Seminars and Colloquia by Series

Sum-product Inequalities and Combinatorial Problems on Sumsets

Series
Dissertation Defense
Time
Friday, July 17, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Albert Bush – School of Mathematics, Georgia Tech
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.

Symmetric ideals and numerical primary decomposition

Series
Dissertation Defense
Time
Tuesday, May 26, 2015 - 11:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Robert Krone – Georgia Tech –
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.

Minimization Problems Involving Policonvex Integrands

Series
Dissertation Defense
Time
Friday, April 24, 2015 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 249
Speaker
Romeo Awi – School of Mathematics, Georgia Tech
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.

The Filippov moments solution on the intersection of two and three manifolds

Series
Dissertation Defense
Time
Thursday, April 2, 2015 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Fabio Difonzo – School of Mathematics, Georgia Tech
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.

Small-time asymptotics of call prices and implied volatilities for exponential Levy models

Series
Dissertation Defense
Time
Tuesday, January 6, 2015 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Allen Hoffmeyer – School of Mathematics, Georgia Tech
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.

Some Results in Sums and Products

Series
Dissertation Defense
Time
Thursday, November 13, 2014 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Chris Pryby – School of Mathematics, Georgia Tech
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.

Low-Rank Estimation of Smooth Kernels on Graphs

Series
Dissertation Defense
Time
Monday, July 21, 2014 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Pedro Rangel – School of Mathematics, Georgia Tech
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.

Linear Systems on Metric graphs and Some Applications to Tropical Geometry and Non-Archimedean Geometry

Series
Dissertation Defense
Time
Thursday, June 26, 2014 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ye Luo – School of Mathematics, Georgia Tech
The work in this dissertation is mainly focused on three subjects which are essentially related to linear systems on metric graphs and its application: (1) rank-determining sets of metric graphs, which can be employed to actually compute the rank function of arbitrary divisors on an arbitrary metric graph, (2) a tropical convexity theory for linear systems on metric graphs, and (3) smoothing of limit linear series of rank one on refined metrized complex (an intermediate object between metric graphs and algebraic curves),

A Numerical Study of Vorticity-Enhanced Heat Transfer

Series
Dissertation Defense
Time
Tuesday, June 24, 2014 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Xiaolin Wang – School of Mathematics, Georgia Tech
In this work, we numerically studied the effect of the vorticity on the enhancement of heat transfer in a channel flow. Based on the model we proposed, we find that the flow exhibits different properties depending on the value of four dimensionless parameters. In particularly, we can classify the flows into two types, active and passive vibration, based on the sign of the incoming vortices. The temperature profiles according to the flow just described also show different characteristics corresponding to the active and passive vibration cases. In active vibration cases, we find that the heat transfer performance is directly related to the strength of the incoming vortices and the speed of the background flow. In passive vibration cases, the corresponding heat transfer process is complicated and varies dramatically as the flow changes its properties. Compared to the fluid parameters, we also find that the thermal parameters have much less effect on the heat transfer enhancement. Finally, we propose a more realistic optimization problem which is to minimize the maximum temperature of the solids with a given input energy. We find that the best heat transfer performance is obtained in the active vibration case with zero background flow.

Graph Structures and Well-Quasi-Ordering

Series
Dissertation Defense
Time
Thursday, June 12, 2014 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Chun-Hung Liu – Georgia Tech
Robertson and Seymour proved that graphs are well-quasi-ordered by the minor relation and the weak immersion relation. In other words, given infinitely many graphs, one graph contains another as a minor (or a weak immersion, respectively). An application of these theorems is that every property that is closed under deleting vertices, edges, and contracting (or "splitting off", respectively) edges can be characterized by finitely many graphs, and hence can be decided in polynomial time. In this thesis we are concerned with the topological minor relation. We say that a graph G contains another graph H as a topological minor if H can be obtained from a subgraph of G by repeatedly deleting a vertex of degree two and adding an edge incident with the neighbors of the deleted vertex. Unlike the relation of minor and weak immersion, the topological minor relation does not well-quasi-order graphs in general. However, Robertson conjectured in the late 1980's that for every positive integer k, the topological minor relation well-quasi-orders graphs that do not contain a topological minor isomorphic to the path of length k with each edge duplicated. This thesis consists of two main results. The first one is a structure theorem for excluding a fixed graph as a topological minor, which is analogous to a cornerstone result of Robertson and Seymour, who gave such structure for graphs that exclude a fixed minor. Results for topological minors were previously obtained by Grohe and Marx and by Dvorak, but we push one of the bounds in their theorems to the optimal value. This improvement is needed for the next theorem. The second main result is a proof of Robertson's conjecture. As a corollary, properties on certain graphs closed under deleting vertices, edges, and "suppressing" vertices of degree two can be characterized by finitely many graphs, and hence can be decided in polynomial time.

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