Friday, January 26, 2018 - 10:00 , Location: Skiles 254 , Trevor Gunn , Georgia Tech , firstname.lastname@example.org , Organizer: Kisun Lee
We will first give a quick introduction to automatic sequences. We will then outine an algebro-geometric proof of Christol's theorem discovered by David Speyer. Christol's theorem states that a formal power series f(t) over GF(p) is algebraic over GF(p)(t) if and only if there is some finite state automaton such that the n-th coefficent of f(t) is obtained by feeding in the base-p representation of n into the automaton. Time permitting, we will explain how to use the Riemann-Roch theorem to obtain bounds on the number of states in the automaton in terms of the degree, height and genus of f(t).
Series: Stochastics Seminar
Today's era of cloud computing is powered by massive data centers. A data center network enables the exchange of data in the form of packets among the servers within these data centers. Given the size of today's data centers, it is desirable to design low-complexity scheduling algorithms which result in a fixed average packet delay, independent of the size of the data center. We consider the scheduling problem in an input-queued switch, which is a good abstraction for a data center network. In particular, we study the queue length (equivalently, delay) behavior under the so-called MaxWeight scheduling algorithm, which has low computational complexity. Under various traffic patterns, we show that the algorithm achieves optimal scaling of the heavy-traffic scaled queue length with respect to the size of the switch. This settles one version of an open conjecture that has been a central question in the area of stochastic networks. We obtain this result by using a Lyapunov-type drift technique to characterize the heavy-traffic behavior of the expected total queue length in the network, in steady-state.
Series: Analysis Seminar
The investigation on Brascamp-Lieb data - their structure, their extremizability, their stability and regularity of their constants - has been an active one in Harmonic Analysis. In this talk, I'll present an example of a Brascamp-Lieb structure: a so-called Gowers structure on Euclidean spaces, together with the related Gowers-Host-Kra norms - these were originally tools in additive combinatorics context. I'll dissertate on what happens when a function nearly achieves its Gowers-Host-Kra norm in a Euclidean context - this can be seen as continuation of the work of Eisner-Tao - and a related stability result of the Gowers structure on Euclidean spaces.
Wednesday, January 24, 2018 - 13:55 , Location: Skiles 006 , Justin Lanier , GaTech , Organizer: Anubhav Mukherjee
Take a map from the interval [0,1] to itself. Such a map can be iterated, and many phenomena (such as periodic points) arise. An interval self-map is an example of a topological dynamical system that is simple enough to set up, but wildly complex to analyze. In the late 1970s, Milnor and Thurston developed a combinatorial framework for studying interval self-maps in their paper "Iterated maps of the interval". In this talk, we will give an introduction to the central questions in the study of iterated interval maps, share some illustrative examples, and lay out some of the techniques and results of Milnor and Thurston.
Series: Job Candidate Talk
Convex-geometric methods, involving random projection operators and coverings, have been successfully used in the study of the largest and smallest singular values, delocalization of eigenvectors, and in establishing the limiting spectral distribution for certain random matrix models. Among further applications of those methods in computer science and statistics are restricted invertibility and dimension reduction, as well as approximation of covariance matrices of multidimensional distributions. Conversely, random linear operators play a very important role in geometric functional analysis. In this talk, I will discuss some recent results (by my collaborators and myself) within convex geometry and the theory of random matrices, focusing on invertibility of square non-Hermitian random matrices (with applications to numerical analysis and the study of the limiting spectral distribution of directed d-regular graphs), approximation of covariance matrices (in particular, a strengthening of the Bai–Yin theorem), as well as some applications of random operators in convex geometry.
Series: Geometry Topology Seminar
Model-Based Multichannel Blind Deconvolution: Mathematical Analysis and Nonconvex Optimization AlgorithmsMonday, January 22, 2018 - 13:55 , Location: Skiles 005 , Dr. Lee, Kiryung , GT ECE , Organizer: Sung Ha Kang
There are numerous modern applications in data science that involve inference from incomplete data. Various geometric prior models such as sparse vectors or low-rank matrices have been employed to address the ill-posed inverse problems arising in these applications. Recently, similar ideas were adopted to tackle more challenging nonlinear inverse problems such as phase retrieval and blind deconvolution. In this talk, we consider the blind deconvolution problem where the desired information as a time series is accessed as indirect observations through a time-invariant system with uncertainty. The measurements in this case is given in the form of the convolution with an unknown kernel. Particularly, we study the mathematical theory of multichannel blind deconvolution where we observe the output of multiple channels that are all excited with the same unknown input source. From these observations, we wish to estimate the source and the impulse responses of each of the channels simultaneously. We show that this problem is well-posed if the channel impulse responses follow a simple geometric model. Under these models, we show how the channel estimates can be found by solving corresponding non-convex optimization problems. We analyze methods for solving these non-convex programs, and provide performance guarantees for each.
Series: CDSNS Colloquium
Some relevant Hamiltonian systems in Celestial Mechanics have first integrals in involution. A classic technique to study such systems, known as symplectic reduction, is based in reducing the number of degrees of freedom by using the first integrals. In this talk we present two a posteriori KAM theorems for Hamiltonian systems with first integrals in involution, including the isoenergetic case, without using symplectic reduction. The approach leads to efficient numerical methods and validating techniques.This is a joint work with Alejandro Luque.
Series: ACO Student Seminar
Studying random samples drawn from large, complex sets is one way to begin to learn about typical properties and behaviors. However, it is important that the samples examined are random enough: studying samples that are unexpectedly correlated or drawn from the wrong distribution can produce misleading conclusions. Sampling processes using Markov chains have been utilized in physics, chemistry, and computer science, among other fields, but they are often applied without careful analysis of their reliability. Making sure widely-used sampling processes produce reliably representative samples is a main focus of my research, and in this talk I'll touch on two specific applications from statistical physics and combinatorics.I'll also discuss work applying these same Markov chain processes used for sampling in a novel way to address research questions in programmable matter and swarm robotics, where a main goal is to understand how simple computational elements can accomplish complicated system-level goals. In a constrained setting, we've answered this question by showing that groups of abstract particles executing our simple processes (which are derived from Markov chains) can provably accomplish remarkable global objectives. In the long run, one goal is to understand the minimum computational abilities elements need in order to exhibit complex global behavior, with an eye towards developing systems where individual components are as simple as possible.This talk includes joint work with Marta Andrés Arroyo, Joshua J. Daymude, Daniel I. Goldman, David A. Levin, Shengkai Li, Dana Randall, Andréa Richa, William Savoie, Alexandre Stauffer, and Ross Warkentin.
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