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Series: Graph Theory Seminar

Györi and Lovasz independently proved that a k-connected graph can be partitioned into k subgraphs, with each subgraph connected, containing a prescribed vertex, and with a prescribed
vertex count. Lovasz used topological methods, while Györi found a purely graph theoretical approach. Chen et al. later generalized the topological proof to graphs with weighted
vertices, where the subgraphs have prescribed weight sum rather than vertex count. The weighted result was recently proven using Györi's approach by Chandran et al. We will use the
Györi approach to generalize the weighted result slightly further. Joint work with Robin Thomas.

Wednesday, April 18, 2018 - 14:10 ,
Location: Skiles 006 ,
Sarah Davis ,
GaTech ,
Organizer: Anubhav Mukherjee

The theorem of Dehn-Nielsen-Baer says the extended mapping class group is isomorphic to the outer automorphism group of the fundamental group of a surface. This theorem is a beautiful example of the interconnection between purely topological and purely algebraic concepts. This talk will discuss the background of the theorem and give a sketch of the proof.

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: 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.

Series: Other Talks

[CV: Prof. Oded Margalit, PhD in Computer Science from Tel-Aviv University under the

supervision of Prof. Zvi Galil has worked at IBM's Haifa research lab on

machine learning, constraint satisfaction, verification and more. Currently he is the CTO

of the IBM Cyber security center of excellence at Ben Gurion University

of the Negev. Oded participates in organising several computer science

competitions (like the international IEEEXtreme and the national CodeGuru). He loves riddles and authors the monthly

challenge corner of IBM research: "Ponder-This".]

IBM
research runs a mathematical challenge site. Every month a new
challenge is posted; as well as a solution for the previous month's
riddle. Prof. Oded Margalit
is the puzzlemaster, for the last decade.
In the talk, he will survey some of the riddles over the years, and tell some anecdotes about the challenges and the solvers.
For example:
A PRL paper born from a riddle on random walks; ITA-2014 paper on water hose model (using quantum entanglement to break location based encryption); Games: 2048, Kakuro, Infinite chess game, the probability of a backgammon to end with a double, Fisher Foul Chess and more. Minimal hash function, Combinatorial Test Design; A solver from Intensive Care Unit and other stories; Finding a natural number n such that round ((1+2 cos(20))^n) is divisible by 10^9; We'll leave you with a still open question about Permutation-firing cannon...
Don't worry - no high math knowledge is assumed.