Seminars and Colloquia Schedule

Dynamical Structures near the Solitons of the Supercritical gKDV Equations

Series
CDSNS Colloquium
Time
Monday, March 6, 2017 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dr. Jiayin Jin – Georgia Tech
We classify the local dynamics near the solitons of the supercritical gKDV equations. We prove that there exists a co-dim 1 center-stable (unstable) manifold, such that if the initial data is not on the center-stable (unstable) manifold then the corresponding forward(backward) flow will get away from the solitons exponentially fast; There exists a co-dim 2 center manifold, such that if the intial data is not on the center manifold, then the flow will get away from the solitons exponentially fast either in positive time or in negative time. Moreover, we show the orbital stability of the solitons on the center manifold, which also implies the global existence of the solutions on the center manifold and the local uniqueness of the center manifold. Furthermore, applying a theorem of Martel and Merle, we have that the solitons are asymptotically stable on the center manifold in some local sense. This is a joint work with Zhiwu Lin and Chongchun Zeng.

Cellular Legendrian contact homology for surfaces

Series
Geometry Topology Seminar
Time
Monday, March 6, 2017 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Dan Rrutherford – Ball State University
This is joint work with Mike Sullivan. We consider a Legendrian surface L in R5 or more generally in the 1-jet space of a surface. Such a Legendrian can be conveniently presented via its front projection which is a surface in R3 that is immersed except for certain standard singularities. We associate a differential graded algebra (DGA) to L by starting with a cellular decomposition of the base projection to R2 of L that contains the projection of the singular set of L in its 1-skeleton. A collection of generators is associated to each cell, and the differential is determined in a formulaic manner by the nature of the singular set above the boundary of a cell. Our cellular DGA is equivalent to the Legendrian contact homology DGA of L whose construction was carried out in this setting by Etnyre-Ekholm-Sullivan with the differential defined by counting holomorphic disks in C2 with boundary on the Lagrangian projection of L. Equivalence of our DGA with LCH is established using work of Ekholm on gradient flow trees. Time permitting, we will discuss constructions of augmentations of the cellular DGA from two parameter families of functions.

A Brill-Noether theorem for curves of a fixed gonality

Series
Algebra Seminar
Time
Monday, March 6, 2017 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dhruv Raganathan – IAS
The Brill-Noether varieties of a curve C parametrize embeddings of C of prescribed degree into a projective space of prescribed dimension. When C is general in moduli, these varieties are well understood: they are smooth, irreducible, and have the “expected” dimension. As one ventures deeper into the moduli space of curves, past the locus of general curves, these varieties exhibit intricate, even pathological, behaviour: they can be highly singular and their dimensions are unknown. A first measure of the failure of a curve to be general is its gonality. I will present a generalization of the Brill—Noether theorem, which determines the dimensions of the Brill—Noether varieties on a general curve of fixed gonality, i.e. “general inside a chosen special locus". The proof blends a study of Berkovich skeletons of maps from curves to toric varieties with tropical linear series theory. The deformation theory of logarithmic stable maps acts as the bridge between these ideas. This is joint work with Dave Jensen.

Channel of energy for outgoing waves and universality of blow up for wave maps

Series
PDE Seminar
Time
Tuesday, March 7, 2017 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hao Jia – IAS –
We will introduce a recently found channel of energy inequality for outgoing waves, which has been useful for semi-linear wave equations at energy criticality. Then we will explain an application of this channel of energy argument to the energy critical wave maps into the sphere. The main issue is to eliminate the so-called "null concentration of energy". We will explain why this is an important issue in the wave maps. Combining the absence of null concentration with suitable coercive property of energy near traveling waves, we show a universality property for the blow up of wave maps with energy that are just above the co-rotational wave maps. Difficulties with extending to arbitrarily large wave maps will also be discussed. This is joint work with Duyckaerts, Kenig and Merle.

Loose Legendrians in high dimensional contact manifolds (II)

Series
Geometry Topology Student Seminar
Time
Wednesday, March 8, 2017 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hyun Ki Min – Georgia Tech
There is no general h-principle for Legendrian embeddings in contact manifolds. In dimension 3, however, Legendrian knots in the complement of an overtwisted disc, which are called loose, satisfy an h-principle. We will discuss the high dimensional analog of loose knots.

Sparse operators and the sparse T1 Theorem

Series
Analysis Seminar
Time
Wednesday, March 8, 2017 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dario Mena – Georgia Tech
We impose standard $T1$-type assumptions on a Calderón-Zygmund operator $T$, and deduce that for bounded compactly supported functions $f,g$ there is a sparse bilinear form $\Lambda$ so that $$ |\langle T f, g \rangle | \lesssim \Lambda (f,g). $$ The proof is short and elementary. The sparse bound quickly implies all the standard mapping properties of a Calderón-Zygmund on a (weighted) $L^p$ space.

Likelihood geometry of determinantal point processes

Series
Stochastics Seminar
Time
Thursday, March 9, 2017 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Victor-Emmanuel Brunel – MIT
Determinantal point processes (DPPs) have attracted a lot of attention in probability theory, because they arise naturally in many integrable systems. In statistical physics, machine learning, statistics and other fields, they have become increasingly popular as an elegant mathematical tool used to describe or to model repulsive interactions. In this talk, we study the geometry of the likelihood associated with such processes on finite spaces. Interestingly, the local behavior of the likelihood function around its global maxima can be very different according to the structure of a specific graph that we define for each DPP. Finally, we discuss some statistical consequences of this fact, namely, the asymptotic accuracy of a maximum likelihood estimator.

Similarities and differences: faculty positions at research universities versus highly selective liberal arts colleges

Series
Professional Development Seminar
Time
Thursday, March 9, 2017 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Julianna Tymoczko – Smith College
A conversation with Julianna Tymoczko, associate professor and chair of the Department of Mathematics & Statistics at Smith, who received her BS from Harvard and PhD from Princeton and was a postdoc at the University of Michigan and assistant professor at the University of Iowa.

Hardness Results for Solving Graph-Structured Linear Systems

Series
ACO Student Seminar
Time
Friday, March 10, 2017 - 13:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Peng Zhang – College of Computing, Georgia Tech
Spielman and Teng (2004) showed that linear systems in Laplacian matrices can be solved in nearly linear time. Since then, a major research goal has been to develop fast solvers for linear systems in other classes of matrices. Recently, this has led to fast solvers for directed Laplacians (CKPPRSV'17) and connection Laplacians (KLPSS'16).Connection Laplacians are a special case of PSD-Graph-Structured Block Matrices (PGSBMs), block matrices whose non-zero structure correspond to a graph, and which additionally can be expressed as a sum of positive semi-definite matrices each corresponding to an edge in the graph. A major open question is whether fast solvers can be obtained for all PGSBMs (Spielman, 2016). Fast solvers for Connection Laplacians provided some hope for this. Other important families of matrices in the PGSBM class include truss stiffness matrices, which have many applications in engineering, and multi-commodity Laplacians, which arise when solving multi-commodity flow problems. In this talk, we show that multi-commodity and truss linear systems are unlikely to be solvable in nearly linear time, unless general linear systems (with integral coefficients) can be solved in nearly linear time. Joint work with Rasmus Kyng.

Lagrangian Floer Theory I

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

This will be a 1.5 hour seminar.

Following up on the previous series of talks we will show how to construct Lagrangian Floer homology and discuss it properties.

Conjugacy of circle maps to rotations

Series
Dynamical Systems Working Seminar
Time
Friday, March 10, 2017 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 254
Speaker
Rafael de la Llave – GT Math
A classical theorem of Arnold, Moser shows that in analytic families of maps close to a rotation we can find maps which are smoothly conjugate to rotations. This is one of the first examples of the KAM theory. We aim to present a self-contained version of Moser's proof and also to present some efficient numerical algorithms.

An Application of Combinatorics on Posets to Topological Graph Theory

Series
Combinatorics Seminar
Time
Friday, March 10, 2017 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Tom Trotter – Georgia Tech
Researchers here at Georgia Tech initiated a "Ramsey Theory" on binary trees and used the resulting tools to show that the local dimension of a poset is not bounded in terms of the tree-width of its cover graph. Subsequently, in collaboration with colleagues in Germany and Poland, we extended these Ramsey theoretic tools to solve a problem posed by Seymour. In particular, we showed that there is an infinite sequence of graphs with bounded tree-chromatic number and unbounded path-chromatic number. An interesting detail is that our research showed that a family conjectured by Seymour to have this property did not. However, the insights gained in this work pointed out how an appropriate modification worked as intended. The Atlanta team consists of Fidel Barrera-Cruz, Heather Smith, Libby Taylor and Tom Trotter The European colleagues are Stefan Felsner, Tamas Meszaros, and Piotr Micek.