THE MIDWEST GEOMETRY CONFERENCE 1998

FIRST ANNOUNCEMENT

LOUISIANA STATE UNIVERSITY, BATON ROUGE, LA

OCTOBER 23-25, 1998

The eighth Midwest Geometry Conference will take place October 23--25 at the Louisiana State University, Baton Rouge, LA 70803. There will be four sessions, with the first beginning on Friday Oct. 23 in the early afternoon and the fourth ending on Sunday, Oct. 25, at about noon.
Each section will consist of a one hour talk for a general audience by a main speaker and two talks by selected speakers.
The topics of the sessions, the organizers and featured speakers will be:

INTEGRAL GEOMETRY, Friday Oct. 23:
Organizer: G. Olafsson, LSU. e-mail: olafsson@math.lsu.edu):
2:00-3:00 S. Helgason (MIT)
3:00-3:30 Discussion and break
3:30-4:30 B. Hall (University of California)
4:30-5:30 E.T. Quinto (Tufts University)

GEOMETRIC INEQUALITIES, Saturday Oct. 24:
Organizer: T. Branson, Univ of Iowa. e-mail: branson@math.uiowa.edu
08:00-08:30 Coffee
08:30-09:30 E. Lieb (Princeton)
09:30-10:00 Discussion and break
10:00-11:00 M. Loss (Georgia Tech)
11:00-12:00 W. Beckner (University of Texas)

INTEGRABLE SYSTEMS, Friday Oct. 23
Organizer: L. Smolinsky, LSU. e-mail: smolinsk@math.lsu.edu
2:00-3:00 M. Adler (Brandeis University)
3:00-3:30 Discussion and break
3:30-4:30 A. McDaniel (Georgetown University)
4:30-5:30 Chuu-Lian Terng (Northeastern University)

LOW DIMENSIONAL GAUGE THEORY, Sunday Oct. 24
Organizer: A. Sengupta, LSU. e-mail: sengupta@math.lsu.edu
08:00-08:30 Coffee
08:30-09:30 I. M. Singer (MIT)
09:30-10:00 Discussion and break
10:00-11:00 A. Ashtekar (Penn State University)
11:00-12:00 R. Formann (Rice University)

FUNDING:
Funding from NSF and Louisiana State University will allow us to provide a limited amount of support (mainly for lodging, double occupancy) to participants, in particular graduate students, who do not have other sources of support. We would like to encourage all participants to look for other sources of support.
WEB SITE AND E-MAIL:
We will soon set up a web site at:
http://math.lsu.edu/~mgc
We will post all relevant information there and update it on a regular basis. You will find there a detailed schedule, abstracts for the talks, travel instruction, hotel information and forms.
You can also contact the organizers by e-mail at: mgc@math.lsu.edu

INTEGRAL GEOMETRY
The theory of Lie groups links Geometry, symmetries of physical systems, the theory of transformation groups of manifolds. The central geometric objects are the quotient spaces $G/H$, where $H$ is a closed subgroup of $G$. Integral transforms can be interpreted in the terms of the group $G$, its subgroups, and the representation theory of these groups. A good example of this viewpoint is the work of S. Helgason on the Fourier and Radon transform on semisimple Riemannian symmetric spaces. Here the spherical functions extend the notion of the exponential function to symmetric spaces, and the Radon transform is an integral transform over orbits in $G/H$, where $H$ is now a maximal compact subgroup of $G$, of a closed unipotent subgroup of $G$. Classically, the Fourier transform is a tool for analyzing and solving differential equation, and the Radon transform is tightly connected to tomography. In the general setting on symmetric spaces, these transforms have become important tools in the investigation of geometric properties of solutions to invariant differential equations, e.g., the Huygens principle, and equipartion of energy.
The Fock space and its generalizations, the highest weight representations of semisimple Lie groups, have for a long time been related to Physics through quantization, the Bargmann transform, and Toeplitz operators. Lately the Bargmann transform has been generalized to both compact Lie groups and some semisimple symmetric spaces, and spaces of connections modulo gauge transformations. These generalizations allow quantization in a more general setting, and will be important for further developments in both Analysis and Physics.
Tomography is exciting mathematics that allows doctors to image the inside of the body using X-rays or other indirect data. In many problems in medicine and industry, some tomography data are missing, and this gives rise to limited data problems. The author will give an introduction to limited data tomography and describe some of the fundamental roles integral geometry plays in the subject. Microlocal analysis will be used to explain how different types of limited data detect singularities. Reconstructions from industrial data will be presented to illustrate these ideas.

GEOMETRIC INEQUALITIES
The subject of Geometric inequalities has enjoyed accelerated growth, and increased in importance since the early 1980's. One of the centerpieces of Geometric Analysis, the Yamabe problem, is really the study of a sharp inequality of Sobolev embedding type. The solution of the Yamabe problem in its original form was completed by Schoen in 1984. But questions remain, and new problems have opened up; for example, the Yamabe problem on noncompact manifolds; variants of the Yamabe problem on manifolds with boundary; and the existence and meaning of higher critical metrics of the Yamabe functional on compact manifolds. Geometric inequalities are the ``bread and butter'' of work on the major nonlinear problems now confronting geometers; for example, harmonic maps, Ginzburg-Landau vortices, gauge theory, and liquid crystals. They forge a link between spectral invariants of differential operators, like the nonlocal functional determinant, and the harmonic analysis of manifolds.
A fundamental paper which has been crucial to many of these developments that of Lieb, giving the best constants and extremals for Hardy-Littlewood-Sobolev (HLS) inequalities in $R^n$. This led directly to the work of Beckner and of Carlen-Loss on exponential class inequalities of Moser-Trudinger type, and logarithmic HLS inequalities. This, in turn, was of fundamental importance in work on estimating and extremizing the functional determinant on the sphere.

INTEGRABLE SYSTEMS
Integrable systems is one of the classical subjects in mathematics. The subject ties the classical systems of physics to our modern methods and understanding. It forms a portion of ordinary and partial differential equation theory. It is a motivating force for Symplectic and Poisson geometry. Lie algebras and Kac--Moody algebras occur because of symmetries, but the subject also has a subtle relationship to algebraic geometry. This relationship is explicit in the notion of an algebraically completely integrable system. One such system constructed by Hitchen has played a role in string theory. In recent years, the objects of study or application to integrable systems have included physical systems, harmonic maps, random matrices, and generalizations of the Korteweg-de-Vries hierarchy by Drinfeld and Sokolov. Professor Mark Adler will be the Plenary speaker. His early work evolved into the Kostant-Adler-Symes theorem which is one of the main tools for understand! ing symmetries and producing completely intergrable systems. In work with Pierre van Moerbeke, he has shown the importance of the link with algebraic geometry.

LOW DIMENSIONAL GAUGE THEORY
Several beautiful mathematical ideas at the juncture of geometry, topology, and physics have grown out of studies of gauge theories in dimensions two and three. This section will contain lectures explaining some of these ideas.
The gauge theory of interest in dimension 3 is based on the Chern-Simons functional. Witten showed that heuristic functional integrals arising from the quantum field theory of Chern-Simons gauge theory can be used to obtain topological invariants of knots in 3-dimensional manifolds. Although these integrals still defy a fully rigorous formalization, the consequences of Witten's considerations have spawned a vast industry at the juncture of Mathematics and physics.
Classical two dimensional Yang-Mills theory was studied by Atiyah and Bott from a Morse theoretic point of view. In the two deep works by Witten, it was shown, among other things, that quantum Yang-Mills theory on a surface can be used to obtain an understanding of the structure of the moduli space of flat connections over the surface. The relationship of lattice gauge theory on a closed Riemann surface to the symplectic nature of the moduli space was explicated further by Forman. Motivated originally by these issues, but branching out in a different direction, Forman went on to develop a discrete Morse theory on cell-complexes. This theory has recently been the focus of much interest, and has unexpected applications to questions in Combinatorics.