The flag manifold is a
geometric object that generalizes the notion of
the tangent space to a point of a curve. The geometry of the flag
manifold encodes the combinatorics of the permutation group on n
letters, as well as algebra associated to the n x n invertible
matrices. I will discuss some problems about the geometry of the
flag
manifold that arise in numerical analysis, and show some combinatorial
approaches to solve them.
January 30.
Prof. Walter Morris, GMU
Title: Partitioning the edge set of a graph
into the edge sets of complete bipartite graphs.
Graham and Pollak
proved in 1971 that the edge set of the complete graph on n vertices
may not be written as the disjoint union of the edge sets of fewer than
n-1 complete bipartite graphs. We review a simple proof of this
by Tverberg. We will also look at a conjectured generalization of
Alon, Saks and Seymour, and related partitioning problems coming from
commutative algebra.
February
5. SPECIAL
SEMINAR (JOB CANDIDATE):
Room 242
Lenny Fukshansky
3:30-4:30 Title: Frobenius number, covering radius,
and well-rounded lattices
Let N > 1 be an integer, and let 1 < a1 <
... < aN be relatively prime integers. The Frobenius number of this
N-tuple is defined to be the largest positive integer that cannot be
expressed as a linear combination of a1, ..., aN with non-negative
integer coefficients. The condition that a1, ..., aN are relatively
prime implies that such number exists. The general problem of
determining the Frobenius number given N and a1, ..., aN is known to be
NP-hard, but there has been a number of different bounds on the
Frobenius number produced by various authors. We use techniques from
the geometry of numbers to produce a new bound, relating the Frobenius
number to the covering radius of the null-lattice of the linear form
with coefficients a1, ..., aN. Our bound is frequently better than the
previously known ones, in particular when this lattice belongs to the
class of so called well-rounded lattices; we show that this happens
infinitely often. This is joint work with Sinai Robins (Temple
University). I may also briefly discuss some of my recent work on the
distribution of well-rounded lattices, if time allows.
February 6.
Keith's talk has been postponed due to the
following:
3:30-4:20
SPECIAL SEMINAR (JOB CANDIDATE):
Room
242
Padmanabhan
Seshaiyer
Title: Mathematical
and Computational Modeling of biological and bio-inspired systems
Modeling the mechanical behavior of biological and bio-inspired systems in their service configuration is often challenging because of their complicated geometry, material heterogeneity, and non-linear behavior under finite strains. In this talk, we will review analytical, experimental and numerical methods for studying such systems. In particular, we will consider two specific applications (a) the biomechanics of an intracranial saccular aneurysm which is a thin membranous balloon-like widening of the arterial wall, the rupture of which is the most common cause of nontraumatic subarachnoid hemorrhage (bleeding onto the surface of the brain) which results in a stroke and; (b) the development of structural models for studying the dynamics of flexible wings of a micro-air vehicle. A discussion on non-conforming finite element methods that are suitable to solve such applications will also be presented.
February 8.
SPECIAL
SEMINAR (JOB CANDIDATE):
3:30-4:20 Kevin Lin
Room 242
Title: Reliable and unreliable dynamics in driven
oscillator networks
This talk concerns the reliability of coupled oscillator
networks in response to fluctuating inputs. Reliability means
that repeated presentations of an input elicit essentially
identical responses regardless of the system's state at the
onset of the input. This work is motivated by basic questions
from neuroscience.
I will show how the question of reliability can be precisely
formulated in the framework of random dynamical systems theory,
and review the well-known fact that single phase oscillators are
reliable. I will then show that unreliability can occur even in
a 2-oscillator system, and propose a geometric mechanism for the
observed phenomena. The talk will conclude with some
observations concerning larger networks, including a natural
condition which precludes unreliability. No prior knowledge of
neuroscience or random dynamical systems theory is assumed.
This is joint work with Eric Shea-Brown and Lai-Sang Young.
February 13.
SPECIAL SEMINAR (JOB CANDIDATE):
3:30-4:20 Tian Jianjun
Room 242
February 20. Tina Hartley,
working under Dr. Wanner
Discrete Fourier Transforms
February 22. Keith Fox, working under Dr. Kulesza
THURSDAY! y Spaces
February 27.
Robert Allen,
working under Dr. Colonna
On the
Spectrum
of an Isometric Composition Operator on the Bloch Space of the Polydisk
March 6.
Jill Dunham, working under Dr. Agnarsson
Matching Polynomials and the Problem of the Rooks
March 13.
Spring Break
March 20.
Scott Cochran, working under Dr. Wanner
From Simple Walks to Brownian Motion
March 27.
Duncan
Ramsey, working under Dr. Jim Lawrence
Ehrhart Polynomials
April 3.
Richard Tatum, working under Dr. Sander
Spikes and Plateaus
April 10.
Trey Andreani, working under Dr. Goldin
Matrix Groups and Diagonals of a Rotation Matrix
April 17. Andrew
Samuelson, working under Dr. Walnut
Wavelets
April 24.
Javed Siddique, working under Dr. Anderson
Fluid Flow in
Some Simple Geometries
May 1.
Mao-Tsuen Jeng, working under Dr. Walnut
Interpolation
in Paley-Wiener Space