Seminars and Colloquia Schedule

Stable sets and unstable sets in positive entropy systems

Series
CDSNS Colloquium
Time
Monday, November 2, 2009 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Wen Huang – USTC, China and SoM, Georgia Tech
Stable sets and unstable sets of a dynamical system with positive entropy are investigated. It is shown that in any invertible system with positive entropy, there is a measure-theoretically ?rather big? set such that for any point from the set, the intersection of the closure of the stable set and the closure of the unstable set of the point has positive entropy. Moreover, for several kinds of specific systems, the lower bound of Hausdorff dimension of these sets is estimated. Particularly the lower bound of the Hausdorff dimension of such sets appearing in a positive entropy diffeomorphism on a smooth Riemannian manifold is given in terms of the metric entropy and of Lyapunov exponent.

Mathematical Paradigms for Periodic Phase Separation and Self-Assembly of Diblock Copolymers

Series
Applied and Computational Mathematics Seminar
Time
Monday, November 2, 2009 - 13:00 for 30 minutes
Location
Skiles 255
Speaker
Rustum Choksi – Simon Fraser University

A density functional theory of Ohta and Kawasaki gives rise to nonlocal perturbations of the well-studied Cahn-Hilliard and isoperimetric variational problems. In this talk, I will discuss these simple but rich variational problems in the context of diblock copolymers. Via a combination of rigorous analysis and numerical simulations, I will attempt to characterize minimizers without any preassigned bias for their geometry.

Energy-driven pattern formation induced by competing short and long-range interactions is ubiquitous in science, and provides a source of many challenging problems in nonlinear analysis. One example is self-assembly of diblock copolymers. Phase separation of the distinct but bonded chains in dibock copolymers gives rise to an amazingly rich class of nanostructures which allow for the synthesis of materials with tailor made mechanical, chemical and electrical properties. Thus one of the main challenges is to describe and predict the self-assembled nanostructure given a set of material parameters.

Counting contingency tables: algorithms and asymptotics

Series
Joint ACO and ARC Colloquium
Time
Monday, November 2, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Klaus 1116W
Speaker
Alexander Barvinok – University of Michigan

Tea and light refreshments 1:30 in Room 2222. Organizer: Santosh Vempala

I will discuss recent progress on the construction of randomized algorithms for counting non-negative integer matrices with prescribed row and column sums and on finding asymptotic formulas for the number of such matrices (also known as contingency tables). I will also discuss what a random (with respect to the uniform measure) non-negative integer matrix with prescribed row and column sums looks like.

Pricing Catastrophe Put Options Using Methods in Ruin Theory

Series
Mathematical Finance/Financial Engineering Seminar
Time
Tuesday, November 3, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Sheldon Lin – Department of Statistics, University of Toronto
The discounted penalty function proposed in the seminal paper Gerber and Shiu (1998) has been widely used to analyze the time of ruin, the surplus immediately before ruin and the deficit at ruin of insurance risk models in ruin theory. However, few of its applications can be found beyond, except that Gerber and Landry (1998) explored its use for the pricing of perpetual American put options. In this talk, I will discuss the use of the discounted penalty function and mathematical tools developed for the function for perpetual American catastrophe put options. Assuming that catastrophe losses follow a mixture of Erlang distributions, I will show that an analytical (semi-closed) expression for the price of perpetual American catastrophe put options can be obtained. I will then discuss the fitting of a mixture of Erlang distributions to catastrophe loss data using an EM algorithm.

The Linearized System for Isometric Embeddings and Its Characteristic Variety

Series
PDE Seminar
Time
Tuesday, November 3, 2009 - 15:05 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Qing Han – University of Notre Dame
We prove a conjecture of Bryant, Griffiths, and Yang concerning the characteristic variety for the determined isometric embedding system. In particular, we show that the characteristic variety is not smooth for any dimension greater than 3. This is accomplished by introducing a smaller yet equivalent linearized system, in an appropriate way, which facilitates analysis of the characteristic variety.

Universal Gaussian fluctuations of non-Hermitian matrix ensembles

Series
Stochastics Seminar
Time
Tuesday, November 3, 2009 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 255 (Note unusual time and location)
Speaker
Ivan NOURDIN – Paris VI
My aim is to explain how to prove multi-dimensional central limit theorems for the spectral moments (of arbitrary degrees) associated with random matrices with real-valued i.i.d. entries, satisfying some appropriate moment conditions. The techniques I will use rely on a universality principle for the Gaussian Wiener chaos as well as some combinatorial estimates. Unlike other related results in the probabilistic literature, I will not require that the law of the entries has a density with respect to the Lebesgue measure. The talk is based on a joint work with Giovanni Peccati, and use an invariance principle obtained in a joint work with G. P. and Gesine Reinert

Computational Analysis of Dynamic Networks (and its applications to social life of zebras)

Series
Mathematical Biology Seminar
Time
Wednesday, November 4, 2009 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Tanya Berger-Wolf – Department of Computer Science, University of Illinois at Chicago
Computation has fundamentally changed the way we study nature. Recent breakthroughs in data collection technology, such as GPS and other mobile sensors, are giving biologists access to data about wild populations that are orders of magnitude richer than any previously collected. Such data offer the promise of answering some of the big ecological questions about animal populations. The data are not unique to animal domain but is now prevalent in human interactions: emails, blogs, and online social networks. Unfortunately, our ability to analyze these data lags substantially behind our ability to collect it. In particular, interactions among individuals are often modeled as social networks where nodes represent individuals and an edge exists if the corresponding individuals have interacted during the observation period. The model is essentially static in that the interactions are aggregated over time and all information about the time and ordering of social interactions is discarded. We show that suchtraditional social network analysis methods may result in incorrect conclusions on dynamic data about the structure of interactions and the processes that spread over those interactions. We have extended computational methods for social network analysis to explicitly address the dynamic nature of interactions among individuals. We have developed techniques for identifying persistent communities, influential individuals, and extracting patterns of interactions in dynamic social networks. We will present our approach and demonstrate its applicability by analyzing interactions among zebra populations.

Dynamical Systems, Graphs, Entropies, Dynamical Networks, and Statistical Mechanics

Series
Research Horizons Seminar
Time
Wednesday, November 4, 2009 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Leonid Bunimovich – School of Mathematics, Georgia Tech –
Dynamical systems theory is concerned with systems that change in time (where time can be any semigroup). However, it is quite rare that one can find the solutions for such systems or even a "sizable" subset of such solutions. An approach motivated by this fact, that goes back to Poincaré, is to study instead partitions of the (phase) space M of all states of a dynamical system and consider the evolution of the elements of this partition (instead of the evolution of points of M). I'll explain how the objects in the title appear, some relations between them, and formulate a few general as well as more specific open problems suitable for a PhD thesis in dynamical systems, mathematical biology, graph theory and applied and computational mathematics. This talk will also serve to motivate and introduce to the topics to be given in tomorrow's colloquium.

Derived functors and sheaf cohomology

Series
Other Talks
Time
Wednesday, November 4, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Farbod Shokrieh – Ga Tech
We will continue the study of derived functors between abelian categories. I will show why injective objects are needed for the construction. I will then show that, for any ringed space, the abelian category of all sheaves of Modules has enough injectives. The relation with Cech cohomology will also be studied.

On a Bargmann transform and coherent states for the n-sphere

Series
Analysis Seminar
Time
Wednesday, November 4, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Dr Carlos Villegas Blas – Instituto de Matematicas UNAM, Unidad. Cuernavaca –
We will introduce a Bargmann transform from the space of square integrable functions on the n-sphere onto a suitable Hilbert space of holomorphic functions on a null quadric. On base of our Bargmann transform, we will introduce a set of coherent states and study their semiclassical properties. For the particular cases n=2,3,5, we will show the relation with two known regularizations of the Kepler problem: the Kustaanheimo-Stiefel and Moser regularizations.

DYNAMICAL NETWORKS, ISOSPECTRAL GRAPH REDUCTION

Series
School of Mathematics Colloquium
Time
Thursday, November 5, 2009 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Lyonia Bunimovich – Georgia Tech
Real life networks are usually large and have a very complicated structure. It is tempting therefore to simplify or reduce the associated graph of interactions in a network while maintaining its basic structure as well as some characteristic(s) of the original graph. A key question is which characteristic(s) to conserve while reducing a graph. Studies of dynamical networks reveal that an important characteristic of a network's structure is a spectrum of its adjacency matrix. In this talk we present an approach which allows for the reduction of a general weighted graph in such a way that the spectrum of the graph's (weighted) adjacency matrix is maintained up to some finite set that is known in advance. (Here, the possible weights belong to the set of complex rational functions, i.e. to a very general class of weights). A graph can be isospectrally reduced to a graph on any subset of its nodes, which could be an important property for various applications. It is also possible to introduce a new equivalence relation in the set of all networks. Namely, two networks are spectrally equivalent if each of them can be isospectrally reduced onto one and the same (smaller) graph. This result should also be useful for analysis of real networks. As the first application of the isospectral graph reduction we considered a problem of estimation of spectra of matrices. It happens that our procedure allows for improvements of the estimates obtained by all three classical methods given by Gershgorin, Brauer and Brualdi. (Joint work with B.Webb) A talk will be readily accessible to undergraduates familiar with matrices and complex functions.

Integrated random walks: the probability to stay positive

Series
Stochastics Seminar
Time
Thursday, November 5, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Vlad Vysotsky – University of Delaware
Let $S_n$ be a centered random walk with a finite variance, and define the new sequence $\sum_{i=1}^n S_i$, which we call the {\it integrated random walk}. We are interested in the asymptotics of $$p_N:=\P \Bigl \{ \min \limits_{1 \le k \le N} \sum_{i=1}^k S_i \ge 0 \Bigr \}$$ as $N \to \infty$. Sinai (1992) proved that $p_N \asymp N^{-1/4}$ if $S_n$ is a simple random walk. We show that $p_N \asymp N^{-1/4}$ for some other types of random walks that include double-sided exponential and double-sided geometric walks (not necessarily symmetric). We also prove that $p_N \le c N^{-1/4}$ for lattice walks and upper exponential walks, i.e., walks such that $\mbox{Law} (S_1 | S_1>0)$ is an exponential distribution.

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 Keller – School 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.

Small noise limit for dynamics near unstable critical points (Oral Comprehensive Exam).

Series
Other Talks
Time
Friday, November 6, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 154
Speaker
Sergio Almada – Georgia Tech
We consider the Stochastic Differential Equation $dX_\epsilon=b(X_\epsilon)dt + \epsilon dW$ . Given a domain D, we study how the exit time and the distribution of the process at the time it exits D behave as \epsilon goes to 0. In particular, we cover the case in which the unperturbed system $\frac{d}{dt}x=b(x)$ has a unique fixed point of the hyperbolic type. We will illustrate how the behavior of the system is in the linear case. We will remark how our results give improvements to the study of systems admitting heteroclinic or homoclinic connections. We will outline the general proof in two dimensions that requires normal form theory from differential equations. For higher dimensions, we introduce a new kind of non-smooth stochastic calculus.

Constructing 3-Manifolds Using Dehn Surgery on Handlebodies

Series
Geometry Topology Working Seminar
Time
Friday, November 6, 2009 - 15:00 for 2 hours
Location
Skiles 269
Speaker
Meredith Casey – Georgia Tech
The goal of this talk is to describe simple constructions by which we can construct any compact, orientable 3-manifold. It is well-known that every orientable 3-manifold has a Heegaard splitting. We will first define Heegaard splittings, see some examples, and go through a very geometric proof of this therem. We will then focus on the Dehn-Lickorish Theorem, which states that any orientation-preserving homeomorphism of an oriented 2-manifold without boundary can by presented as the composition of Dehn twists and homeomorphisms isotopic to the identity. We will prove this theorm, and then see some applications and examples. With both of these resutls together, we will have shown that using only handlebodies and Dehn twists one can construct any compact, oriented 3-manifold.

Graph Tiling in Bipartite Graphs

Series
Combinatorics Seminar
Time
Friday, November 6, 2009 - 15:05 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Albert Bush – School of Mathematics, Georgia Tech

This is joint work with Dr. Yi Zhao.

Graph tiling problems can be summarized as follows: given a graph H, what conditions do we need to find a spanning subgraph of some larger graph G that consists entirely of disjoint copies of H. The most familiar example of a graph tiling problem is finding a matching in a graph. With the Regularity Lemma and the Blow-up Lemma as our main tools, we prove a degree condition that guarantees an arbitrary bipartite graph G will be tiled by an arbitrary bipartite graph H. We also prove this degree condition is best possible up to a constant. This answers a question of Zhao and proves an asymptotic version of a result of Kuhn and Osthus for bipartite graphs.