Department of Mathematical Sciences

Math colloquium Spring 07
Department Colloquium, Spring 2008
Colloquium Seminars Clec Lectures Public Lectures Distinguished Lectures

Schedule

Date
Speaker
Title
Friday, January 25, 3-4pm, MP3314 meeting
Friday, February 1st, 3-4pm, MP3314 Richard Ball, University of Denver P-spaces and P-frames
Friday, February 8, 3-4pm, MP3314  cancelled
Thursday, February 14, 3-4pm, MP3314 Matt Blair, University of Rochester
 Nonlinear wave equations on exterior domains
Friday, February 22, 3-4pm, MP3314 Gabor Luckacs, University of Manitoba (Canada) Pontryagin duality and number theory
Friday, February 29, 3-4pm, MP3314 Shijun  Zheng, Georgia Southern University
 Spectral calculus, Besov spaces and Dispersive equations
Friday, March 7, 3-4pm, MP3314 Xingping Sun, Missouri State University Approximation of Equilibrium Measures via Radial Basis Functions
Friday, March 14, 3-4pm, MP3314 Willard Miller, University of Minnesota Truth and beauty in science. A celestial mechanics case study.
Spring Break
Friday, March 28, 3-4pm, MP3314 Shanshuang Yang, Emory University Rigidity of conformal embeddings
Monday, March 31st, 5-6pm, MP3314 George Andrews, Pennsylvania State Old and New Thoughts on the Rogers-Ramanujan Identities
Tuesday, April 1st,  6-7pm, IT 1004 DISTINGUISHED LECTURE:
George Andrews, Pennsylvania State		 Euler and the beginning of the theory of partitions
Friday, April 4th, 3-4pm, MP3314 Gavin Seal, Georgia Southern University
 Galois connections and the filter monad
Friday, April 11, 3-4pm, MP3314 Ed Enochs, University of Kentucky Covers and Envelopes
Friday, April 18, 3-4pm, MP3314 Chris Heil, Georgia Tech Music, Time-Frequency Shifts, and Linear Independence
Monday, April 21, 5pm-6pm, MP3314 Homeira Pajoohesh, Medgar Evers College (CUNY)
Friday, April 25, 2pm-3pm, MP3314 Ramona Anton, John Hopkins University Non-linear Schrödinger equations on domains with boundary
Friday, April 25, 3-4pm, MP3314 Grillakis, University of Maryland Impurity and quaternions in nonrelativistic
scattering from quantum memory
Monday, April 28, 5pm-6pm, MP3314 Dirk Hofmann, University of Aveiro (Portugal) Triquotient maps via ultrafilter convergence
Friday, May 2nd, 3-4pm, MP3314 Shao, UCLA The Restriction Conjecture

Details of Spring 2008 lectures

 Friday, February 1st, 3pm-4pm, MP 3314

Speaker: Richard Ball, University of Denver
Title: P-spaces and P-frames

Abstract: 

A space is called “pseudo discrete,” or more commonly, a P-space, provided that every real-valued function is constant in a neighborhood of every point.  Since we assume all spaces Tychanov, i.e., Hausdorff and completely regular, there are plenty of non-constant real-valued functions, enough to separate the points.  Thus it might appear that these spaces are exotic and remote from daily mathematical existence.  In fact, they play a central role in general topology since they carry the epicomplete objects in the most natural categories which axiomatize C(X), the ring of continuous functions on the space X.  We will sketch some of the classical characterizations of P-spaces at the beginning of the talk.

     But the story gets more interesting when we consider the point-free analog, namely a P-frame.  Here the spatial theory carries over for the most part, but diverges for considerations involving subspaces, i.e., quotient frames.  The true objective of this talk is to pose a problem we have been unable to resolve.   


 Thursday, February 14st, 3pm-4pm, MP 3314

Speaker: Matt Blair, Rochester University
Title: Nonlinear wave equations on exterior domains

Abstract: 

We consider certain semilinear wave equations posed on an exterior domain.  While basic questions such as existence, uniqueness, and scattering of solutions have been answered in the Euclidean case, less is known in the case of an exterior domain.  Here the presence of Dirichlet or Neumann boundary conditions can affect the flow of energy, complicating these issues considerably.  We discuss recent progress in the area, including the development and applications of space-time integrability estimates for the wave equation ("Strichartz estimates").  This is a joint work with H. Smith and C. Sogge.

Friday, February 22, 3pm-4pm, MP 3314
 Speaker:  Gabor Lukacs, University of Manitoba (Canada)

Title: Pontryagin duality and number theory
Abstract: pdf file.

Friday, February 29, 3pm-4pm, MP 3314
Speaker: Shijun Zheng, Georgia Southern University

Title: Spectral calculus, Besov spaces and Dispersive equations 

Abstract.  In  this talk we consider Hörmander type spectral multiplier problem for Schrödinger
operators with  a critical potential. It is shown that the multiplier operator  is bounded on $L^p$, Besov spaces and
Triebel-Lizorkin spaces under the same sharp condition.  We then derive 
Strichartz estimates that measure spacetime regularity for the corresponding wave equation.
Our work is partially motivated by the standing wave problem 
for the quintic wave equation in 3+1 dimensions.

Friday, March 7, 3pm-4pm, MP 3314
Speaker: Xingping Sun, Missouri State University

Title: Approximation of Equilibrium Measures via Radial Basis Functions 

Friday, February 14, 3pm-4pm, MP 3314
Speaker: Willard Miller, University of Minnesota

Title: Truth and beauty in science. A celestial mechanics case study.

Abstract.  Beauty, and especially symmetry, are important motivators in the
development of scientific theories. Surely what is mathematically
beautiful and elegant ought to be true, and what is true ought to be
beautiful! Much of the history of the rise of modern science concerns
the power of a few beautiful mathematical ideas to explain observed
physical phenomena. Such is not always the case, however. Sometimes
enforcement of preconceived notions of beauty and symmetry in science
actually impede understanding and progress. We will review the
development of celestial mechanics over the past two and a half
millennia as a case study and learn some rocket science along the way.

Friday, March 28, 3pm-4pm, MP 3314
Speaker: Shanshuang Yang, Emory University

Title: Rigidity of conformal embeddings

Abstract.  A classical Liouville type theorem states that any orientation preserving
conformal self map of the complex plane (or any homeomorphic solution to the
Cauchy- Riemann equations) has the form f(z)=az+b. In this talk, we will
discuss various extensions of this rigidity phenomena to general settings in
higher dimensions and their implications in geometric analysis and differential
geometry.


Monday, March 31st, 5pm-6pm, MP3314
Speaker: George Andrews, Penn State University

Title: Old and New Thoughts on the Rogers-Ramanujan Identities

Abstract:  The Rogers-Ramanujan identities first came to prominence in the
early part of the 20th century when Ramanujan conjectured them and later
found them fully proved in an old and very much neglected paper by L.J.
Rogers.
     In this talk, we shall provide some account of what happened
subsequently and will conclude with a look at recent discoveries and
ongoing research.

Tuesday, April 1st, 6pm-7pm, IT 1004--Distinguished Lecture
Speaker: George Andrews, Penn State University

Title:  Euler and the Beginning of the Theory of Partitions

Abstract:  Chapter 16 of Euler's Introductio in Analysin Infinitorum is titled
De Partitione Numerorum.  We shall examine how Euler began the theory of
partitions, note some surprising connections with subsequent developments
and reveal how Euler's bizarre presentation of an ancient chestnut (every
integer is uniquely the sum or difference of distinct powers of 3, e.g. 73
= 3^4 - 3^2 + 3^0) leads to new insights in the theory of partitions.

Friday, April 4, 3pm-4pm, MP 3314
 Speaker:  Gavin Seal, Georgia Southern University
Title:  Galois connections and the filter monad

Abstract: 
The Eilenberg-Moore category of a monad classically crystallizes the "algebraic content" of the said monad. For example, the Eilenberg-Moore category of the free group monad is exactly the category of groups, a fact that formalizes our intuition that groups are much more "algebraic" than mere sets.

In category theory textbooks, it is traditionally left as an exercise to verify that Eilenberg-Moore category of the powerset monad is the category of suprema-complete lattices. In this talk, we will nevertheless prove this fact using the theory of Galois connections, and show how the same argument can be further exploited to relate Hausdorff's original 1914 definition of topological spaces with a less classical presentation based on convergence of filters.

Friday, April 11, 3pm-4pm, MP 3314
 Speaker:  Ed. Enoch, University of Kentucky
Title:  Covers and Envelopes

Abstract: 
The notion of an injective envelope of a module goes back to Baer in the
case of abelian groups. Later Eckmann and Schopf gave a  general
definition for modules and also proved their existence in general. All this
gave rise to the dual notion of  projective covers and raised the question
of their existence. Gradually these notions have been extended to other
kind of covers and envelopes. 
 I will give some of the history of the development of these topics and
also tell about my involvement in the subject.

Friday, April 18, 3pm-4pm, MP 3314
 Speaker:  Chris Heil, Georgia Tech
Title: Music, Time-Frequency Shifts, and Linear Independence

Abstract: 

Fourier series provide a way of writing almost any signal as a
superposition of pure tones, or musical notes.  But this representation
is not local, and does not reflect the way that music is actually generated
by instruments playing individual notes at different times.  We will discuss
Fourier series, and then present time-frequency representations, which
are a type of local Fourier representation of signals.  This gives us a
mathematical model for representing music.  While the model is crude for
music, it is in fact a powerful mathematical representation that has
appeared widely throughout mathematics (e.g., partial differential equations),
physics (e.g., quantum mechanics), and engineering (e.g., time-varying
filtering).  We ask one very basic question: are the notes in this
representation linearly independent?  This seemingly trivial question
leads to surprising mathematical difficulties.

Monday, April 21, 5pm-6pm, MP 3314
 Speaker:  Homeira Pajoohesh, Medgar Evers College (NYC)

Title: Convergence for partial metrics in value lattices

Abstract: (joint work by R. Kopperman, S. Matthews, and H. Pajoohesh)

In this paper we investigate some notions of completion of partial metric spaces, including the bicompletion, to some extent the Smyth completion, and a new ``spherical completion". Given an auxiliary relation, we find a partial metric that gives rise to it and whose spherical completion is its round ideal completion. A partial metric induces an order; with respect to this order we give an example of a partial metric space which is a continuous dcpo but its bicompletion and Smyth completion are not continuous posets, and an example of a continuous poset whose spherical completion is not a continuous poset.

Friday, April 25, 2pm-3pm, MP 3314
 Speaker: Ramona Anton, John Hopkins University

Title: Non-linear Schrödinger equations on domains with boundary
Abstract: 
We are interested in proving global existence results in the energy space for the
semi-linear Schrödinger equation on domains of dimension 2 or 3. The main ingredients
are generalized Strichartz inequalities adapted to the domains, which have some loss of
derivatives. We present the results and the strategy for three types of domains.

Friday, April 25, 3pm-4pm, MP 3314
 Speaker: M. Grillakis, University of Maryland

Title: Impurity and quaternions in nonrelativistic scattering from quantum memory

Abstract: 
Models in quantum computing rely on transformations of states of
quantum memory. We study mathematical aspects of a model proposed
by Wu in which the memory state is changed via scattering of incoming
particles. This operation causes the memory content to deviate from a
pure state, i.e. induces impurity. For nonrelativistic particles scattered
from a two-state memory and sufficiently general interaction potentials
in 1 + 1 dimensions, we express impurity in terms of quaternionic commutators.
I this context, pure memory states correspond to null hyperbolic
quaternions. In the case of point interactions, the scattering process
amounts to appropriate rotations of quaternions in the frequency domain.
This point of view complements a previous analysis by Margetis and Myers
(2006 J. Phys. A 39 11567-11581) and is in collaboration with D.
Margetis.

Monday, April 28, 5pm-6pm, MP 3314
 Speaker: Dirk Hofmann, University of Aveiro (Portugal)
Title: Triquotient maps via ultrafilter convergence

Abstract: See pdf file.

Friday, May 2nd, 3pm in MP3314
Speaker: Dr. Shao, University of California Los Angeles
 Title: The Restriction Conjecture
Abstract:
Restriction conjecture is one of the central problems in harmonic analysis. It was solved in 
2-dimension but remains open in higher dimensions. In this talk, I will focus on the linear
and bilinear versions of this conjecture for paraboloids in the cylindrically
symmetric case. The main result is that we have further estimates available with this assumption
which are sharp up to endpoints and turn out to be very useful in establishing the global wellposedness
of certain mass-critical NLS in the radial case by Killip, Tao and Visan; 
Another consequence is that the restriction conjecture for the paraboloid is true
in all dimensions in the cylindrically symmetric case.
Department Colloquium Archives

Please direct questions or comments regarding the colloquium to  Frederic Mynard