This is a joint work with Tian. We study the structure of the limit space of a sequence of almost
Einstein manifolds, which are generalizations of Einstein manifolds. Roughly speaking, such manifolds are the
initial manifolds of some normalized Ricci flows whose scalar curvatures are almost constants over space-time in the
$L^1$-sense, Ricci curvatures are bounded from below at the initial time. Under the non-collapsed condition, we show that the limit space of a
sequence of almost Einstein manifolds has most properties which is known for the limit space of Einstein
manifolds. As applications, we can apply our structure results to study the
properties of K\"ahler manifolds.

In the range 4 < \kappa < 8, the intersection of the Schramm-Loewner Curve (one of the central objects in the theory of 2-D Conformally Invariant Systems) with the boundary of its domain is a random fractal set. After reviewing some previous results on the dimension and measure of this set, I will describe recent joint work with Ilia Binder and Fredrik Viklund that partitions this set of points according to the generalized "angle" at which the curve hits the boundary, and computes the Hausdorff dimension of each partition set. The Hausdorff dimension as a function of the angle is what we call the dimension spectrum.

Talk Abstract:
We will talk about the Continuum Hypothesis and how we tackle it in the field of set theory.  Learn about large cardinals, independence and consistency proofs, and elementary embeddings, and why they are important.

Suppose we take a sample of size n from a population and follow the ancestral lines backwards in time.  Under standard assumptions, this process can be modeled by Kingman's coalescent, in which each pair of lineages merges at rate one.  However, if some individuals have large numbers of offspring or if the population is affected by selection, then many ancestral lineages may merge at one time.  In this talk, we will introduce the family of coalescent processes with multiple mergers and discuss some circumstances under which populations can be modeled by these coalescent processes.  We will also describe how genetic data, such as the number of segregating sites and the site frequency spectrum, would be affected by multiple mergers of ancestral lines, and we will discuss the implications for statistical inference.