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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).

Wednesday, April 4, 2018 - 14:00 ,
Location: Skiles 006 ,
Hongyi Zhou (Hugo) ,
GaTech ,
Organizer: Anubhav Mukherjee

Exotic sphere is a smooth manifold that is homeomorphic to, but not diffeomorphic to standard sphere. The simplest known example occurs in 7-dimension. I will recapitulate Milnor’s construction of exotic 7-sphere, by first constructing a candidate bundle M_{h,l}, then show that this manifold is a topological sphere with h+l=-1. There is an 8-dimensional bundle with M_{h,l} its boundary and if we glue an 8-disc to it to obtain a manifold without boundary, it should possess a natural differential structure. Failure to do so indicates that M_{h,l} cannot be mapped diffeomorphically to 7-sphere. Main tools used are Morse theory and characteristic classes.

Series: Geometry Topology Seminar

Heegaard Floer homology has proven to be a useful tool in the study of knot concordance. Ozsvath and Szabo first constructed the tau invariant using the hat version of Heegaard Floer homology and showed it provides a lower bound on the slice genus. Later, Hom and Wu constructed a concordance invariant using the plus version of Heegaard Floer homology; this provides an even better lower-bound on the slice genus. In this talk, I discuss a sequence of concordance invariants that are derived from the truncated version of Heegaard Floer homology. These truncated Floer concordance invariants generalize the Ozsvath-Szabo and Hom-Wu invariants.

Monday, April 2, 2018 - 13:55 ,
Location: Skiles 005 ,
Tuo Zhao ,
Georgia Institute of Technology ,
Organizer: Wenjing Liao

Nonconvex
optimization naturally arises in many machine learning problems.
Machine learning researchers exploit various nonconvex formulations to
gain modeling flexibility, estimation robustness, adaptivity, and
computational scalability. Although classical computational complexity
theory has shown that solving nonconvex optimization is generally
NP-hard in the worst case, practitioners have proposed numerous
heuristic optimization algorithms, which achieve outstanding empirical
performance in real-world applications.To
bridge this gap between practice and theory, we propose a new
generation of model-based optimization algorithms and theory, which
incorporate the statistical thinking into modern optimization.
Specifically, when designing practical computational algorithms, we take
the underlying statistical models into consideration. Our novel
algorithms exploit hidden geometric structures behind many nonconvex
optimization problems, and can obtain global optima with the desired
statistics properties in polynomial time with high probability.