Seminars and Colloquia by Series

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 Tinaztepe – Georgia 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 Baker – School 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 Garoufalidis – School 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 Shokrieh – Georgia 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 Belegradek – Georgia 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 Minsker – School 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 Bishop – Georgia 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.

Arbitrage ­free option pricing models 

Series
Stochastics Seminar
Time
Thursday, October 1, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Denis Bell – University of North Florida
The Black‐Scholes model for stock price as geometric Brownian motion, and the corresponding European option pricing formula, are standard tools in mathematical finance. In the late seventies, Cox and Ross developed a model for stock price based on a stochastic differential equation with fractional diffusion coefficient. Unlike the Black‐Scholes model, the model of Cox and Ross is not solvable in closed form, hence there is no analogue of the Black‐Scholes formula in this context. In this talk, we discuss a new method, based on Stratonovich integration, which yields explicitly solvable arbitrage‐free models analogous to that of Cox and Ross. This method gives rise to a generalized version of the Black‐Scholes partial differential equation. We study solutions of this equation and a related ordinary differential equation.

Some Results on Linear Discrepancy for Partially Ordered Sets

Series
Dissertation Defense
Time
Thursday, October 1, 2009 - 14:00 for 2 hours
Location
Skiles 255
Speaker
Mitch Keller – School of Mathematics, Georgia Tech
Tanenbaum, Trenk, and Fishburn introduced the concept of linear discrepancy in 2001, proposing it as a way to measure a partially ordered set's distance from being a linear order. In addition to proving a number of results about linear discrepancy, they posed eight challenges and questions for future work. This dissertation completely resolves one of those challenges and makes contributions on two others. This dissertation has three principal components: 3-discrepancy irreducible posets of width 3, degree bounds, and online algorithms for linear discrepancy. The first principal component of this dissertation provides a forbidden subposet characterization of the posets with linear discrepancy equal to 2 by completing the determination of the posets that are 3-irreducible with respect to linear discrepancy. The second principal component concerns degree bounds for linear discrepancy and weak discrepancy, a parameter similar to linear discrepancy. Specifically, if every point of a poset is incomparable to at most \Delta other points of the poset, we prove three bounds: the linear discrepancy of an interval order is at most \Delta, with equality if and only if it contains an antichain of size \Delta+1; the linear discrepancy of a disconnected poset is at most \lfloor(3\Delta-1)/2\rfloor; and the weak discrepancy of a poset is at most \Delta-1. The third principal component of this dissertation incorporates another large area of research, that of online algorithms. We show that no online algorithm for linear discrepancy can be better than 3-competitive, even for the class of interval orders. We also give a 2-competitive online algorithm for linear discrepancy on semiorders and show that this algorithm is optimal.

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