Title: "Paradoxical" sets with no well-ordering of the reals Abstract: By a Hamel basis we mean a basis for the reals, R, construed as a vecor space over the field of rationals. In 1905, G. Hamel constructed such a basis from a well-ordering of R. In 1975, D. Pincus and K. Prikry asked "whether a Hamel basis exists in any model in which R cannot be well ordered." About two years ago, we answered this positively in a joint paper with M. Beriashvili, L. Wu, and L. Yu. In more recent joint work, additionally with J. Brendle and F. Castiblanco we constructed a model of

Title: Projections of Tree-Prikry forcing. Abstract: Gitik, Kanovei and Koepke proved that if U is a normal measure over \kappa then the projections of Prikry forcing with U is essentially Prikry forcing with U. The questions remains regarding to the Tree-Prikry forcing. Gitik and B. showed that without normality, it is possible that a Tree-Prikry generic sequence adds a Add(\kappa,1) generic function. In this talk we wish to examine which forcing notions can be projections of Tree-Prikry forcing under different large cardinals assumptions.

Abstract: In their 1985 paper, the above three authors introduced a consistent generalization of Ramsey's theorem to pairs of countable ordinals, which we abbreviate as $OCA_{ARS}$. This axiom asserts that for any continuous coloring (with respect to an appropriate topology) of pairs of countable ordinals, there is a decomposition of $\omega_1$ into countably-many homogeneous sets. The key to their argument is to construct Preassignments of Colors.

Following "Dynamics of some piecewise smooth Fermi-Ulam Models” by De Simoi and Dolgopyat.

A powerful theory due to Ornstein and his collaborators has been successfully applied to many random systems to show that they are isomorphic to independent and identically distributed systems. The isomorphisms provided by Ornstein's theory may not be finitary, that is, effectively realizable in practice. Despite the large number of systems known to be Bernoulli, there are only a handful of cases where explicit finitary isomorphisms have been constructed. In this talk, we will discuss classical and recent constructions, and some long standing open problems.

Title: Cwikel's bound reloaded Abstract: There are a couple of proofs by now for the famous Cwikel--Lieb--Rozenblum (CLR) bound, which is a semiclassical bound on the number of bound states for a Schr\"odinger operator, proven in the 1970s.

Title: Mesoscopic universality for orthogonal polynomial ensembles on the unit circle

Title: Fluctuations of linear statistics for Schroedinger operators with a random decaying potential Abstract: Linear statistics provide a tool for the analysis of fluctuations of random measures and have been extensively studied for various models in random matrix theory. In this talk we discuss the application of the same philosophy to the analysis of the finite volume eigenvalue counting measure of one dimensional Schroedinger operators and demonstrate it with some interesting results in the case of a random decaying potential. This is joint work with Jonathan Breuer and Moshe White.

Title: How large can Hardy-weight be? Abstract: In the first part of the talk we will discuss the existence of optimal Hardy-type inequalities with 'as large as possible' Hardy-weight for a general second-order elliptic operator defined on a noncompact Riemannian manifold, while the second part of the talk will be devoted to a sharp answer to the question: "How large can Hardy-weight be?"

We describe several examples of tame subgroups of finitely presented groups and prove that the fundamental groups of certain finite graphs of groups are locally tame.

We consider the problem of extending a representation of the fundamental group of 3-manifolds from part of the boundary surfaces. Applications to links will be discussed. Combining this with some cohomology classes of Atiyah and Bott leads to new multivariable polynomial invariants of 3-manifolds with boundary. This is joint work with Edward Miller. No background in 3-dimensional topology will be assumed in this survey and research talk.

Mean dimension is a topological invariant of dynamical systems introduced by Gromov that measures the number of parameters per iteration needed to describe a trajectory in the system. We characterize this invariant (at least for dynamical systems with the marker property, such as infinite minimal systems) using a min-max principle, where choices of both a metric on the topological space and an invariant probability measure on the system are varied. The work I will report on is joint work with M. Tsukamoto.

Given two Hamiltonian isotopic curves in a surface, one would like to tell whether they are "close" or "far apart". A natural way to do that is to consider Hofer's metric which computes mechanical energy needed to deform one curve into the other. However due to lack of tools the large-scale Hofer geometry is only partially understood. On some surfaces (e.g. S^2) literally nothing is known.

Title: Path integral representations for magnetic Schroedinger operators on graphs Abstract: We consider the semigroup and the unitary group of magnetic Schrödinger operators on graphs. Using the ideas of the Feynman Kac formula, we develop a representation of the semigroup and the unitary group in terms of the stochastic process associated with the free Laplacian. As a consequence we derive Kato-Simon estimates for the unitary group. This is joint work with Batu Güneysu (Bonn).

Abstract: We answer the following questions: 1. Consider a Borel set $X \subset \R^N$ equipped with a probability measure $\mu$. For fixed $k