Graduate Information

Several types of financial assistance are available to graduate students in the Department of Mathematics at LSU. These include teaching assistantships, research assistantship, fellowships, and tuition awards. Most graduate students receive some form of financial support; however, priority for support goes to PhD students, which includes all students who have applied to earn the PhD in Mathematics as their highest degree at LSU.

Qualifying Exam policy is that at least 50% of the credit on each exam will come from the test problem banks below. There will normally be approximately 6 to 8 problems offered on each exam, and students will typically need to turn in approximately 5 of these.

Prof. Sengupta's research work has centered around the following topics:

Prof. Kuo works on three areas in stochastic analysis::

1. White noise analysis,
2. Stochastic differential equations, and
3. Probability theory on infinite dimensional spaces.

The probability group has broad research activities in progress.

Advances in technology are often due to fundamental advances in our understanding of the materials of which things are made. Obvious examples include the transistor and superconductors. Promising new technologies have been spawned by the recent discovery of Photonic Band Gap Crystals. Less spectacular but equally important developments in materials science are providing steady improvements in infrastructure, aerospace, and micro-electronics. The study of materials science has evolved to the point at which scientists from physics, engineering and applied mathematics are working on problems of common interest.

Harmonic Analysis has a long history. In the 19th century it was much concerned with the expansion of functions in terms of series or integrals of trigonometric functions — i.e., Fourier series and Fourier integrals. Such expansions were particularly important because of their effectiveness in the solution of differential equations. Efforts to understand the phenomena of convergence versus divergence of Fourier series were a motivating force behind many developments, including Lebesgue integration, and even Cantor's theory of sets.

Combinatorics is one of the oldest branches of mathematics. It has been influenced by almost all areas of mathematics, including number theory, algebra, topology, mathematical logic, and many more. In recent years, stimulated by the development of computer science, combinatorics has found new applications in algorithm analysis, network designs, and so on. Now it is one of the fastest growing areas in mathematics. The research of the LSU combinatorics group focuses on the study of graphs, matroids, and hyperplane arrangements.

The origin of graph theory can be traced back to Euler when he studied the Königsberg bridges problem in 1736. However, the development of graph theory as a separate subject occurred predominantly in the twentieth century. In the last twenty years, the study of graph structures has attracted more and more researchers. In this study, the objective is to decompose a large graph into small pieces according to certain very specific rules. Results of this study often have a wide range of applications in computer science and operations research. Several members of the LSU Combinatorics group are currently working in this area.

Members of the control group include G. Ferreyra, J. Lawson, M. Malisoff, P. Sundar, and P. Wolenski. The group cooperates with the larger LSU Systems and Control Group which includes faculty from the LSU departments of electrical and mechanical engineering. The concept of control can be summarized by saying that one seeks to influence a dynamical system in order to achieve some desired goal. The dynamical system in mathematical control theory is usually a system of differential or difference equations that depends on a set of parameters, where the parameters are the "control" variables. The idea then is to find these control variables so as to minimize (or maximize) a given objective function, to stabilize the system, or to move the system to a desired destination.

Within Algebra, there are two main research groups at LSU: Algebraic Number Theory and Algebraic Geometry. An algebraic number field is a finite extension field of the field of rational numbers. Within an algebraic number field is a ring of algebraic integers, which plays a role similar to the usual integers in the rational numbers. The study of algebraic number theory goes back to the nineteenth century, and was initiated by mathematicians such as Kronecker, Kummer, Dedekind, and Dirichlet. Gauss called Algebraic Number Theory the ``Queen of Mathematics.'' One motivation for this study was an attempt to establish Fermat's Last Theorem (proved a few years ago by Andrew Wiles). It turns out that, while rings of algebraic integers bear many similarities to the usual integers, there can be significant differences. For example, one may lose the unique factorization of elements into products of powers of primes (the Fundamental Theorem of Arithmetic). However, there is still a unique factorization on the level of ideals in these rings of integers. The researchers in Algebraic Number Theory at LSU also are studying Quadratic Forms both from the algebraic point of view (function fields of quadrics, generic splitting, Milnor K-theory) and from the arithmetic point of view (integral properties, class numbers, zeta functions). The methods used in these studies include techniques from diverse areas of algebra.