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.
Series: Dissertation Defense
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.
Series: Analysis Seminar
The trigonometric Grassmannian parametrizes specific solutions of the KP hierarchy which correspond to rank one solutions of a differential-difference bispectral problem. It can be considered as a completion of the phase spaces of the trigonometric Calogero-Moser particle system or the rational Ruijsenaars-Schneider system. I will describe the characterization of this Grassmannian in terms of representation theory of a suitable difference W-algebra. Based on joint work with L. Haine and E. Horozov.
Series: Other Talks
After a few remarks to tie up some loose ends from last week's talk on locally ringed spaces, I will discuss exact sequences of sheaves and give some natural examples coming from real, complex, and algebraic geometry. In the context of these examples, we'll see that a surjective map of sheaves (meaning a morphism of sheaves which is surjective on the level of stalks) need not be surjective on global sections. This observation will be used to motivate the need for "sheaf cohomology" (which will be discussed in detail in subsequent talks).
Series: Research Horizons Seminar
In the last 10 years there has been a resurgence of interest in questions about certain spaces of analytic functions. In this talk we will discuss various advances in the study of these spaces of functions and highlight questions of current interest in analytic function theory. We will give an overview of recent advances in the Corona Problem, bilinear forms on spaces of analytic functions, and highlight some methods to studying these questions that use more discrete techniques.