Department of Mathematics and Computer
Science
Faculty Seminars
2010 - 2011
Swords 302
Spring 2011
Monday, February 21 at 3:30 PM
Speaker: John Little
Title: Was
Pythagoras a Babylonian?
Monday, March 21 at 3:30 PM
Speaker: Keith Ouellette
Title: On the Fourier inversion theorem
for PGL(2,Qp)
Abstract: We
will demonstrate our proof of the Fourier inversion and Plancherel
formulae for spherical unramified principal series
for the p-adic group PGL(2, Qp)
where Qp is the quotient field of Zp,
the p-adic integers. If time remains, we will apply
the Fourier inversion theorem to transforms of some smooth compactly supported
functions of PGL(2, Qp) for
illustration.
Monday, April 18 at 3:30 PM
Speaker: Alisa DeStefano
Title: Universal Observability
of Dynamical Systems
Abstract: A general
question in control theory is whether the solution of a dynamical system is
uniquely determined by a set of measurements of the system. If this is the
case, the system is said to be observable. There are different ways to observe
a dynamical system and some systems may be observable using only certain types
of measurements. One might ask if there is a dynamical system that can be
observed by any continuous function from the state space to the real numbers? If such a system exists, it is said to
be universally observable. It
might seem unlikely that such a system could exist. However, there are two known examples of such systems. We will discuss the history of the
problem and then give the motivation and a brief outline of the construction of
an elementary universally observable system on the two-dimensional torus. The
ideas involved come from topological dynamics and small divisor problems. We
also discuss an equivalence between universal observability and certain properties in the field of
topological dynamics.
Monday, May 2 at 3:30 PM
Speaker: John Anderson
Title: The Hull of Rudin's Klein Bottle
Abstract: In 1981 Walter Rudin exhibited a totally real imbedding of the Klein
bottle into C2. I'll talk about the problem of finding holomorphic images of the unit disk in C with boundaries on
Rudin's Klein bottle, and show that there are enough
such disks so that their interiors fill an open subset of C2.
I'll also attempt to explain why this is interesting.
Fall 2010
Monday, September 13 at 3:00 PM
Speaker: John Little
Title: An
Euler-Maclaurin Formula for Polytopes
and Hirzebruch-Riemann-Roch for Toric
Varieties - This is an expository talk and it is the first in a series of three
talks.
Abstract: In its
basic form, the classical Euler-Maclaurin summation
formula relates the integral of a smooth function over an interval to the sum
of the values of the function at points in the interval, with an error term
involving Bernoulli numbers and derivatives of the function at the
endpoints. We will start out by looking at this elementary result
and some of its applications. In the 1990's, two (transplanted) Russians, Khovanskii and Pukhlikov, published a generalization of Euler-Maclaurin where the interval above is replaced by a
convex polytope in a higher dimensional
Euclidean space. We will look at their generalization, again in
essentially elementary terms. All of this is essentially generating
function "magic." But the generating function of the
Bernoulli numbers, f(x) = x/(1 - e^{-x}),
also quickly leads to Todd classes and the Hirzebruch-Riemann-Roch theorem.
Polytopes lead to projective toric varieties. Khovanskii-Pukhlikov
gives an amazing proof of the Hirzebruch-Riemann-Roch
theorem in this case. I hope to show you how it all works!
Monday, September 20 at 3:00 PM
Speaker: John Little
Title: An
Euler-Maclaurin Formula for Polytopes
and Hirzebruch-Riemann-Roch for Toric
Varieties (part II)
Monday, September 27 at 3:00 PM
Speaker: John Little
Title: An
Euler-Maclaurin Formula for Polytopes
and Hirzebruch-Riemann-Roch for Toric
Varieties (part III)
Monday, October 4 at 3:00 PM
Speaker: John Little
Title: An
Euler-Maclaurin Formula for Polytopes
and Hirzebruch-Riemann-Roch for Toric
Varieties (Epilogue)
Monday, October 11 - no talk (fall
break)
Monday, October 18 - no talk
(colloquium on Tuesday, Oct. 19)
Monday, October 25 at 3:00 PM
Speaker: Keith Ouellette
Title: On
the Fourier Inversion Formula for the Full Modular Group
Abstract: We offer a new
proof of the Fourier inversion and Plancherel
formulae for Maass-Eisenstein wave packets. The proof
(presented in the case of the full modular group) uses
truncation, basic analysis, and classical Fourier theory.
Brief
sketches of the proofs due to Langlands, Lapid, and Casselman are then
presented for comparison.
Monday, November 1 at 3:00 PM
Speaker: Keith Ouellette
Title: On
the Fourier Inversion Formula for the Full Modular Group (Part II)
Monday, November 8 at 3:00 PM
Speaker: Cristina Ballantine
Title: Powers of the Vandermonde
determinant, Schur functions and the dimension game
Abstract: Vandermonde determinants are ubiquitous in mathematics.
Since
every even power of the Vandermonde determinant is a symmetric polynomial, we would
like to understand its decomposition in terms of the basis of Schur functions. Besides its obvious importance in
mathematics, such a decomposition would shed light on
the quantum Hall effect, in particular on the Slater decomposition of the
Laughlin wave function. While we will not explore the problem from the point of
view of physics, we will investigate several combinatorial properties of the
coefficients in the decomposition. In particular, I will give an inductive
approach of computing some of the coefficients by building them up from tetris type shapes.
Monday, November 15 at 3:00 PM
Speaker: Sarah Wright
Title: Graph C*-Algebras and Aperiodic Paths (Part I)
Abstract: Associating a visual
object with a complicated mathematical structure is a popular and successful
technique for teaching and discovering new ideas. We'll introduce the
idea of a graph-algebra, where we associate a C*-algebra to some type of
"graph". We'll discuss the "how's",
"who's" and "why we cares" of this field of study.
Once we have the basics, we can move on to the idea of aperiodicity.
The condition "every cycle has an entry" first appeared in the
literature in Kumjian, Pask,
and Raeburn's paper on Cuntz-Krieger algebras of
directed graphs, where it was called Condition (L). It provides a
necessary condition for simplicity of the associated graph algebra. This
condition has been generalized to aperiodicity conditions in the theory of
topological graphs (Katsura), k-graphs (Kumjian, Pask), and the unifying
theory of topological k-graphs (Yeend). We'll
discuss the details of these generalizations as well as the theorems associated
with them. We'll then introduce a Condition (F) on the finite paths of a
topological k-graph that is equivalent to the corresponding aperiodicity
condition. Hence we obtain a condition which is
much easier to check than the aperiodicity of infinite paths.
Monday, November 22 at 3:00 PM
Speaker: Sarah Wright
Title: Graph C*-Algebras and Aperiodic Paths (Part II)