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.

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: The Modal Logic of Forcing (Part III) Abstract: Modal logic is used to study various modalities, i.e. various ways in which statements can be true, the most notable of which are the modalities of necessity and possibility. In set-theory, a natural interpretation is to consider a statement as necessary if it holds in any forcing extension of the world, and possible if it holds in some forcing extension. One can now ask what are the modal principles which captures this interpretation, or in other words - what is the "Modal Logic of Forcing"?

I will review homological mirror symmetry for the torus, which describes Lagrangian Floer theory on T^2 in terms of vector bundles on the Tate elliptic curve --- a version of Lekili and Perutz's works "over Z", where t is the Novikov parameter. Then I will describe a modified form of this story, joint with Lekili, where the Floer theory is altered by a locally constant sheaf of rings on T^2 (an "F-field").

Title: Interpolation sets and arithmetic progressions Abstract: Given a set S of positive measure on the unit circle, a set of integers K is an interpolation set (IS) for S if for any data {c(k)} in l^2(K) there exists a function f in L^2(S) such that its Fourier coefficients satisfy f^(k)=c(k) for all k in K. In the talk I will discuss the relationship between the concept of IS and the existence arithmetic structure in the set K, I will focus primarily on the case where K contains arbitrarily long arithmetic progressions with specified lengths and step sizes.

Abstract: Substitution schemes provide a classical method for constructing tilings of Euclidean space. Allowing multiple scales in the scheme, we introduce a rich family of sequences of tile partitions generated by the substitution rule, which include the sequence of partitions of the unit interval considered by Kakutani as a special case. In this talk we will use new path counting results for directed weighted graphs to show that such sequences of partitions are uniformly distributed, thus extending Kakutani's original result. Furthermore, we will describe certain limiting frequencies

Quantum computation ================== You can hardly open the newspaper nowadays without seeing something about Quantum computation. But aside from the hype and the industry interest, this deceivingly simple model offers a surprisingly rich set of mathematical, physical and conceptual questions, which seem to touch upon almost any area of mathematics: from group representations, to Markov chains, Knot invariants, expanders, cryptography, lattices, differential geometry, and many more. I will give some definitions, provide some basic results, and sketch some open problems.