The Mathematics Department consists of 24 regular faculty members, several postdoctoral researchers and frequent visiting faculty in many areas of mathematics.

Its faculty enjoys national and international recognition in Algebra, Analysis, Differential Geometry, Mathematical Physics, Probability and Statistics, Scientific Computation, Theoretical Computer Science, and Topology. The researchers in Scientific Computation and in Statistics, and an increasing number of faculty in other areas, collaborate actively with colleagues in other units of the university such as the Schools of Engineering, Liberal Arts and Sciences, Medicine, and Public Health.

During the past five years our regular faculty have published over 100 research articles and several books. The regular faculty direct theses in very diverse areas which range through all of Pure Mathematics, Applied Mathematics, and Statistics. You can read brief descriptions of our specialties below. More detailed information can be found on the faculty page.

#### Algebra

Tulane has had a pioneering role in the research on semigroups, abelian groups, modules over valuation domains, and ordered algebraic systems. Some of the standard reference books in these areas are from Tulane authors. Recent work of the faculty is on problems in commutative algebra, the theory of rings and modules, and in the structure and cohomology of commutative semigroups (especially in the finite case). Weekly research seminars lead students to the frontiers of current research.

**Faculty:** Tai Huy Ha

**Adjunct:** Karl Hofmann

#### Domain Theory and Theoretical Computer Science

Domains are structures that are equipped with partial orders having special properties. Interest in these structures arose in the late 1960s when it was realized they could be used to produce models of the untyped lambda calculus of Church and Curry. This calculus can be viewed as a prototypical programming language without assignment. A whole area of research has sprung up since then, the focus of which is to investigate the structure of domains as well as their applications to areas ranging from theoretical computer science to pure mathematics. For example, recent work has shown that domains can be used to model fractals and to provide a novel approach to Riemann integration that extends established theorems in that area. This last work relies on a construction that introduces probability theory into domain theory. Domain theory is characterized by the relatively simply constructions that are available, but that have surprisingly general applications. The focus of the research taking place here at Tulane encompasses both the theory of domains and their applications to areas such as theoretical computer science. In recent years, the interest has been on modeling crypto-protocols used in the security community to establish secure communication between users on systems such as the Internet. Tulane enjoys collaborations with researchers at a number of other institutions in this area, including the University of Oxford, the Naval Research Laboratory in Washington, D.C., and the University of Paris VII. Collaborators from these institutions regularly visit Tulane, and we currently have an Oxford Ph.D. who holds a postdoctoral fellowship in the department.

**Faculty:** Michael W. Mislove

#### Geometry and Topology

Tulane has a sizable group working in a diverse array of topics in topology and geometry. Research in topology includes algebraic topology, 3 and 4 dimensional manifold theory, continuum theory and holomorphic dynamics. Research in geometry includes complex, differential, and algebraic geometry. The overlap and interplay among various fields of geometry and topology is apparent in this group: the researchers often use a variety of techniques from different areas to attack a problem, for example techniques from gauge theory and algebraic geometry have been used to study topological questions about 4-dimensional manifolds. This group currently has two weekly seminars (topology and geometry) that are well-attended by Tulane students and faculty as well as faculty from Loyola University.

**Faculty:**

Morris Kalka (Chairman)

Slawomir Kwasik

James T. Rogers, Jr.

Albert L. Vitter III

David Yang

#### Applied Mathematics and Partial Differential Equations

The work of the group focuses on the qualitative behavior of solutions to partial differential equations, such as concentration, spatial and temporal patterns, stability, long time behavior, blow-up, phase separation and interfacial motion. Recently, we have studied anisotropic heat transfer. The equations are often nonlinear and are motivated by applications of mathematics to natural sciences. The goal is to understand a phenomenon and to develop mathematical methods that apply to more general situations.

**Faculty:** Xuefeng Wang.

**Post Docs:** Michael Nicholas, Sarah Olson

#### Symbolic Analysis

This area relates to mathematical problems that have their origin in Symbolic Computation. These include the closed-form evaluation of series and integrals, many of which have connections to Dynamical Systems, Combinatorics and Number Theory. The goal is to develop algorithms that will obtain such closed forms or prove their impossibility. Recent work of this group includes the development of Landen transformations that are the analogue of the classical Arithmetic Geometric mean for elliptic integrals.

**Faculty:** Victor Moll

**Visiting Faculty:** Tewodros Amdeberhan

#### Computational Mathematics

This group works on numerical methods for partial differential equations and maintains a strong interest in applications to general fluid dynamics with emphasis on biofluid dynamics. Recent work of this group includes accurate computational methods for flows in bounded domains and developing improved immersed boundary methods for aquatic animal locomotion, bacterial chemotaxis and bioremediation. Members of this group were co-founders of the Center for Computational Science (CCS) where they collaborate in research with faculty from the School of Engineering, the sciences, Health sciences, and more. Through the CCS, this group holds grants that support postdoctoral researchers, graduate students, undergraduates, and distinguished visitors.

**Faculty:**

Ricardo Cortez

Lisa J. Fauci

Alexander Kurganov

#### Probability and Statistics

Group members work in applied probability, probability, statistics, and stochastic processes. Specific research interests include the bootstrap, censored data and survival analysis, dynamical systems with random perturbations, likelihood methods, linear models, Markov chains and Markov processes, and statistical inference. This group cooperates in research, applications, and seminars with the Department of Biostatistics.

**Faculty:**

Gustavo Didier

Michelle Lacey

John R. Liukkonen

Alexander D. Wentzell

#### Mathematical Physics

The research in this area ranges from broad topics in Mathematical Physics such as cosmology and general relativity to geometric aspects of super-symmetric string theories and M-theory.

**Faculty:** Frank Tipler