- You are here:
- GT Home
- Home
- News & Events

Monday, November 12, 2018 - 13:00 ,
Location: Skiles 006 ,
Tom Bachmann ,
MIT ,
Organizer: Kirsten Wickelgren

I will review various ways of modeling the homotopy theory of spaces:
several model categories of simplicial sheaves and simplicial
presheaves, and related infinity categorical constructions.

Friday, November 9, 2018 - 15:05 ,
Location: Skiles 156 ,
Rui Han ,
Georgia Tech ,
Organizer: Jiaqi Yang

We prove an elementary formula about the average expansion of certain products of 2 by 2 matrices. This permits us to quickly re-obtain an inequality by M. Herman and a theorem by Dedieu and Shub, both concerning Lyapunov exponents. This is a work of A. Avila and J. Bochi. https://link.springer.com/article/10.1007/BF02785853

Series: Combinatorics Seminar

(1) A set D of natural numbers is called t-intersective if every positive upper density subset A of natural numbers contains a (t+1)-length arithmetic progression (AP) whose common differences is in D. Szemeredi's theorem states that the set of all natural numbers is t-intersective for every t. But there are other non-trivial examples like {p-1: p prime}, {1^k,2^k,3^k,\dots} for any k etc. which are t-intersective for every t. A natural question to study is at what density random subsets of natural numbers become t-intersective?
(2) Let X_t be the number of t-APs in a random subset of Z/NZ where each element is selected with probability p independently. Can we prove precise estimates on the probability that X_t is much larger than its expectation?
(3) Locally decodable codes (LDCs) are error correcting codes which allow ultra fast decoding of any message bit from a corrupted encoding of the message. What is the smallest encoding length of such codes?
These seemingly unrelated problems can be addressed by studying the Gaussian width of images of low degree polynomial mappings, which seems to be a fundamental tool applicable to many such problems. Adapting ideas from existing LDC lower bounds, we can prove a general bound on Gaussian width of such sets which reproves the known LDC lower bounds and also implies new bounds for the above mentioned problems. Our bounds are still far from conjectured bounds which suggests that there is plenty of room for improvement. If time permits, we will discuss connections to type constants of injective tensor products of Banach spaces (or chernoff bounds for tensors in simpler terms).
Joint work with Jop Briet.

Series: ACO Student Seminar

Often
the popular press talks about the power of quantum computing coming
from its ability to perform several computations
simultaneously. We’ve already had a similar capability from
probabilistic machines. This talk will explore the relationship between
quantum and randomized computation, how they are similar and how they
differ, and why quantum can work exponentially faster on
some but far from all computational problems. We’ll talk about some open
problems in quantum complexity that can help shed light on the
similarities and differences between randomness and “quantumness”.
This talk will not assume any previous knowledge of quantum information or quantum computing.

Thursday, November 8, 2018 - 13:30 ,
Location: Skiles 006 ,
Stephen McKean ,
Georgia Tech ,
Organizer: Trevor Gunn

In 1986, Herb Clemens conjectured that on a general quintic threefold, there are finitely many rational curves of any given degree. In this talk, we will give a survey of what is known about this conjecture. We will also highlight the connections between enumerative geometry and physics that arise in studying the quintic threefold.

Series: Graph Theory Working Seminar

For a graph on $n$ vertices, a vertex partition $A,B,C$ is a $f(n)$-vertex separator if $|C| \le f(n)$ and $|A|,|B| \le \frac{2}{3}n$ and $(A,B) = \emptyset$. A theorem from Gary Miller states for an embedded 2-connected planar graph with maximum face size $d$ there exists a simple cycle such that it is vertex separator of size at most $2\sqrt{dn}$. This has applications in divide and conquer algorithms.

Series: Math Physics Seminar

I will talk about what happens on the spectral transition lines for the almost Mathieu operator. This talk is based on joint works with Svetlana Jitomirskaya and Qi Zhou. For both transition lines \{\beta(\alpha)=\ln{\lambda}\} and \{\gamma(\alpha,\theta)=\ln{\lambda}\} in the positive Lyapunov exponent regime, we show purely point spectrum/purely singular continuous spectrum for dense subsets of frequencies/phases.

Series: High Dimensional Seminar

High-dimensional data arise in many fields of contemporary science and introduce new challenges in statistical learning and data recovery. Many datasets in image analysis and signal processing are in a high-dimensional space but exhibit a low-dimensional structure. We are interested in building efficient representations of these data for the purpose of compression and inference, and giving performance guarantees depending on the intrinsic dimension of data. I will present two sets of problems: one is related with manifold learning; the other arises from imaging and signal processing where we want to recover a high-dimensional, sparse vector from few linear measurements. In the first problem, we model a data set in $R^D$ as samples from a probability measure concentrated on or near an unknown $d$-dimensional manifold with $d$ much smaller than $D$. We develop a multiscale adaptive scheme to build low-dimensional geometric approximations of the manifold, as well as approximating functions on the manifold. The second problem arises from source localization in signal processing where a uniform array of sensors is set to collect propagating waves from a small number of sources. I will present some theory and algorithms for the recovery of the point sources with high precision.

Series: Research Horizons Seminar

We all know that the air in a room is made up by a huge number of atoms that zip around at high velocity colliding continuously. How is this consistent with our observation of air as a thin and calm fluid surrounding us? This is what Statistical Mechanics try to understand. I'll introduce the basic examples and ideas of equilibrium and non equilibrium Statistical Mechanics showing that they apply well beyond atoms and air.

Series: Analysis Seminar

This
talk concerns two-variable rational inner functions phi with
singularities on the two-torus T^2, the notion of contact order (and
related quantities), and its various uses. Intuitively, contact order is
the rate at which phi’s zero set approaches T^2 along a coordinate
direction, but it can also be defined via phi's well-behaved unimodular
level sets. Quantities like contact order are important because they
encode information about the numerical stability of phi, for example
when it belongs to Dirichlet-type spaces and when its partial
derivatives belong to Hardy spaces. The unimodular set definition is
also useful because it allows one to “see” contact order and in some
sense, deduce numerical stability from pictures. This is joint work with
James Pascoe and Alan Sola.