Department of Mathematics

Co-organizers:

Vladimir Retakh (retakh {at} math [dot] rutgers [dot] edu)

Anthony Zaleski (az202 {at} math [dot] rutgers [dot] edu)

Lennart Carleson's celebrated theorem of 1966 asserts the pointwise convergence of the partial Fourier sums of square integrable functions, giving the positive answer to Luzin's conjecture from 1915. The aim of this talk is to provide yet another proof of this fact. In particular, we will see a new simplified approach to this result, which can be presented in a brief self-contained manner. A number of related results can be seen by variants of the same argument. We survey the historical background and some complements to Carleson's theorem, as well as open problems.

The so-called Stefan problem describes the temperature distribution in a homogeneous medium undergoing a phase change, for example ice passing to water, and one aims to describe the regularity of the interface separating the two phases. In its stationary version, the Stefan problem can be reduced to the classical obstacle problem, which consists in finding the equilibrium position of an elastic membrane whose boundary is held fixed, and that is constrained to lie above a given obstacle. The aim of this talk is to give a general overview of the classical theory of the obstacle problem, and then discuss some very recent developments on the optimal regularity of the free boundary both in the static and the parabolic setting.

The ultraproduct construction gives a way of averaging an infinite sequence of mathematical structures, such as fields, graphs, or linear orders. The talk will be about the strength of such a construction.

CANCELED

The first part of the talk will be general background on 7-dimensional Riemannian manifolds with the exceptional holonomy group *G _{2}*, going back to Berger's classification from the 1950's. Then we will explain that there is a natural boundary value problem for these structures, involving fixing a 3-form on the boundary, and discuss some of the existence questions that arise. We will consider various reductions of the equations, imposing symmetry, to lower dimensions which lead to interesting PDE problems—some of which are familiar and some new.

What is the minimum possible number of vertices of a graph that contains
every *k*-vertex graph as an induced subgraph? What is the minimum possible
number of edges in a graph that contains every *k*-vertex graph with maximum
degree 3 as a subgraph? These questions and related ones were initiated
by Rado in the 60s, and received a considerable amount of attention over
the years, partly motivated by algorithmic applications. The study of
the subject combines probabilistic arguments and explicit, structured
constructions. I will survey the topic focusing on a recent asymptotic
solution of the first question, where an asymptotic formula, improving
earlier estimates by several researchers, is obtained by combining
combinatorial and probabilistic arguments with group theoretic tools.

CANCELED due to Passover

This talk will be a (hopefully) gentle introduction to applications of gauge theory to some questions in low dimensional topology. I will focus on some methods with origins in the mathematics behind gauge theory for detecting the unknot and hint at how extensions of these might give a route to a new proof of the four-color map theorem.

A complex variety is rational if it can be obtained from projective space by modifications, i.e., algebraic surgeries like blow-ups. Is rationality a deformation invariant for smooth projective varieties? This is the case for curves and surfaces but not when the dimension is at least four. The case of threefolds remains mysterious but we now know that stable rationality—rationality after taking products with projective spaces—is not a deformation invariant. (joint with Kresch, Pirutka, and Tschinkel)

Contact geometry is a beautiful subject that has important interactions with topology in dimension three. In this talk I will give a brief introduction to contact geometry and discuss its interactions with Riemannian geometry. In particular I will discuss a contact geometry analog of the famous sphere theorem and more generally indicate how the curvature of a Riemannian metric can influence properties of a contact structure adapted to it. This is joint work with Rafal Komendarczyk and Patrick Massot.

CANCELED

To understand a finitely presented group it is natural to explore its finite quotients. If the groups at hand are residually finite, a natural question is the extent to which the groups are determined by the totality of their finite quotients. This talk will discuss recent progress on constructing residually finite groups completely determined by their finite quotients.

CANCELED

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