is so crucial for their future.
There are no prerequisites, and no previous programming knowledge is assumed. Also, no overlap with previous years. The final projects will, with high probability, lead to publishable articles, and with strictly positive probability, to seminal ones.
Prerequisites: Topics the student should know, together with the courses in which they are taught at Rutgers, are as follows: Introductory Linear Algebra (640:250); Multivariable Calculus (640:251); Elementary Differential Equations (640:244 or 640:252). The course Advanced Calculus for Engineering (640:421), which covers Laplace transforms, trigonometric series, and introductory partial differential equations, is a valuable preparation for Math 527, but is not required. Students uncertain of their preparation for this course should consider taking 640:421, or consult with the instructor.
Description: This is a first-semester graduate course, intended primarily for students in mechanical and aerospace engineering, biomedical engineering, and other engineering programs. Topics include power series and the method of Frobenius for solving differential equations; Laplace transforms; nonlinear differential equations and phase plane methods; vector spaces of functions and orthonormal bases; Fourier series and
Sturm-Liouville theory; Fourier transforms; and separation of variables and other elementary solution methods for linear differential equations of physics, including the heat, wave, and Laplace equations. More information can be found on the course webpage.
Text: Michael D. Greenberg, Advanced Engineering Mathematics (second Edition; 0-13-221431-1), Prentice-Hall, 1998.
Robert Weinstock, Calculus of Variations, with Applications to Physics and Engineering (Dover, New York, 1974) ISBN 0-486-63069-2.
Prerequisites: Math 527, or else permission of the instructor
Description: This is a second-semester graduate course, appropriate for students of mechanical and aerospace engineering, biomedical, electrical, or other engineering areas, materials science, or physics. It begins with the algebra of complex numbers, complex-valued functions of complex variables, analytic functions and the Cauchy-Riemann conditions, poles and branch cuts, and conformal mappings, with applications in physics and engineering to the solution of differential equations and to fluid mechanics. Finally, we address some topics in the calculus of variations with applications.
Subtitle: Collective Phenomena in Equilibriu, and Nonequilibrium Systems
Text: Choice of specific topics will be based on student interest.
For background material see the following books:
1. Mathematical Biology I and II, J.D. Murray (Springer)
2. Large Scale Dynamics of Interacting Particles, H. Spohn (Springer)
3. A Kinetic View of Statistical Physics, P. Krapivsky, S. Redner and E. Ben-Naim (Cambridge)
4. Dynamics of Self-Organized and Self-Assembled Structures, R. Desai and R. Krapral (Cambridge)
5. Evolutionary Games and population Dynamics, J. Hofbauer and K. Sigmund (Cambridge)
The course will be informal and interactive.
Prerequisites: For information about prerequisites please contact me: Joel Lebowitz, email@example.com.
Description: Course will start with a broad overview of the physics and mathematics of equilibrium and nonequilibrium statistical mechanics: This will focus on the elucidation and derivation of collective behavior of macroscopic systems made up of very many individual components from the microscopic dynamics of the individual components.
I will then consider application of statistical mechanics to real world problems of current interest. An example of such an application is pattern formation. This occurs in both equilibrium and nonequilibrium systems. The former generally represent low temperature phases in materials and can be studied via ensembles. The latter involve dynamical microscopic considerations and are generally described on the macroscopic level by reaction-biological systems. They range in scale from microns for cells forming an organism to hundreds of meters for flocking birds.
Text: (Recommended) K. Atkinson “An Introduction to Numerical Analysis" 0-471-62489-6; A. Quarteroni, R. Sacco, and F. Saleri, /Numerical Mathematics, (second edition), Springer, 2004, 0-387-98959-5.
Prerequisites: Numerical Analysis I - 16:642:573 is desirable, but not required
Description: This is the first part of a general survey of the basic topics in numerical analysis -- the study and analysis of numerical algorithms for approximating the solution of a variety of generic problems which occur in applications. In the fall semester, we will consider the approximation of functions by polynomials and piecewise polynomials, numerical integration, and the numerical solution of initial value problems for ordinary differential equations, and see how all these problems are related. In the spring semester (642:574), we will study the numerical solution of linear systems of equations, the approximation of matrix eigenvalues and eigenvectors, the numerical solution of nonlinear systems of equations, numerical techniques for unconstrained function minimization, finite difference and finite element methods for two-point boundary value problems, and finite difference methods for some model problems in partial differential equations. Despite the many solution techniques presented in elementary calculus and differential equations courses, mathematical models used in applications often do not have the simple forms required for using these methods. Hence, a quantitative understanding of the models requires the use of numerical approximation schemes. This course provides the mathematical background for understanding how such schemes are derived and when they are likely to work. To illustrate the theory, in addition to the usual pencil and paper problems, some short computer programs will be assigned. To minimize the effort involved, however, the use of /Matlab/ will be encouraged. This program has many built in features which make programming easy, even for those with very little prior programming experience.
This course is also part of the Mathematical Finance Master's Degree Program
Text: Dietrich Braess, Finite Elements: Theory, fast solvers, and applications in solid mechanics, 3rd ed., Cambridge University, 2007.
Note: Most of lectures will be based on hand outs prepared by the instructor. Students may have one of the aforementioned textbooks depending on their preference.
Prerequisites: At least one of Numerical Analysis I (16:642:573) or Numerical Analysis II (16:642:574).
Description: In this course, we study finite difference, finite element, and finite volume methods for the numerical solution of elliptic, parabolic, and hyperbolic partial differential equations. The course will concentrate on the key ideas underlying the derivation of numerical schemes and a study of their stability and accuracy. Students will have the opportunity to gain computational experience with numerical methods with a minimal of programming by the use of Matlab's PDE Toolbox software. Since there are many sophisticated computer packages available for solving partial differential equations, one might think that a thorough understanding of the numerical methods employed is no longer necessary. A striking example of why naive use of such codes can lead to disastrous results is the sinking of the Sleipner A offshore oil platform in Norway in 1991, resulting in an economic loss of about $700 million. The post accident investigation traced the problem to inaccurate finite element approximation of the linear elastic model of the structure (using the popular finite element program NASTRAN). The shear stresses were underestimated by 47%, leading to insufficient design. More careful finite element analysis, made after the accident, predicted that failure would occur with this design at a depth of 62m, which matches well with the actual occurrence at 65m.
This course is also part of the Mathematical Finance Master's Degree Program
Prerequisites: Basic combinatorics, basic linear algebra, mathematical maturity
Description: This course will serve as a graduate course in graph theory. We will follow the text by Bela Bollobas on Modern Graph Theory. Some of the topics we will cover include: Matchings, cuts, flows, connectivity, planar graphs, graph colorings, random graphs, extremal graph theory, Ramsey theory, linear algebra methods, and expander graphs
Text: Babai-Frankl, Linear Algebra Methods in Combinatorics. (This is actually only a manuscript. It's not mandatory: we won't follow it, but will overlap it to some extent, and it has lots of nice material.)
Prerequisites: I'll try to make the course self-contained except for basic combinatorics and linear algebra. A course in each of these would be helpful. See me if in doubt.
Description: This course will survey applications of ideas from algebra (mostly linear)
to problems in discrete mathematics and related areas. Areas of application
include extremal problems for finite sets and the n-cube; theoretical computer
science; discrete geometry; graph theory; probability; additive number theory
and group theory; etc. theory. Various open problems will be discussed.