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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: 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.

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: 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 - 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: Combinatorics Seminar

Spectral algorithms, such as principal component analysis and spectral
clustering, typically require careful data transformations to be
effective: upon observing a matrix A, one may look at the spectrum of
ψ(A) for a properly chosen ψ. We propose a simple and generic
construction for sparse graphs based on graph powering. It is shown
that graph powering regularizes the graph and decontaminates its
spectrum in the following sense: (i) If the graph is drawn from the
sparse Erd˝os-R´enyi ensemble, which has no spectral gap, it is shown
that graph powering produces a “maximal” spectral gap, with the latter
justified by establishing an Alon-Boppana result for powered graphs;
(ii) If the graph is drawn from the sparse SBM, graph powering is
shown to achieve the fundamental limit for weak recovery.
(Joint work with E. Abbe, E. Boix, C. Sandon.)

Friday, October 12, 2018 - 14:00 ,
Location: Skiles 006 ,
Sudipta Kolay ,
Georgia Tech ,
Organizer: Sudipta Kolay

One of the general methods of proving h-principle is holonomic
aprroximation. In this series of talks, I will give a proof of holonomic
approximation theorem,
and talk about some of its applications.

Series: ACO Student Seminar

At the heart of most algorithms today there is an optimization engine trying to learn online
and provide the best decision, for e.g.
rankings of objects, at any time with the partial information observed
thus far in time. Often it becomes difficult to find near optimal
solutions to many problems due to their inherent combinatorial structure that
leads to certain computational bottlenecks. Submodularity is a discrete
analogue of convexity and is a key property often exploited in tackling combinatorial optimization
problems.
In the first part of the talk, we will focus on computational
bottlenecks that involve submodular functions: (a) convex function
minimization over submodular base polytopes (for e.g. permutahedron) and
(b) movement along a line inside submodular base polytopes.
We give a conceptually simple and strongly polynomial algorithm Inc-Fix
for the former, which is useful in computing Bregman projections in
first-order projection-based methods like online mirror descent. For the
latter, we will bound the iterations of the
discrete Newton method which gives a running time improvement of at
least n^6 over the state of the art. This is joint work with Michel
Goemans and Patrick Jaillet. In the second part of the talk, we will
consider the dual problem of (a), i.e. minimization
of composite convex and submodular objectives. We will resolve Bach's
conjecture from 2015 about the running time of a popular Kelley's
cutting plane variant to minimize these composite objectives. This is
joint work with Madeleine Udell and Song Zhou.

Series: Stochastics Seminar

In the continuous-time majority vote model, each vertex of a graph is initially assigned an ``opinion,'' either 0 or 1. At exponential times, vertices update their values by assuming the majority value of their neighbors. This model has been studied extensively on Z^d, where it is known as the zero-temperature limit of Ising Glauber dynamics. I will review some of the major questions and conjectures on lattices, and then explain some new work with Arnab Sen (Minnesota) on the 3-regular tree. We relate the majority vote model to a new model, which we call the median process, and use this process to answer questions about the limiting state of opinions. For example, we show that when the initial state is given by a Bernoulli(p) product measure, the probability that a vertex's limiting opinion is 1 is a continuous function of p.

Series: PDE Seminar

In this talk, we introduce several models of the so-called forward-forward Mean-Field Games (MFGs). The forward-forward models arise in the study of numerical schemes to approximate stationary MFGs. We establish a link between these models and a class of hyperbolic conservation laws. Furthermore, we investigate the existence of solutions and examine long-time limit properties. Joint work with Diogo Gomes and Levon Nurbekyan.