Seminars and Colloquia by Series

The limit distribution the longest significance path(LSP) in point cloud

Series
SIAM Student Seminar
Time
Friday, January 29, 2010 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Kai NiSchool of Mathematics, Georgia Tech
In 2006, my coadvisor Xiaoming Huo and his colleague published an annal of statistics paper which designs an asymptotically powerful testing algorithm to detect the potential curvilinear structure in a messy point cloud image. However, such an algorithm involves a membership threshold and a decision threshold which are not well defined in that paper because the distribution of LSP was unknown. Later on, Xiaoming's student Chen, Jihong found some connections between the distribution of LSP and the so-called Erdos-Renyi law. In some sense, the distribution of LSP is just a generalization of the Erdos-Renyi law. However this JASA paper of Chen, Jihong had some restrictions and only partially found out the distribution of LSP. In this talk, I will show the result of the JASA paper is actually very close to the distribution of LSP. However, these is still much potential work to do in order to strengthen this algorithm.

Reducing the Size of a Matrix While Maintaining its Spectrum

Series
SIAM Student Seminar
Time
Friday, January 22, 2010 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Benjamin WebbSchool of Mathematics, Georgia Tech
The Fundamental Theorem of Algebra implies that a complex valued nxn matrix has n eigenvalues (including multiplicities). In this talk we introduce a general method for reducing the size of a square matrix while preserving this spectrum. This can then be used to improve on the classic eigenvalue estimates of Gershgorin, Brauer, and Brualdi. As this process has a natural graph theoretic interpretation this talk should be accessible to most anyone with a basic understanding of matrices and graphs. These results are based on joint work with Dr. Bunimovich.

Simulation Study of the Length of Longest Increasing Subsequence of Finite Alphabets

Series
SIAM Student Seminar
Time
Friday, November 20, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Huy HuynhGeorgia Tech
Let X_1, X_2,...,X_n be a sequence of i.i.d random variables with values in a finite alphabet {1,...,m}. Let LI_n be the length of the longest increasing subsequence of X_1,...,X_n. We shall express the limiting distribution of LI_n as functionals of m and (m-1)- dimensional Brownian motions as well as the largest eigenvalue of Gaussian Unitary Ensemble (GUE) matrix. Then I shall illustrate simulation study of these results

From Gibbs free energy to the dynamical system with random perturbation

Series
SIAM Student Seminar
Time
Friday, November 13, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Yao LiSchool of Mathematics, Georgia Tech
Gibbs free energy plays an important role in thermodynamics and has strong connection with PDE, Dynamical system. The results about Gibbsfree energy in 2-Wasserstein metric space are known recently.First I will introduce some basic things, so the background knowledge isnot required. I will begin from the classic definition of Gibbs freeenergy functional and then move to the connection between Gibbs freeenergy and the Fokker-Planck equation, random perturbation of gradientsystems. Second, I will go reversely: from a dynamical system to thegeneralized Gibbs free energy functional. I will also talk about animportant property of the Gibbs free energy functional: TheFokker-Planck equation is the gradient flux of Gibbs free energyfunctional in 2-Wasserstein metric.So it is natural to consider a question: In topological dynamical systemand lattice dynamical system, could we give the similar definition ofGibbs free energy, Fokker-Planck equation and so on? If time allowed, Iwill basicly introduce some of my results in these topics.

Online Algorithms for Graphs and Partially Ordered Sets

Series
SIAM Student Seminar
Time
Friday, November 6, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Mitch KellerSchool of Mathematics, Georgia Tech
Suppose that Amtrak runs a train from Miami, Florida, to Bangor, Maine. The train makes stops at many locations along the way to drop off passengers and pick up new ones. The computer system that sells seats on the train wants to use the smallest number of seats possible to transport the passengers along the route. If the computer knew before it made any seat assignments when all the passengers would get on and off, this would be an easy task. However, passengers must be given seat assignments when they buy their tickets, and tickets are sold over a period of many weeks. The computer system must use an online algorithm to make seat assignments in this case, meaning it can use only the information it knows up to that point and cannot change seat assignments for passengers who purchased tickets earlier. In this situation, the computer cannot guarantee it will use the smallest number of seats possible. However, we are able to bound the number of seats the algorithm will use as a linear function of the minimum number of seats that could be used if assignments were made after all passengers had bought their tickets. In this talk, we'll formulate this problem as a question involving coloring interval graphs and discuss online algorithms for other questions on graphs and posets. We'll introduce or review the needed concepts from graph theory and posets as they arise, minimizing the background knowledge required.

Asymptotic behavior of Müntz orthogonal polynomials

Series
SIAM Student Seminar
Time
Friday, October 30, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Ulfar StefanssonSchool of Mathematics, Georgia Tech
After a brief introduction of the theory of orthogonal polynomials, where we touch on some history and applications, we present results on Müntz orthogonal polynomials. Müntz polynomials arise from consideration of the Müntz Theorem, which is a beautiful generalization of the Weierstrass Theorem. We prove a new surprisingly simple representation for the Müntz orthogonal polynomials which holds on the interval of orthogonality, and in particular we get new formulas for some of the classical orthogonal polynomials (e.g. Legendre, Jacobi, Laguerre). This allows us to determine the strong asymptotics on the interval, and the zero spacing behavior follows. We also look at the asymptotic behavior outside the interval, where we apply the method of stationary phase.

Shuffling biological sequences

Series
SIAM Student Seminar
Time
Friday, October 16, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Tianjun YeGeorgia Tech
This talk considers the following sequence shufling problem: Given a biological sequence (either DNA or protein) s, generate a random instance among all the permutations of s that exhibit the same frequencies of k-lets (e.g. dinucleotides, doublets of amino acids, triplets, etc.). Since certain biases in the usage of k-lets are fundamental to biological sequences, effective generation of such sequences is essential for the evaluation of the results of many sequence analysis tools. This talk introduces two sequence shuffling algorithms: A simple swapping-based algorithm is shown to generate a near-random instance and appears to work well, although its efficiency is unproven; a generation algorithm based on Euler tours is proven to produce a precisely uniforminstance, and hence solve the sequence shuffling problem, in time not much more than linear in the sequence length.

Approximations of Short Term Options Pricing Under Stochastic Volatility Models with Jumps

Series
SIAM Student Seminar
Time
Friday, October 9, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Allen HoffmeyerSchool of Mathematics, Georgia Tech
This talk is based on a paper by Medvedev and Scaillet which derives closed form asymptotic expansions for option implied volatilities (and option prices). The model is a two-factor jump-diffusion stochastic volatility one with short time to maturity. The authors derive a power series expansion (in log-moneyness and time to maturity) for the implied volatility of near-the-money options with small time to maturity. In this talk, I will discuss their techniques and results.

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.

Dynamics of Functions with an Eventual Negative Schwarzian Derivative

Series
SIAM Student Seminar
Time
Friday, September 25, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Benjamin WebbSchool of Mathematics, Georgia Tech
In the study of one dimensional dynamical systems one often assumes that the functions involved have a negative Schwarzian derivative. In this talk we consider a generalization of this condition. Specifically, we consider the interval functions of a real variable having some iterate with a negative Schwarzian derivative and show that many known results generalize to this larger class of functions. The introduction of this class was motivated by some maps arising in neuroscience

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