Seminars and Colloquia by Series

Rank-determining sets of metric graphs

Series
Graph Theory Seminar
Time
Thursday, October 8, 2009 - 12:05 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Ye LuoElectrical and Computer Engineering, Georgia Tech
A metric graph is a geometric realization of a finite graph by identifying each edge with a real interval. A divisor on a metric graph Gamma is an element of the free abelian group on Gamma. The rank of a divisor on a metric graph is a concept appearing in the Riemann-Roch theorem for metric graphs (or tropical curves) due to Gathmann and Kerber, and Mikhalkin and Zharkov. A rank-determining set of a metric graph Gamma is defined to be a subset A of Gamma such that the rank of a divisor D on Gamma is always equal to the rank of D restricted on A. I will present an algorithm to derive the reduced divisor from any effective divisor in the same linear system, and show constructively that there exist finite rank-determining sets. Based on this discovery, we can compute the rank of an arbitrary divisor on any metric graph. In addition, I will discuss the properties of rank-determining sets in general and formulate a criterion for rank-determining sets.

The dynamical shape of a complex polynomial

Series
School of Mathematics Colloquium
Time
Thursday, October 8, 2009 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Laura DeMarcoDepartment of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago
A classification of the dynamics of polynomials in one complex variable has remained elusive, even when considering only the simpler "structurally stable" polynomials. In this talk, I will describe the basics of polynomial iteration, leading up to recent results in the direction of a complete classification. In particular, I will describe a (singular) metric on the complex plane induced by the iteration of a polynomial. I will explain how this geometric structure relates to topological conjugacy classes within the moduli space of polynomials.

Embeddings of Modulation Spaces into BMO

Series
Analysis Seminar
Time
Wednesday, October 7, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Ramazan TinaztepeGeorgia Tech
Modulation spaces are a class of Banach spaces which provide a quantitative time-frequency analysis of functions via the Short-Time Fourier Transform. The modulation spaces are the "right" spaces for time-frequency analysis andthey occur in many problems in the same way that Besov Spaces are attached to wavelet theory and issues of smoothness. In this seminar, I will talk about embeddings of modulation Spaces into BMO or VMO (the space of functions of bounded or vanishing mean oscillation, respectively ). Membership in VMO is central to the Balian-Low Theorem, which is a cornerstone of time-frequency analysis.

Cech Cohomology of Sheaves

Series
Other Talks
Time
Wednesday, October 7, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Matt BakerSchool of Mathematics, Georgia Tech
We will define the Cech cohomology groups of a sheaf and discuss some basic properties of the Cech construction.

What is a totally positive matrix?

Series
Research Horizons Seminar
Time
Wednesday, October 7, 2009 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Stavros GaroufalidisSchool of Mathematics, Georgia Tech
In linear algebra classes we learn that a symmetic matrix with real entries has real eigenvalues. But many times we deal with nonsymmetric matrices that we want them to have real eigenvalues and be stable under a small perturbation. In the 1930's totally positive matrices were discovered in mechanical problems of vibtrations, then lost for over 50 years. They were rediscovered in the 1990's as esoteric objects in quantum groups and crystal bases. In the 2000's these matrices appeared in relation to Teichmuller space and its quantization. I plan to give a high school introduction to totally positive matrices.

Computing reduced divisors on finite graphs, and some applications

Series
Combinatorics Seminar
Time
Friday, October 2, 2009 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Farbod ShokriehGeorgia Tech
It is known that, relative to any fixed vertex q of a finite graph, there exists a unique q-reduced divisor (G-Parking function based at q) in each linear equivalence class of divisors. In this talk, I will give an efficient algorithm for finding such reduced divisors. Using this, I will give an explicit and efficient bijection between the Jacobian group and the set of spanning trees of the graph. Then I will outline some applications of the main results, including a new approach to the Random Spanning Tree problem, efficient computation of the group law in the critical and sandpile group, efficient algorithm for the chip-firing game of Baker and Norine, and the relation to the Riemann-Roch theory on finite graphs.

On classification of of high-dimensional manifolds

Series
Geometry Topology Working Seminar
Time
Friday, October 2, 2009 - 15:00 for 2 hours
Location
Skiles 269
Speaker
Igor BelegradekGeorgia Tech
This 2 hour talk is a gentle introduction to simply-connected sugery theory (following classical work by Browder, Novikov, and Wall). The emphasis will be on classification of high-dimensional manifolds and understanding concrete examples.

Central Limit Theorem for Convex Sets

Series
Probability Working Seminar
Time
Friday, October 2, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 154
Speaker
Stas MinskerSchool of Mathematics, Georgia Tech
The talk is based on the paper by B. Klartag. It will be shown that there exists a sequence \eps_n\to 0 for which the following holds: let K be a compact convex subset in R^n with nonempty interior and X a random vector uniformly distributed in K. Then there exists a unit vector v, a real number \theta and \sigma^2>0 such that d_TV(, Z)\leq \eps_n where Z has Normal(\theta,\sigma^2) distribution and d_TV - the total variation distance. Under some additional assumptions on X, the statement is true for most vectors v \in R^n.

Frames and integral operators

Series
SIAM Student Seminar
Time
Friday, October 2, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Shannon BishopGeorgia Tech
I will describe some interesting properties of frames and Gabor frames in particular. Then we will examine how frames may lead to interesting decompositions of integral operators. In particular, I will share some theorems for pseudodifferential operators and Fourier integral operators arising from Gabor frames.

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