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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.

Friday, October 13, 2017 - 15:00 ,
Location: Skiles 154 ,
Bhanu Kumar ,
GT Math ,
Organizer: Jiaqi Yang

Birkhoff's Theorem is a result useful in characterizing the boundary of certain open sets U ⊂ T^1 x [0, inf) which are invariant under "vertical-tilting" homeomorphisms H. We present the method used by A. Fathi to prove Birkhoff's theorem, which develops a series of lemmas using topological arguments to prove that this boundary is a graph.

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)

Friday, October 13, 2017 - 13:55 ,
Location: Skiles 006 ,
John Etnyre ,
Georgia Tech ,
Organizer: John Etnyre

In this series of talks I will introduce branched coverings of manifolds and sketch proofs of most the known results in low dimensions (such as every 3 manifold is a 3-fold branched cover over a knot in the 3-sphere and the existence of universal knots). Along the way several open problems will be discussed.

Series: ACO Student Seminar

We show variants of spectral sparsification routines can preserve thetotal spanning tree counts of graphs, which by Kirchhoff's matrix-treetheorem, is equivalent to determinant of a graph Laplacian minor, orequivalently, of any SDDM matrix. Our analyses utilizes thiscombinatorial connection to bridge between statistical leverage scores/ effective resistances and the analysis of random graphs by [Janson,Combinatorics, Probability and Computing `94]. This leads to a routinethat in quadratic time, sparsifies a graph down to about $n^{1.5}$edges in ways that preserve both the determinant and the distributionof spanning trees (provided the sparsified graph is viewed as a randomobject). Extending this algorithm to work with Schur complements andapproximate Choleksy factorizations leads to algorithms for countingand sampling spanning trees which are nearly optimal for dense graphs.We give an algorithm that computes a $(1\pm \delta)$ approximation tothe determinant of any SDDM matrix with constant probability in about$n^2\delta^{−2}$ time. This is the first routine for graphs thatoutperforms general-purpose routines for computing determinants ofarbitrary matrices. We also give an algorithm that generates in about$n^2\delta^{−2}$ time a spanning tree of a weighted undirected graphfrom a distribution with total variation distance of $\delta$ fromthe w-uniform distribution.This is joint work with John Peebles, Richard Peng and Anup B. Rao.

Friday, October 13, 2017 - 10:00 ,
Location: Skiles 114 ,
Libby Taylor ,
GA Tech ,
Organizer: Timothy Duff

We will give an overview of divisor theory on curves and give
definitions of the Picard group and the Jacobian of a compact Riemann
surface. We will use these notions to prove Plucker’s formula for the
genus of a smooth projective curve. In addition,
we will discuss the various ways of defining the Jacobian of a curve and
why these definitions are equivalent. We will also give an extension
of these notions to schemes, in which we define the Picard group of a
scheme in terms of the group of invertible sheaves
and in terms of sheaf cohomology.