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Series: Geometry Topology Seminar

We use the conjugation symmetry on the Heegaard Floer complexes to define a three-manifold invariant called involutive Heegaard Floer homology, which is meant to correspond to Z_4-equivariant Seiberg-Witten Floer homology. From this we obtain two new invariants of homology cobordism, explicitly computable for surgeries on L-space knots and quasi-alternating knots, and two new concordance invariants of knots, one of which (unlike other invariants arising from Heegaard Floer homology) detects non-sliceness of the figure-eight knot. We also give a formula for how this theory behaves under connected sum, and use it to give examples not homology cobordant to anything computable via our surgery formula. This is joint work with C. Manolescu; the last part of is also joint with I. Zemke.

Monday, March 13, 2017 - 14:00 ,
Location: Skiles 005 ,
Prof. Yao Li ,
University of Massachusetts Amherst ,
yaoli@math.umass.edu ,
Organizer: Molei Tao

In
this talk I will present my recent result about the ergodic properties
of nonequilibrium steady-state (NESS) for a stochastic energy exchange
model. The energy exchange model is numerically reduced from a
billiards-like deterministic particle system that models the microscopic
heat conduction in a 1D chain. By using a technique called the induced
chain method, I proved the existence, uniqueness, polynomial speed of
convergence to the NESS, and polynomial speed of mixing for the
stochastic energy exchange model. All of these are consistent with the
numerical simulation results of the original deterministic
billiards-like system.

Series: Combinatorics Seminar

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.

Friday, March 10, 2017 - 15:00 ,
Location: Skiles 254 ,
Rafael de la Llave ,
GT Math ,
Organizer: Rafael de la Llave

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.

Friday, March 10, 2017 - 14:00 ,
Location: Skiles 006 ,
John Etnyre ,
Georgia Tech ,
Organizer: John Etnyre

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.

Series: ACO Student Seminar

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.

Series: Professional Development Seminar

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.

Series: Stochastics Seminar

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.

Series: Analysis Seminar

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.

Wednesday, March 8, 2017 - 14:05 ,
Location: Skiles 006 ,
Hyun Ki Min ,
Georgia Tech ,
Organizer: Justin Lanier

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.