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Series: Dissertation Defense

The main goal of the thesis is to study integro-differential equations. Integro-differential equations arise naturally in the study of stochastic processes with jumps. These types of processes are of particular interest in finance, physics and ecology. In the thesis, we study uniqueness, existence and regularity of solutions of integro-PDE in domains of R^n.

Series: Dissertation Defense

This thesis studies the effect of transverse surgery on open books, the Heegaard Floer contact invariant, and tightness. We show that surgery on the connected binding of a genus g open book that supports a tight contact structure preserves tightness if the surgery coefficient is greater than 2g-1. We also give criteria for when positive contact surgery on Legendrian knots will result in an overtwisted manifold.

Series: Dissertation Defense

Advisor: Dr. Federico Bonetto

We consider a model of N particles interacting through a Kac-style collision process, with m particles among them interacting, in addition, with a thermostat. When m = N, we show exponential approach to the equilibrium canonical distribution in terms of the L2 norm, in relative entropy, and in the Gabetta-Toscani-Wennberg (GTW) metric, at a rate independent of N. When m < N , the exponential rate of approach to equilibrium in L2 is shown to behave as m/N for N large, while the relative entropy and the GTW distance from equilibrium exhibit (at least) an "eventually exponential” decay, with a rate scaling as m/N^2 for large N. As an allied project, we obtain a rigorous microscopic description of the thermostat used, based on a model of a tagged particle colliding with an infinite gas in equilibrium at the thermostat temperature. These results are based on joint work with Federico Bonetto, Michael Loss and Hagop Tossounian.

Series: Dissertation Defense

The thesis investigates a version of the sum-product inequality studied by
Chang in which one tries to prove the h-fold sumset is large under the
assumption that the 2-fold product set is small. Previous bounds were
logarithmic in the exponent, and we prove the first super-logarithmic
bound. We will also discuss a new technique inspired by convex geometry to
find an order-preserving Freiman 2-isomorphism between a set with small
doubling and a small interval. Time permitting, we will discuss some
combinatorial applications of this result.

Series: Dissertation Defense

The thesis considers two distinct strategies for algebraic computation with polynomials in high dimension. The first concerns ideals and varieties with symmetry, which often arise in applications from areas such as algebraic statistics and optimization. We explore the commutative algebra properties of such objects, and work towards classifying when symmetric ideals admit finite descriptions including equivariant Gröbner bases and generating sets. Several algorithms are given for computing such descriptions. Specific focus is given to the case of symmetric toric ideals. A second area of research is on problems in numerical algebraic geometry. Numerical algorithms such as homotopy continuation can efficiently compute the approximate solutions of systems of polynomials, but generally have trouble with multiplicity. We develop techniques to compute local information about the scheme structure of an ideal at approximate zeros. This is used to create a hybrid numeric-symbolic algorithm for computing a primary decomposition of the ideal.

Series: Dissertation Defense

This thesis is mainly concerned with problems in the areas of the
Calculus of Variations and Partial Differential Equations (PDEs). The
properties of the functional to minimize play an important role in
the existence of minimizers of integral problems. We will introduce the
important concepts of quasiconvexity and polyconvexity. Inspired by
finite element methods from Numerical Analysis, we introduce a
perturbed problem which has some surprising uniqueness properties.

Series: Dissertation Defense

We consider several possibilities on how to select a Filippov sliding
vector field on a co-dimension 2 singularity manifold, intersection of two
co-dimension 1 manifolds, under the assumption of general attractivity. Of specific
interest is the selection of a smoothly varying Filippov sliding vector field. As a
result of our analysis and experiments, the best candidates of the many possibilities
explored are based on so-called barycentric coordinates: in particular, we choose
what we call the moments solution. We then examine the behavior of the moments vector
field at the first order exit points, and show that it aligns smoothly with the exit
vector field. Numerical experiments illustrate our results and contrast the present
method with other choices of Filippov sliding vector field. We further generalize
this construction to co-dimension 3 and higher.

Series: Dissertation Defense

We derive at-the-money call-price and implied volatility asymptotic
expansions in time to maturity for a selection of exponential Levy models, restricting our
attention to asset-price models whose log returns structure is a Levy process. We consider
two main problems. First, we consider very general Levy models that are in the domain of
attraction of a stable random variable. Under some relatively minor assumptions, we give
first-order at-the-money call-price and implied volatility asymptotics. In the case where
our Levy process has Brownian component, we discover new orders of convergence by showing
that the rate of convergence can be of the form t^{1/\alpha} \ell( t ) where \ell is a slowly
varying function and \alpha \in (1,2). We also give an example of a Levy model which
exhibits this new type of behavior where \ell is not asymptotically constant. In the case of
a Levy process with Brownian component, we find that the order of convergence of the call
price is \sqrt{t}. Second, we investigate the CGMY process whose call-price asymptotics are
known to third order. Previously, measure transformation and technical estimation methods were
the only tools available for proving the order of convergence. We give a new method that relies
on the Lipton-Lewis formula, guaranteeing that we can estimate the call-price asymptotics using
only the characteristic function of the Levy process. While this method does not provide a
less technical approach, it is novel and is promising for obtaining second-order call-price
asymptotics for at-the-money options for a more general class of Levy processes.

Series: Dissertation Defense

We demonstrate new results in additive combinatorics, including a proof of
the following conjecture by J. Solymosi: for every epsilon > 0, there
exists delta > 0 such that, given n^2 points in a grid formation in R^2, if
L is a set of lines in general position such that each line intersects at
least n^{1-delta} points of the grid, then |L| < n^epsilon. This result
implies a conjecture of Gy. Elekes regarding a uniform statistical version
of Freiman's theorem for linear functions with small image sets.

Series: Dissertation Defense

This dissertation investigates the problem of estimating a kernel over a
large graph based on a sample of noisy observations of linear measurements
of the kernel. We are interested in solving this estimation problem in the
case when the sample size is much smaller than the ambient dimension of the
kernel. As is typical in high-dimensional statistics, we are able to design
a suitable estimator based on a small number of samples only when the
target kernel belongs to a subset of restricted complexity. In our study,
we restrict the complexity by considering scenarios where the target kernel
is both low-rank and smooth over a graph. The motivations for studying such
problems come from various real-world applications like recommender systems
and social network analysis.
We study the problem of estimating smooth kernels on graphs. Using standard
tools of non-parametric estimation, we derive a minimax lower bound on the
least squares error in terms of the rank and the degree of smoothness of
the target kernel. To prove the optimality of our lower-bound, we proceed
to develop upper bounds on the error for a least-square estimator based on
a non-convex penalty. The proof of these upper bounds depends on bounds for
estimators over uniformly bounded function classes in terms of Rademacher
complexities. We also propose a computationally tractable estimator based
on least-squares with convex penalty. We derive an upper bound for the
computationally tractable estimator in terms of a coherence function
introduced in this work. Finally, we present some scenarios wherein this
upper bound achieves a near-optimal rate.