Seminars and Colloquia by Series

The Boundary Method and General Auction for Optimal Mass Transportation and Wasserstein Distance Computation

Series
Dissertation Defense
Time
Tuesday, April 4, 2017 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
J.D. Walsh IIIGeorgia Tech School of Math

Please Note: Dissertation advisor: Luca Dieci

Numerical optimal transport is an important area of research, but most problems are too large and complex for easy computation. Because continuous transport problems are generally solved by conversion to either discrete or semi-discrete forms, I focused on methods for those two. I developed a discrete algorithm specifically for fast approximation with controlled error bounds: the general auction method. It works directly on real-valued transport problems, with guaranteed termination and a priori error bounds. I also developed the boundary method for semi-discrete transport. It works on unaltered ground cost functions, rapidly identifying locations in the continuous space where transport destinations change. Because the method computes over region boundaries, rather than the entire continuous space, it reduces the effective dimension of the discretization. The general auction is the first relaxation method designed for compatibility with real-valued costs and weights. The boundary method is the first transport technique designed explicitly around the semi-discrete problem and the first to use the shift characterization to reduce dimensionality. No truly comparable methods exist. The general auction and boundary method are able to solve many transport problems that are intractible using other approaches. Even where other solution methods exist, in testing it appears that the general auction and boundary method outperform them.

Weighted Inequalities via Dyadic Operators and A Learning Theory Approach to Compressive Sensing

Series
Dissertation Defense
Time
Thursday, March 30, 2017 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 202
Speaker
Scott SpencerGeorgia Institute of Technology
This thesis explores topics from two distinct fields of mathematics. The first part addresses a theme in abstract harmonic analysis, while the focus of the second part is a topic in compressive sensing. The first part of this dissertation explores the application of dominating operators in harmonic analysis by sparse operators. We make use of pointwise sparse dominations weighted inequalities for Calder\'on-Zygmund operators, Hardy-Littlewood maximal operator, and their fractional analogues. Dominating bilinear forms by sparse forms allows us to derive weighted inequalities for oscillatory integral operators (polynomially modulated CZOs) and random discrete Hilbert transforms. The later is defined on sets of initegers with asymptotic density zero, making these weighted inequalitites particulalry attractive. We also discuss a characterization of a certain weighted BMO space by commutators of multiplication operators with fractional integral operators. Compressed sensing illustrates the possibility of acquiring and reconstructing sparse signals via underdetermined (linear) systems. It is believed that iid Gaussian measurement vectors give near optimal results, with the necessary number of measurements on the order of slog⁡(n/s) -- n is ambient dimension and s is sparsity threshhold.The recovery algorithm used above relies on a certain quasi-isometry property of the measurement matrix. A surprising result is that the same order of measurements gives an analogous quasi-isometry in the extreme quantization of one-bit sensing. Bylik and Lacey deliver this result as a consequence of a certain stochastic process on the sphere. We will discuss an alternative method that relies heavily on the VC-dimension of a class of subsets on the sphere.

Special TK_5 in graphs containing K_4^-

Series
Dissertation Defense
Time
Friday, September 2, 2016 - 14:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 005
Speaker
Dawei HeSchool of Mathematics, Georgia Tech
The well-known Kelmans-Seymour conjecture states that every nonplanar 5-connected graph contains TK_5. Ma and Yu prove the conjecture for graphs containing K_4^- . In the thesis, we will find special TK_5 in graphs containing K_4^-, i.e. two versions of generalization of their result will be dealt with separately.

Uniqueness, existence and regularity of solutions of integro-PDE in domains of R^n

Series
Dissertation Defense
Time
Monday, April 25, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Chenchen MouGeorgia Institute of Technology
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.

Transverse Surgery on Knots in Contact 3-Manifolds

Series
Dissertation Defense
Time
Tuesday, April 19, 2016 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
James ConwayGeorgia Tech
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.

Thermostated Kac Models

Series
Dissertation Defense
Time
Friday, October 30, 2015 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ranjini VaidyanathanSchool of Mathematics, Georgia Tech

Please Note: 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.

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 BushSchool 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 KroneGeorgia 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 AwiSchool 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 DifonzoSchool 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.

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