Seminars and Colloquia by Series

Two Problems in Mathematical Physics: Villani's Conjecture and a Trace Inequality for the Fractional Laplacian

Series
Dissertation Defense
Time
Monday, August 29, 2011 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Amit EinavSchool of Mathematics, Georgia Tech
The presented work deals with two distinct problems in the field of Mathematical Physics, and as such will have two parts addressing each problem. The first part is dedicated to an 'almost' solution of Villani's conjecture, a known conjecture related to a Statistical Mechanics model invented by Kac in 1956, giving a rigorous explanation of some simple cases of the Boltzman equation. In 2003 Villani conjectured that the time it will take the system of particles in Kac's model to equalibriate is proportional to the number of particles in the system. Our main result in this part is an 'almost proof' of that conjecture, showing that for all practical purposes we can consider it to be true. The second part of the presentation is dedicated to a newly developed trace inequality for the fractional Laplacian, connecting between the fractional Laplacian of a function and its restriction to the intersection of the hyperplanes x_n =...= x_n-j+1 = 0 , where 1 <= j < n. The newly found inequality is sharp and the functions that attain inequality in it are completely classified.

Topics in Spatial and Dynamical Phase Transitions of Interacting Particle Systems

Series
Dissertation Defense
Time
Monday, August 15, 2011 - 11:00 for 2 hours
Location
Skiles 005
Speaker
Ricardo Restrepo LopezSchool of Mathematics, Georgia Tech
In this work we provide several improvements in the study of phase transitions of interacting particle systems: 1. We determine a quantitative relation between non-extremality of the limiting Gibbs measure of a tree-based spin system, and the temporal mixing of the Glauber Dynamics over its finite projections. We define the concept of `sensitivity' of a reconstruction scheme to establish such a relation. In particular, we focus in the independent sets model, determining a phase transition for the mixing time of the Glauber dynamics at the same location of the extremality threshold of the simple invariant Gibbs version of the model. 2. We develop the technical analysis of the so-called spatial mixing conditions for interacting particle systems to account for the connectivity structure of the underlying graph. This analysis leads to improvements regarding the location of the uniqueness/non-uniqueness phase transition for the independent sets model over amenable graphs; among them, the elusive hard-square model in lattice statistics, which has received attention since Baxter's solution of the analogue hard-hexagon in 1980. 3. We build on the work of Montanari and Gerschenfeld to determine the existence of correlations for the coloring model in sparse random graphs. In particular, we prove that correlations exist above the `clustering' threshold of such model; thus providing further evidence for the conjectural algorithmic `hardness' occurring at such point.

Normally Elliptic Singular Perturbation Problems: Local Invariant Manifolds and Applications

Series
Dissertation Defense
Time
Monday, May 16, 2011 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Nan LuSchool of Mathematics, Georgia Tech

Please Note: Advisor Chongchun Zeng

We study the normally elliptic singular perturbation problems including both finite and infinite dimensional cases, which could also be nonautonomous. In particular, we establish the existence and smoothness of O(1) local invariant manifolds and provide various estimates which are independent of small singular parameters. We also use our results on local invariant manifolds to study the persistence of homoclinic solutions under weakly dissipative and conservative perturbations.

Judicious Partitions of Graphs and Hypergraphs

Series
Dissertation Defense
Time
Tuesday, April 26, 2011 - 12:30 for 2 hours
Location
Skiles 005
Speaker
Jie MaSchool of Mathematics, Georgia Tech
Classical partitioning problems, like the Max-Cut problem, ask for partitions that optimize one quantity, which are important to such fields as VLSI design, combinatorial optimization, and computer science. Judicious partitioning problems on graphs or hypergraphs ask for partitions that optimize several quantities simultaneously. In this dissertation, we work on judicious partitions of graphs and hypergraphs, and solve or asymptotically solve several open problems of Bollobas and Scott on judicious partitions, using the probabilistic method and extremal techniques.

Hardy-Sobolev-Maz'ya Inequalities for Fractional Integrals on Halfspaces and Convex Domains

Series
Dissertation Defense
Time
Tuesday, April 19, 2011 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Craig A. SloaneSchool of Mathematics, Georgia Tech
Classical Hardy, Sobolev, and Hardy-Sobolev-Maz'ya inequalities are well known results that have been studied for awhile. In recent years, these results have been been generalized to fractional integrals. This Dissertation proves a new Hardy inequality on general domains, an improved Hardy inequality on bounded convex domains, and that the sharp constant for any convex domain is the same as that known for the upper halfspace. We also prove, using a new type of rearrangement on the upper halfspace, based in part on Carlen and Loss' concept of competing symmetries, the existence of the fractional Hardy-Sobolev-Maz'ya inequality in the case p = 2, as well as proving the existence of minimizers, at least in limited cases.

Isospectral Graph Reductions, Estimates of Matrices' Spectra, and Eventually Negative Schwarzian Systems

Series
Dissertation Defense
Time
Tuesday, March 8, 2011 - 09:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Benjamin WebbSchool of Mathematics, Georgia Tech
Real world networks typically consist of a large number of dynamical units with a complicated structure of interactions. Until recently such networks were most often studied independently as either graphs or as coupled dynamical systems. To integrate these two approaches we introduce the concept of an isospectral graph transformation which allows one to modify the network at the level of a graph while maintaining the eigenvalues of its adjacency matrix. This theory can then be used to rewire dynamical networks, considered as dynamical systems, in order to gain improved estimates for whether the network has a unique global attractor. Moreover, this theory leads to improved eigenvalue estimates of Gershgorin-type. Lastly, we will discuss the use of Schwarzian derivatives in the theory of 1-d dynamical systems.

Scaling limit for the diffusion exit problem

Series
Dissertation Defense
Time
Thursday, February 3, 2011 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sergio Angel AlmadaSchool of Mathematics, Georgia Tech
A stochastic differential equation with vanishing martingale term is studied. Specifically, given a domain D, the asymptotic scaling properties of both the exit time from the domain and the exit distribution are considered under the additional (nonstandard) hypothesis that the initial condition also has a scaling limit. Methods from dynamical systems are applied to get more complete estimates than the ones obtained by the probabilistic large deviation theory. Two situations are completely analyzed. When there is a unique critical saddle point of the deterministic system (the system without random effects), and when the unperturbed system escapes the domain D in finite time. Applications to these results are in order. In particular, the study of 2-dimensional heteroclinic networks is closed with these results and shows the existence of possible asymmetries. Also, 1-dimensional diffusions conditioned to rare events are further studied using these results as building blocks.

Master's Thesis. Limit theorems for a one dimensional system with random switchings.

Series
Dissertation Defense
Time
Tuesday, October 5, 2010 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Tobias HurthSchool of Mathematics, Georgia Tech
We consider a simple one-dimensional random dynamical system with two driving vector fields and random switchings between them. We show that this system satisfies a one force - one solution principle and compute the unique invariant density explicitly. We study the limiting behavior of the invariant density as the switching rate approaches zero or infinity and derive analogues of classical probability theory results such as central limit theorem and large deviation principle.

Pages