A curve is a one dimensional space cut out by polynomial equations. In particular, one can consider curves over finite fields, which means the polynomial equations should have coefficients in some finite field and that points on the curve are given by values of the variables in the finite field that satisfy the given polynomials. A basic question is how many points such a curve has, and for a family of curves one can study the distribution of this statistic. We will give concrete examples of families in which this distribution is known or predicted, and give a sense of the different kinds of mathematics that are used to study different families.

It has been a challenging problem to studying the existence of Kahler-Einstein metrics on Fano manifolds. A Fano manifold is a compact Kahler manifold with positive first Chern class. There are obstructions to the existence of Kahler-Einstein metrics on Fano manifolds. In these lectures, I will report on recent progresses on the study of Kahler-Einstein metrics on Fano manifolds. The first lecture will be a general one. I will discuss approaches to studying the existence problem. I will discuss the difficulties and tools in these approaches and results we have for studying them. In the second lecture, I will discuss the partial C^0-estimate which plays a crucial role in recent progresses on the existence of Kahler-Einstein metrics. I will show main technical aspects of proving such an estimate.

The triangulation conjecture stated that any n-dimensional
topological manifold is homeomorphic to a simplicial complex. It is
true in dimensions at most 3, but false in dimension 4 by the work of
Casson and Freedman. In this talk I will explain the proof that the
conjecture is also false in higher dimensions. This result is based
on previous work of Galewski-Stern and Matumoto, who reduced the
problem to a question in low dimensions (the existence of elements of
order 2 and Rokhlin invariant one in the 3-dimensional homology
cobordism group). The low-dimensional question can be answered in the
negative using a variant of Floer homology, Pin(2)-equivariant
Seiberg-Witten Floer homology.

We establish uniform $L^p$ estimates for resolvents of
elliptic self-adjoint differential operators on compact manifolds
without boundary.  We also show that the spectral regions in our
resolvent estimates are optimal in general. Applications to spectral
theory of periodic Schr\"odinger operators and to inverse boundary
problems will be given. This is joint work with Gunther Uhlmann.

a) Structure of the population inside the propagating front in KPP-model

b) Contact processes in d R (Kondratiev – Skorokhod model)

c) Global limit theorems

d) Intermittency for the contact model