Kempner Colloquium Abstracts

Fall 2005

The circle method originates with work of Hardy and Ramanujan on the partition function. It has been developed into a refined set of number theory tools by Hardy and Littlewood, Vinogradov, Davenport and others. This talk contains a look back over this history, and a sketch of an important recent variant of the method, due to Heath-Brown.

In 1957 a race was underway, and the outcome was almost a dead heat. Consider a homogeneous cubic polynomial with rational integral coefficients. The competition was to show, assuming that the cubic has "sufficiently many" variables, that the polynomial necessarily possesses a non-trivial integral zero (and more generally, linear spaces of rational solutions). The three contestants in the photo-finish (Birch, Davenport and Lewis) applied quite different methods, and later developments demonstrated these methods to be applicable in a wider context. In this talk we discuss what can be said for the analogous problem for pairs of cubic equations. The methods we present are based on simple geometry, linear algebra and harmonic analysis, and should be largely accessible to those less familiar with number theory.

Ricci flow is an evolution equation for a Riemannian metric on a manifold. It is the basis of Perelman's recent work on Thurston's geometrization conjecture. The large-time behavior of Ricci flow is largely unknown. I'll discuss the relevance of a special type of solution, called an expanding soliton. In order to get large-time limits in general, it turns out that one has to extend the notion of Ricci flow from Riemannian manifolds to Riemannian groupoids.

No prior knowledge of Ricci flow will be assumed.

The Riemann ζ-function ζ(z) = ∏ (1 - 1/p^z)^-1 encapsulates in complex analytic form information about the distribution of the prime numbers. Analogously, we may define the ζ-function of a smooth flow in terms of the set of prime periods of the flow. Complex analytic properties of this zeta function reflect topological and statistical properties of the flow. In this talk we give a little history going back to the Selberg and Weil zeta functions, discuss the outstanding problem in the field (of great interest to physicists), and talk about some recent results.

The talk is intended for a general audience - in particular, it is suitable for graduate students and does not require a background in dynamical systems (or complex analysis).

The theory and practice of error-correcting coding has recently witnessed a major paradigm shift, due to the emergence of a simple, yet extremely powerful technique for near-optimal decoding based on probabilistic inference on graphical models. This technique, termed belief-propagation decoding, represents a straightforward adaptation of well-known methods borrowed from artificial intelligence, mathematical statistics and statistical physics to the setting of error-control decoding. Of special interest is the fact that belief propagation can be viewed as an instance of solving linear programming (LP) problems by using relaxation techniques. The insight gained from the LP formulation for decoding of low-density parity-check codes (LDPC) allows one to deduce the code parameters and characteristics bearing influence on the performance of the scheme under investigation. We will be concerned with several combinatorial aspects of LP decoding that pertain to the structure of the code polytope and its "relaxation", including the number of edges and facets of the polytopes. These turn out to be closely related to minimal codewords of the code and its dual code. We also describe some new approaches to finding bounds on the diameter of a code polytope in terms of minimal codewords and propose how to enumerate minimal codewords in LDPC code ensembles. We conclude the talk by describing some interesting relationships between the group algebra of a code, its Newton polytope and iterative decoding.

This is a joint work with N. Kashyap (Queen's University) and E. Soljanin and P. Whiting (Bell Labs, Lucent Technologies).

The complete characterization of orthonormal wavelets is known in L²(R). G. Gripenperg and X. Wang proved independently that a square-integrable function of norm one is an orthonormal wavelet if and only if two (somewhat unexpected) equations are satisfied. Their results have been generalized to multiwavelets, to higher dimensional square-integrable function spaces, to dilation by a real expansive matrix with translation by a general lattice, and to dual frames.

We will discuss an analogous characterization of semi-orthogonal Parseval wavelets in an abstract Hilbert space. We will examine the structural assumptions that must be made about this space in order to compensate for the loss of known facts about L²(R).