Seminars and Colloquia by Series

Series

Physical Billiards and Finite Time Predictions for First Passage Probabilities

Series: Dissertation Defense
Time: Thursday, March 15, 2018 - 13:00 for 1 hour (actually 50 minutes)
Location: Skiles 006
Speaker: Mark Bolding – Georgia Tech – mbolding3@gatech.edu

I will discuss two topics in Dynamical Systems. A uniformly hyperbolic dynamical system preserving Borel probability measure μ is called fair dice like or FDL if there exists a finite Markov partition ξ of its phase space M such that for any integers m and j(i), 1 ≤ j(i) ≤ q one has μ ( C(ξ, j(0)) ∩ T^(-1) C(ξ, j(1)) ∩ ... ∩ T^(-m+1)C(ξ, j(m-1)) ) = q^(-m) where q is the number of elements in the partition ξ and C(ξ, j) is element number j of ξ. I discuss several results about such systems concerning finite time prediction regarding the first hitting probabilities of the members of ξ. Then I will discuss a natural modification to all billiard models which is called the Physical Billiard. For some classes of billiard, this modification completely changes their dynamics. I will discuss a particular example derived from the Ehrenfests' Wind-Tree model. The Physical Wind-Tree model displays interesting new dynamical behavior that is at least as rich as some of the most well studied examples that have come before.

Nonnegative matrix factorization for Text, Graph, and Hybrid Data Analytics

Series: Dissertation Defense
Time: Monday, March 12, 2018 - 10:00 for 1 hour (actually 50 minutes)
Location: Klaus 2108
Speaker: Rundong Du – Georgia Tech – rdu@gatech.edu

Constrained low rank approximation is a general framework for data analysis, which usually has the advantage of being simple, fast, scalable and domain general. One of the most known constrained low rank approximation method is nonnegative matrix factorization (NMF). This research studies the design and implementation of several variants of NMF for text, graph and hybrid data analytics. It will address challenges including solving new data analytics problems and improving the scalability of existing NMF algorithms. There are two major types of matrix representation of data: feature-data matrix and similarity matrix. Previous work showed successful application of standard NMF for feature-data matrix to areas such as text mining and image analysis, and Symmetric NMF (SymNMF) for similarity matrix to areas such as graph clustering and community detection. In this work, a divide-and-conquer strategy is applied to both methods to improve their time complexity from cubic growth with respect to the reduced low rank to linear growth, resulting in DC-NMF and HierSymNMF2 method. Extensive experiments on large scale real world data shows improved performance of these two methods.Furthermore, in this work NMF and SymNMF are combined into one formulation called JointNMF, to analyze hybrid data that contains both text content and connection structure information. Typical hybrid data where JointNMF can be applied includes paper/patent data where there are citation connections among content and email data where the sender/receipts relation is represented by a hypergraph and the email content is associated with hypergraph edges. An additional capability of the JointNMF is prediction of unknown network information which is illustrated using several real world problems such as citation recommendations of papers and activity/leader detection in organizations.The dissertation also includes brief discussions of relationship among different variants of NMF.

Minors of graphs of large path-width

Series: Dissertation Defense
Time: Tuesday, January 9, 2018 - 14:00 for 1 hour (actually 50 minutes)
Location: Skiles 005
Speaker: Thanh Dang – Math, GT

Let P be a graph with a vertex v such that P-v is a forest and let Q be an outerplanar graph. In 1993 Paul Seymour asked if every two-connected graph of sufficiently large path-width contains P or Q as a minor.mDefine g(H) as the minimum number for which there exists a positive integer p(H) such that every g(H)-connected H-minor-free graph has path-width at most p(H). Then g(H) = 0 if and only if H is a forest and there is no graph H with g(H) = 1, because path-width of a graph G is the maximum of the path-widths of its connected components.Let A be the graph that consists of a cycle (a_1,a_2,a_3,a_4,a_5,a_6,a_1) and extra edges a_1a_3, a_3a_5, a_5a_1. Let C_{3,2} be a graph of 2 disjoint triangles. In 2014 Marshall and Wood conjectured that a graph H does not have K_{4}, K_{2,3}, C_{3,2} or A as a minor if and only if g(H) >= 2. In this thesis we answer Paul Seymour's question in the affirmative and prove Marshall and Wood's conjecture, as well as extend the result to three-connected and four-connected graphs of large path-width. We introduce ``cascades", our main tool, and prove that in any tree-decomposition with no duplicate bags of bounded width of a graph of big path-width there is an ``injective" cascade of large height. Then we prove that every 2-connected graph of big path-width and bounded tree-width admits a tree-decomposition of bounded width and a cascade with linkages that are minimal. We analyze those minimal linkages and prove that there are essentially only two types of linkage. Then we convert the two types of linkage into the two families of graphs P and Q. In this process we have to choose the ``right'' tree decomposition to deal with special cases like a long cycle. Similar techniques are used for three-connected and four-connected graphs with high path-width.

Curvature and Isoperimetry in Graphs

Series: Dissertation Defense
Time: Thursday, September 28, 2017 - 09:30 for 1 hour (actually 50 minutes)
Location: Skiles 005
Speaker: Peter Ralli – School of Mathematics, Georgia Tech – pralli3@math.gatech.edu

This dissertation concerns isoperimetric and functional inequalities in discrete spaces. The majority of the work concerns discrete notions of curvature. There isalso discussion of volume growth in graphs and of expansion in hypergraphs. [The dissertation committee consists of Profs. J. Romberg (ECE), P. Tetali (chair of the committee), W.T. Trotter, X. Yu and H. Zhou.]

Multilinear Dyadic Operators and Their Commutators

Series: Dissertation Defense
Time: Tuesday, July 18, 2017 - 16:00 for 1 hour (actually 50 minutes)
Location: Skiles 005
Speaker: Ishwari Kunwar – Georgia Tech

In this thesis, we introduce multilinear dyadic paraproducts and Haar multipliers, and discuss boundedness properties of these operators and their commutators with locally integrable functions in various settings. We also present pointwise domination of these operators by multilinear sparse operators, which we use to prove multilinear Bloom’s inequality for the commutators of multilinear Haar multipliers. Along the way, we provide several characterizations of dyadic BMO functions.

Results on the construction of whiskered invariant tori for fibered holomorphic dynamics and on compensated domains.

Series: Dissertation Defense
Time: Monday, May 15, 2017 - 10:05 for 1 hour (actually 50 minutes)
Location: Skiles 006
Speaker: Mikel Viana – Georgia Tech

We first discuss the construction of whiskered invariant tori for fibered holomorphic dynamics using a Nash-Moser iteration. The results are in a-posteriori form. The iterative procedure we present has numerical applications (it lends itself to efficient numerical implementations) since it is not based on transformation theory but rather in applying corrections which ameliorate notably the curse of dimensionality. Then we will discuss results on compensated domains in a Banach space.

Analysis and Numerical Methods in Solid State Physics and Chemistry

Series: Dissertation Defense
Time: Thursday, April 27, 2017 - 10:00 for 2 hours
Location: Skiles 005
Speaker: Lei Zhang – Georgia Institute of Technology

We present two distinct problems in the field of dynamical systems.I the first part, we cosider an atomic model of deposition of materials over a quasi-periodic medium, that is, a quasi-periodic version of the well-known Frenkel-Kontorova model. We consider the problem of whether there are quasi-periodic equilibria with a frequency that resonates with the frequencies of the medium. We show that there are always perturbative expansions. We also prove a KAM theorem in a-posteriori form.In the second part, we consider a simple model of chemical reaction and present a numerical method calculating the invariant manifolds and their stable/unstable bundles based on parameterization method.

Approach to equilibrium in Mark Kac's model

Series: Dissertation Defense
Time: Wednesday, April 26, 2017 - 09:00 for 1 hour (actually 50 minutes)
Location: Skiles 005
Speaker: Hagop Tossounian – Georgia Tech – hagopt87@gatech.edu

Kinetic theory is the branch of mathematical physics that studies the motion of gas particles that undergo collisions. A central theme is the study of systems out of equilibrium and approach of equilibrium, especially in the context of Boltzmann's equation. In this talk I will present Mark Kac's stochastic N-particle model, briefly show its connection to Boltzmann's equation, and present known and new results about the rate of approach to equilibrium, and about a finite-reservoir realization of an ideal thermostat.

Statistical Inference for Some Risk Measures

Series: Dissertation Defense
Time: Wednesday, April 12, 2017 - 14:30 for 1 hour (actually 50 minutes)
Location: Skiles 114
Speaker: Yanxi Hou – Georgia Institute of Technology – yhou44@gatech.edu

This thesis addresses asymptotic behaviors and statistical inference methods for several newly proposed risk measures, including relative risk and conditional value-at-risk. These risk metrics are intended to measure the tail risks and/or systemic risk in financial markets. We consider conditional Value-at-Risk based on a linear regression model. We extend the assumptions on predictors and errors of the model, which make the model more flexible for the financial data. We then consider a relative risk measure based on a benchmark variable. The relative risk measure is proposed as a monitoring index for systemic risk of financial system. We also propose a new tail dependence measure based on the limit of conditional Kendall’s tau. The new tail dependence can be used to distinguish between the asymptotic independence and dependence in extreme value theory. For asymptotic results of these measures, we derive both normal and Chi-squared approximations. These approximations are a basis for inference methods. For normal approximation, the asymptotic variances are too complicated to estimate due to the complex forms of risk measures. Quantifying uncertainty is a practical and important issue in risk management. We propose several empirical likelihood methods to construct interval estimation based on Chi-squared approximation.

Subdivisions of complete graphs

Series: Dissertation Defense
Time: Monday, April 10, 2017 - 15:00 for 2 hours
Location: Skiles 006
Speaker: Yan Wang – Georgia Institute of Technology

A subdivision of a graph G, also known as a topological G and denoted by TG, is a graph obtained from G by replacing certain edges of G with internally vertex-disjoint paths. This dissertation has two parts. The first part studies a structural problem and the second part studies an extremal problem. In the first part of this dissertation, we focus on TK_5, or subdivisions of K_5. A well-known theorem of Kuratowski in 1932 states that a graph is planar if, and only if, it does not contain a subdivision of K_5 or K_{3,3}. Wagner proved in 1937 that if a graph other than K_5 does not contain any subdivision of K_{3,3} then it is planar or it admits a cut of size at most 2. Kelmans and, independently, Seymour conjectured in the 1970s that if a graph does not contain any subdivision of K_5 then it is planar or it admits a cut of size at most 4. In this dissertation, we give a proof of the Kelmans-Seymour conjecture. We also discuss several related results and problems. The second part of this dissertation concerns subdivisions of large cliques in C_4-free graphs. Mader conjectured that every C_4-free graph with average degree d contains TK_l with l = \Omega(d). Komlos and Szemeredi reduced the problem to expanders and proved Mader's conjecture for n-vertex expanders with average degree d < exp( (log n)^(1/8) ). In this dissertation, we show that Mader's conjecture is true for n-vertex expanders with average degree d < n^0.3, which improves Komlos and Szemeredi's quasi-polynomial bound to a polynomial bound. As a consequence, we show that every C_4-free graph with average degree d contains a TK_l with l = \Omega(d/(log d)^c) for any c > 3/2. We note that Mader's conjecture has been recently verified by Liu and Montgomery.

Georgia Institute of Technology College of Sciences

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