Seminars and Colloquia by Series

How to find counterfeit coins? An algorithmic version

Series
Combinatorics Seminar
Time
Friday, August 31, 2012 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jeong Han Kim – Professor, Yonsei University, South Korea
In this talk, we consider a well-known combinatorial search problem. Suppose that there are n identical looking coins and some of them are counterfeit. The weights of all authentic coins are the same and known a priori. The weights of counterfeit coins vary but different from the weight of an authentic coin. Without loss of generality, we may assume the weight of authentic coins is 0. The problem is to find all counterfeit coins by weighing (queries) sets of coins on a spring scale. Finding the optimal number of queries is difficult even when there are only 2 counterfeit coins. We introduce a polynomial time randomized algorithm to find all counterfeit coins when the number of them is known to be at most m \geq 2 and the weight w(c) of each counterfeit coin c satisfies \alpha \leq |w(c)| \leq \beta for fixed constants \alpha, \beta > 0. The query complexity of the algorithm is O(\frac{m \log n }{\log m}), which is optimal up to a constant factor. The algorithm uses, in part, random walks. The algorithm may be generalized to find all edges of a hidden weighted graph using a similar type of queries. This graph finding algorithm has various applications including DNA sequencing.

Uchiyama's lemma and the John-Nirenberg inequality

Series
Analysis Seminar
Time
Wednesday, August 29, 2012 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Greg Knese – University of Alabama
Using integral formulas based on Green's theorem and in particular a lemma of Uchiyama, we give simple proofs of comparisons of different BMO norms without using the John-Nirenberg inequality while we also give a simple proof of the strong John-Nirenberg inequality. Along the way we prove the inclusions of BMOA in the dual of H^1 and BMO in the dual of real H^1. Some difficulties of the method and possible future directions to take it will be suggested at the end.

Train track and automorphisms of free groups

Series
Geometry Topology Student Seminar
Time
Wednesday, August 29, 2012 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hyunshik Shin – School of Mathematics, Georgia Tech
We will discuss the basics of automorphisms of free groups and train track structure. We will define the growth rate which is a topological entropy of the train track map.

Asymptotic Hilbert series

Series
Algebra Seminar
Time
Monday, August 27, 2012 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Gregory G. Smith – Queens University
How does one study the asymptotic properties for the Hilbert series of a module? In this talk, we will examine the function which sends the numerator of the rational function representing the Hilbert series of a module to that of its r-th Veronese submodule. As r tends to infinity, the behaviour of this function depends only on the multidegree of the module and the underlying multigraded polynomial ring. More importantly, we will give a polyhedral description for the asymptotic polynomial and show that the coefficients are log-concave.

The cohomological dimension of the hyperelliptic Torelli group

Series
Geometry Topology Seminar
Time
Monday, August 27, 2012 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tara Brendle – U Glasgow
The hyperelliptic Torelli group SI(S) is the subgroup of the mapping class group of a surface S consisting of elements which commute with a fixed hyperelliptic involution and which act trivially on homology. The group SI(S) appears in a variety of settings, for example in the context of the period mapping on the Torelli space of a Riemann surface and also as a kernel of the classical Burau representation of the braid group. We will show that the cohomological dimension of SI(S) is g-1; this result fits nicely into a pattern with other subgroups of the mapping class group, particularly those of the Johnson filtration. This is joint work with Leah Childers and Dan Margalit.

Exponential Time Differencing (ETD) Schemes for Nonlinear Reaction-Diffusion Systems

Series
Applied and Computational Mathematics Seminar
Time
Monday, August 27, 2012 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Bruce A. Wade – Department of Mathematical Sciences, University of Wisconsin-Milwaukee
We discuss various exponential time differencing (ETD) schemes designed to handle nonlinear parabolic systems. The ETD schemes use certain Pade approximations of the matrix exponential function. These ETD schemes have potential to be implemented in parallel and their performance is very robust with respect to the type of PDE. They are unconditionally stable and computationally very fast due to the technique of computing the nonlinear part explicitly. To handle the problem of irregular initial or boundary data an adaptive ETD scheme is utilized, which adds sufficient damping of spurious oscillations. We discuss algorithm development, theory and applications.

Computer assisted proofs for normally hyperbolic invariant manifolds

Series
CDSNS Colloquium
Time
Monday, August 27, 2012 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 06
Speaker
M. Capinski – AGH Univ. Krakow and SOM, Gatech
We shall present a method which establishes existence of normally hyperbolic invariant manifolds for maps within a specified domain. The method can be applied in a non-perturbative setting. The required conditions follow from bounds on the first derivative of the map, and are verifiable using rigorous numerics. We show how the method can be applied for a driven logistic map, and also present examples of proofs of invariant manifolds in the restricted three body problem.

Pages