Seminars and Colloquia by Series

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 BunimovichGeorgia 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.

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 BlasInstituto 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.

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 ShokriehGa 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.

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 BunimovichSchool 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.

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-WolfDepartment 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.

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

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 HanUniversity 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.

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 LinDepartment 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.

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 BarvinokUniversity of Michigan

Please Note: 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.

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