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Mathematics & Statistics

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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.

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Scarce sets with the Green function possessing the highest smoothness

Vladimir Andrievskii

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.

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**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.

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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.

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Regular projections of spatial graphs

Ryo Nikkuni

Waseda University

Japan

3:00pm-4:00pm

PHY 120

Masahiko Saito

**Abstract**

Study of regular projections of spatial graphs is more rich in content as compared with the regular projection of knots and links. Actually some interesting phenomena in the regular projection of spatial graphs which are not appeared in the one of knots and links have been discovered since the 1990's. In this talk we introduce recent progress in this area.