Seminars and Colloquia by Series

Scalings and saturation in infinite-dimensional control problems with applications to stochastic partial differential equations

Series
Stochastics Seminar
Time
Friday, April 7, 2017 - 13:05 for 1 hour (actually 50 minutes)
Location
Skiles 270
Speaker
David Herzog – Iowa State University –
We discuss scaling methods which can be used to solve low mode control problems for nonlinear partial differential equations. These methods lead naturally to a infinite-dimensional generalization of the notion of saturation, originally due to Jurdjevic and Kupka in the finite-dimensional setting of ODEs. The methods will be highlighted by applying them to specific equations, including reaction-diffusion equations, the 2d/3d Euler/Navier-Stokes equations and the 2d Boussinesq equations. Applications to support properties of the laws solving randomly-forced versions of each of these equations will be noted.

Strategic Stable Marriage

Series
ACO Student Seminar
Time
Friday, April 7, 2017 - 13:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
James Bailey – Georgia Tech
We study stable marriage where individuals strategically submit private preference information to a publicly known stable marriage algorithm. We prove that no stable marriage algorithm ensures actual stability at every Nash equilibrium when individuals are strategic. More specifically, we show that any rational marriage, stable or otherwise, can be obtained at a Nash equilibrium. Thus the set of Nash equilibria provides no predictive value nor guidance for mechanism design. We propose the following new minimal dishonesty equilibrium refinement, supported by experimental economics results: an individual will not strategically submit preference list L if there exists a more honest L' that yields as preferred an outcome. Then for all marriage algorithms satisfying monotonicity and IIA, every minimally dishonest equilibrium yields a sincerely stable marriage. This result supports the use of algorithms less biased than the (Gale-Shapley) man-optimal, which we prove yields the woman-optimal marriage in every minimally dishonest equilibrium. However, bias cannot be totally eliminated, in the sense that no monotonic IIA stable marriage algorithm is certain to yield the egalitarian-optimal marriage in a minimally dishonest equilibrium – thus answering a 28-year old open question of Gusfield and Irving's in the negative. Finally, we show that these results extend to student placement problems, where women are polygamous and honest, but not to admissions problems, where women are both polygamous and strategic. Based on joint work with Craig Tovey at Georgia Tech.

Probabilistic analysis of a semidefinite program on sparse Erdos-Renyi graphs

Series
Stochastics Seminar
Time
Thursday, April 6, 2017 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Zhou Fan – Stanford University
Spectral algorithms are a powerful method for detecting low-rank structure in dense random matrices and random graphs. However, in certain problems involving sparse random graphs with bounded average vertex degree, a naive spectral analysis of the graph adjacency matrix fails to detect this structure. In this talk, I will discuss a semidefinite programming (SDP) approach to address this problem, which may be viewed both as imposing a delocalization constraint on the maximum eigenvalue problem and as a natural convex relaxation of minimum graph bisection. I will discuss probabilistic results that bound the value of this SDP for sparse Erdos-Renyi random graphs with fixed average vertex degree, as well as an extension of the lower bound to the two-group stochastic block model. Our upper bound uses a dual witness construction that is related to the non-backtracking matrix of the graph. Our lower bounds analyze the behavior of local algorithms, and in particular imply that such algorithms can approximately solve the SDP in the Erdos-Renyi setting. This is joint work with Andrea Montanari.

Bounding marginals of product measures

Series
Analysis Seminar
Time
Wednesday, April 5, 2017 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Galyna Livshyts – Georgia Tech
It was shown by Keith Ball that the maximal section of an n-dimensional cube is \sqrt{2}. We show the analogous sharp bound for a maximal marginal of a product measure with bounded density. We also show an optimal bound for all k-codimensional marginals in this setting, conjectured by Rudelson and Vershynin. This bound yields a sharp small ball inequality for the length of a projection of a random vector. This talk is based on the joint work with G. Paouris and P. Pivovarov.

Branched covers of spheres II

Series
Geometry Topology Student Seminar
Time
Wednesday, April 5, 2017 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sudipta Kolay – Georgia Tech
Continuing from last time, we will discuss Hilden and Montesinos' result that every smooth closed oriented three manifold is a three fold branched cover over the three sphere, and also there is a representation by bands.

Groups Actions on Spanning Trees II

Series
Research Horizons Seminar
Time
Wednesday, April 5, 2017 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Chi Ho Yuen – Georgia Tech
I will continue the discussion on the group actions of the graph Jacobian on the set of spanning trees. After reviewing the basic definitions, I will explain how polyhedral geometry leads to a new family of such actions. These actions can be described combinatorially, but proving that they are simply transitive uses geometry in an essential way. If time permits, I will also explain the following surprising connection: the canonical group action for a plane graph (via rotor-routing or Bernardi process) is related to the canonical tropical geometric structure of its dual graph. This is joint work with Spencer Backman and Matt Baker.

The Boundary Method and General Auction for Optimal Mass Transportation and Wasserstein Distance Computation

Series
Dissertation Defense
Time
Tuesday, April 4, 2017 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
J.D. Walsh III – Georgia Tech School of Math –

Please Note: Dissertation advisor: Luca Dieci

Numerical optimal transport is an important area of research, but most problems are too large and complex for easy computation. Because continuous transport problems are generally solved by conversion to either discrete or semi-discrete forms, I focused on methods for those two. I developed a discrete algorithm specifically for fast approximation with controlled error bounds: the general auction method. It works directly on real-valued transport problems, with guaranteed termination and a priori error bounds. I also developed the boundary method for semi-discrete transport. It works on unaltered ground cost functions, rapidly identifying locations in the continuous space where transport destinations change. Because the method computes over region boundaries, rather than the entire continuous space, it reduces the effective dimension of the discretization. The general auction is the first relaxation method designed for compatibility with real-valued costs and weights. The boundary method is the first transport technique designed explicitly around the semi-discrete problem and the first to use the shift characterization to reduce dimensionality. No truly comparable methods exist. The general auction and boundary method are able to solve many transport problems that are intractible using other approaches. Even where other solution methods exist, in testing it appears that the general auction and boundary method outperform them.

Joint GT-UGA Seminar at GT: L-space surgeries and satellites by algebraic links

Series
Geometry Topology Seminar
Time
Monday, April 3, 2017 - 16:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sarah Rasmussen – University of Cambridge
Exploring when a closed oriented 3-manifold has vanishing reduced Heegaard Floer homology---hence is a so-called L-space---lends insight into the deeper question of how Heegaard Floer homology can be used to enumerate and classify interesting geometric structures. Two years ago, J. Rasmussen and I developed a tool to classify the L-space Dehn surgery slopes for knots in 3-manifolds, and I later built on these methods to classify all graph manifold L-spaces. After briefly discussing these tools, I will describe my more recent computation of the region of rational L-space surgeries on any torus-link satellite of an L-space knot, with a result that precisely extends Hedden’s and Hom’s analogous result for cables. More generally, I will discuss the region of L-space surgeries on iterated torus-link satellites and algebraic link satellites, along with implications for conjectures involving co-oriented taut foliations and left-orderable fundamental groups.

Joint GT-UGA Seminar at GT: An integral lift of contact homology

Series
Geometry Topology Seminar
Time
Monday, April 3, 2017 - 15:15 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jo Nelson – Barnard College, Columbia University
I will discuss joint work with Hutchings which gives a rigorousconstruction of cylindrical contact homology via geometric methods. Thistalk will highlight our use of non-equivariant constructions, automatictransversality, and obstruction bundle gluing. Together these yield anonequivariant homological contact invariant which is expected to beisomorphic to SH^+ under suitable assumptions. By making use of familyFloer theory we obtain an S^1-equivariant theory defined with coefficientsin Z, which when tensored with Q recovers the classical cylindrical contacthomology, now with the guarantee of well-definedness and invariance. Thisintegral lift of contact homology also contains interesting torsioninformation.

Analysis of an ice-structure interaction model with a dynamic nonlinearity and random resetting

Series
Applied and Computational Mathematics Seminar
Time
Monday, April 3, 2017 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Michael Muskulus – NTNU: Norwegian University of Science and Technology –
This talk addresses an important problem in arctic engineering due to interesting dynamic phenomena in a forced linear system. The nonautonomous system considered is representative of a whole class of engineering problems that are not approachable by standard techniques from dynamical system theory.The background are ice-induced vibrations of structures (e.g. wind turbines or measurement masts) in regions with active sea ice. Ice is a complex material and the mechanism for ice-induced vibrations is not fully clear at present. In particular, the conditions under which the observed, qualitatively different vibration regimes are active cannot be predicted accurately so far. A recent mathematical model developed by Delft University of Technology assumes that a number of parallel ice strips are pushing with a constant velocity against a flexible structure. The structure is modelled as a single degree of freedom harmonic oscillator. The contact force acts on the structure, but at the same time slows down the advancement of the ice, thereby introducing a dynamic nonlinearity in the otherwise linear system. When the local contact force becomes large enough, the ice crushes and the corresponding strip is reset to a random offset in front of the structure.This is the first mathematical model that exhibits all three different dynamic regimes that are observed in reality: for slow ice velocities the structure undergoes quasi-static sawtooth responses where all ice strips fail at the same time (a kind of synchronization phenomenon), for large ice velocities the structure response appears random, and for intermediate ice velocities the system exhibits vibrations at the structure eigenfrequency, commonly called frequency lock-in behavior. The latter type of vibrations causes a lot of damage to the structure and poses a safety and economic risk, so its occurrence needs to be predicted accurately.As I will show in this talk, the descriptive terms for the three vibration regimes are slightly misleading, as the mechanisms behind the observed behaviors are somewhat different than intuition suggests. I will present first results in analyzing the system and offer some explanations of the observed behaviors, as well as some simple criteria for the switch between the different vibration regimes.

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