Seminars and Colloquia by Series

Monday, April 25, 2016 - 15:05 , Location: Skiles 005 , Chenchen Mou , Georgia Institute of Technology , cmou3@math.gatech.edu , Organizer:
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.
Tuesday, April 19, 2016 - 10:00 , Location: Skiles 006 , James Conway , Georgia Tech , Organizer: James Conway
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.
Friday, October 30, 2015 - 12:05 , Location: Skiles 006 , Ranjini Vaidyanathan , School of Mathematics, Georgia Tech , Organizer: Ranjini Vaidyanathan

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.
Friday, July 17, 2015 - 14:00 , Location: Skiles 006 , Albert Bush , School of Mathematics, Georgia Tech , Organizer:
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.
Tuesday, May 26, 2015 - 11:30 , Location: Skiles 006 , Robert Krone , Georgia Tech , krone@math.gatech.edu , Organizer: Robert Krone
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.
Friday, April 24, 2015 - 13:30 , Location: Skiles 249 , Romeo Awi , School of Mathematics, Georgia Tech , Organizer: Wilfrid Gangbo
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.
Thursday, April 2, 2015 - 12:05 , Location: Skiles 006 , Fabio Difonzo , School of Mathematics, Georgia Tech , Organizer:
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.
Tuesday, January 6, 2015 - 15:00 , Location: Skiles 005 , Allen Hoffmeyer , School of Mathematics, Georgia Tech , Organizer: Allen Hoffmeyer
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.
Thursday, November 13, 2014 - 10:00 , Location: Skiles 114 , Chris Pryby , School of Mathematics, Georgia Tech , Organizer:
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.
Monday, July 21, 2014 - 14:05 , Location: Skiles 006 , Pedro Rangel , School of Mathematics, Georgia Tech , Organizer:
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.

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