| Date
|
Speaker
|
Title
|
| 11 September |
Tom McNamara (SLU) |
Decomposition and Admissibility for the Quasiregular Representation of Generalized Oscillator Groups, Part 1 |
| Abstract:
We analyze the co-normally induced quasiregular representation for
two families of Lie groups: the $2d$-oscillator groups $N \rtimes SO(2d)$
where $N$ is the free two-step nilpotent group on $2d$ generators, and the
dilated $2d$-oscillator groups $N \rtimes (SO(2d) \times\Bbb R^*_+)$.
We construct irreducible decompositions in both cases with explicit spectrum
and intertwining operators, and in both cases we prove a Cald\'eron-type
admissibility condition for multiplicity-free, quasi-equivalent
subrepresentations. We prove that in the case of the $2d$-oscillator groups,
the quasiregular representation has no admissible vectors, and for the
dilated $2d$-oscillator groups, we give an explicit construction for
admissible vectors.
|
| 18 September |
Tom McNamara (SLU) |
Decomposition and Admissibility for the Quasiregular Representation of Generalized Oscillator Groups, Part 2 |
| Abstract:
Continuation of previous topic.
|
| 25 September |
Brad Currey (SLU) |
Decomposition and Admissibility for the Quasiregular Representation of Generalized Oscillator Groups, Part 3 |
| Abstract:
Continuation of previous topic.
|
| 2 October |
Miron Bekker (University of Missouri, Rolla) |
Automorphic-Invariant Non-Densely Defined Hermitian Contractive Operators |
| Abstract: We consider operators with norms not greater than 1, defined on proper subspaces of Hilbert spaces that have Hermitian property (non-densely defined Hermitian contractions). In addition we assume that such operators are unitarily equivalent to their linear-fractional transformations (automorphic-invariant operators). We show that any such operator A always admits a self-adjoint extension $\hat{A}$ with the same norm that is also automorphic-invariant. In particular, extreme extensions $\hat{A}_{M}$ and $\hat{A}_{\mu}$ are always automorphic-invariant. A functional characterization of an automorphic-invariant pair $(A,\hat{A})$ is given in terms of a resolvent of the operator $\hat{A}$. Special attention is paid to the case when the codimension of the domain of the operator A is one. Examples of automorphic-invariant operators are considered.
|
| 9 October |
Brody Johnson (SLU) |
Trigonometric Interpolation |
| Abstract: This talk will explore elementary concepts and results related to the interpolation of trigonometric polynomials. The relationship between interpolation over equidistant nodal points and the Dirichlet kernel will be discussed.
|
| 16 October |
Philip Huling (SLU) |
Extension of Hilbert space maps and the Moving Lemma |
| Abstract: There are many extension theorems in mathematics: Tietze Extension, Whitney's Extension, and Hahn-Banach just to name a few. I will show that it is a surprisingly easy task to extend uniformly continuous functions between two Hilbert spaces. I will focus mainly on the Lipschitz case and show how the extension question is tied directly to the geometry of the closed balls in the space. A short discussion about the geometries of non-Hilbert spaces that do and do not give the conditions will also be given.
|
| 23 October |
Fall Break |
No Seminar |
| Abstract:
|
2 November
(Friday) |
Paul Koester
(Indiana University) |
The Polynomial Freiman-Ruzsa Conjecture |
| Special Time/Place: 3:30pm, Ritter Hall 119. Abstract: Freiman's Theorem describes the structure of sets of integers which are almost closed under addition. While his theorem was a major
breakthrough, examples show that his description is inefficient. The Polynomial Freiman-Ruzsa Conjecture attempts to fix the inefficiencies in Freiman's Theorem. I will discuss the analogue of the Polynomial Freiman-Ruzsa Conjecture in $\mathbb{F}_{2}^{n}$, the n-dimensional vector space over the field of order 2, along with the best partial result on this conjecture, which heavily uses the $\mathbb{F}_{2}^{n}$ Fourier transform.
|
| 6 November |
Darrin Speegle (SLU) |
Characterizing equations of orthonormal wavelets, Part 1 |
| Abstract: Orthonormal wavelets can be characterized by 2 equations and one norm condition in the Fourier domain. I will motivate the equations and give applications to constructing orthonormal wavelets, including Meyers' wavelet. Generalizations to the non-dyadic case will also be considered. These talks are intended to be an introduction to one of my research programs, and will be aimed at the level of an advanced graduate student.
|
| 13 November |
Darrin Speegle (SLU) |
Characterizing equations of orthonormal wavelets, Part 2 |
| Abstract: Continuation from November 6.
|
| 20 November |
Thanksgiving |
No Seminar |
| Abstract:
|
| 27 November |
Tom McNamara (SLU) |
The Calderón Condition in Various Settings |
| Abstract: Wavelets have generated considerable research activity. We will begin with a discussion of what these wavelets are and what they do for us. From there we shall investigate the conditions a function must satisfy to be a wavelet. It will be shown that a succinct criteria can be achieved by looking at the Fourier transform of a function, rather than the function itself. Our focus will then move from wavelets to continuous wavelet transforms in various setting, where we will look at the Calderón condition. The talk will wrap up by examining some current research that makes use of this condition.
|