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Series: Research Horizons Seminar

We examine some problems in the coupled motions of fluids and flexible solid bodies. We first present some basic equations in fluid dynamics and solid mechanics, and then show some recent asymptotic results and numerical simulations. No prior experience with fluid dynamics is necessary.

Series: PDE Seminar

We consider the existence of periodic solutions to the Euler equations of gas dynamics. Such solutions have long been thought not to exist due to shock formation, and this is confirmed by the celebrated Glimm-Lax decay theory for 2x2 systems. However, in the full 3x3 system, multiple interaction effects can combine to slow down and prevent shock formation. In this talk I shall describe the physical mechanism supporting periodicity, describe combinatorics of simple wave interactions, and develop periodic solutions to a "linearized" problem. These linearized solutions have a beautiful structure and exhibit several surprising and fascinating phenomena. I shall also discuss partial progress on the perturbation problem: this leads us to problems of small divisors and KAM theory. This is joint work with Blake Temple.

Series: CDSNS Colloquium

Nonlinear wave phenomena are of great importance in the physical world and have been for a long time a challenging topic of research for both pure and applied mathematicians. There are numerous nonlinear evolution equations for which we need to analyze the properties of the solutions for time evolution of the system. As the first step, we should understand the dynamics of their traveling wave solutions. There exists an enormous literature on the study of nonlinear wave equations, in which exact explicit solitary wave, kink wave, periodic wave solutions and their dynamical stabilities are discussed. To find exact traveling wave solutions for a given nonlinear wave system, a lot of methods have been developed. What is the dynamical behavior of these exact traveling wave solutions? How do the travelling wave solutions depend on the parameters of the system? What is the reason of the smoothness change of traveling wave solutions? How to understand the dynamics of the so-called compacton and peakon solutions? These are very interesting and important problems. The aim of this talk is to give a more systematic account for the bifurcation theory method of dynamical systems to find traveling wave solutions and understand their dynamics for two classes of singular nonlinear traveling systems.

Monday, November 17, 2008 - 13:00 ,
Location: Skiles 255 ,
Maoan Han ,
Shanghai Normal University ,
Organizer: Haomin Zhou

Let H(m) denote the maximal number of limit cycles of polynomial systems of degree m. It is called the Hilbert number. The main part of Hilbert's 16th problem posed in 1902 is to find its value. The problem is still open even for m=2. However, there have been many interesting results on the lower bound of it for m\geq 2. In this paper, we give some new lower bounds of this number. The results obtained in this paper improve all existing results for all m\geq 7 based on some known results for m=3,4,5,6. In particular, we confirm the conjecture H(2k+1) \geq (2k+1)^2-1 and obtain that H(m) grows at least as rapidly as \frac{1}{2\ln2}(m+2)^2\ln(m+2) for all large m.

Friday, November 14, 2008 - 14:00 ,
Location: Skiles 269 ,
Thang Le ,
School of Mathematics, Georgia Tech ,
Organizer: Thang Le

We will explain the famous result of Luck and Schick which says that for a large class of 3-manifolds, including all knot complements, the hyperbolic volume is equal to the l^2-torsion. Then we speculate about the growth of homology torsions of finite covers of knot complements. The talk will be elementary and should be accessible to those interested in geometry/topology.

Series: Stochastics Seminar

Let X=(X_1,\ldots,X_n) be a n-dimensional random vector for which the distribution has Markov structure corresponding to a junction forest, assuming functional forms for the marginal distributions associated with the cliques of the underlying graph. We propose a latent variable approach based on computing junction forests from filtrations. This methodology establishes connections between efficient algorithms from Computational Topology and Graphical Models, which lead to parametrizations for the space of decomposable graphs so that: i) the dimension grows linearly with respect to n, ii) they are convenient for MCMC sampling.

Series: ACO Distinguished Lecture

RECEPTION TO FOLLOW

Fifty five years ago, two results of the Hungarian mathematicians, Koenig and Egervary, were combined using the duality theory of linear programming to construct the Hungarian Method for the Assignment Problem. In a recent reexamination of the geometric interpretation of the algorithm (proposed by Schmid in 1978) as a steepest descent method, several variations on the algorithm have been uncovered, which seem to deserve further study.
The lecture will be self-contained, assuming little beyond the duality theory of linear programming. The Hungarian Method will be explained at an elementary level and will be illustrated by several examples.
We shall conclude with account of a posthumous paper of Jacobi containing an algorithm developed by him prior to 1851 that is essentially identical to the Hungarian Method, thus anticipating the results of Koenig (1931), Egervary (1931), and Kuhn (1955) by many decades.

Series: Stochastics Seminar

Many context-free formalisms based on transitive properties of trees and strings have been converted to probabilitic models. We have Probabilistic Finite Automaton, Probabilistic Context Free Grammar and Probabilistic Tree Adjoining Grammars and many other probabilistic models of grammars. Typically such formalisms employ context-free productions that are transitively closed. Context-free grammars can be represented declaratively through context-sensitive grammars that analyse or check wellformedness of trees. When this direction is elaborated further, we obtain constraint-based representations for regular, context-free and mildly-context sensitive languages and their associated structures. Such representations can also be Probabilistic and this could be achieved by combining weighted rational operations and Dyck languages. More intuitively, the rational operations are packed to a new form of conditional rule: Generalized Restriction or GR in short (Yli-Jyrä and Koskenniemi 2004), or a predicate logic over strings. The conditional rule, GR, is flexible and provides total contexts, which is very useful e.g. when compiling rewriting rules for e.g. phonological alternations or speech or text normalization. However, the total contexts of different conditional rewriting rules can overlap. This implies that the conditions of different rules are not independent and the probabilities do not combine like in the case of context-free derivations. The non-transitivity causes problems for the general use of probabilistic Generalized Restriction e.g. when adding probabilities to phonological rewriting grammars that define regular relations.

Wednesday, November 12, 2008 - 14:00 ,
Location: Skiles 255 ,
Christian Houdré ,
School of Mathematics, Georgia Tech ,
Organizer: Christian Houdre

In connection with the class Stochastic Processes in Finance II, we will have a supplementary lecture where a first, 50 minutes long, movie on Doeblin's life will be shown. This will be followed by a second movie, 30 minutes long, where Yor explains on the blackboard Doeblin's contribution to what Shreeve calls the Ito-Doeblin's lemma.

Series: ACO Student Seminar

We consider the following Maximum Budgeted Allocation(MBA) problem: Given a set of m indivisible items and n agents; each agent i is willing to pay b_ij amount of money on item j, and in addition he species the maximum amount (budget of B_i) he is willing to pay in total over all the items he receives. Goal is to allocate items to agents so as to maximize the total payment received from all the agents. The problem naturally arises as auctioneer revenue maximization in first price budget-constrained Auctions (For e.g. auctioning of TV/Radio ads by Google). Our main results are: 1) We give a 3/4-approximation algorithm for MBA improving upon the previous best of 0.632 [Anelman-Mansour, 04]. Our factor matches the integrality gap of the LP used by the previous results. 2) We prove it is NP-hard to approximate MBA to any factor better than 15/16, previously only NP-hardness was known. Our result also implies NP-hardness of approximating maximum submodular welfare with demand oracle to a factor better than 15/16, improving upon the best known hardness of 275/276 [Feige-Vondrak, 07]. Our hardness techniques can be modified to prove that it is NP-hard to approximate the Generalized Assignment Problem (GAP) to any factor better than 10/11. This improves upon the 422/423 hardness of [Chekuri-Kumar, 04]. We use iterative rounding on a natural LP relaxation of MBA to obtain the 3/4-approximation. Recently iterative rounding has achieved considerable success in designing approximation algorithms. However, these successes have been limited to minimization problems, and as per our knowledge, this work is the first iterative rounding based approximation algorithm for a natural maximization problem. We also give a (3/4 - \epsilon)-factor algorithm based on the primal-dual schema which runs in O(nm) time, for any constant \epsilon > 0. In this talk, I will present the iterative rounding based algorithm, show the hardness reductions, and put forward some directions which can help in solving the natural open question of closing the approximation gap. Joint work with Deeparnab Chakrabarty.