# Research

As the term "mathematical sciences" correctly suggests, pure mathematics, applied mathematics, statistics, and mathematics education coexist harmoniously within the Department. This is also reflective of the varied research interests which exist in our Department.As with any quality mathematics department in the U.S., the strength of our Department depends inextricably on the strength of the research programs of the faculty and the soundness of our academic programs. Increasingly, the tools-the concepts and techniques-of mathematics are playing a vital role in the advancement of science and technology. As these tools are discovered, developed, and refined through mathematical research, they also feed into a broader effort in the training of technical personnel in which all mathematics Departments participate. Faculty research is essential in fostering excellence in our training of undergraduate and graduate students.

Owing to our extraordinarily successful recruitment efforts during the past decade, we have been able to significantly upgrade the quality of our faculty. The Department now has a strong and talented faculty with research programs covering many important areas in pure and applied mathematics, as well as in statistics. Currently, these areas of strength include: Index Theory, Operator Algebras, Noncommutative Geometry, and Differential Geometry; Dynamical Systems; Mathematical Physics; Applied and Computational Mathematics; and Statistics. The diversity of our faculty's research indeed mirrors the intellectual trends of mathematics in a time of burgeoning applications.

### Research Expertise

## Algebra and Number Theory

### Patrick Morton

Prof. Morton works in number theory, algebra, and geometry. In number theory he has mainly studied Diophantine equations that are related to algebraic structure, including the -1 Pell equation, elliptic curves, and equations that arise from dynamical systems. Most recently he has discovered connections between subtle number theoretic properties of the Legendre polynomials and complex multiplication on supersingular elliptic curves. He has studied algebraic approaches to finding the Galois groups of equations whose roots are the periodic points of polynomial and rational maps over a field, and helped to open up the new field of algebraic and arithmetic dynamics, which studies dynamical systems from an algebraic or number-theoretic point of view. Galois theory is one of his deep interests, along with advanced Euclidean and hyperbolic geometry.

## Dynamical Systems

The theory of dynamical systems is a branch of mathematics that has been created in order to describe, explain, and predict the behavior of mathematical models of various phenomena from the experimental sciences. Those models are systems that evolve in time. This evolution is determined by the state of the system at a given moment, and the laws for the evolution, specific for the system. Those laws may be given by a system of differential equations, or just by one transformation that is iterated.

In order to discover and examine possible interesting features of various systems, mathematicians have introduced many systems that are not models of any concrete real phenomena. One can study them by purely theoretical methods or investigate them on computers, or, very often, both. This approach has turned out to be successful. For instance, it has led to the discovery of chaotic phenomena. Those phenomena were subsequently found in most of the experimental sciences and this has had a tremendous impact on the way scientists think. It is now widely recognized that the deterministic nature of a system does not imply long-term validity (the best example is weather prediction), except in a statistical sense.

## Mathematical Physics

The research work of this group involves statistical and asymptotic properties of classical and quantum systems; the theory of integrable systems and exactly solvable quantum field and statistical mechanics models; random matrices and orthogonal polynomials; modern theory of special functions; representation theory of Lie algebras and quantum group; noncommutative geometry; quantum chaos; along with related number theoretic aspects of spectral theory.

The Mathematical Physics Group maintains very strong working contacts with all the leading world centers in the Group's research areas. The members of the Group have been repeatedly invited as short-term visiting scholars by several top Universities, and the Group has hosted several high-profile visitors-Bernard Malgrange, Tetsuji Miwa, Sergei Novikov, just to name a few. The Group also intends to strengthen its potential by inviting long-term visitors for carrying out important joint research projects.

## Modern Analysis/Geometry

We have a strong active group in Modern Analysis/Geometry. The topics covered by this include noncommutative geometry (i.e. operator algebras, index theory), geometric theory of boundary value problems, Riemannian and Finsler geometry, symplectic geometry, algebraic topology, and K-theory.

This general area is at the center of the recent flourishing interaction of mathematics and physics. We are fortunate to have strength in this active area since it builds on the connections between the different subjects represented and, hence, encourages and stimulates interaction between all the areas in the Department. Discover activities in the Modern Analysis and Geometry Seminar.

## Applied and Computational Mathematics

The diverse research interests among the members of this group have a unified theme-namely, the development of mathematical modeling and computational techniques for the study of nonlinear phenomena.

## Statistics & Biostatistics

The members of the Statistics Group are Benzion Boukai, Fang Li, Hanxiang Peng, Jyotirmoy Sarkar, Fei Tan, Honglang Wang and Wei Zheng. Their research covers a wide array of areas such as

- Applied Probability
- Bayesian Statistics
- Biostatistics
- Design of Experiments
- Empirical Likelihood
- High Dimensional Statistical Inference
- Longitudinal/Functional Data Analysis
- Mathematical Statistics
- Reliability Theory
- Sequential Analysis
- Spatial Statistics
- Time Series

The Statistics Group offers two Ph.D. degrees: a Ph.D. in Mathematics with specialization in Statistics, and a joint Ph.D. degree in Biostatistics with the Department of Biostatistics in the IU School of Public Health (see Appendix IV-2 for more details). It also offers an MS degree in Applied Statistics. In addition to the responsibilities of teaching, student advising and mentoring for all these Statistics programs, the Statistics Faculty also teach some of the courses in the Actuarial Sciences program of the Department. The Statistics Group has a seminar in which local and visiting researchers give expository talks on their current work. These talks are also attended by graduate students and faculty from other Departments within the University.

The Statistics Group also maintains an active Statistics Consulting Center (SCC). SCC provides consultation, collaboration, and educational outreach to all IUPUI researchers and other non-university clients including local governments to meet and exceed their need in statistics and biostatistics. SCC provides a synergistic environment for training up graduate students in the art of statistical consulting, for identifying research problems with high utility to applied investigators, and for finding practical examples for teaching and testing. Thus, the activities of SCC blend seamlessly with the mission of the department.

## Math Education

## Applied Mathematics

The diverse research interests among the members of this group have a unified theme-namely, the development of mathematical modeling and computational techniques for the study of nonlinear phenomena.

Distinguished professor finds beauty in mathematics

Alexander Its, Ph.D. Distinguished Professor