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Mathematics & Statistics

# Colloquia — Fall 2004

## Wednesday, December 8, 2004

Title
Speaker

Time
Place

Minimal Generalized Interpolating Projections and the $$P$$-lambda Problem
Bruce Chalmers
University of California-Riverside
3:30pm-4:30pm
EDU 411
Boris Shekhtman

Abstract

Given an $$n$$-dimensional Banach space $$V$$, the $$P$$-lambda problem asks for a relationship between the projection constant of $$V$$ and the (Banach-Mazur) distance of $$V$$ to the space with ball the $$n$$-cube. This talk will show how this latter quantity can be recognized as the norm of a minimal generalized interpolating projection onto $$V$$ and discuss some recent progress on the $$P$$-lambda problem.

## Friday, November 12, 2004

Title
Speaker

Time
Place

Scarce sets with the Green function possessing the highest smoothness
Kent State University
3:00pm-4:00pm
PHY 120
Vilmos Totik

Abstract

Let $$E$$ be a regular compact subset of the real line. We relate the Green function $$g$$ for the complement of $$E$$ with respect to the extended complex plane to some conformal mapping $$f$$. We discuss the relationship between the geometry of $$E$$ and $$f(E)$$.

As an application we construct examples of sets of the minimal possible Hausdorff dimension with $$g$$ satisfying the Hoelder $$1/2$$ condition localy or uniformly.

## Friday, October 29, 2004

Title
Speaker

Time
Place
Note

DNA Topology: Experiments and Analysis
DeWitt L. Sumners
Distinguished Professor
Florida State University
2:00pm-3:00pm
PHY 130
Mohamed Elhamdadi and Masahiko Saito
This colloquium is joint with the Biology Department.

Abstract

Cellular DNA is a long, thread-like molecule with remarkably complex topology. Enzymes which manipulate the geometry and topology of cellular DNA perform many important cellular processes (including segregation of daughter chromosomes, gene regulation, DNA repair, and generation of antibody diversity). Some enzymes pass DNA through itself via enzyme-bridged transient breaks in the DNA; other enzymes break the DNA apart and reconnect it to different ends. In the topological approach to enzymology, circular DNA is incubated with an enzyme, producing an enzyme signature in the form of DNA knots and links. By observing the changes in DNA geometry (supercoiling) and topology (knotting and linking) due to enzyme action, the enzyme binding and mechanism can often be characterized. This talk will discuss topological models for DNA strand passage and exchange in site-specific DNA recombination, and use of the spectrum of DNA knots to infer bacteriophage DNA packing in viral capsids.

## Friday, October 1, 2004

Title
Speaker

Time
Place

Resonance and web structure in integrable systems
Gino Biondini
State University of New York at Buffalo
3:00-4:00 p.m
PHY 120
Wen-Xiu Ma

Abstract

We discuss a family of non-singular soliton solutions of the Kadomtsev-Petviashvili equation. We show that all of these solutions are of resonant type, consisting of an arbitrary number of line solitons in both aymptotics: namely, an arbitrary number $$M$$ of incoming solitons interacts to form an arbitrary number $$N$$ of outgoing solitons. We also describe the interaction pattern, and we show that the resonant interactions create a web-like structure having $$(M-1)(N-1)$$ holes. Finally, we show that the class of elastic $$N$$-soliton solutions of the Kadomtsev-Petviashvili equation is much broader than previously thought and includes partially and fully resonant solutions. We end by presenting a classification of all these elastic $$N$$-soliton solutions in terms of the individual soliton parameters.

Title
Speaker

Time
Place