Seminars and Colloquia Schedule

Geometric Small Cancellation

Series
Geometry Topology Working Seminar
Time
Monday, September 26, 2016 - 10:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Justin Lanier – Georgia Tech
In this lecture series, held jointly (via video conference) with the University of Buffalo and the University of Arkansas, we aim to understand the lecture notes by Vincent Guirardel on geometric small cancellation. The lecture notes can be found here: https://perso.univ-rennes1.fr/vincent.guirardel/papiers/lecture_notes_pcmi.pdf

Fillings of unit cotangent bundles of nonorientable surfaces

Series
Geometry Topology Seminar
Time
Monday, September 26, 2016 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Burak Ozbagci – UCLA and Koc University
We prove that any minimal weak symplectic filling of the canonical contact structure on the unit cotangent bundle of a nonorientable closed surface other than the real projective plane is s-cobordant rel boundary to the disk cotangent bundle of the surface. If the nonorientable surface is the Klein bottle, then we show that the minimal weak symplectic filling is unique up to homeomorphism. (This is a joint work with Youlin Li.)

Algebraic Geometry for Computer Vision

Series
Algebra Seminar
Time
Monday, September 26, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Joe Kileel – UC Berkeley
This talks presents two projects at the interface of computer vision and algebraic geometry. Work with Zuzana Kukelova, Tomas Pajdla and Bernd Sturmfels introduces the distortion varieties of a given projective variety. These are parametrized by duplicating coordinates and multiplying them with monomials. We study their degrees and defining equations. Exact formulas are obtained for the case of one-parameter distortions, the case of most interest for modeling cameras with image distortion. Single-authored work determines the algebraic degree of minimal problems for the calibrated trifocal variety. Our techniques rely on numerical algebraic geometry, and the homotopy continuation software Bertini.

Basics and generalities leading to Boltzmann's kinetic equation

Series
Non-Equilibrium Statistical Mechanics Reading Group
Time
Monday, September 26, 2016 - 16:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Zaher Hani – GeorgiaTech –
We will start explaining and formulating the mathematical questions involved in justifying statistical physics from dynamical first principles. We will particularly discuss the approach, suggested by Boltzmann, based on deriving effective equations for the distribution function of a particle system. This will lead us to Boltzmann kinetic equation and its H-principle. This corresponds to Chapters 1 and 2 of Dorfman "An introduction to Chaos in Non-equilibrium Statistical Mechanics".

Target identification in sonar imagery via simulations of Helmholtz equations

Series
Research Horizons Seminar
Time
Wednesday, September 28, 2016 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Christina Frederick – Department of Mathematics, Georgia Institute of Technology –

Food and Drinks will be provided before the seminar.

We present a multiscale approach for identifying objects submerged in ocean beds by solving inverse problems in high frequency seafloor acoustics. The setting is based on Sound Navigation And Ranging (SONAR) imaging used in scientific, commercial, and military applications. The forward model incorporates simulations, by solving Helmholtz equations, on a wide range of spatial scales in the seafloor geometry. This allows for detailed recovery of seafloor parameters including the material type. Simulated backscattered data is generated using microlocal analysis techniques. In order to lower the computational cost of large-scale simulations, we take advantage of a library of representative acoustic responses from various seafloor parametrizations.

Inequalities for eigenvalues of sums of self-adjoint operators and related intersection problems (Part II)

Series
Analysis Seminar
Time
Wednesday, September 28, 2016 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Wing Li – Georgia Tech
Consider Hermitian matrices A, B, C on an n-dimensional Hilbert space such that C=A+B. Let a={a_1,a_2,...,a_n}, b={b_1, b_2,...,b_n}, and c={c_1, c_2,...,c_n} be sequences of eigenvalues of A, B, and C counting multiplicity, arranged in decreasing order. Such a triple of real numbers (a,b,c) that satisfies the so-called Horn inequalities, describes the eigenvalues of the sum of n by n Hermitian matrices. The Horn inequalities is a set of inequalities conjectured by A. Horn in 1960 and later proved by the work of Klyachko and Knutson-Tao. In these two talks, I will start by discussing some of the history of Horn's conjecture and then move on to its more recent developments. We will show that these inequalities are also valid for selfadjoint elements in a finite factor, for types of torsion modules over division rings, and for singular values for products of matrices, and how additional information can be obtained whenever a Horn inequality saturates. The major difficulty in our argument is the proof that certain generalized Schubert cells have nonempty intersection. In the finite dimensional case, it follows from the classical intersection theory. However, there is no readily available intersection theory for von Neumann algebras. Our argument requires a good understanding of the combinatorial structure of honeycombs, and produces an actual element in the intersection algorithmically, and it seems to be new even in finite dimensions. If time permits, we will also discuss some of the intricate combinatorics involved here. In addition, some recent work and open questions will also be presented.

Imaging Science meets Compressed Sensing

Series
School of Mathematics Colloquium
Time
Thursday, September 29, 2016 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Gitta Kutyniok – Technical University of Berlin –
Modern imaging data are often composed of several geometrically distinct constituents. For instance, neurobiological images could consist of a superposition of spines (pointlike objects) and dendrites (curvelike objects) of a neuron. A neurobiologist might then seek to extract both components to analyze their structure separately for the study of Alzheimer specific characteristics. However, this task seems impossible, since there are two unknowns for every datum. Compressed sensing is a novel research area, which was introduced in 2006, and since then has already become a key concept in various areas of applied mathematics, computer science, and electrical engineering. It surprisingly predicts that high-dimensional signals, which allow a sparse representation by a suitable basis or, more generally, a frame, can be recovered from what was previously considered highly incomplete linear measurements, by using efficient algorithms. Utilizing the methodology of Compressed Sensing, the geometric separation problem can indeed be solved both numerically and theoretically. For the separation of point- and curvelike objects, we choose a deliberately overcomplete representation system made of wavelets (suited to pointlike structures) and shearlets (suited to curvelike structures). The decomposition principle is to minimize the $\ell_1$ norm of the representation coefficients. Our theoretical results, which are based on microlocal analysis considerations, show that at all sufficiently fine scales, nearly-perfect separation is indeed achieved. This project was done in collaboration with David Donoho (Stanford University) and Wang-Q Lim (TU Berlin).

Decomposition of graphs under average degree condition

Series
Graph Theory Seminar
Time
Thursday, September 29, 2016 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Yan Wang – Math, GT
Stiebitz showed that a graph with minimum degree s+t+1 can be decomposed into vertex disjoint subgraphs G_1 and G_2 such that G_1 has minimum degree at least s and G_2 has minimum degree at least t. Motivated by this result, Norin conjectured that a graph with average degree s+t+2 can be decomposed into vertex disjoint subgraphs G_1 and G_2 such that G_1 has average degree at least s and G_2 has average degree at least t. Recently, we prove that a graph with average degree s+t+2 contains vertex disjoint subgraphs G_1 and G_2 such that G_1 has average degree at least s and G_2 has average degree at least t. In this talk, I will discuss the proof technique. This is joint work with Hehui Wu.

Can one hear the shape of a random walk?

Series
Stochastics Seminar
Time
Thursday, September 29, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Eviatar Procaccia – Texas A&M University –
We consider a Gibbs distribution over random walk paths on the square lattice, proportional to a random weight of the path’s boundary. We show that in the zero temperature limit, the paths condensate around an asymptotic shape. This limit shape is characterized as the minimizer of the functional, mapping open connected subsets of the plane to the sum of their principle eigenvalue and perimeter (with respect to the first passage percolation norm). A prime novel feature of this limit shape is that it is not in the class of Wulff shapes. This is joint work with Marek Biskup.

Anisotropic Structures and Sparse Regularization of Inverse Problems

Series
Analysis Seminar
Time
Friday, September 30, 2016 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Gitta Kutyniok – Technical University of Berlin –

Note the unusual time.

Many important problem classes are governed by anisotropic structures such as singularities concentrated on lower dimensional embedded manifolds, for instance, edges in images or shear layers in solutions of transport dominated equations. While the ability to reliably capture and sparsely represent anisotropic features for regularization of inverse problems is obviously the more important the higher the number of spatial variables is, principal difficulties arise already in two spatial dimensions. Since it was shown that the well-known (isotropic) wavelet systems are not capable of efficiently approximating such anisotropic features, the need arose to introduce appropriate anisotropic representation systems. Among various suggestions, shearlets are the most widely used today. Main reasons for this are their optimal sparse approximation properties within a model situation in combination with their unified treatment of the continuum and digital realm, leading to faithful implementations. In this talk, we will first provide an introduction to sparse regularization of inverse problems, followed by an introduction to the anisotropic representation system of shearlets and presenting the main theoretical results. We will then analyze the effectiveness of using shearlets for sparse regularization of exemplary inverse problems such as recovery of missing data and magnetic resonance imaging (MRI) both theoretically and numerically.

Topology Optimization of Structures and Materials

Series
GT-MAP Seminar
Time
Friday, September 30, 2016 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 257
Speaker
Tomas Zegard – GT CE

Bio: Tomas Zegard is a postdoctoral fellow in the School of Civil and Environmental Engineering at Georgia Tech. He received a PhD in Structural Engineering from the University of Illinois at Urbana-Champaign in 2014. Afterwards, he took a position at SOM LLP in Chicago, an Architecture + Engineering firm specializing in skyscrapers. He has made significant contributions to the field of topology optimization through research papers and free open-source tools. Xiaojia Zhang is a doctoral candidate in the School of Civil and Environmental Engineering at Georgia Tech. She received her bachelor’s and master’s degrees in structural engineering from the University of Illinois at Urbana-Champaign. Her major research interests are structural topology optimization with material and geometric nonlinearity, stochastic programming, and additive manufacturing.

Topology optimization, an agnostic design method, proposes new and innovative solutions to structural problems. The previously established methodology of sizing a defined geometry and connectivity is not sufficient; in these lie the potential for big improvements. However, topology optimization is not without its problems, some of which can be controlled or mitigated. The seminar will introduce two topology optimization techniques: one targeted at continuum, and one targeted at discrete (lattice-like) solutions. Both will be presented using state-of-the-art formulations and implementations. The stress singularity problem (vanishing constraints), the ill-posedness of the problem, the large number of variables involved, and others, continue to challenge researchers and practitioners. The presented concepts find potential applications in super-tall building designs, aircrafts, and the human body. The issue of multiple load cases in a structure, a deterministic problem, will be addressed using probabilistic methodologies. The proposed solution is built around a suitable damping scheme based on simulated annealing. A randomized approach with stochastic sampling is proposed, which requires a fraction of the computational cost compared to the standard methodologies.

NumericalImplicitization for Macaulay2

Series
Algebra Seminar
Time
Friday, September 30, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Justin Chen – UC Berkeley

Many varieties of interest in algebraic geometry and applications<br />
are given as images of regular maps, i.e. via a parametrization.<br />
Implicitization is the process of converting a parametric description of a<br />
variety into an intrinsic (i.e. implicit) one. Theoretically,<br />
implicitization is done by computing (a Grobner basis for) the kernel of a<br />
ring map, but this can be extremely time-consuming -- even so, one would<br />
often like to know basic information about the image variety. The purpose<br />
of the NumericalImplicitization package is to allow for user-friendly<br />
computation of the basic numerical invariants of a parametrized variety,<br />
such as dimension, degree, and Hilbert function values, especially when<br />
Grobner basis methods take prohibitively long.