Seminars and Colloquia by Series

Equiangular tight frames from association schemes

Series
Analysis Seminar
Time
Wednesday, November 9, 2016 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
John Jasper – University of Cincinnati –
An equiangular tight frame (ETF) is a set of unit vectors whose coherence achieves the Welch bound. Though they arise in many applications, there are only a few known methods for constructing ETFs. One of the most popular classes of ETFs, called harmonic ETFs, is constructed using the structure of finite abelian groups. In this talk we will discuss a broad generalization of harmonic ETFs. This generalization allows us to construct ETFs using many different structures in the place of abelian groups, including nonabelian groups, Gelfand pairs of finite groups, and more. We apply this theory to construct an infinite family of ETFs using the group schemes associated with certain Suzuki 2-groups. Notably, this is the first known infinite family of equiangular lines arising from nonabelian groups.

Skein algebras and quantum topology

Series
Research Horizons Seminar
Time
Wednesday, November 9, 2016 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jonathan Paprocki – Georgia Institute of Technology
Quantum topology is a collection of ideas and techniques for studying knots and manifolds using ideas coming from quantum mechanics and quantum field theory. We present a gentle introduction to this topic via Kauffman bracket skein algebras of surfaces, an algebraic object that relates "quantum information" about knots embedded in the surface to the representation theory of the fundamental group of the surface. In general, skein algebras are difficult to compute. We associate to every triangulation of the surface a simple algebra called a "quantum torus" into which the skein algebra embeds. In joint work with Thang Le, we make use of this embedding to give a simple proof of a difficult theorem.

Global existence for quasilinear wave equations close to Schwarzschild

Series
PDE Seminar
Time
Tuesday, November 8, 2016 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Prof. Mihai Tohaneanu – University of Kentucky –
We study the quasilinear wave equation $\Box_{g} u = 0$, where the metric $g$ depends on $u$ and equals the Schwarzschild metric when u is identically 0. Under a couple of assumptions on the metric $g$ near the trapped set and the light cone, we prove global existence of solutions. This is joint work with Hans Lindblad.

Introduction to ergodic problems in statistical mechanics (part 2).

Series
Non-Equilibrium Statistical Mechanics Reading Group
Time
Monday, November 7, 2016 - 16:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Mikel Viana – Georgia Tech
In this introductory talk we present some basic results in ergodic theory, due to Poincare, von Neumann, and Birkhoff. We will also discuss many examples of dynamical systems where the theory can be applied. As motivation for a broad audience, we will go over the connection of the theory with some classical problems in statistical mechanics (part 2 of 3).

The boundary method for numerical optimal transport

Series
Applied and Computational Mathematics Seminar
Time
Monday, November 7, 2016 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
JD Walsh – GA Tech Mathematics, doctoral candidate
The boundary method is a new algorithm for solving semi-discrete transport problems involving a variety of ground cost functions. By reformulating a transport problem as an optimal coupling problem, one can construct a partition of its continuous space whose boundaries allow accurate determination of the transport map and its associated Wasserstein distance. The boundary method approximates region boundaries using the general auction algorithm, controlling problem size with a multigrid discard approach. This talk describes numerical and mathematical results obtained when the ground cost is a convex combination of lp norms, and shares preliminary work involving other ground cost functions.

C^0-characterization of contact embeddings (via coisotropic embeddings)

Series
Geometry Topology Seminar
Time
Monday, November 7, 2016 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Stefan Mueller – Georgia Southern University.
We show that an embedding of a (small) ball into a contact manifold is contact if and only if it preserves the (modified) shape invariant. The latter is, in brief, the set of all cohomology classes that can be represented by the pull-back (to a closed one-form) of a contact form by a coisotropic embedding of a fixed manifold (of maximal dimension) and of a given homotopy type. The proof is based on displacement information about (non)-Lagrangian submanifolds that comes from J-holomorphic curve methods (and gives topological invariants), and the construction of a coisotropic torus whose image (under a given embedding that is not contact) admits a transverse contact vector field (i.e. a convex surface in dimension 3). The definition of shape preserving does not involve derivatives and is preserved by uniform convergence (on compact subsets). As a consequence, we prove C^0-rigidity of contact embeddings (and diffeomorphisms). The underlying ideas are adaptations of symplectic techniques to contact manifolds that, in contrast to symplectic capacities, work well in the contact setting; the heart of the proof however uses purely contact topological methods.

A discrete version of Koldobsky's slicing inequality

Series
Combinatorics Seminar
Time
Friday, November 4, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Matt Alexander – Kent State University –
In this talk we will discuss an answer to a question of Alexander Koldobsky and present a discrete version of his slicing inequality. We let $\# K$ be a number of integer lattice points contained in a set $K$. We show that for each $d\in \mathbb{N}$ there exists a constant $C(d)$ depending on $d$ only, such that for any origin-symmetric convex body $K \subset \mathbb{R}^d$ containing $d$ linearly independent lattice points $$ \# K \leq C(d)\text{max}_{\xi \in S^{d-1}}(\# (K\cap \xi^\perp))\, \text{vol}_d(K)^{\frac{1}{d}},$$where $\xi^\perp$ is the hyperplane orthogonal to a unit vector $\xi$ .We show that $C(d)$ can be chosen asymptotically of order $O(1)^d$ for hyperplane slices. Additionally, we will discuss some special cases and generalizations for this inequality. This is a joint work with Martin Henk and Artem Zvavitch.

Numerical calculation of domains of analyticity for Lindstedt expansions of KAM Tori (Part III)

Series
Dynamical Systems Working Seminar
Time
Friday, November 4, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 170
Speaker
Adrián P. Bustamante – Georgia Tech
In the first part of the talks we are going to present a way to study numerically the complex domains of invariant Tori for the standar map. The numerical method is based on Padé approximants. For this part we are going to follow the work of C. Falcolini and R. de la LLave.In the second part we are going to present how the numerical method, developed earlier, can be used to study the complex domains of analyticity of invariant KAM Tori for the dissipative standar map. This part is work in progress jointly with R. Calleja (continuation of last talk).

Legendrian Contact Homology

Series
Geometry Topology Working Seminar
Time
Friday, November 4, 2016 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
John Etnyre – Georgia Tech
I will give 2 or 3 lectures on Legendrian contact homology. This invariant has played a big role in our understanding of Legendrian submanifolds of contact manifolds in all dimensions. We will discuss the general definition but focus on the 3-dimensional setting where it easiest to compute (and describe Legendrian knots). I will also discuss the A^\infty structure associated to the linearized co-chain groups of contact homology.

Hierarchical clustering via spreading metrics

Series
ACO Student Seminar
Time
Friday, November 4, 2016 - 13:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Aurko Roy – Georgia Tech
We study the cost function for hierarchical clusterings introduced by Dasgupta where hierarchies are treated as first-class objects rather than deriving their cost from projections into flat clusters. It was also shown that a top-down algorithm returns a hierarchical clustering of cost at most O (α_n log n) times the cost of the optimal hierarchical clustering, where α_n is the approximation ratio of the Sparsest Cut subroutine used. Thus using the best known approximation algorithm for Sparsest Cut due to Arora-Rao-Vazirani, the top down algorithm returns a hierarchical clustering of cost at most O(log^{3/2} n) times the cost of the optimal solution. We improve this by giving an O(log n)- approximation algorithm for this problem. Our main technical ingredients are a combinatorial characterization of ultrametrics induced by this cost function, deriving an Integer Linear Programming (ILP) formulation for this family of ultrametrics, and showing how to iteratively round an LP relaxation of this formulation by using the idea of sphere growing which has been extensively used in the context of graph partitioning. We also prove that our algorithm returns an O(log n)-approximate hierarchical clustering for a generalization of this cost function also studied in Dasgupta. This joint work with Sebastian Pokutta is to appear in NIPS 2016 (oral presentation).

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