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Series: Geometry Topology Seminar

Augmentations and exact Lagrangian fillings are closely related. However, not all the augmentations of a Legendrian knot come from embedded exact Lagrangian fillings. In this talk, we show that all the augmentations come from possibly immersed exact Lagrangian fillings. In particular, let ∑ be an immersed exact Lagrangian filling of a Legendrian knot in $J^1(M)$ and suppose it can be lifted to an embedded Legendrian L in J^1(R \times M). For any augmentation of L, we associate an induced augmentation of the Legendrian knot, whose homotopy class only depends on the compactly supported Legendrian isotopy type of L and the homotopy class of its augmentation of L. This is a joint work with Dan Rutherford.

Series: Geometry Topology Seminar

Based on the known examples, it had been conjectured that all L-space knots in S3 are strongly invertible. We show this conjecture is false by constructing large families of asymmetric hyperbolic knots in S3 that admit a non-trivial surgery to the double branched cover of an alternating link. The construction easily adapts to produce such knots in any lens space, including S1xS2. This is joint work with John Luecke.

Monday, April 16, 2018 - 13:55 ,
Location: Skiles 005 ,
Xiuyuan Cheng ,
Duke University ,
xiuyuan.cheng@duke.edu ,
Organizer: Wenjing Liao

Filters in a Convolutional Neural Network
(CNN) contain model parameters learned from enormous amounts of data.
The properties of convolutional filters in a trained network directly
affect the quality of the data representation being produced. In this
talk, we introduce a framework for decomposing convolutional filters
over a truncated expansion under pre-fixed bases, where the expansion coefficients are learned from data. Such a structure not only reduces the number of trainable parameters and computation load but
also explicitly imposes filter regularity by bases truncation. Apart
from maintaining prediction accuracy across image classification
datasets, the decomposed-filter CNN also produces a stable
representation with respect to input variations, which is proved under generic assumptions on the basis expansion. Joint work with Qiang Qiu, Robert Calderbank, and Guillermo Sapiro.

Series: CDSNS Colloquium

Transition State Theory describes how a reactive system crosses an energy barrier that is marked by a saddle point of the potential energy. The transition from the reactant to the product side of the barrier is regulated by a system of invariant manifolds that separate trajectories with qualitatively different behaviour.

The situation becomes more complex if there are more than two reaction channels, or possible outcomes of the reaction. Indeed, the monkey saddle potential, with three channels, is known to exhibit chaotic dynamics at any energy. We investigate the boundaries between initial conditions with different outcomes in an attempt to obtain a qualitative and quantitative description of the relevant invariant structures.

TBA

Series: GT-MAP Seminars

There are 5 short presentations in this mini-workshop. Please go to http://gtmap.gatech.edu or http://gtmap.gatech.edu/events/mini-workshop-mathematics-and-control for schedule, title and abstract.

Friday, April 13, 2018 - 10:00 ,
Location: Skiles 006 ,
Tim Duff ,
Georgia Tech ,
Organizer: Kisun Lee

The fundamental data structures for numerical methods in algebraic geometry are called "witness sets." The term "trace test" refers to certain numerical methods which verify the completeness of such witness
sets. It is natural to ask questions about the complexity of such a test and in what sense its output may be regarded as "proof." I will give a basic exposition of the trace test(s) with a view towards these questions

Series: Stochastics Seminar

In this talk I will explore the subject of Bernoulli percolation on
Galton-Watson trees. Letting $g(T,p)$ represent the probability a tree
$T$ survives Bernoulli percolation with parameter $p$, we establish
several results relating to the behavior of $g$ in the supercritical
region. These include an expression for the right derivative of $g$ at
criticality in terms of the martingale limit of $T$, a proof that $g$ is
infinitely continuously differentiable in the supercritical region, and
a proof that $g'$ extends continuously to the boundary of the
supercritical region. Allowing for some mild moment constraints on the
offspring distribution, each of these results is shown to hold for
almost surely every Galton-Watson tree. This is based on joint work
with Marcus Michelen and Robin Pemantle.

Series: Graph Theory Seminar

A classic theorem of Mader gives the extremal functions for graphs that
do not contain the complete graph on p vertices as a minor for p up to
7. Motivated by the study of linklessly embeddable graphs, we present
some results
on the extremal functions of apex graphs with respect to the number of
triangles, and on triangle-free graphs with excluded minors. Joint work with Robin Thomas.

Series: Analysis Seminar

Consider an n by n square matrix with i.i.d. zero mean unit variance entries. Rudelson and Vershynin showed that its smallest singular value is bounded from above by 1/sqrt{n} with high probability, under the assumption of the bounded fourth moment of the entries. We remove the assumption of the bounded fourth moment, thereby extending the result of Rudelson and Vershynin to a wide range of distributions.

Series: Research Horizons Seminar

The goal of this lecture is to explain and motivate the connection between Aubry-Mather theory (Dynamical Systems), and viscosity solutions of the Hamilton-Jacobi equation (PDE).This connection is the content of weak KAM Theory.The talk should be accessible to the “generic” mathematician. No a priori knowledge of any of the two subjects is assumed.The set-up of this theory is classical mechanical systems, in its Lagrangian formulation to take advantage of the action principle. This is the natural setting for Celestial Mechanics. Today it is also the setting for motions of satellites in the solar system.Hamilton found a reformulation of Lagrangian mechanics in terms of position and momentum instead of position and speed. In this formulation appears the Hamilton-Jacobi equation. Although this is a partial differential equation, its solutions allow to find solutions of the Hamiltonian (or Lagrangian) systems which are, in fact, governed by an ordinary differential equation.KAM (Kolmogorov-Arnold-Moser) theorem addressed at its beginning (Kolomogorov) the problem of stability of the solar system. It came as a surprise, since Poincare ́’s earlier work pointed to instability. In fact, some initial conditions lead to instability (Poincare ́) and some others lead to stability(Kolomogorov).Aubry-Mather theory finds some more substantial stable motion that survives outside the region where KAM theorem applies.The KAM theorem also provides global differentiable solutions to the Hamilton-Jacobi equation.It is known that the Hamilton-Jacobi equation usually does not have smooth global solutions. Lions & Crandall developed a theory of weak solutions of the Hamilton-Jacobi equation.Weak KAM theory explains how the Aubry-Mather sets can be obtained from the points where weak solutions of the Hamilton-Jacobi equation are differentiable.