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Mathematics & Statistics

**Title**

**Speaker**

**Time**

**Place**

**Sponsor**

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.

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**Speaker**

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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.

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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 \(A\).

**Title**

**Speaker**

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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.

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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.

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Homological Symbols

Marian Anton

University of Kentucky

3:00pm-4:00pm

TBA

Mohamed Elhamdadi

**Abstract**

We define some symbols called “homological symbols” and explain their relationship with group theory and number theory. The talk will be accessible to graduate students.