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

Series: Combinatorics Seminar

We prove that every triangle-free graph with maximum degree $D$ has list chromatic number at most $(1+o(1))\frac{D}{\ln D}$. This matches the best-known bound for graphs of girth at least 5. We also provide a new proof that for any $r \geq 4$ every $K_r$-free graph has list-chromatic number at most $200r\frac{D\ln\ln D}{\ln D}$.

Friday, October 20, 2017 - 10:00 ,
Location: Skiles 114 ,
Kisun Lee ,
Georgia Institute of Technology ,
Organizer: Timothy Duff

We will introduce a class of
nonnegative real matrices which are called slack matrices. Slack
matrices provide the distance from equality of a vertex and a facet. We
go over concepts of polytopes and polyhedrons briefly,
and define slack matrices using those objects. Also, we will
give several necessary and sufficient conditions for slack matrices of
polyhedrons. We will also restrict our conditions for slack matrices for
polytopes. Finally, we introduce the polyhedral verification
problem, and some combinatorial characterizations of slack matrices.

Series: Stochastics Seminar

We present a unified framework for sequential low-rank matrix completion and estimation, address the joint goals of uncertainty quantification (UQ) and statistical design. The first goal of UQ aims to provide a measure of uncertainty of estimated entries in the unknown low-rank matrix X, while the second goal of statistical design provides an informed sampling or measurement scheme for observing the entries in X. For UQ, we adopt a Bayesian approach and assume a singular matrix-variate Gaussian prior the low-rank matrix X which enjoys conjugacy. For design, we explore deterministic design from information-theoretic coding theory. The effectiveness of our proposed methodology is then illustrated on applications to collaborative filtering.

Wednesday, October 18, 2017 - 13:55 ,
Location: Skiles 006 ,
Sudipta Kolay ,
Georgia Tech ,
Organizer: Jennifer Hom

I will talk about the Berge conjecture, and Josh Greene's resolution of a related problem, about which lens spaces can be obtained by integer surgery on a knot in S^3.

Series: Analysis Seminar

We are going to prove that indicator functions of convex sets with a
smooth boundary cannot serve as window functions for orthogonal Gabor
bases.

Series: Research Horizons Seminar

This
talk will cover some recent and preliminary results in the area of
non-smooth dynamics, with connections to applications that have been
overlooked.
Much of the talk will present open questions for research projects related to this area.

Series: Algebra Seminar

In this talk, I will summarize forthcoming work with Griffin, Ono, and Zagier. In 1927 Pólya proved that the Riemann Hypothesis is equivalent to the hyperbolicity of Jensen polynomials for Riemann's Xi-function. This hyperbolicity has been proved for degrees $d\leq 3$. We obtain an arbitrary precision asymptotic formula for the derivatives $\Xi^{(2n)}(0)$, which allows us to prove thehyperbolicity of 100% of the Jensen polynomials of each degree. We obtain a general theorem which models such polynomials by Hermite polynomials. This general condition also confirms a conjecture of Chen, Jia, and Wang.

Monday, October 16, 2017 - 14:00 ,
Location: Skiles 005 ,
Dr. Barak Sober ,
Tel Aviv University ,
barakino@gmail.com ,
Organizer: Doron Lubinsky

We approximate a function defined over a $d$-dimensional manifold $M
⊂R^n$ utilizing only noisy function values at noisy locations on the manifold. To produce
the approximation we do not require any knowledge regarding the manifold
other than its dimension $d$. The approximation scheme is based upon the
Manifold Moving Least-Squares (MMLS) and is therefore resistant to noise in
the domain $M$ as well. Furthermore, the approximant is shown to be smooth
and of approximation order of $O(h^{m+1})$ for non-noisy data, where $h$ is
the mesh size w.r.t $M,$ and $m$ is the degree of the local polynomial
approximation. In addition, the proposed algorithm is linear in time with
respect to the ambient space dimension $n$, making it useful for cases
where d is much less than n. This assumption, that the high dimensional data is situated
on (or near) a significantly lower dimensional manifold, is prevalent in
many high dimensional problems. Thus, we put our algorithm to numerical
tests against state-of-the-art algorithms for regression over manifolds and
show its dominance and potential.

Series: Geometry Topology Seminar

Every four-dimensional Stein domain has a Morse function whoseregular level sets are contact three-manifolds. This allows us to studycomplex curves in the Stein domain via their intersection with thesecontact level sets, where we can comfortably apply three-dimensional tools.We use this perspective to understand links in Stein-fillable contactmanifolds that bound complex curves in their Stein fillings.

Series: Combinatorics Seminar

The original notion of poset dimension is due to Dushnik and Miller (1941). Last year, Uerckerdt (2016) proposed a variant, called local dimension, which has garnered considerable interest. A local realizer of a poset P is a collection of partial linear extensions of P that cover the comparabilities and incomparabilities of P. The local dimension of P is the minimum frequency of a local realizer where frequency is the maximum multiplicity of an element of P.
Hiraguchi (1955) proved that any poset with n points has dimension at most n/2, which is sharp. We prove that the local dimension of a poset with n points is O(n/log n). To show that this bound is best possible, we use probabilistic methods to prove the following stronger result which extends a theorem of Chung, Erdős, and Spencer (1983): There is an n-vertex bipartite graph in which each difference graph cover of the edges will cover one of the vertices Θ(n/log n) times. (This is joint work with Jinha Kim, Ryan R. Martin, Tomáš Masařı́k, Warren Shull, Andrew Uzzell, and Zhiyu Wang)