Vanderbilt Mathematics
Colloquia
Spring 1999

    April 27. Beifang Chen, of the Hong Kong University of Science and Technology, Kowloon. Counting Cohesive Geometric Objects. Mathematics started with counting. Cantor knew how to count an infinite number of objects, but was finally mad when counting uncountable sets. Fortunately, Euler and Poincare taught us how to count cohesive objects, including some of those uncoutable sets. In this talk I will present a combinatorial theory on geometric measures, a typical connection between combinatorics and other branches of mathematics from the counting point of view.

    April 26. John Lavery, of the Army Research Office (Research Triangle Park, NC). First, a 10-minute talk about The Applied Analysis Program at ARO. The Applied Analysis Program at the Army Research Office supports research in mathematical analysis for advanced solid materials for structures, armor and sensors, soil and granular materials, fluid flow for chem/bio defense and diesel combustion, photonic bandgaps, nonlinear dynamics for chaotic communication devices, inverse scattering for landmine detection, data fusion and modeling of irregular surfaces. --- Then, a 40- or 50-minute talk on Nonoscillatory Splines: Shape-preserving, Multiresolution, Piecewise Polynomial Geometric Modeling. We discuss a new class of "nonoscillatory" polynomial interpolating and approximating splines that preserve the shape both of smooth data and of data with abrupt changes in magnitude or spacing. These splines do not require constraints, penalties, a posteriori filtering or interaction with the user. One nonlinear minimization principle with no ad hoc components and with only discretization and "regularization" needing to be chosen works for all cases. Univariate and multivariate cases and cubic and higher-degree splines are treated in one and the same framework. Finally, these splines can be implemented using efficient interior-point methods for convex nonlinear optimization.

    April 15. David R. Larson, of Texas A&M University. Operator Algebras, Wavelets and Frames. Orthonormal wavelets can be regarded as complete wandering vectors for a system of bilateral shifts acting on a separable infinite dimensional Hilbert space. Operator algebraic considerations lead naturally to certain results for orthonormal wavelets, Riesz wavelets and their kindred frame wavelets, that might not seem so natural from a purely function-theoretic point of view. These considerations also yield alternate derivations of some well-known function-theoretic results. There are connections between this functional analytic approach to wavelet theory and the harmonic analysis approach developed by Guido Weiss and his group over the past five years.

    April 8. Alexander Kostochka, of the Russian Academy of Sciences, Institute of Mathematics, Novosibirsk, Russia. On the number of edges in color-critical graphs and hypergraphs. Studying color-critical graphs initiated by G. Dirac in the fifties helps to understand some reasons why the chromatic number of a graph is high. One of the interesting questions on critical (hyper)graphs is: how small can the number of edges in a k-critical (hyper)graph be with a given number of edges? Basic results are due to Dirac (1957 and 1974) and T. Gallai (1963). Somehow, between 1974 and 1994, there was little progress in this direction. But in the last 4 years new interesting bounds were proved. The aim of the talk is to review the recent progress on the topic.

    April 6. Zeph Grunschlag of the University of California at Berkeley. Connections between word hyperbolic groups and formal languages. Let G be a group with a finite set X generating G as a monoid. The word problem for G is the set of words on X which represent the identity element of G. Thus the word problem is a certain language (i.e. it is a subset of the free monoid X^*). For example, if G is the cyclic group Z₅ = <x|x⁵ = 1> then x⁵, x¹⁰, x¹⁵, ... are in the word problem for G. -- There are well known characterizations of certain classes of groups in terms of the word problem. Indeed, G has a regular word problem if and only if it is finite, and a context free word problem if and only if it is virtually free. It is also possible to classify word hyperbolic groups in this manner. In a different direction, one can define a notion of regularity for subsets of word hyperbolic groups. The problem of computing the Gersten-Stallings angle between subgroups of a hyperbolic group can often be reduced to the membership problem of a certain regular subset. This implies the following theorem:

    March 25. Heinz-Juergen Voss, of the Technical University of Dresden. Light subgraphs of multigraphs embedded in compact 2-manifolds. This is joint work with Stanislaw Jendrol, P.J.Safárik University Kosice, Slovakia. Fabrici and Jendrol' proved that each 3-connected plane graph with a k-path (a path of k vertices), contains a k-path such that each vertex has a degree at most 5k. Further they showed that each 3-connected plane graph with at least k vertices contains a connected subgraph on k vertices such that each vertex has a degree at most 4k+3 for each k>3. Both bounds are sharp. We generalized these results to compact 2-manifolds M of Euler characteristic x(M)<0. Three of our results are: each polyhedral map of M with a k-path contains a k-path such that each vertex has a degree at most . Equality for even k>2. If G is a 3-connected multigraph on M without loops and 2-faces then the degree bound is (6k-2e)(1+|x(M)|/3), where e=0 if k is even, and e=1 if k is odd. If G is a 3-connected graph on M, then the degree bound is for k>4, x(M)<0 and for k>2, x(M)<-3. All the bounds for G are the best possible.

    March 23. Joachim Stöckler, of the University of Missouri at St. Louis. On Wavelet Frames Generated by Multiresolution Analysis. Several approaches to wavelet frames generated from multiresolution analysis were proposed in the literature. Let us recall that a family Y={y_i; i=1,...,n} is said to generate a wavelet frame if the dilates and translates of all functions y_i, defined by , satisfy the relation

    March 19. Boris Okun, of Ohio State University. l₂-homology of Coxeter groups. A famous conjecture of Hopf states that the Euler characteristic of a closed aspherical manifold of even dimension satisfies the inequality (-1)ⁿx(M²ⁿ) > 0. In recent work with Mike Davis, we calculated the l₂-homology groups of (some) right-angled Coxeter groups. This calculation implies the Hopf conjecture for non-positively curved cubical manifolds of dimension 4 and also leads to a new obstruction for graph embedding problems.

    March 17. Jonathan Corbett, of the University of Washington in Saint Louis. A Brief Introduction to Spatio-Temporal Wavelets. A function is called an orthonormal wavelet in a Hilbert Space, H, if the family of integer translations of its dyadic dilations forms an orthonormal basis for H. A family of orthonormal wavelets is actually a discretization of a family of continuous wavelets which is the orbit of a single "mother wavelet" under the action of the affine, or "ax + b," group. This family is tuned to the translations and dilations of the affine group and can identify these features in signals. We may construct wavelets on other physical groups wavelet families tuned to the actions of those groups. Previously, such wavelets had been created for the Galilei group of constant velocity motions. We shall extend these constructions to Newtonian groups, which add arbitrary levels of acceleration to the Galilei group, and to Rotational motion groups. We shall also propose an algorithm for estimating such motion in digital image sequences and consider simultaneously estimating translational and rotational motion parameters.

    March 5. Alexander Dranishnikov, of the University of Florida. Alexandroff's Problem and the Novikov Conjecture. In late 20s P.S. Alexandroff came with a definition of the cohomological dimension and he posed the natural problem: Does the cohomological dimension agree with the standard (covering) dimension for compact metric spaces? This problem has a long history. It was finally solved only in late 80s. Recently a shadow of the Alexandroff problem appeared in the coarse approach to the Novikov higher signatures conjecture.

    March 4. Peter Casazza, of the University of Missouri at Columbia. The Modern Theory of Frames. Frames in Hilbert spaces play a major role in many areas such as signal processing, image and data compression etc. We will survey many of the uses (both concrete and abstract) of modern frame theory. In the process, we will look at some recent advances in the area as well as some of the important open questions.

    February 25. Ervin Gyori, of the Renyi Institute of Mathematics, in the Hungarian Academy of Sciences, Budapest. Applications of Graph Theory to Analysis and Number Theory. Have you seen a monotone increasing real function with a discontinuity at every rational number? As you probably know, it is a non-trivial problem to construct such a function. In this talk, a function will be presented what was not constructed but was discovered in graph theory. It (hopefully) describes the behaviour of a numerical invariant of graphs depending on their minimum degree. ... How many numbers can we choose from among the integers 1,2,...,n so that no six of them have a square number as their product. Why is six in the question? What is the answer? And how does graph theory come into the picture? We will answer these questions in the second part of the lecture.

    February 18. Robert Carr, of Sandia National Laboratories. A New Bound for the 2-Edge Connected Subgraph Problem. Given a complete undirected graph with nonnegative costs on the edges, the 2-Edge Connected Subgraph Problem consists in finding the minimum cost spanning 2-edge connected subgraph (where multiedges are allowed in the solution). A lower bound for the minimum cost 2-edge connected subgraph is obtained by solving the linear programming relaxation for this problem, which coincides with the subtour relaxation of the traveling salesman problem with the costs satisfy the triangle inequality. ... The simplest fractional solutions to the subtour relaxation are the half-integral solutions in which every edge variable has a value which is a multiple of 1/2. We show that the minimum cost of a 2-edge connectes subgraph is at most 4/3 the cost of the minimum cost half-integral solution of the subtour relaxation. This Hamilton cycle which is within 4/3 the cost of the optimal subtour relaxation solution when the costs satisfy the triangle inequality. ... This is a joint work with R. Ravi.

    February 16. Bin Han, of Oklahoma State University. Analysis and Construction of Optimal Multivariate Wavelets. In this talk, we shall talk about some basic concepts and some developments in wavelet theory. Refinable functions play an important role in wavelet theory since wavelets are built from them. The success of wavelet theory lies in the three most important properties of refinable functions: short support, high approximation order and reasonably high smoothness. -- First of all, we shall talk about how to measure and characterize the smoothness of a refinable function. It is known that short support, high approximation order and high smoothness of a wavelet function are competing objectives in design of wavelets. Next, we shall talk about optimal wavelets and investigate the highest possible approximation order and highest possible smoothness order of a refinable function with respect to its support. -- Finally, a newly developed TCBC algorithm to construct biorthogonal wavelets will be presented.

    February 9. Bojan Popov, of the University of South Carolina. Conservation Laws and Transport Equations. We are interested in the analytical properties and numerical approximations of the entropy solution operator to scalar convex conservation laws. In this context, by duality, one encounters linear equations with discontinuous transport velocities. We discuss our new results on existence, uniqueness and regularity of solutions to linear transport equations. We use these regularity results to describe a new contractivity space for conservation laws. We discuss the applications of these results in proving error estimates for adaptive numerical methods for conservation laws. (Refreshments afterward in SC-1425.)

    February 2. Guergana Petrova, of the University of South Carolina (Columbia, South Carolina). Transport Equations and Velocity Averages.

    January 14. Catherine H. Yan, of New York University (Courant Institute). A Recurrence Associated with Extremal Problems. In this talk we study sequences of integers {g(n)} defined by recurrence relations of the following form: