Seminars and Colloquia Schedule

Joint GT-UGA Seminar at UGA

Series
Geometry Topology Seminar
Time
Monday, February 6, 2017 - 14:30 for 2.5 hours
Location
UGA Room 303
Speaker
Dan Cristofaro-Gardiner and John Etnyre – Harvard and Georgia Tech
John Etnyre: "Embeddings of contact manifolds" Abstract: I will discuss recent results concerning embeddings and isotopies of one contact manifold into another. Such embeddings should be thought of as generalizations of transverse knots in 3-dimensional contact manifolds (where they have been instrumental in the development of our understanding of contact geometry). I will mainly focus on embeddings of contact 3-manifolds into contact 5-manifolds. In this talk I will discuss joint work with Ryo Furukawa aimed at using braiding techniques to study contact embeddings. Braided embeddings give an explicit way to represent some (maybe all) smooth embeddings and should be useful in computing various invariants. If time permits I will also discuss other methods for embedding and constructions one may perform on contact submanifolds. Dan Cristofaro-Gardiner: "Beyond the Weinstein conjecture" Abstract: The Weinstein conjecture states that any Reeb vector field on a closed manifold has at least one closed orbit. The three-dimensional case of this conjecture was proved by Taubes in 2007, and Hutchings and I later showed that in this case there are always at least 2 orbits. While examples exist with exactly two orbits, one expects that this lower bound can be significantly improved with additional assumptions. For example, a theorem of Hofer, Wysocki, and Zehnder states that a generic nondegenerate Reeb vector field associated to the standard contact structure on $S^3$ has either 2, or infinitely many, closed orbits. We prove that any nondegenerate Reeb vector field has 2 or infinitely many closed orbits as long as the associated contact structure has torsion first Chern class. This is joint work with Mike Hutchings and Dan Pomerleano.

Dynamics and Analysis of some Degenerate 4th order PDEs related to crystal evolution

Series
PDE Seminar
Time
Tuesday, February 7, 2017 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jeremy Marzuola – University of North Carolina, Chapel Hill –
We discuss the derivation and analysis of a family of 4th order nonlinear PDEs that arise in the study of crystal evolution. This is joint work with Jon Weare, Jianfeng Lu, Dio Margetis, Jian-Guo Liu and Anya Katsevich.

Normal rulings of Legendrian links

Series
Geometry Topology Student Seminar
Time
Wednesday, February 8, 2017 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Caitlin Leverson – Georgia Tech
Normal rulings are decompositions of a projection of a Legendrian knot or link. Not every link has a normal ruling, so existence of a normal ruling gives a Legendrian link invariant. However, one can use the normal rulings of a link to define the ruling polynomial of a link, which is a more useful Legendrian knot invariant. In this talk, we will discuss normal rulings of Legendrian links in various manifolds and prove that the ruling polynomial is a Legendrian link invariant.

Interpolation sets and arithmetic progressions

Series
Analysis Seminar
Time
Wednesday, February 8, 2017 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Itay Londner – Tel-Aviv University
Given a set S of positive measure on the unit circle, a set of integers K is an interpolation set (IS) for S if for any data {c(k)} in l^2(K) there exists a function f in L^2(S) such that its Fourier coefficients satisfy f^(k)=c(k) for all k in K. In the talk I will discuss the relationship between the concept of IS and the existence of arbitrarily long arithmetic progressions with specified lengths and step sizes in K. Multidimensional analogues of this subject will also be considered.This talk is based on joint work with Alexander Olevskii.

Geodesics in First-Passage Percolation

Series
Stochastics Seminar
Time
Thursday, February 9, 2017 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Christopher Hoffman – University of Washington –
First-passage percolation is a classical random growth model which comes from statistical physics. We will discuss recent results about the relationship between the limiting shape in first passage percolation and the structure of the infinite geodesics. This incudes a solution to the midpoint problem of Benjamini, Kalai and Schramm. This is joint work with Gerandy Brito and Daniel Ahlberg.

Large Even Factors of Graphs

Series
Graph Theory Seminar
Time
Thursday, February 9, 2017 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Genghua Fan – Center for Discrete Mathematics, Fuzhou University
A spanning subgraph $F$ of a graph $G$ iscalled an even factor of $G$ if each vertex of $F$ has even degreeat least 2 in $F$. It was proved that if a graph $G$ has an evenfactor, then it has an even factor $F$ with $|E(F)|\geq {4\over7}(|E(G)| + 1)$, which is best possible. Recently, Cheng et al.extended the result by considering vertices of degree 2. They provedthat if a graph $G$ has an even factor, then it has an even factor$F$ with $|E(F)|\geq {4\over 7}(|E(G)| + 1)+{1\over 7}|V_2(G)|$,where $V_2(G)$ is the set of vertices of degree 2 in $G$. They alsogave examples showing that the second coefficient cannot be largerthan ${2\over 7}$ and conjectured that if a graph $G$ has an evenfactor, then it has an even factor $F$ with $|E(F)|\geq {4\over7}(|E(G)| + 1)+ {2\over 7}|V_2(G)|$. We note that the conjecture isfalse if $G$ is a triangle. We confirm the conjecture for all graphson at least 4 vertices. Moreover, if $|E(H)|\leq {4\over 7}(|E(G)| +1)+ {2\over 7}|V_2(G)|$ for every even factor $H$ of $G$, then everymaximum even factor of $G$ is a 2-factor in which each component isan even circuit.

Polynomial mixing of the edge-flip Markov chain for unbiased dyadic tilings

Series
ACO Student Seminar
Time
Friday, February 10, 2017 - 13:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Sarah Cannon – College of Computing, Georgia Tech
We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall, and Spencer in 2002. The technique used, adapted from spin system analysis in statistical physics and not widely used in computer science literature, involves a multilevel decomposition of the state space and is of independent interest. A dyadic tiling of size n is a tiling of the unit square by n non-overlapping dyadic rectangles, each of area 1/n, where a dyadic rectangle is any rectangle that can be written in the form [a2^{-s}, (a+1)2^{-s}] x [b2^{-t}, (b+1)2^{-t}] for non-negative integers a,b,s,t. The edge-flip Markov chain selects a random edge of the tiling and replaces it with its perpendicular bisector if doing so yields a valid dyadic tiling. Specifically, we show that the relaxation time of the edge-flip Markov chain for dyadic tilings is at most O(n^{4.09}), which implies that the mixing time is at most O(n^{5.09}). We complement this by showing that the relaxation time is at least \Omega(n^{1.38}), improving upon the previously best lower bound of \Omega(n log n) coming from the diameter of the chain. This is joint work with David Levin and Alexandre Stauffer.

Building Morse/Floer type homology theories II

Series
Geometry Topology Working Seminar
Time
Friday, February 10, 2017 - 14:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 006
Speaker
John Etnyre – Georgia Tech

Note the semianr scheduled for 1.5 hours. (We might take a short break in the middle and then go slightly longer.)

In this series of talks I will descibe a general proceedure to construct homology theories using analytic/geometric techiques. We will then consider Morse homology in some detail and a simple example of this process. Afterwords we will consider other situations like Floer theory and possibly contact homology.

Energy identity for a sequence of Sacks-Uhlenbeck maps to a sphere

Series
PDE Seminar
Time
Friday, February 10, 2017 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jiayu Li – University of Science and Technology of China –
For a map u from a Riemann surface M to a Riemannian manifold and a>1, the a-energy functional is defined as E_a(u)=\int_M |\nabla u|^{2a}dx. We call u_a a sequence of Sacks-Uhlenbeck maps if u_a is a critical point of E_a, and sup_{a>1} E_a(u_a)<\infty. In this talk, when the target manifold is a standard sphere S^K, we prove the energy identity for a sequence of Sacks-Uhlenbeck maps during blowing up.

Coloring curves that cross a fixed curve

Series
Combinatorics Seminar
Time
Friday, February 10, 2017 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Bartosz Walczak – Jagiellonian University in Kraków –
A class of graphs is *χ-bounded* if the chromatic number of all graphs in the class is bounded by some function of their clique number. *String graphs* are intersection graphs of curves in the plane. Significant research in combinatorial geometry has been devoted to understanding the classes of string graphs that are *χ*-bounded. In particular, it is known since 2012 that the class of all string graphs is not *χ*-bounded. We prove that for every integer *t*≥1, the class of intersection graphs of curves in the plane each of which crosses a fixed curve *c* in at least one and at most *t* points is *χ*-bounded. This result is best possible in several aspects; for example, the upper bound *t* on the number of crossings of each curve with *c* cannot be dropped. As a corollary, we obtain new upper bounds on the number of edges in so-called *k*-quasi-planar topological graphs. This is joint work with Alexandre Rok.