Consider a crystal formed of two types of atoms placed at the nodes of the
integer lattice. The type of each atom is chosen at random, but the crystal
is statistically shift-invariant. Consider next an electron hopping from atom
to atom. This electron performs a random walk on the integer lattice with
randomly chosen transition probabilities (since the configuration seen by
the electron is different at each lattice site). This process is highly
non-Markovian, due to the interaction between the walk and the
environment.

We will present a martingale approach to proving the invariance principle
(i.e. Gaussian fluctuations from the mean) for (irreversible) Markov chains
and show how this can be transferred to a result for the above process
(called random walk in random environment).

This is joint work with Timo Sepp\"al\"ainen.

We discuss 3-Sasakian 7-manifolds carrying metrics of positive curvature and an example of this kind on a new diffeomorphism type.

It has been over two decades since M. Gromov initiated the study of pseudo-holomorphic curves in symplectic manifolds. In the past decade we have witnessed mathematical constructions of Gromov-Witten theory for algebraic varieties, as well as many major advances in understanding their properties. Recent works in string theory have motivated us to extend our interests to Gromov-Witten theory for Deligne-Mumford stacks. Such a theory has been constructed, but many of its properties remain to be understood. In this talk I will explain the main ingredients of Gromov-Witten theory of Deligne-Mumford stacks, and I will discuss some recent progress regarding main questions in Gromov-Witten theory of Deligne-Mumford stacks.

Using cone metrics on S2, W. Thurston proved that the triangulations of the sphere of non-negative combinatorial curvature are parameterized by the points of positive norm in a certain Eisenstein lattice. In this talk, I will discuss a different approach to this result based on the study of the degenerations of K3 surfaces. I will also discuss the connection to the compactification problem for the moduli space of polarized K3 surfaces.