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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.

Series: PDE Seminar

In 1904, Prandtl introduced his famous boundary layer theory to describe the behavior of solutions of incompressible Navier Stokes equations near a boundary in the inviscid limit. His Ansatz was that the solution of Navier Stokes can be described as a solution of Euler, plus a boundary layer corrector, plus a vanishing error term in $L^\infty$. In this talk, I will present a recent joint work with E. Grenier (ENS Lyon), proving that, for a class of regular solutions of Navier Stokes equations, namely for shear profiles that are unstable to Rayleigh equations, this Prandtl's Ansatz is false. In addition, for shear profiles that are monotone and stable to Rayleigh equations, the Prandtl's asymptotic expansions are invalid.

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.

Series: School of Mathematics Colloquium

[CV: Prof. Oded Margalit has a PhD in computer science from Tel Aviv University under the supervision of Prof. Zvi Galil. He has worked at IBM Research – Haifa in the areas of machine learning, constraint satisfaction, verification, and more. Currently, he is the CTO of the IBM Cybersecurity Center of Excellence in Beer Sheva, Israel. Oded helps organize several computer science competitions, like the international IEEEXtreme and the Israeli national CodeGuru competition. He loves riddles and authors the IBM Research monthly challenge corner Ponder This.]

For the sake of puzzle-lovers worldwide, IBM Research offers a monthly mathematical challenge known as Ponder This. Every month, a new challenge is posted together with the solution for the previous month's riddle. Prof. Oded Margalit has served as the Ponder This puzzlemaster for the last decade. In this talk, he’ll survey some of most interesting riddles posted over the years, and tell some anecdotes about various challenges and regular solvers, such as one person who sent in his solution from an intensive care unit. Several challenges have led to conference and journal papers, such as a PRL paper born from a riddle on random walks, and an ITA 2014 paper on a water hose model (using quantum entanglement to break location-based encryption). Other monthly challenges have riffed on games such as 2048, Kakuro, an infinite chess game, the probability of backgammon ending with a double, Fischer Random Chess, and more. Other challenges have been more purely mathematic, focusing on minimal hash functions, combinatorial test design, or finding a natural number n such that round ((1+2 cos(20))^n) is divisible by 10^9.
The talk will present a still-open question about a permutation-firing cannon. The talk will be self contained.

Series: Algebra Seminar

Given data and a statistical model, the maximum likelihood estimate is
the point of the statistical model that maximizes the probability of
observing the data. In this talk, I will address three different
approaches to maximum likelihood estimation using algebraic methods.
These three approaches use boundary stratification of the statistical
model, numerical algebraic geometry and the EM fixed point ideal. This
talk is based on joint work with Allman, Cervantes, Evans, Hoşten,
Kosta, Lemke, Rhodes, Robeva, Sturmfels, and Zwiernik.

Series: Geometry Topology Seminar

Planar contact manifolds have been intensively studied to understand several aspects of 3-dimensional contact geometry. In this talk, we define "iterated planar contact manifolds", a higher-dimensional analog of planar contact manifolds, by using topological tools such as "open book decompositions" and "Lefschetz fibrations”. We provide some history on existing low-dimensional results regarding Reeb dynamics, symplectic fillings/caps of contact manifolds and explain some generalization of those results to higher dimensions via iterated planar structure. This is partly based on joint work in progress with J. Etnyre and B. Ozbagci.