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Monday, October 15, 2018 - 00:45 ,
Location: Skiles 006 ,
Lev Tovstopyat-Nelip ,
Boston College ,
Organizer: John Etnyre

We explain the (classical) transverse Markov Theorem which relates transverse links in the tight three sphere to classical braid closures. We review an invariant of such transverse links coming from knot Floer homology and discuss some applications which appear in the literature.

Series: Other Talks

Thanks are due to our colleague, Vladimir Koltchinskii, for arranging this visit. Please write to Vladimir if you would like to meet with Professor Gabor Lugosi during his visit, or for additional information.

In these lectures we discuss some statistical problems with an interesting combinatorial structure behind. We start by reviewing the "hidden clique" problem, a simple prototypical example with a surprisingly rich structure. We also discuss various "combinatorial" testing problems and their connections to high-dimensional random geometric graphs. Time permitting, we study the problem of estimating the mean of a random variable.

Monday, October 15, 2018 - 13:55 ,
Location: Skiles 005 ,
Prof. Yun Jing ,
NCSU ,
Organizer: Molei Tao

In recent years, metamaterials have drawn a great deal of attention in the scientific community due to their unusual properties and useful applications. Metamaterials are artificial materials made of subwavelength microstructures. They are well known to exhibit exotic properties and could manipulate wave propagation in a way that is impossible by using nature materials.In this talk, I will present our recent works on membrane-type acoustic metamaterials (AMMs). First, I will talk about how to achieve near-zero density/index AMMs using membranes. We numerically show that such an AMM can be utilized to achieve angular filtering and manipulate wave-fronts. Next, I will talk about the design of an acoustic complimentary metamaterial (CMM). Such a CMM can be used to acoustically cancel out aberrating layers so that sound transmission can be greatly enhanced. This material could find usage in transcranial ultrasound beam focusing and non-destructive testing through metal layers. I will then talk about our recent work on using membrane-type AMMs for low frequency noise reduction. We integrated membranes with honeycomb structures to design simultaneously lightweight, strong, and sound-proof AMMs. Experimental results will be shown to demonstrate the effectiveness of such an AMM. Finally, I will talk about how to achieve a broad-band hyperbolic AMM using membranes.

Series: Geometry Topology Seminar

Let K be a link braided about an open book (B,p) supporting a contact manifold (Y,x). K and B are naturally transverse links. We prove that the hat version of the transverse link invariant defined by Baldwin, Vela-Vick and Vertesi is non-zero for the union of K with B. As an application, we prove that the transverse invariant of any braid having fractional Dehn twist coefficient greater than one is non-zero. This generalizes a theorem of Plamenevskaya for classical braid closures.

Series: PDE Seminar

We give derivative estimates for solutions to divergence form elliptic equations with piecewise
smooth coefficients. The novelty of these estimates is that, even though they depend on the shape
and on the size of the surfaces of discontinuity of the coefficients, they are independent of the
distance between these surfaces.

Series: Research Horizons Seminar

Tropical geometry provides a combinatorial approach for studying geometric objects by reducing them to graphs and polytopes. In recent years, tropical techniques have been applied in numerous areas such as optimization, number theory, phylogenetic trees in biology, and auction systems in economics. My talk will focus on geometric counting problems and their tropical counterpart. By considering these combinatorial gadgets, we gain newinsights into old problems, and tools for approaching new problems.

Series: High Dimensional Seminar

We already know that the Euclidean unit ball is at the center of the Banach-Mazur compactum, however its structure is still being explored to this day. In 1987, Szarek and Talagrand proved that the maximum distance $R_{\infty} ^n$ between an arbitrary $n$-dimensional normed space and $\ell _{\infty} ^n$, or equivalently the maximum distance between a symmetric convex body in $\mathbb{R} ^n$ and the $n$-dimensional unit cube is bounded above by $c n^{7/8}$. In this talk, we will discuss a related paper by A. Giannopoulos, "A note to the Banach-Mazur distance to the cube", where he proves that $R_{\infty} ^n < c n^{5/6}$.

Series: Analysis Seminar

Dynamical sampling is a new area in
sampling theory that deals with signals that evolve over time under the
action of a linear operator. There are lots of studies on various
aspects of the dynamical sampling problem. However, they all focus on
uniform
discrete time-sets $\mathcal T\subset\{0,1,2,\ldots, \}$. In our study,
we concentrate on the case $\mathcal T=[0,L]$. The goal of the
present work is to study the frame property of the systems
$\{A^tg:g\in\mathcal G, t\in[0,L] \}$. To this end, we also
characterize the completeness and Besselness properties of these
systems.

Series: Graph Theory Working Seminar

Erdős and Nešetřil conjectured in 1985 that every graph with maximum degree Δ can be strong edge colored using at most (5/4)Δ^2 colors. In this talk, we focus on a conjecture by R.J. Faudree et al, that Δ^2 holds as a bound for strong chromatic index in bipartite graphs, and related results where a bound is known.

Series: Other Talks

What to do if the measurements that you took were
corrupted by a malicious spy? We will see how the natural geometric
approach to the problem leads to a geometry where lines are crooked, and
triangles are square.

Thursday, October 18, 2018 - 13:30 ,
Location: Skiles 006 ,
Trevor Gunn ,
Georgia Tech ,
Organizer: Trevor Gunn

We will go over a short proof of the Littlewood-Richardson Rule due to Stembridge as well as some related combinatorics of tableaux.

Series: Other Talks

Thanks are due to our colleague, Vladimir Koltchinskii, for arranging this visit. Please write to Vladimir if you would like to meet with Professor Gabor Lugosi during his visit, or for additional information.

Series: Stochastics Seminar

In these lectures we discuss some statistical problems with an interesting combinatorial structure behind. We start by reviewing the "hidden clique" problem, a simple prototypical example with a surprisingly rich structure. We also discuss various "combinatorial" testing problems and their connections to high-dimensional random geometric graphs. Time permitting, we study the problem of estimating the mean of a random variable

Series: ACO Student Seminar

We investigate whether the standard dimensionality reduction techniques
inadvertently produce data representations with different fidelity for
two different populations. We show on several real-world datasets, PCA
has higher reconstruction error on population
A than B (for example, women versus men or lower versus higher-educated
individuals). This can happen even when the dataset has similar number
of samples from A and B . This motivates our study of dimensionality
reduction techniques which maintain similar fidelity
for A as B . We give an efficient algorithm for finding a projection
which is nearly-optimal with respect to this measure, and evaluate it on
several datasets. This is a joint work with Uthaipon
Tantipongpipat, Jamie Morgenstern, Mohit Singh, and Santosh Vempala.

Series: Algebra Seminar

I will begin with a gentle introduction to hyperrings and hyperfields (originally introduced by Krasner for number-theoretic reasons), and then discuss a far-reaching generalization, Oliver Lorscheid’s theory of ordered blueprints. Two key examples of hyperfields are the hyperfield of signs S and the tropical hyperfield T. An ordering on a field K is the same thing as a homomorphism to S, and a (real) valuation on K is the same thing as a homomorphism to T. In particular, the T-points of an ordered blue scheme over K are closely related to Berkovich’s theory of analytic spaces.I will discuss a common generalization, in this language, of Descartes' Rule of Signs (which involves polynomials over S) and the theory of Newton Polygons (which involves polynomials over T). I will then introduce matroids over hyperfields (as well as certain more general kinds of ordered blueprints). Matroids over S are classically called oriented matroids, and matroids over T are also known as valuated matroids. I will explain how the theory of ordered blueprints and ordered blue schemes allow us to construct a "moduli space of matroids”, which is the analogue in the theory of ordered blue schemes of the usual Grassmannian variety in algebraic geometry. This is joint work with Nathan Bowler and Oliver Lorscheid.

Friday, October 19, 2018 - 14:00 ,
Location: Skiles 006 ,
Sudipta Kolay ,
Georgia Tech ,
Organizer: Sudipta Kolay

I will discuss some applications of the holonomic approximation theorem to questions about immersions, embeddings, and singularities.

Series: Combinatorics Seminar

How can d+k vectors in R^d be arranged so that they are as close
to orthogonal as possible? We show intimate connection of this problem to
the problem of equiangular lines, and to the problem of bounding the first
moment of isotropic measures. Using these connections, we pin down the
answer precisely for several values of k and establish asymptotics for all k.
Joint work with Chris Cox.

Friday, October 19, 2018 - 15:05 ,
Location: Skiles 156 ,
Jiaqi Yang ,
GT Math ,
Organizer: Jiaqi Yang

We show that, if the linearization of a map at a fixed point leaves invariant a spectral subspace, and some non-resonance conditions are satisfied. Then the map leaves invariant a smooth (as smooth as the map) manifold, which is unique among C^L invariant manifolds. Here, L only depends on the spectrum of the linearization. This is based on a work of Prof. Rafael de la Llave.