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Mathematics & Statistics

# Colloquia — Fall 2006

## Friday, December 8, 2006

Title
Speaker

Time
Place

Multiwavelets on the Interval
Patrick Van Fleet
University of St. Thomas
3:00pm-4:00pm
PHY 130
Catherine Bénéteau

Abstract

In this talk, we present a method for constructing scaling vectors and multiwavelets on the interval. While other constructions exist (for example, Dahmen/Micchelli 1997, Goh/Jiang/Xia 2000, and Lakey/Pereya 2000), ours is different in that we take an existing scaling vector for $$L^2(R)$$ and modify it to construct a scaling vector on $$L^2([a,b])$$. The process is similar to that employed by Daubechies 1992 and Meyer 1991. Moreover, our construction allows us to build nonnegative scaling functions if so desired. The talk will conclude with some examples of our method.

## Friday, December 1, 2006

Title
Speaker

Time
Place

Inverse scattering transform for the vector NLS equation with non-vanishing boundary conditions
Barbara Prinari
Dipartimento di Fisica and Sezione INFN
Universita di Lecce, Italy
3:00pm-4:00pm
PHY 130
Wen-Xiu Ma

Abstract

The inverse scattering transform for the vector defocusing vector nonlinear Schrodinger NLS equation with non-vanishing boundary values at infinity is constructed. The direct scattering problem is formulated on a two-sheeted covering of the complex plane. On the direct side, two out of the six scattering eigenfunctions do not admit an analytic extension on either sheet of the surface. Two additional analytic solutions are constructed by considering *adjoint* eigenfunctions. The discrete spectrum, bound states and symmetries of the direct problem are discussed. In general a discrete eigenvalue corresponds to a quartet of zeros (poles) of certain scattering data. The inverse scattering problem is formulated in terms of a Riemann-Hilbert (RH) problem in the upper/lower half planes of a suitable uniformization variable. Special soliton solutions, which have dark solitonic behavior in both components, and ones which have one dark and one bright component, are constructed from the poles in the RH problem. The linear limit is obtained from the RH problem and is shown to correspond to the Fourier solution obtained from the linearized vector NLS system.

## Friday, November 17, 2006

Title
Speaker
Time

Place

Applications of Potential Theory to Problems in Numerical Linear Algebra
John Rossi
Virginia Tech
3:00pm-4:00pm
TBA
Dmitry Khavinson

Abstract

It is shown that convergence rates of iterative methods based on Krylov subspace techniques can be quantified via elementary results in $$2$$-dimensional potential theory involving Green's functions and capacity. We will demonstrate our results by looking at three different problems involving a very large matrix $$A$$:

• Solving $$Ax=b$$
• Finding the eigenspaces of $$A$$
• Computing $$f(A)$$ where $$f$$ is analytic in a domain containing the eigenvalues of $$A$$.

## Wednesday, November 15, 2006

Title
Speaker

Time
Place

On Braid Index of Connected Sum of $$2$$-Knots
Shin Satoh
Kobe University
Kobe, Japan
2:00pm-3:00pm
NES 103
Masahiko Saito

Abstract

Every $$1$$-dimensional knot can be deformed into the closure of a $$1$$-dimensional braid (Alexander's theorem). The braid index of a $$1$$-knot $$K$$, denoted by $$\mathrm{Braid}(K)$$, is defined to be the minimal number of the strings for all such $$1$$-braids. For the connected sum $$K\# L$$ of two 1-knots $$K$$ and $$L$$, Birman and Menasco prove the equality $$\mathrm{Braid}(K\# L)=\mathrm{Braid}(K)+\mathrm{Braid}(L)-1$$.

For a $$2$$-dimensional knot (a knotted $$2$$-sphere in $$4$$-space), the braid index is similarly defined by introducing the notion of a $$2$$-dimensional braid. However, in the study with Kamada and Takabayashi, we prove the inequality $$\mathrm{Braid}(K\# L)<\mathrm{Braid}(K)+\mathrm{Braid}(L)-1$$ for any non-trivial $$2$$-knots $$K$$ and $$L$$.

The aim of this talk is to give a new proof which is simpler than the original one.

## Friday, October 27, 2006

Title
Speaker

Time
Place

Symmetric Groups, General Linear Groups, and a Greek Island
Bhama Srinivasan
University of Illinois at Chicago
2:00pm-3:00pm
TBA
Xiang-dong Hou

Abstract

The classical representation theory of symmetric groups involves rich combinatorics such as partitions and Young diagrams. Similar concepts arise also in the representation theory of general linear groups over finite fields. The talk will first give a description of these two theories, and then mention more recent work on classical groups such as orthogonal groups. Finally the talk will mention some mysterious objects which arose from the theory and have been named after a Greek island.

Title
Speaker

Time
Place