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Series: ACO Student Seminar

The concentration of measure phenomenon is of great importance in probabilistic combinatorics and theoretical computer science. For example, in the theory of random graphs, we are often interested in showing that certain random variables are concentrated around their expected values. In this talk we consider a variation of this theme, where we are interested in proving that certain random variables remain concentrated around their expected trajectories as an underlying random process (or random algorithm) evolves. In particular, we shall give a gentle introduction to the differential equation method popularized by Wormald, which allows for proving such dynamic concentration results. This method systematically relates the evolution of a given random process with an associated system of differential equations, and the basic idea is that the solution of the differential equations can be used to approximate the dynamics of the random process. If time permits, we shall also sketch a new simple proof of Wormalds method.

Series: Algebra Seminar

We introduce a novel representation of structured polynomial ideals, which we refer to as chordal networks. The sparsity structure of a polynomial system is often described by a graph that captures the interactions among the variables. Chordal networks provide a computationally convenient decomposition of a polynomial ideal into simpler (triangular) polynomial sets, while preserving its underlying graphical structure. We show that many interesting families of polynomial ideals admit compact chordal network representations (of size linear in the number of variables), even though the number of components could be exponentially large. Chordal networks can be computed for arbitrary polynomial systems, and they can be effectively used to obtain several properties of the variety, such as its dimension, cardinality, equidimensional components, and radical ideal membership. We apply our methods to examples from algebraic statistics and vector addition systems; for these instances, algorithms based on chordal networks outperform existing techniques by orders of magnitude.

Series: Professional Development Seminar

A conversation with Amanda Streib, a 2012 GT ACO PhD, who is now working at the Institute for Defense Analyses - Center for Computing Sciences (IDA/CCS) and who was previously a National Research Council (NRC) postdoc at the Applied and Computational Mathematics Division of the National Institute of Standards and Technology (NIST).

Series: Stochastics Seminar

I will revisit the classical Stein's method, for normal random variables, as well as its version for Poisson random variables and show how both (as well as many other examples) can be incorporated in a single framework.

Series: Dissertation Defense

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.

Series: Analysis Seminar

Finding and understanding patterns in data sets is of significant
importance in many applications. One example of a simple pattern is the
distance between data points, which can be thought of as a 2-point
configuration. Two classic questions, the Erdos distinct
distance problem, which asks about the least number of distinct
distances determined by N points in the plane, and its continuous
analog, the Falconer distance problem, explore that simple pattern.
Questions similar to the Erdos distinct distance problem and
the Falconer distance problem can also be posed for more complicated
patterns such as triangles, which can be viewed as 3-point
configurations. In this talk I will present recent progress on Falconer
type problems for simplices. The main techniques used come
from analysis and geometric measure theory.

Series: Geometry Topology Seminar

I will present the recent result with P.Albers and D.Hein that every graphical hypersurface in a prequantization bundle over a symplectic manifold M pinched between two circle bundles whose ratio of radii is less than \sqrt{2} carries either one short simple periodic orbit or carries at least cuplength(M)+1 simple periodic Reeb orbits.

Series: Research Horizons Seminar

Defined in the early 2000's by Ozsvath and Szabo,
Heegaard Floer homology is a package of invariants for three-manifolds,
as well as knots inside of them. In this talk, we will describe how work
from Poul Heegaard's 1898 PhD thesis,
namely the idea of a Heegaard splitting, relates to the definition of
this invariant. We will also provide examples of the kinds of questions
that Heegaard Floer homology can answer. These ideas will be the subject
of the topics course that I am teaching in
Fall 2017.