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# Faculty Research Areas

Our faculty perform original research in many different areas of mathematics.

**Abstract Algebra** is the branch of mathematics concerned with such structures as groups, rings, fields, vector spaces, and modules.
Each of these algebraic structures are abstractions of the number
systems and arithmetic that we deal with in everyday life. Research in
the field of Abstract Algebra is often aimed at trying to understand the
deep underlying properties of these abstract algebraic structures.

Faculty who perform research in the area of abstract algebra:

- Allan Berele - associative rings and algebras
- Sarah Bockting-Conrad - algebraic combinatorics, representation theory, quantum groups
- Andrew Carroll - representation theory, invariant theory
- Stefan Catoiu - noncommutative algebra, representation theory
- William Chin - Hopf algebras, representation theory
- Christopher Drupieski - representation theory, Lie algebras, algebraic groups
- Yevgenia Kashina - Hopf algebras

**Analysis**
is the branch of mathematics that includes the theories of limits,
derivatives, integrals, infinite series, and continuous functions. It
is an extension of the ideas and techniques that were first developed in
the study of Calculus.
The modern foundations of mathematical analysis were established in the
17th century by such noted figures as Isaac Newton and Gottfried
Leibniz. Today, the techniques of mathematical analysis are applied
throughout the physical sciences, in such fields of engineering as signal processing, and in quantitative finance.

Faculty who perform research in the area of Analysis:

- Enrico Au-Yeung - pure and applied harmonic analysis, functional analysis
- Karl Liechty - analysis, probability, mathematical physics
- Stefanos Orfanos - operator algebras
- David Sher - spectral geometry
- Gang Wang - Fourier analysis, probability theory, stochastic processes
- Ahmed Zayed - applied harmonic analysis

**Combinatorics**
is the art and science of counting. In all branches of mathematics, the
question "How many?" often arises, and the tools of combinatorics can
be used to try to answer this question. Fundamentally, combinatorics
studies discrete structures as opposed to continuous ones, making the
feel of this branch of mathematics quite different from, say,
differential equations. Examples of discrete structures include graphs, permutations, finite (or countably infinite) groups, and cell decompositions of topological spaces.

Faculty who perform research in the area of Combinatorics:

- T. Kyle Petersen - enumerative, algebraic, and topological combinatorics
- Bridget Tenner - enumerative, algebraic, and topological combinatorics

**Dynamical Systems **is
the study of systems that change or evolve over time in accordance with
some fixed set of rules. Examples of dynamical systems include
populations of bacteria that grow at a particular rate, or the motion of
a pendulum, which is governed by the rules of Newtonian mechanics.

Faculty who perform research in the area of Dynamical Systems:

- Ilie Ugarcovici - dynamical systems, ergodic theory, population dynamics

**Number Theory****
** is the study of the integers and of their arithmetic. Number theory
also includes the study of other related number systems where it also
makes sense to talk about such concepts as divisibility and primes. The subject of number theory is renowned for its many old and simply-stated problems
that have resisted the best efforts of generations of mathematicians.
As a result, modern number theory has come to borrow tools from
virtually every branch of mathematics, and has become an incredibly
broad and deep field with scores of specializations and applications.
Some of the most well-known applications of elementary theorems from
number theory are in the field of cryptography.

Faculty who perform research in Number Theory:

- Nick Ramsey -
*p-*adic properties of modular forms, Euclidean rings and ideals

**Probability Theory** is concerned with the mathematical analysis of random phenomena, and is the mathematical foundation for statistics.

Faculty who perform research in Probability Theory:

- Karl Liechty - analysis, probability, mathematical physics
- Gang Wang - probability theory, stochastic processes

**Statistics**
is a discipline that uses mathematical and statistical theories to
learn from data. Statistics concerns every aspect of data, from the
design of experiments, to collecting data, to analyzing data, to
presenting findings. Statistical techniques can be applied in a wide
variety of disciplines such as biology, business, economics, psychology,
public health, and education.

Faculty who perform research in Statistics:

- Desale Habtzghi - survival analysis, biostatistics, statistical computing
- Juan Hu - geospatial data modeling, massive data analysis, nonparametric modeling
- Hung-Chih Ku - statistical genetics, genome-wide association studies
- Yiou Li - methodologies in statistics and computational mathematics
- Claudia Schmegner - mathematical statistics, Bayesian analysis, sequential procedures
- Philip Yates - applications of statistics in environmental science, hydrology, and sports; statistics education
- Gang Wang - probability theory, stochastic processes, statistics