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Series: CDSNS Colloquium

A special class of dynamical systems that we will focus on are substitutions. This class of systems provides a variety of ergodic theoretic behavior and is connected to self-similar interval exchange transformations. During this talk we will explore rigidity sequences for these systems. A sequence $\left( n_m \right)$ is a rigidity sequence for the dynamical system $(X,T,\mu)$ if $\mu(T^{n_m}A\cap A)\rightarrow \mu(A)$ for all positive measure sets $A$. We will discuss the structure of rigidity sequences for substitutions that are rank-one and substitutions that have constant length. This is joint work with Jon Fickenscher.

Series: Job Candidate Talk

We study the convergence rate of the least squares estimator (LSE) in a regression model with possibly heavy-tailed errors. Despite its importance in practical applications, theoretical understanding of this problem has been limited. We first show that from a worst-case perspective, the convergence rate of the LSE in a general non-parametric regression model is given by the maximum of the Gaussian regression rate and the noise rate induced by the errors. In the more difficult statistical model where the errors only have a second moment, we further show that the sizes of the 'localized envelopes' of the model give a sharp interpolation for the convergence rate of the LSE between the worst-case rate and the (optimal) parametric rate. These results indicate both certain positive and negative aspects of the LSE as an estimation procedure in a heavy-tailed regression setting. The key technical innovation is a new multiplier inequality that sharply controls the size of the multiplier empirical process associated with the LSE, which also finds applications in shape-restricted and sparse linear regression problems.

Monday, December 4, 2017 - 14:00 ,
Location: Skiles 005 ,
Tao Pang ,
Department of Mathematics, North Carolina State University ,
Organizer: Luca Dieci

In the real world, the historical performance of a stock may have
impacts on its dynamics and this suggests us to consider models with
delays. We consider a portfolio optimization problem of Merton’s type
in which the risky asset is described by a stochastic delay model. We
derive the Hamilton-Jacobi-Bellman (HJB) equation, which turns out to
be a nonlinear degenerate partial differential equation of the
elliptic type. Despite the challenge caused by the nonlinearity and
the degeneration, we establish the existence result and the
verification results.

Series: Geometry Topology Seminar

The general linear groups GL_n(A) can be defined for any ring A, and Quillen's definition of K-theory of A takes these groups as its starting point. If A is commutative, one may define symplectic K-theory in a very similar fashion, but starting with the symplectic groups Sp_{2n}(A), the subgroup of GL_{2n}(A) preserving a non-degenerate skew-symmetric bilinear form. The result is a sequence of groups denoted KSp_i(A) for i = 0, 1, .... For the ring of integers, there is an interesting action of the absolute Galois group of Q on the groups KSp_i(Z), arising from the moduli space of polarized abelian varieties. In joint work with T. Feng and A. Venkatesh we study this action, which turns out to be an interesting extension between a trivial representation and a cyclotomic representation.

Series: Other Talks

This is a brief (15 minute) presentation of an undergraduate project that took place in the 2017 Fall semester.

Series: Combinatorics Seminar

Suppose we want to find the largest independent set or maximal cut in a sparse Erdos-Renyi graph, where the average degree is constant. Many algorithms proceed by way of local decision rules, for instance, the "nibbling" procedure. I will explain a form of local algorithms that captures many of these. I will then explain how these fail to find optimal independent sets or cuts once the average degree of the graph gets large. There are some nice connections to entropy and spin glasses.

Series: GT-MAP Seminars

Please go to http://gtmap.gatech.edu or http://gtmap.gatech.edu/events/workshop-mathematics-dynamical-systems for schedule, title and abstract.

Series: Stochastics Seminar

Cars are placed with density p on the lattice. The remaining vertices are parking spots that can fit one car. Cars then drive around at random until finding a parking spot. We study the effect of p on the availability of parking spots and observe some intriguing behavior at criticality. Joint work with Michael Damron, Janko Gravner, Hanbeck Lyu, and David Sivakoff. arXiv id: 1710.10529.

Series: Graph Theory Seminar

Let G be a graph containing 5 different vertices a0, a1, a2, b1 and b2.
We say that (G, a0, a1, a2, b1, b2) is feasible if G contains disjoint
connected subgraphs G1, G2, such that {a0, a1, a2}⊆V(G1) and {b1,
b2}⊆V(G2). In
this talk, we will complete a sketch of our arguments for characterizing when (G, a0, a1, a2, b1, b2) is feasible. Joint work with Changong Li, Robin Thomas, and Xingxing Yu.

Series: Job Candidate Talk

Random effects models are commonly used to measure genetic
variance-covariance matrices of quantitative phenotypic traits. The
population eigenvalues of these matrices describe the evolutionary
response to selection. However, they may be difficult to estimate from
limited samples when the number of traits is large. In this talk, I will
present several results describing the eigenvalues of classical MANOVA
estimators of these matrices, including dispersion of the bulk
eigenvalue distribution, bias and aliasing of large "spike" eigenvalues,
and distributional limits of eigenvalues at the spectral edges. I will
then discuss a new procedure that uses these results to obtain better
estimates of the large population eigenvalues when there are many
traits, and a Tracy-Widom test for detecting true principal components
in these models. The theoretical results extend proof techniques in
random matrix theory and free probability, which I will also briefly
describe.This is joint work with Iain Johnstone, Yi Sun, Mark Blows, and Emma Hine.