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Series: Geometry Topology Seminar

Lagrangian fillings of Legendrian knots are interesting objects that are related, on one hand, to the 4-genus of the underlying smooth knot and, on the other hand, to Floer-type invariants of Legendrian knots. Most work on Lagrangian fillings to date has concentrated on orientable fillings. I will present some first steps in constructions of and obstructions to the existence of (decomposable exact) non-orientable Lagrangian fillings. In addition, I will discuss links between the 4-dimensional crosscap number of a knot and the non-orientable Lagrangian fillings of its Legendrian representatives. This is joint work in progress with Linyi Chen, Grant Crider-Philips, Braeden Reinoso, and Natalie Yao.

Series: Math Physics Seminar

I'll report on a project, developed in collaboration with Michael Loss, to extend a very simple model of rarefied gas due to Mark Kac and use it to understand some basic issues of Equilibrium and Non-Equilibrium Statistical Mechanics.

Friday, February 9, 2018 - 15:00 ,
Location: Skiles 271 ,
Joan Gimeno ,
BGSMath-UB ,
Organizer: Jiaqi Yang

We are going to explain how invariant dynamical objects, such as (quasi)periodic orbits, can numerically be computed for Delay Differential Equations as well as their stability. To this end, we will use Automatic Differentiation techniques and iterative linear solvers with appropiate preconditioners. Additionally some numerical experiments will be presented to illustrate the approaches for each of those objects.This is joint work with A. Jorba.

Series: ACO Student Seminar

A lot of well-studied problems in CS Theory are about making
“sketches” of graphs that occupy much less space than the graph itself,
but where the shortest path distances of the graph can still be
approximately recovered from the sketch. For example, in the literature
on Spanners, we seek a sparse subgraph whose distance metric
approximates that of the original graph. In Emulator literature, we
relax the requirement that the approximating graph is a subgraph. Most
generally, in Distance Oracles, the sketch can be an arbitrary data
structure, so long as it can approximately answer queries about the
pairwise distance between nodes in the original graph.
Research on these objects typically focuses on optimizing the
worst-case tradeoff between the quality of the approximation and the
amount of space that the sketch occupies. In this talk, we will survey a
recent leap in understanding about this tradeoff, overturning the
conventional wisdom on the problem. Specifically, the tradeoff is not
smooth, but rather it follows a new discrete hierarchy in which the
quality of the approximation that can be obtained jumps considerably at
certain predictable size thresholds. The proof is graph-theoretic and
relies on building large families of graphs with large discrepancies in
their metrics.

Friday, February 9, 2018 - 10:10 ,
Location: Skiles 254 ,
Marc Härkönen ,
Georgia Tech ,
harkonen@gatech.edu ,
Organizer: Kisun Lee

As a continuation to last week's talk, we introduce the ring D of differential operators with complex coefficients, or the Weyl algebra. As we saw last week, the theory of the ring R, the ring of differential operators with rational function coefficients, is in many ways almost the same as the regular polynomial ring. The ring D however will look slightly different as its structure is much finer. We will look at filtrations, graded rings and Gröbner bases induced by weight vectors. Finally we will present an overview on the integration algorithm of holonomic D-modules and mention some applications.

Series: Stochastics Seminar

It has been conjectured that phenomena as diverse as the behavior of large "self-organizing" neural networks, and causality in standard model particle physics, can be explained by suitably rich algebras acting on themselves. In this talk I discuss the asymptotics of large causal tree diagrams that combine freely independent elements of such algebras. The Marchenko-Pastur law and Wigner's semicircle law are shown to emerge as limits of a normalized sum-over-paths of non-negative elements assigned to the edges of causal trees. The results are established in the setting of non-commutative probability. Trees with classically independent positive edge weights (random multiplicative cascades) were originally proposed by Mandelbrot as a model displaying the fractal features of turbulence. The novelty of the present work is the use of non-commutative (free) probability to allow the edge weights to take values in an algebra.

Series: Graph Theory Seminar

The celebrated Erdos-Hajnal conjecture states that for every graph H, there is a constant c > 0 such that every graph G that does not contain H as an induced subgraph has a clique or stable set of size at least n^c, where n = |V(G)|.
One approach for proving this conjecture is to prove that in every H-free graph G, there are two linear-size sets A and B such that either there are no edges between A and B, or every vertex in A is adjacent to every vertex in B. As is turns out, this is not true unless both H and its complement are trees. In the case when G contains neither H nor its complement as an induced subgraph, the conclusion above was conjectured to be true for all trees (Liebenau & Pilipczuk), and I will discuss a proof of this for a class of tree called "caterpillars".
I will also talk about results and open questions for some variants, including allowing one or both of A and B to have size n^c instead of linear size, and requiring the bipartite graph between A and B to have high or low density instead of being complete or empty. In particular, our results improve the bound on the size of the largest clique or stable that must be present in a graph with no induced five-cycle.
Joint work with Maria Chudnovsky, Jacob Fox, Anita Liebenau, Marcin Pilipczuk, Alex Scott, and Paul Seymour.

Wednesday, February 7, 2018 - 13:55 ,
Location: Skiles 006 ,
Hyun Ki Min ,
GaTech ,
Organizer: Anubhav Mukherjee

The figure 8 knot is the simplest hyperbolic knot. In the late 1970s, Thurston studied how to construct hyperbolic manifolds from ideal tetrahedra. In this talk, I present the Thurston’s theory and apply this to the figure 8 knot. It turns out that every Dehn surgery on the figure 8 knot results in a hyperbolic manifold except for 10 exceptional surgery coefficients. If time permits, I will also introduce the classification of tight contact structures on these manifolds. This is a joint work with James Conway.

Series: Analysis Seminar

Among functions $f$ majorized by indicator functions $1_E$, which functions have maximal ratio $\|\widehat{f}\|_q/|E|^{1/p}$? I will briefly describe how to establish the existence of such functions via a precompactness argument for maximizing sequences. Then for exponents $q\in(3,\infty)$ sufficiently close to even integers, we identify the maximizers and prove a quantitative stability theorem.

Series: PDE Seminar

This is a joint work with Piermarco Cannarsa and Wei Cheng. We study the properties of the set S of non-differentiable points of viscosity solutions of the Hamilton-Jacobi equation, for a Tonelli Hamiltonian. The main surprise is the fact that this set is locally arc connected—it is even locally contractible. This last property is far from generic in the class of semi-concave functions. We also “identify” the connected components of this set S. This work relies on the idea of Cannarsa and Cheng to use the positive Lax-Oleinik operator to construct a global propagation of singularities (without necessarily obtaining uniqueness of the propagation).