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Series: School of Mathematics Colloquium

[CV: Prof. Oded Margalit has a PhD in computer science from Tel Aviv University under the supervision of Prof. Zvi Galil. He has worked at IBM Research – Haifa in the areas of machine learning, constraint satisfaction, verification, and more. Currently, he is the CTO of the IBM Cybersecurity Center of Excellence in Beer Sheva, Israel. Oded helps organize several computer science competitions, like the international IEEEXtreme and the Israeli national CodeGuru competition. He loves riddles and authors the IBM Research monthly challenge corner Ponder This.]

For the sake of puzzle-lovers worldwide, IBM Research offers a monthly mathematical challenge known as Ponder This. Every month, a new challenge is posted together with the solution for the previous month's riddle. Prof. Oded Margalit has served as the Ponder This puzzlemaster for the last decade. In this talk, he’ll survey some of most interesting riddles posted over the years, and tell some anecdotes about various challenges and regular solvers, such as one person who sent in his solution from an intensive care unit. Several challenges have led to conference and journal papers, such as a PRL paper born from a riddle on random walks, and an ITA 2014 paper on a water hose model (using quantum entanglement to break location-based encryption). Other monthly challenges have riffed on games such as 2048, Kakuro, an infinite chess game, the probability of backgammon ending with a double, Fischer Random Chess, and more. Other challenges have been more purely mathematic, focusing on minimal hash functions, combinatorial test design, or finding a natural number n such that round ((1+2 cos(20))^n) is divisible by 10^9.
The talk will present a still-open question about a permutation-firing cannon. The talk will be self contained.

Series: Algebra Seminar

Given data and a statistical model, the maximum likelihood estimate is
the point of the statistical model that maximizes the probability of
observing the data. In this talk, I will address three different
approaches to maximum likelihood estimation using algebraic methods.
These three approaches use boundary stratification of the statistical
model, numerical algebraic geometry and the EM fixed point ideal. This
talk is based on joint work with Allman, Cervantes, Evans, Hoşten,
Kosta, Lemke, Rhodes, Robeva, Sturmfels, and Zwiernik.

Series: Geometry Topology Seminar

Planar contact manifolds have been intensively studied to understand several aspects of 3-dimensional contact geometry. In this talk, we define "iterated planar contact manifolds", a higher-dimensional analog of planar contact manifolds, by using topological tools such as "open book decompositions" and "Lefschetz fibrations”. We provide some history on existing low-dimensional results regarding Reeb dynamics, symplectic fillings/caps of contact manifolds and explain some generalization of those results to higher dimensions via iterated planar structure. This is partly based on joint work in progress with J. Etnyre and B. Ozbagci.

Monday, April 9, 2018 - 13:55 ,
Location: Skiles 005 ,
Prof. Qingshan Chen ,
Department of Mathematical Sciences, Clemson University ,
qsc@clemson.edu ,
Organizer: Yingjie Liu

Large-scale geophysical flows, i.e. the ocean and
atmosphere, evolve on spatial scales ranging from meters to thousands
of kilometers, and on temporal scales ranging from seconds to
decades. These scales interact in a highly nonlinear fashion, making
it extremely challenging to reliably and accurately capture the
long-term dynamics of these flows on numerical models. In fact, this
problem is closely associated with the grand challenges of long-term
weather and climate predictions. Unstructured meshes have been gaining
popularity
in recent years on geophysical models, thanks to its being almost free
of polar singularities, and remaining highly scalable even at eddy
resolving resolutions. However, to unleash the full potential of these
meshes, new schemes are needed. This talk starts with a brief
introduction to large-scale geophysical flows. Then it goes
over the main considerations, i.e. various numerical and algorithmic
choices, that one needs to make in deisgning numerical schemes for these
flows. Finally, a new vorticity-divergence based
finite volume scheme will be introduced. Its strength and challenges,
together with some numerical results, will be presented and discussed.

Series: SIAM Student Seminar

This joint SIAM student conference is organized by the SIAM Student Chapter at School of Mathematics, Georgia Tech together with SIAM chapters at Clemson University, Emory University and University of Alabama at Birmingham. Detailed schedule and information can be found at jssc.math.gatech.edu.

Series: Math Physics Seminar

Localization properties of quantum many-body systems have been a very active subject in theoretical physics in the most recent decade. At the same time, finding rigorous approaches to understanding many-body localization remains a wide open challenge. We will report on some recent progress obtained for the case of quantum spin chains, where joint work with A. Elgart and A. Klein has provided a proof of several manifestations of MBL for the droplet spectrum of the disordered XXZ chain.

Series: ACO Student Seminar

We study the $A$-optimal design problem where we are given vectors $v_1,\ldots, v_n\in \R^d$, an integer $k\geq d$, and the goal is to select a set $S$ of $k$ vectors that minimizes the trace of $\left(\sum_{i\in S} v_i v_i^{\top}\right)^{-1}$. Traditionally, the problem is an instance of optimal design of experiments in statistics (\cite{pukelsheim2006optimal}) where each vector corresponds to a linear measurement of an unknown vector and the goal is to pick $k$ of them that minimize the average variance of the error in the maximum likelihood estimate of the vector being measured. The problem also finds applications in sensor placement in wireless networks~(\cite{joshi2009sensor}), sparse least squares regression~(\cite{BoutsidisDM11}), feature selection for $k$-means clustering~(\cite{boutsidis2013deterministic}), and matrix approximation~(\cite{de2007subset,de2011note,avron2013faster}). In this paper, we introduce \emph{proportional volume sampling} to obtain improved approximation algorithms for $A$-optimal design.Given a matrix, proportional volume sampling involves picking a set of columns $S$ of size $k$ with probability proportional to $\mu(S)$ times $\det(\sum_{i \in S}v_i v_i^\top)$ for some measure $\mu$. Our main result is to show the approximability of the $A$-optimal design problem can be reduced to \emph{approximate} independence properties of the measure $\mu$. We appeal to hard-core distributions as candidate distributions $\mu$ that allow us to obtain improved approximation algorithms for the $A$-optimal design. Our results include a $d$-approximation when $k=d$, an $(1+\epsilon)$-approximation when $k=\Omega\left(\frac{d}{\epsilon}+\frac{1}{\epsilon^2}\log\frac{1}{\epsilon}\right)$ and $\frac{k}{k-d+1}$-approximation when repetitions of vectors are allowed in the solution. We also consider generalization of the problem for $k\leq d$ and obtain a $k$-approximation. The last result also implies a restricted invertibility principle for the harmonic mean of singular values.We also show that the $A$-optimal design problem is$\NP$-hard to approximate within a fixed constant when $k=d$.

Series: Algebra Seminar

The nerve complex of an open covering is a well-studied notion. Motivated by the so-called Lyubeznik complex in local algebra, and other sources, a notion of higher nerves of a collection of subspaces can be defined. The definition becomes particularly transparent over a simplicial complex. These higher nerves can be used to compute depth, and the h-vector of the original complex, among other things. If time permits, I will discuss new questions arises from these notions in commutative algebra, in particular a recent example of Varbaro on connectivity of hyperplane sections of a variety. This is joint work with J. Doolittle, K. Duna, B. Goeckner, B. Holmes and J. Lyle.

Friday, April 6, 2018 - 10:00 ,
Location: Skiles 006 ,
Jaewoo Jung ,
Georgia Tech ,
jaewoojung@gatech.edu ,
Organizer: Kisun Lee

H. Dao, C. Huneke, and J. Schweig provided a bound of the regularity of edge-ideals in their paper “Bounds on the regularity and projective dimension of ideals associated to graphs”. In this talk, we introduced their result briefly and talk about a bound of the regularity of Stanley-Reisner ideals using similar approach.

Series: Stochastics Seminar

How should one estimate a signal, given only access to noisy versions of the signal corrupted by unknown cyclic shifts? This simple problem has surprisingly broad applications, in fields from aircraft radar imaging to structural biology with the ultimate goal of understanding the sample complexity of Cryo-EM. We describe how this model can be viewed as a multivariate Gaussian mixture model whose centers belong to an orbit of a group of orthogonal transformations. This enables us to derive matching lower and upper bounds for the optimal rate of statistical estimation for the underlying signal. These bounds show a striking dependence on the signal-to-noise ratio of the problem. We also show how a tensor based method of moments can solve the problem efficiently. Based on joint work with Afonso Bandeira (NYU), Amelia Perry (MIT), Amit Singer (Princeton) and Jonathan Weed (MIT).