Series: School of Mathematics Colloquium
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.
Series: Analysis Seminar
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.
Series: Other Talks
We will define the Cech cohomology groups of a sheaf and discuss some basic properties of the Cech construction.
Series: Research Horizons Seminar
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.
Series: Combinatorics Seminar
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.
Friday, October 2, 2009 - 15:00 , Location: Skiles 269 , Igor Belegradek , Georgia Tech , Organizer:
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.
Series: Probability Working Seminar
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.
Series: SIAM Student Seminar
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.
Series: Stochastics Seminar
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.