The research within the graduate faculty of mathematics spans a wide range of areas of mathematics including:

- Algebra and Algebraic Geometry
- Applied Analysis

(Including Mathematical Biology, Mathematical Finance, Numerical Analysis, Control Theory) - Geometry and Topology
- Discrete Mathematics
- Lie Theory and Representation Theory
- Logic
- Mathematical Physics
- Nonlinear Analysis
- Number Theory and Algebraic Geometry
- Partial Differential Equations

## ALGEBRA AND ALGEBRAIC GEOMETRY

Algebra and algebraic geometry have always been strongly represented in the Department of Mathematics. Faculty members with active interests in these areas include:**Lev Borisov **(algebraic geometry) **Anders Buch** (algebraic geometry, Schubert calculus, combinatorics)**Lisa Carbone** (geometric group theory, Kac-Moody groups, applications to high-energy physics)**Yi-Zhi Huang** (representation theory of vertex operator algebras, conformal field theory and connections with geometry and topology) **James Lepowsky** vertex operator algebra theory, conformal field theory, Lie theory and representation theory, mathematics related to string theory**Vladimir Retakh** (noncommutative algebra, algebraic aspects of analysis)**Siddhartha Sahi **(representation theory, harmonic analysis, algebraic combinatorics)**Charles Weibel **(algebraic K-theory, algebraic geometry and homological algebra)

Other areas of faculty research with roots in or connections to algebra include Discrete Mathematics, Lie Theory and Representation Theory, Logic, Number Theory. These areas offer numerous graduate courses and support various weekly seminars.

There are typically three to five graduate courses in algebra and algebraic geometry each semester (and additional courses in related fields such as number theory.) Various seminars in algebra and related areas are held regularly.

## APPLIED ANALYSIS

A number of mathematics graduate faculty members work in the general area of applied analysis, with applications in a diverse set of fields including mechanics and materials science, biology, and finance:**Fioralba Cakoni** Inverse Problems, PDEs, Integral Equations, Inverse Scattering Theory

**Richard Falk**(Numerical Analysis, partial differential equations)

**Paul Feehan**(Nonlinear elliptic and parabolic PDE, geometric analysis, mathematical physics, probability theory)

**Liping Liu**(Mechanics and Mathematics of Materials)

**Konstantin Mischaikow**(Dynamical systems, mathematical biology, computational topology)

**Daniel Ocone**(Stochastic processes, Stochastic control)

**Eduardo Sontag**(Control theory, mathematical biology)

**Hector Sussmann**(System and Control theory)

**Michael Vogelius**(Numerical Analysis, partial differential equations)

**Wujun Zhang**(Numerical Analysis, computational mathematics)

Faculty in this group participate in various interdisciplinary centers: SYCON(the Rutgers Center for Systems and Control), BioMaPS (Institute for Quantitative Biology), and IAMD (the Institute for Advanced Materials and Devices).

## GEOMETRY AND TOPOLOGY

The department has several faculty members working in a variety of areas of geometry and topology.**Paul Feehan **(Nonlinear elliptic and parabolic PDE, geometric analysis, mathematical physics, probability theory)

**Simon Gindikin**(theory of representations, integral geometry, several complex variables),

**Xiaojun Huang**(Complex Geometry)

**Alex Kontorovich**Automorphic Forms and Representations, Homogeneous Dynamics, Harmonic Analysis and Spectral Theory

**Feng Luo**(Low dimensional topology, geometric structures on manifolds),

**Xiaochun Rong**(Riemannian geometry),

**Natasa Sesum**(geometric flows),

**Jian Song**(geometric analysis)

**Hongbin Sun**(low-dimensional topology and hyperbolic geometry)

**Charles Weibel**(algebraic topology, K-theory),

**Chris Woodward**(symplectic geometry, moduli spaces, Lie groups).

Rutgers has a diverse group of faculty in topology/geometry covering many different areas. The group interacts with a variety of other faculty including **Sagun Chanillo, Zheng-Chao Han, YanYan Li**, and **Hector Sussmann**. Members of the New Brunswick graduate faculty on other Rutgers campuses include: **Jane Gilman **(Reimann surfaces), **Harold Jacobowitz **(Differential geometry), **Lee Mosher **(CR geometry), **John Randall** (Low dimensional topology and geometry, geometric group theory) **Gabor Toth** (differential geometry).

Each year, the department offers several graduate courses in these areas and various. Inaddtion, two D'Atri lectures are given each year by an eminent geometer.

## DISCRETE MATHEMATICS

Discrete mathematics is a rapidly growing branch of modern mathematics, which includes such fields as combinatorics, graph theory, and operations research. It is at the heart of many recent applications of mathematics which relate to computer science, engineering, communications, transportation, decision making by industry and government, and problems of the social, biological, and environmental sciences.

The Department of Mathematics at Rutgers University has a substantial group in discrete mathematics.**József Beck** (Diophantine approximation and uniform distribution (number theory), probabilistic methods, combinatorial games, combinatorial geometry)

**Shubhangi Saraf**(Theoretical Computer Science: complexity theory and property testing)

**Jeffry N. Kahn**(discrete mathematics)

**János Komlós**(discrete mathematics, probability)

**Alex Kontorovich**Automorphic Forms and Representations, Homogeneous Dynamics, Harmonic Analysis and Spectral Theory

**Swastik Kopparty**(Computational Complexity, Error-Correcting Codes, Randomness and Pseudorandomness)

**Fred Roberts**(discrete mathematical models of social, environmental, and biological problems, graph theory, decision making, measurement theory)

**Michael Saks**(theory of computation, partially ordered sets, graph theory)

**Doron Zeilberger**(algebraic and enumerative combinatorics, experimental mathematics).

The group is augmented by faculty in the Computer Science Department, including

**Endre Szemeredi**(Theoretical computer science, graph theory, combinatorial number theory and geometry)---recently retired, and

**Mario Szegedy**(Quantum computing, computational complexity and combinatorics), and by faculty in RUTCOR. Other faculty working in algebra, including

**Anders Buch**,

**James Lepowsky**,

**Vladimir Retakh**, and

**Siddhartha Sahi**

**have interests related to combinatorics.**

Rutgers is also the principal site for DIMACS, the center for discrete mathematics and theoretical computer science, which has many visitors and holds numerous workshops each year.

There are various regularly scheduled seminars in discrete mathematics and related areas. During Spring 2009, there was a weekly Discrete Math Seminar and an Experimental Mathematics Seminar in the mathematics department. Other related series include the Theoretical Computer Science Seminar and RUTCOR seminars.

Each year, the graduate program typically offers a full year course sequence in Combinatorics, an introductory graph theory course, the Experimental Mathematics course, and two selected topics courses. Various courses of interest to discrete mathematics students are offered by the Computer Science Department and RUTCOR.

## LIE THEORY AND REPRESENTATION THEORY

The structure and representation theories of Lie algebras and of Lie groups, and natural analogues and generalizations, are very active areas of research. They include the theory of finite-dimensional and infinite-dimensional Lie algebras and Lie groups as well as vertex operator algebra theory and the theory of quantum groups. They have important connections with many other fields, both classical and modern, including algebraic groups, finite groups, geometry, harmonic analysis, differential equations, topology, number theory, combinatorics, and string theory and conformal field theory in theoretical physics. Faculty members at Rutgers working in these research areas include:**Lisa Carbone** (geometric group theory, Kac-Moody groups, applications to high-energy physics)**Simon Gindikin** (Lie theory, theory of representations, integral geometry, several complex variables)**Yi-Zhi Huang** (representation theory of vertex operator algebras, conformal field theory and connections with geometry and topology) **Alex Kontorovich** Automorphic Forms and Representations, Homogeneous Dynamics, Harmonic Analysis and Spectral Theory**James Lepowsky** vertex operator algebra theory, conformal field theory, Lie theory and representation theory, mathematics related to string theory**Stephen Miller **(automorphic forms, L-functions) **Vladimir Retakh** (noncommutative Lie theory)**Siddhartha Sahi **(representation theory, harmonic analysis, algebraic combinatorics)

Members of the New Brunswick graduate faculty on other Rutgers campuses include **Jane Gilman** and **Diana Shelstad**, whose work involves Lie theory, and **Haisheng Li**, who works in vertex operator algebra theory. In addition, several other faculty members at Rutgers work in directions that have important interactions with Lie theory, including **Jerrold Tunnell** (number theory, automorphic forms).

Students usually begin the study of the theory of Lie algebras, Lie groups and their representations by taking frequently offered introductory graduate courses. Most instruction after that is by further courses, seminars and directed reading, in areas that might involve algebra, analysis, geometry, topology, combinatorics and/or physics, depending on the student's interests. Quite a number of students have received Ph.D.s for work in Lie theory and related fields over a period of many years, and many advanced students are currently working in these areas. Several ongoing seminars are devoted to the discussion of recent research in a wide range of areas related to Lie theory and representation theory. Seminar talks are presented by Rutgers faculty and students and by outside speakers. The Rutgers Physics Department has a very strong research group in areas related to string theory and conformal field theory. Advanced students sometimes attend courses and seminars in the Physics Department, as well as seminars at nearby institutions including the Institute for Advanced Study and Princeton University.

## LOGIC

Rutgers has a small but very strong group in logic:

Grigor Sargsyan (set theory)

Saharon Shelah (model theory and set theory)

Simon Thomas (set theory and group theory)

There are typically one or two graduate courses in logic each year. There is a lively weekly Logic Seminar. Rutgers also hosts the September meeting of the Mid-Atlantic Mathematical Logical Seminar(MAMLS). Faculty and graduate students frequently attend the CUNY Logic Workshop in Manhattan.

For further information, please visit the web page for Professor Cherlin.

## MATHEMATICAL PHYSICS

The Mathematical Physics group at Rutgers University includes:**Tadeusz Balaban** (Mathematical physics) **Eric Carlen** (Functional analysis, probability, mathematical physics) **Giovanni Gallavotti** (mathematical physics) **Gerald Goldin** (infinite dimensional Lie groups and quantum theory) **Sheldon Goldstein** (statistical mechanics, probability theory, foundations of quantum mechanics) **Michael Kiessling** (mathematical physics, statistical mechanics, nonlinear partial differential equations) **Joel L. Lebowitz** (statistical mechanics, material science, dynamical systems) **David Ruelle** (statistical mechanics; dynamical systems) **Avraham Soffer** (partial differential equations, scattering theory) **Eugene Speer** (statistical mechanics, quantum field theory)

This group interacts strongly with the analysis group, Lie Theory group, and with members of the Physics Department interested in statistical mechanics, condensed matter physics, and theoretical high energy physics. (**G. Goldin, S. Goldstein** and **J. Lebowitz** have joint appointments in the Physics Department).

The group is augmented by a large number of post-docs and visitors. It conducts a weekly seminars and a widely attended semi-annual conferences in Statistical Mechanics. There are also weekly condensed matter and high energy seminars, and the departmental colloquia in mathematics and physics are often related to mathematical physics.

The relationship between faculty and students is informal and pleasant with frequent joint lunches; in particular the weekly mathematical physics seminar is followed by a brown bag lunch at which there is much informal discussion of all kinds of problems - both scientific and non-scientific.

## NONLINEAR ANALYSIS

Faculty working in nonlinear analysis include: **Haim Brezis** (Nonlinear analysis; PDE) **Sagun Chanillo** (Classical analysis, PDE)**Zheng-Chao Han** (Nonlinear analysis, PDE)**Yanyan Li** (Nonlinear analysis, PDE)**Roger Nussbaum** (Nonlinear functional analysis )

Nonlinear functional analysis comprises a body of techniques which have been developed since the early 1900's in order to study various nonlinear equations from analysis, geometry, physics and applied mathematics. Typically, these techniques have had a strong functional analytic flavor, but ideas from many other parts of mathematics (notably algebraic and differential topology) have played an important role. The past twenty-five years have seen a particularly explosive growth of nonlinear functional analysis. A partial list includes such developments as the theories of monotone, A-proper and condensing operators, global theories of bifurcation, new methods for finding critical points of real-valued maps from Banach spaces or Banach manifolds, extensions of the classical Leray-Schauder degree and applications of these ideas to concrete nonlinear problems.

The research interests of members of this group include nonlinear functional analysis and its applications to particular problems (e.g., questions about periodic solutions of Hamiltonian systems, existence and qualitative properties of periodic solutions of differential-delay equations, solvability of certain nonlinear boundary value problems from ordinary and partial differential equations, global problems in symplectic geometry, and curvature equations in geometry). The two D'Atri lectures, given each year by an eminent geometer, are sponsored by this group.

A general course in nonlinear functional analysis is usually given every other year, while more specialized courses are also offered. Students are strongly encouraged to take courses in ordinary differential equations and partial differential equations, while knowledge of differential topology and algebraic topology is useful. Of course, reading courses for advanced graduate students are also available. The analysis seminar frequently has speakers on nonlinear problems.

## NUMBER THEORY

Faculty members at Rutgers with research interests in number theory include:**Jozsef Beck **(combinatorics; combinatorial number theory) **Henryk Iwaniec **(Analytic Number Theory) **Stephen Miller **(Automorphic forms; L-functions) **Alex Kontorovich** Automorphic Forms and Representations, Homogeneous Dynamics, Harmonic Analysis and Spectral Theory**Jerrold Tunnell** (Number theory; automorphic forms)=

Research interests are varied including the study of analytic, algebraic and combinatorical number theory. The study of automorphic forms is prominent both in its analytic and algebraic aspects, and there is some connection with the faculty members studying Lie groups.

Several graduate courses in number theory or algebraic geometry are offered each year.

## PARTIAL DIFFERENTIAL EQUATIONS

Specialists in PDE among the faculty include:**R. Michael Beals** (harmonic analysis, fourier integral operators, PDE) **Haim Brezis** (nonlinear analysis, PDE) **Fioralba Cakoni** Inverse Problems, PDEs, Integral Equations, Inverse Scattering Theory**Sagun Chanillo** (Classical analysis, PDE)**Richard Falk** (numerical analysis, finite element method) **Paul Feehan** (Nonlinear elliptic and parabolic PDE, geometric analysis, mathematical physics, probability theory) **Zheng-Chao Han** (nonlinear analysis, PDE) **Xiaojun Huang **(several complex variables)**Michael Kiessling **(statistical mechanics, nonlinear PDE)**Yanyan Li** (nonlinear analysis, PDE) **Roger Nussbaum** (ODE, bifurcation theory, elliptic PDE) **Vladimir Scheffer** (fluid dynamics, Navier-Stokes equation) **Natasa Sesum** (Geometric flows; PDE)**Avy Soffer** (scattering theory, PDE) **A. Shadi Tahvildar-Zadeh** (nonlinear hyperbolic partial differential equations) **Michael Vogelius **(numerical analysis and partial differential equations)

As one can see from the list above, research in PDE at Rutgers is extensive. Topics from both linear and nonlinear PDE are included, and research ranges from the study of properties of general classes of equations to work with particular equations that occur in the physical sciences as well as their numerical solution.

The graduate curriculum in PDE builds on the standard courses in real, complex, and functional analysis. The introductory course, which is given almost every year, treats such topics as elementary distribution theory, fundamental solutions of the heat, wave, and Laplace equations, the Cauchy problem for linear PDE's, elliptic equations, and Sobolev spaces. In recent years other courses have dealt with distribution theory, pseudodifferential and Fourier integral operators, microlocal analysis, paradifferential operators, measure theoretic methods for variational equations, and nonlinear propagation of singularities. Courses in nonlinear functional analysis and functions of several complex variables are usually offered every other year. Some study of the numerical solution of PDE's is included in the basic one year survey course in numerical analysis given each year. Every other year, a more specialized course is offered on this topic.

Graduate students at Rutgers are able to attend courses and seminars at Princeton University, where research in linear PDE and several complex variables is also very active. The Courant Institute of New York University, a leading research center for pure and applied analysis, is about one hour away from the Rutgers campus.