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Series: Dissertation Defense

Graduate Advisor: Eberhard Voit

Series: Dissertation Defense

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.

Series: Dissertation Defense

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.

Series: Dissertation Defense

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.

Series: Dissertation Defense

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.

Series: Dissertation Defense

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.

Series: Dissertation Defense

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.

Series: Dissertation Defense

Series: Dissertation Defense

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.

Series: Dissertation Defense

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.