**Text:**G. Folland, Real Analysis: Modern Techniques and Their Applications, second edition, Wiley Interscience, 1999. ISBN 0-471-31716-0.

**Prerequisites:**This course assumes familiarity with real analysis at the level, roughly, of W. Rudin, Principles of Mathematical Analysis.

**Description:**Basic real variable function theory, measure and integration theory prerequisite to pure and applied analysis. Topics: Riemann and Lebesgue-Stieltjes integration; measure spaces, measurable functions and measure; Lebesgue measure and integration; convergence theorems for integrals; Lusin and Egorov theorems; product measures and Fubini-Tonelli theorem; signed measures, Radon-Nikodym theorem, and Lebesgue's differentiation theorem.

**Text:**Function Theory of One Complex Variable: Third Edition (Graduate Studies in Mathematics) *by Robert E. Greene

**Publisher:**American Mathematical Society; 3rd edition (March 29, 2006)

**ISBN**-13: 978-0821839621

**Prerequisites:**

**Description:**

The beginning of the study of one complex variable is certainly one of
the loveliest mathematical subjects. It's the magnificent result of
several centuries of investigation into what happens when **R** is
replaced by **C** in "calculus". Among the consequences were the
creation of numerous areas of modern pure and applied mathematics, and
the clarification of many foundational issues in analysis and
geometry. Gauss, Cauchy, Weierstrass, Riemann and others found this
all intensely absorbing and wonderfully rewarding. The theorems and
techniques developed in modern complex analysis are of great use in
all parts of mathematics.

The course will be a rigorous introduction with examples and proofs
foreshadowing modern connections of complex analysis with differential
and algebraic geometry and partial differential equations. Acquaintance
with analytic arguments at the level of Rudin's *Principles of
Modern Analysis* is necessary. Some knowledge of algebra and
point-set topology is useful.

The course will include some appropriate review of relevant topics,
but this review will *not* be enough to educate the uninformed
student adequately. A previous "undergraduate" course in
complex analysis would also be useful though not necessary.

There are many excellent books about this subject. The official text
will be *Function Theory of One Complex Variable*, by Greene and
Krantz (American Math Society, 3^{rd} edition, 2006). The
course will cover most of Chapters 1 through 5 of the text, parts of
Chapters 6 and 7, and possibly other topics. The titles of these
chapters follow.

**1:** Fundamental Concepts; **2:** Complex Lines Integrals;** 3:** Applications of the Cauchy Integral;**4:** Meromorphic functions and Residues; **5:** The Zeros of a Holomorphic Function; **6:** Holomorphic Functions as Geometric Mappings.

**Text:**Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer 2010, ISBN-10: 9780387709130, 614 pages.

**Prerequisites:**

**Description:**TBA

**Text:**Partial Differential Equations by Emmanuele DiBenedetto, 2nd Ed., Birkhauser 2009, ISBN-10: 9780817645519

**Prerequisites:**A strong background on advanced calculus involving multivariables (esp. Green's Theorem and Divergence Theorem). We will also use some basic facts of Lp function spaces and the usual integral inequalities (mostly completeness and Holder inequalities in L2 setting).

These topics are covered in the first semester graduate real variable course (640:501).

**Description:**This is the first half of a year-long introductory graduate course on PDE. PDE is an enormously vast field. PDEs arise from very diverse fields: from classical to modern physics, to more applied sciences such as material sciences, mathematical biology, and signal processing, etc, and from the more pure aspects of mathematics such as complex analysis and geometric analysis.

This introductory course should be useful for students with a variety of research interests: physics and mathematical physics, applied analysis, numerical analysis, complex analysis, differential geometry, and, of course, partial differential equations.

For an introductory course, it is more important to examine some important examples to certain depth, to introduce the formulation, concepts, most useful methods and techniques through such examples, than to concentrating on presentation and proof of results in their most general form.

**This is the way the course will be conducted.**

The beginning weeks of the course aim to develop enough familiarity and experience to the basic phenomena, approaches, and methods in solving initial/boundary value problems in the contexts of the classical prototype linear PDEs of constant coefficients: the Laplace equation, the D'Alembert wave equation, the heat equation and the Schroedinger equation.

Fourier series/eigenfunction expansions, Fourier transforms, energy methods, and maximum principles will be introduced. More importantly, appropriate methods are introduced for the purpose of establishing qualitative, characteristic properties of solutions to each class of equations. It is these properties that we will focus on later in extending our beginning theories to more general situations, such as variable coefficient equations and nonlinear equations.

Next we will discuss some notions and results that are relevant in treating general PDEs: characteristics, non-characteristic Cauchy problems and Cauchy-Kowalevskaya theorem, wellposedness.

Towards the end of the semester, we will begin some introductory discussion on the extension of the energy methods to variable coefficient wave/heat equations, and/or the Dirichlet principle in the calculus of variations.

The purpose here is to motivate and introduce the notion of weak solutions and Sobolev spaces, which will be more fully developed in the second semester.

**Subtitle:**Elliptic and parabolic equations: Symmetries and isolated singularities

**Text:**

**Prerequisites:**640:517 or permission by the instructor

**Description:**I will discuss nonlinear second order elliptic and parabolic equations, about symmetries of solutions which are inherited from symmetries of equations, various maximum principles, and about isolated singularities of solutions. Effort will be made to always present results from the very beginning, for harmonic functions and solutions of the heat equation, and for solutions of linear equations with smooth coefficients. Then we will discuss new research results for nonlinear equations, as well as presenting open problems.

More concretely, we plan to do the following:

1. Quickly review strong maximum principle, Hopf Lemma, Harnack inequality, isolated singularity of solutions, comparison principles, for Harmonic functions. We will then introduce the concept of viscosity solutions, and present some recent joint and ongoing work with Caffarelli and Nirenberg on strong maximum principle for singular solutions to fully nonlinear elliptic equations of second order, and to present removable singularity results for viscosity solutions of such equations. We will also present works of Caffarelli-Gidas-Spruck on isolated singularity for semilinear elliptic equations of critical exponent, as well as some recent work with Z.C. Han and E. Teixeira on a fully nonlinear version of it. We will present Harnack inequalities, Liouville theorems, derivative estimates for conformally invariant elliptic equations, and present Bernstein type arguments in obtaining a priori estimates.

2. Introduce the method of moving planes, proving a theorem of A.D. Alexandrov asserting that embeded compact hypersufaces of constant mean curvature in Euclidean space must be round spheres. Present some variant of this theorem in joint work with Nirenberg and some open problems.

3. We will give a survey of existence theorems for the $\sigma_k$-Yamabe problem and will present some open problems in this direction.

**Text:**TBA

**Prerequisites:**TBA

**Description:**"This course is an introduction to differential geometry. Possible topics0 include:

- Differential forms on manifolds
- Vector bundles & curvature
- Riemannian metrics & geodesics
- Symplectic forms & moment maps

**Subtitle:**Introduction to Minimal Surfaces

**Text:**There is no textbook for the course.

**Prerequisites:**Math. 501, Math. 503, Math. 432 (undergraduate Differential Geometry)

**Description:**

** Topics that will be covered in the course**

- First and Second Variation of Area.
- Monotonicity Formula for minimal surfaces.
- Isothermal Coordinates and Bernstein's theorem.
- Weierstrass Parametrization of Minimal Surfaces.
- Omission of points by the Gauss Map, Fujimoto's theorem.
- Heinz curvature Estimate.
- Plateau Problem.
- Stability of minimal surfaces and Schoen-Fischer-Colbrie theorem.
- Rado's theorem on branch points.

**Subtitle:**Quantization of Teichmueller Space

**Text:**No textbook. A good reference for hyperbolic geometry is Bonahon's textbook:

1) Francis Bonahon, "Low-dimensional Geometry: From Euclidean Surfaces to Hyperbolic Knots", Student Mathematical Library, 49.

Understanding the following articles will be our first goal:

2) Leonid O. Chekhov, Vladimir V. Fock, "Quantum Teichmuller spaces", Theoret. and Math. Phys. 120 (1999), no. 3, 1245--1259.

3) Xiaobo Liu, "The quantum Teichmuller space as a noncommutative algebraic object", J. Knot Theory Ramifications 18 (2009), no. 5, 705--726.

4) Francis Bonahon, Xiaobo Liu, "Representations of the quantum Teichmuller space and invariants of surface diffeomorphisms", Geom. Topol. 11 (2007), 889--937.

**Prerequisites:**Most of the mathematical concepts will be introduced as needed. Some background in topology and differential geometry is recommended.

**Description:**The Teichmueller space has been used extensively to study the geometry and topology of surfaces and 3-manifolds. It also arises in physics as a key object in the study of (2+1)-gravity, which provides the first motivation for finding its "quantized" analogue. The construction of the quantum Teichmueller space was achieved in the late 90's by L. Chekhov and V. V. Fock, and independently by R. Kashaev.

In the first part of the course, I will give a brief account of symplectic geometry and quantization, and introduce the necessary concepts in hyperbolic geometry. I will then carefully describe the (non-quantum) Teichmueller space and its associated symplectic structure. At this point we will have the necessary ingredients to introduce the quantum Teichmueller space, following the construction of Chekhov and Fock. I will explain how it can be studied using methods from representation theory, as described in the work of F. Bonahon and X. Liu.

Depending on students interest and if time permits, I will also survey its possible relationship with some of the following topics: quantum invariants of knots and 3-manifolds, conformal field theory and TQFT, the study of the mapping class groups.

**Text:**Publisher: Cambridge University Press; 1 edition (December 3, 2001) • Language: English • ISBN-10: 9780521795401 • ISBN-13: 978-0521795401 • ASIN: 0521795400

**Prerequisites:**

**Description:**This course will be an introduction to the fundamental group and homology theory. The text will be Allen Hatcher's excellent book Algebraic Topology, available for $32 in paperback from Cambridge University Press, as well as (for free) online.

The plan is to cover chapters 1 and 2 of Hatcher's book. Topics include fundamental group, Van Kampen's Theorem, covering spaces, simplicial and singular homology, Brouwer's fixed-point theorem, the Borsuk-Ulam theorem, and the Jordan-Brouwer separation theorem.

**Text:**Introduction to Lie Algebras and Representation Theory, by James E. Humphreys

**Prerequisites:**First semester of Abstract Algebra for graduate students, permission of instructor required for students not enrolled in the mathematics Ph.D. program.

**Description:**This course will be an introduction to Lie algebras in the context of linear algebraic groups, with emphasis on the classical complex matrix groups (the general and special linear group, orthogonal group, and symplectic group).

Topics will include elementary properties of linear algebraic groups, their finite-dimensional representations, and their Lie algebras. The Lie algebras of the classical groups will be studied using root systems and Weyl groups relative to a maximal torus. The complete reducibility of finite-dimensional representations will be proved and the Cartan-Weyl highest weight theory of irreducible finite-dimensional representations will be developed. For the classical simple Lie algebras explicit models for the irreducible representations will be constructed. The structure and classification of semisimple Lie algebras will be also covered.

If time permitted, more advanced topics (Kac-Moody algebras, etc) will be discussed.

**Text:**

**Main**N. Jacobson,

*Basic Algebra I, II (2nd edition, 1985)*

These books are available in paperback for under $20 from Dover Publications (2009). (ISBN: 0486471896 and 048647187X)

There are supplementary handouts for: bilinear forms over fields, simple/semisimple algebras, and group representations.

**Prerequisites:**

**Description:**

This is a standard course for beginning graduate students. It covers
Group Theory, basic Ring & Module theory, and bilinear forms.

**Group Theory:** Basic concepts, isomorphism theorems,
normal subgroups, Sylow theorems, direct products and free products of groups.
Groups acting on sets: orbits, cosets, stabilizers.
Alternating/Symmetric groups.

**Basic Ring Theory:** Fields, Principal Ideal Domains (PIDs),
matrix rings, division algebras, field of fractions.

**Modules over a PID:** Fundamental Theorem for abelian groups,
application to linear algebra: rational and Jordan canonical form.

**Bilinear Forms:** Alternating and symmetric forms, determinants.
Spectral theorem for normal matrices, classification over R and C.
(Class supplement provided)

**Modules:** Artinian and Noetherian modules.
Krull-Schmidt Theorem for modules of finite length.
Simple modules and Schur's Lemma, semisimple modules.
(from *Basic Algebra II*)

**Finite-dimensional algebras:** Simple and semisimple
algebras, Artin-Wedderburn Theorem, group rings, Maschke's Theorem.
(Class supplement provided)

**Subtitle:**Introduction to Tensor Category Theory for Vertex Operator Algebras

**Text:**I. Frenkel, Y.-Z. Huang and J. Lepowsky, On Axiomatic Approaches to Vertex Operator Algebras and Modules, Memoirs Amer. Math. Soc., Vol. 104, Number 494, 1993.

Certain papers by Y.-Z. Huang, J. Lepowsky and L. Zhang developing logarithmic tensor category theory for modules for a vertex algebra. Parts of the introductory monograph listed above, covered in the course, provide sufficient background for these papers.

**Supplementary texts:**

J. Lepowsky and H. Li, Introduction to Vertex Operator Algebras and Their Representations, Progress in Math., Vol. 227, Birkhauser, Boston, 2003.

I. Frenkel, J. Lepowsky and A. Meurman, Vertex Operator Algebras and the Monster, Pure and Applied Mathematics, Vol. 134, Academic Press, 1988.

**Prerequisites:**Basic algebra and some familiarity with Lie algebra theory. While a little experience with vertex operator algebra theory would make this course easily accessible, the basics of the theory will be developed, and students without such experience and potentially interested in this course are encouraged to consult me.

**Description:**The course will develop tensor product theory and tensor category theory for modules for a vertex operator algebra. We will start with the basics of vertex operator algebra theory and also the basics of tensor category theory. The construction and study of tensor categories of modules for a vertex operator algebra is analogous to the corresponding notions for a Lie algebra, but is much more elaborate, and is deeply related to a range of developments in both mathematics and conformal field theory in physics.

**Please note:** The Lie Groups/Quantum Mathematics Seminar, which will
meet Fridays at 11:45, will sometimes be related to the subjects of
the course. Students planning to take the course should also try to
arrange to attend the seminar, although the seminar will not be
required for the course.

**Text:**An introduction to homological algebra, by C.~Weibel, Cambridge U. Press, paperback edition (1995).

**Prerequisites:**First-year knowledge of groups and modules.

**Description:**This will be an introduction to the subject of Homological Algebra. Homological Algebra is a tool used in many branches of mathematics, especially in Algebra, Topology and Algebraic Geometry.

The first part of the course will cover Chain Complexes, Projective and Injective Modules, Derived Functors, Ext and Tor. In addition, some basic notions of Category Theory will be presented: adjoint functors, abelian categories, natural transformations, limits and colimits.

The second part of the course will study Spectral Sequences, and apply this to several topics such as Homology of Groups and Lie Algebras. Which topics we cover will be determined by the interests of the students in the class.

**Text:**No textbook

**Prerequisites:**No prerequisites

**Description:**

This is an introductory course in Mathematical Logic aimed at graduate students in mathematics rather than prospective logicians.

In Set Theory, we will discuss basic topics such as ordinals,
cardinals and the various equivalents of the Axiom of Choice; as well
as more advanced topics such as the club filter and stationary sets.
(The latter are needed to prove results such as the existence of
2^{ℵ1} nonisomorphic dense linear
orders of cardinality ℵ_{1}.)

In Model Theory, we will begin with basic results such as the Completeness and Compactness Theorems. Then we will cover some more advanced topics focusing on the structure of the countable models of a complete theory in a countable language.

The course has no formal prerequisites.

**Subtitle:**Automorphic Forms. Spectral Theory

**Text:**I shall follow my book "Spectral Methods of Automorphic Forms", Graduate Studies in Mathematics, Vol 53, AMS 2002.

**Prerequisites:**The participants are assumed to have a solid knowledge of the complex function theory, some skill in group theory and differential calculus.

**Description:**Modular forms are in the center of number theory. The classical theory of holomorphic forms has been used in arithmetic for a long time. More recent developments in the real analytic forms made a significant impact on modern analytic number theory. The main goal of this course is to present from scratch the spectral theory of automorphic forms, including the Fourier analysis of the coefficients of cusp forms. Here are some topics to be covered;

- Geometry of the hyperbolic plane
- Discontinuous groups
- Eisenstein series
- Spectral resolution of automorphic forms
- Selberg's trace formula
- Spectral theory of Kloosterman sums
- Applications to L-functions

**Text:**

*M.Greenberg*,

**Advanced Engineering Mathematics**(second edition); Prentice, 1998 (ISBN# 0-13-321431-1))

**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); Advanced Calculus for Engineering (Laplace transforms, sine and cosine series, introductory partial differential equations) (640:421).

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; nonlinear differential equations and phase plane methods; vector spaces of functions, Hilbert spaces, and orthonormal bases; Fourier series and Sturm-Liouville theory; Fourier and Laplace 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 is on the www.math.rutgers.edu/courses/527/

**Text:**Gilbert Strang, "Linear Algebra and its Applications", 4

^{th}edition, ISBN #0030105676, Brooks/Cole Publishing, 2007

**Prerequisites:**Familiarity with matrices, vectors, and mathematical reasoning at the level of advanced undergraduate applied mathematics courses.

**Description:**Note: This course is intended for graduate students in science, engineering and statistics.

This is an introductory course on vector spaces, linear transformations, determinants, and canonical forms for matrices (Row Echelon form and Jordan canonical form). Matrix factorization methods (LU and QR factorizations, Singular Value Decomposition) will be
emphasized and applied to solve linear systems, find eigenvalues, and diagonalize quadratic forms. These methods will be developed in class and through homework assignments using MATLAB. Applications of linear algebra will include Least Squares Approximations, Discrete Fourier Transform, Differential Equations, Image Compression, and Data-base searching.

**Grading:** Written mid-term exam, homework, MATLAB projects, and a
written final exam.

**Text:**Bollobas: Modern graph theory

**Prerequisites:**Permission of instructor required for students not enrolled in mathematics PhD program.

**Description:**

The course will cover fundamentals of graph theory and the following topics:

- Electrical networks
- Flows, Connectivity and Matching
- Extremal Problems
- Coloring
- Ramsey Theory
- Random Graphs
- Graphs, Groups and Matrices
- Random Walks on Graphs
- The Tutte Polynomial

**Text:**There is no one text; various relevant books will be on reserve in the library.

**Prerequisites:**There are no formal prerequisites, but the course assumes a level of mathematical maturity consistent with having had good courses in linear algebra (such as 640:350) and real analysis (such as 640:411) at the undergraduate level. It will help to have seen at least a little prior combinatorics, and (very) rudimentary probability will also occasionally be useful. See me if in doubt.

**Description:**This is the first part of a two-semester course surveying basic topics in combinatorics. Topics for the full year should incude most of:

- Enumeration (basics, generating functions, recurrence relations, inclusion-exclusion, asymptotics)
- Matching theory, polyhedral and fractional issues
- Partially ordered sets and lattices, Mobius functions
- Theory of finite sets, hypergraphs, combinatorial discrepancy, Ramsey theory, correlation inequalities
- Probabilistic methods
- Algebraic and Fourier methods
- Entropy methods

**Subtitle:**Analysis of boolean functions

**Text:**

**Prerequisites:**642:582 (Combinatorics I) is adequate preparation for the course, though is not required. Others may take it with permission of the instructor. Students in computer science with a good background in theory of computing are generally encouraged to take the course but should talk to me to discuss whether they are mathematically prepared for the course.

**Description:**See the description here

**Text:**There is no official text. Notes will be provided. There are many excellent texts. Any of the following is recommended: L. Breiman,

*Probability*; R. Durret,

*Probability: Theory and Examples*, second edition; D. Williams,

*Probability with Martingales.*

**Prerequisites:**Graduate analysis on the level of 640:501 (basic measure and integration theory, point set topology) is essential. An understanding of undergraduate probability as presented in Ross,

*A First Course in Probability*is very desirable, but not absolutely necessary.

**Description:**This course will survey topics fundamental to probability and stochastic process theory. The purpose is to give the student As much as possible of the following outline will be covered.

- Probability Spaces: construction; monotone class theorems; independence.
- Random variables: distribution; Kolmogorov extension theorem; expectation; types of convergence.
- Laws of large numbers.
- Convergence in distribution and the central limit theorem.
- Brownian motion: construction; basic properties.
- L'evy processes and infinitely divisible distributions: construction.
- Martingales in discrete time.
- Martingales in continuous time.

**Subtitle:**Variational Inequalities, Obstacle and Free Boundary Problems in Mathematical Finance

**Text:**

**Prerequisites:**

**Description:**Please click here for the course text, prerequisites, co-requisites, and description.

**Subtitle:**Infinite in All Directions

**Text:**No textbook. Instead, selected material from various texts will be handed out.

**Prerequisites:**Basic knowledge in physics and partial differential equations. Permission of instructor required for students not enrolled in the mathematics Ph.D. program.

**Description:**Our "best" physical theories are all plagued by infinities. Already in classical special-relativistic physics, the electromagnetic force field surrounding a point charge diverges in magnitude as the location of the point charge is approached, but its direction depends on how you approach it: the force is "infinite in all directions." This problem is not resolved by general relativity, on the contrary! Naked singularities abound, and the lame response has been to "censor them" (or rather the inquisitive minds that want to explore them). And, to be sure, the problem resurfaces in quantum field theory under the name "ultraviolet divergences." But we can do better than that!

This course introduces interested students to these fundamental conceptual problems, explains the deficiencies of the currently prescribed remedies, and then develops in detail a mathematical framework which promises to restore a well-defined finiteness to the whole affair.

**Text:**

**Prerequisites:**Real Analysis and basic Linear algebra; ordinary differential equations at a basic level will help.

**Description:**We begin with review of basic topics from Functional Analysis, which are relevant to spectral and scattering theory. It includes basics like Hilbert spaces, and Hilbert space operators. Then, compact operators and some classical applications to PDE and spectarl theory, including the Fredholm Alternative Theorem.

Then, we discuss more advanced topics, like construction of self-adjoint operators and finding their spectral properties; the relation to PDE and Quantum Mechanics will be treated in detail. Some problems, motivated by current research in nonlinear dispersive equations will be described.

**Subtitle:**Collective Phenomena in Equilibrium and Nonequilibrium Systems

**Text:**

**Prerequisites:**For information about prerequisites contact Joel Lebowitz, lebowitz@math.rutgers.edu

**Description:**The 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 equilibrium ensembles. The latter involve dynamical microscopic considerations and are generally described on the microscopic level by reaction- diffusion type equations. The resulting patterns are visible everywhere in biological systems. They range in scale from microns for cells forming an organism to hundreds of meters for flocking birds.

Choice of specific topics will be based on student interest. For background material see the following books:

- Mathematical Biology I and II, J.D. Murray (Springer)
- Large Scale Dynamics of Interacting Particles, H. Spohn (Springer)
- A Kinetic View of Statistical Physics, P. Krapivsky, S. Redner and E. Ben-Naim (Cambridge)
- Dynamics of Self-Organized and Self-Assembled Structures, R. Desai and R. Krapral (Cambridge)
- Evolutionary Games and Population Dynamics, J. Hofbauer and K. Sigmund (Cambridge)

The course will be informal and interactive. For information about prerequisites please contact me: Joel Lebowitz, lebowitz@math.rutgers.edu