Seminars and Colloquia by Series

The dynamics of two classes of singular traveling wave systems

Series
CDSNS Colloquium
Time
Monday, November 17, 2008 - 16:30 for 2 hours
Location
Skiles 255
Speaker
Jibin LiKunming Univeristy of Science and Technology and Zhejiang Normal University
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.

Lower bounds for the Hilbert number of polynomial systems

Series
Applied and Computational Mathematics Seminar
Time
Monday, November 17, 2008 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Maoan HanShanghai Normal University
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.

Hyperbolic volume and torsions of 3-manifolds

Series
Geometry Topology Working Seminar
Time
Friday, November 14, 2008 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Thang LeSchool of Mathematics, Georgia Tech
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.

Computing Junction Forests from Filtrations of Simplicial Complexes

Series
Stochastics Seminar
Time
Friday, November 14, 2008 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Sayan MukherjeeDepartment of Statistical Science, Duke University
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.

Reflections on a favorite child

Series
ACO Distinguished Lecture
Time
Thursday, November 13, 2008 - 16:30 for 2 hours
Location
Klaus 1116
Speaker
Harold W. KuhnPrinceton University

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.

Conditionals on structural properties of strings and their stochastic counterparts in a declarative formalism

Series
Stochastics Seminar
Time
Thursday, November 13, 2008 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Anssi Yli-JyräHelsink University
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.

Wolfgang Doeblin: A Mathematician Rediscovered

Series
Mathematical Finance/Financial Engineering Seminar
Time
Wednesday, November 12, 2008 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Christian HoudréSchool of Mathematics, Georgia Tech
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.

On the Approximability of Budgeted Allocations and Improved Lower Bounds for Submodular Welfare Maximization and GAP

Series
ACO Student Seminar
Time
Wednesday, November 12, 2008 - 13:30 for 2 hours
Location
Skiles 269
Speaker
Gagan GoelACO Computer Science, Georgia Tech
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.

A Simple Quantitative Genetic Model of Parent-Offspring Interactions

Series
Mathematical Biology Seminar
Time
Wednesday, November 12, 2008 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Benjamin Ridenhour CDC/CCID/NCIRD, CTR
Parent-offspring interactions lead to natural conflicts. Offspring want as many resources as possible from parents in order to gain maximal fitness levels. On the other hand, parents desire to invest only enough to guarantee survival to reproduction. The resolution of the parent-offspring conflict has been a topic of much debate in evolutionary biology and typically invoke the concept of 'costs' to begging by offspring. Here I present the analysis of a simple quantitative genetic model of parent-offspring interactions that does not costs to resolve parent-offspring conflicts.

Uniqueness in the boundary inverse problem for elasticity

Series
PDE Seminar
Time
Tuesday, November 11, 2008 - 15:15 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Anna MazzucatoPenn State University, State College
We discuss the inverse problem of determining elastic parameters in the interior of an anisotropic elastic media from dynamic measurements made at the surface. This problem has applications in medical imaging and seismology. The boundary data is modeled by the Dirichlet-to-Neumann map, which gives the correspondence between surface displacements and surface tractions. We first show that, without a priori information on the anisotropy type, uniqueness can hold only up to change of coordinates fixing the boundary. In particular, we study orbits of elasticity tensors under diffeomorphisms. Then, we obtain partial uniqueness for special classes of transversely isotropic media. This is joint work with L. Rachele (RPI).