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Series: Other Talks

This is a workshop designed to provide an introduction to the use of
modern tools from Dynamical Systems in the design of space exploration
missions. More details and a detailed schedule is found in http://people.math.gatech.edu/~rll6/JPL/jpl.html

Series: Other Talks

Series: Other Talks

Series: Job Candidate Talk

Semiparametric regressions enjoy the flexibility of nonparametric models as well as the in-terpretability of linear models. These advantages can be further leveraged with recent ad-vance in high dimensional statistics. This talk begins with a simple partially linear model,Yi = Xi β ∗ + g ∗ (Zi ) + εi , where the parameter vector of interest, β ∗ , is high dimensional butsufficiently sparse, and g ∗ is an unknown nuisance function. In spite of its simple form, this highdimensional partially linear model plays a crucial role in counterfactual studies of heterogeneoustreatment effects. In the first half of this talk, I present an inference procedure for any sub-vector (regardless of its dimension) of the high dimensional β ∗ . This method does not requirethe “beta-min” condition and also works when the vector of covariates, Zi , is high dimensional,provided that the function classes E(Xij |Zi )s and E(Yi |Zi ) belong to exhibit certain sparsityfeatures, e.g., a sparse additive decomposition structure. In the second half of this talk, I discussthe connections between semiparametric modeling and Rubin’s Causal Framework, as well asthe applications of various methods (including the one from the first half of this talk and thosefrom my other papers) in counterfactual studies that are enriched by “big data”.Abstract as a .pdf

Series: School of Mathematics Colloquium

The probability of outcomes of repeated
fair coin tosses can be computed exactly using binomial coefficients.
Performing asymptotics on these formulas uncovers the Gaussian
distribution and the first instance of the central limit theorem. This
talk will focus on higher version of this story. We will consider random
motion subject to random forcing. By leveraging structures from representation theory and quantum integrable systems
we can compute the analogs of binomial coefficients and extract new and
different asymptotic behaviors than those of the Gaussian. This model
and its analysis fall into the general theory of "integrable
probability".

Series: Other Talks

Series: Geometry Topology Seminar

Series: Other Talks

This is a preliminary talk for the Workshop "Introduction to Dynamical Systems Methods for Mission Design" that will take place Jan 16-19 in the school of Mathematics. In this talk, we will present the basics of Hamiltonian dynamics and why it is useful. It ishoped that it will be accesible for people with background in undergraduate differential equations who want to participate in the workshop.

Series: Dissertation Defense

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.

Series: PDE Seminar

In a recent work Sideris constructed a finite-parameter family of compactly supported affine solutions to the free boundary compressible Euler equations satisfying the physical vacuum condition. The support of these solutions expands at a linear rate in time. We show that if the adiabatic exponent gamma belongs to the interval(1, 5/3] then these affine motions are globally-in-time nonlinearly stable. If time permits we shall also discuss several classes of global solutions to the compressible Euler-Poisson system. This is a joint work with Juhi Jang.