Seminars and Colloquia by Series

Boltzmann's equation and its entropy inequality

Series
Non-Equilibrium Statistical Mechanics Reading Group
Time
Monday, October 3, 2016 - 16:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Zaher Hani – GeorgiaTech –
We continue our discussion, started last week, on what we called the "Boltzmann approach" to non-equilibrium statistical physics. We shall start with some remarks concerning the derivation and regimes of validity of the Boltzmann equation for rarefied gases (the Boltzmann-Grad limit). Then we will consider Boltzmann kinetic equation, and prove its H-principle. This corresponds mainly to Chapters 1 and 2 of Dorfman "An introduction to Chaos in Non-equilibrium Statistical Mechanics".

Hypersurfaces with central convex cross sections

Series
Geometry Topology Seminar
Time
Monday, October 3, 2016 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Alper Gur – Indiana University –
The compact transverse cross-sections of a cylinder over a central ovaloid in Rn, n ≥ 3, with hyperplanes are central ovaloids. A similar result holds for quadrics (level sets of quadratic polynomials in Rn, n ≥ 3). Their compact transverse cross-sections with hyperplanes are ellipsoids, which are central ovaloids. In R3, Blaschke, Brunn, and Olovjanischnikoff found results for compact convex surfaces that motivated B. Solomon to prove that these two kinds of examples provide the only complete, connected, smooth surfaces in R3, whose ovaloid cross sections are central. We generalize that result to all higher dimensions, proving: If M^(n-1), n >= 4, is a complete, connected smooth hypersurface of R^n, which intersects at least one hyperplane transversally along an ovaloid, and every such ovaloid on M is central, then M is either a cylinder over a central ovaloid or a quadric.

Knudsen layer: coupling fluids with kinetics

Series
Applied and Computational Mathematics Seminar
Time
Monday, October 3, 2016 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Qin Li – UW-Madison –
Many kinetic equations have the corresponding fluid limits. In the zero limit of the Knudsen number, one derives the Euler equation out of the Boltzmann equation and the heat equation out of the radiative transfer equation. While there are good numerical solvers for both kinetic and fluid equations, it is not quite well-understood when the two regimes co-exist. In this talk, we model the layer between the fluid and the kinetic using a half-space equation, study the well-posedness, design a numerical solver, and utilize it to couple the two sets of equations that govern separate domains. It is a joint work with Jianfeng Lu and Weiran Sun.

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

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

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

Please Note: 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.

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 –

Please Note: 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.

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.

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.

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).

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