Abstract: This talk describes two classes of symbolic topological systems, the odometer based and the circular systems. The odometer based systems are ubiquitous--when equipped with invariant measures they form an upwards closed cone in the space of ergodic transformations (in the pre-ordering induced by factor maps). The circular systems are a small class, but represent the diffeomorphisms of the 2-torus built using the Anosov-Katok technique of approximation by conjugacy.
For fixed t>1 and L>3 we establish sharp asymptotic formula for
the log-probability that the number of cycles of length L in the Erdos - Renyi
random graph G(N,p) exceeds its expectation by a factor t, assuming only that
p >> log N/sqrt(N). In a narrower range of p=p(N) we obtain such sharp upper tail
for general subgraph counts and for the Schatten norms of the corresponding adjacency
In this talk, based on a joint work with Nick Cook, I will explain our approach,
based on convex-covering and a new quantitative refinement of
This is a continuation of the talk on October 29. After finishing a brief review of basic facts about Berkovich curves, I will associate a reduction datum to differential forms on such curves and explain how a lifting theorem for such data is proved and why it reproves the lifting theorem of [BCGGM].
Abstract: The stable fields conjecture asserts that every infinite stable field is separably closed. We will talk a bit about the history of this conjecture, its connection to an analogous conjecture on dependent fields and some of their consequences. Finally, we will end by proving the conjecture for fields of finite dp-rank.
Title: The Morris model
Abstract: Douglass Morris was a student of Keisler, and in 1970 he announced the
following result: It is consistent with ZF that for every \alpha, there is a set
A_\alpha which is the countable union of countable sets, and the power set of
A_\alpha can be partitioned into \aleph_\alpha non-empty sets.
The result was never published, and survived only in the form of a short
announcement and an exercise in Jech's "The Axiom of Choice". We go over the
proof of this theorem using modern tools, as well as some of its odd
implications about "size" and countability.
The moduli space of smooth curves with a chosen differential form has a natural stratification by the pattern of zeros of the form. In a recent paper of Bainbridge-Chen-Gendron-Grushevsky-Moeller, one used a complicated complex-analytic technique to explicitly describe a compactification of these strata. In a joint work in progress with I. Tyomkin we provide an algebraic proof of these results based on studying differential forms on Berkovich curves over fields of residual characteristic zero.
We would like present several results in descriptive set theory involving definable equivalence relations on Polish spaces.
Given an equivalence relation E on a polish space X, we would like to study the classification problem of determining whether two objects x,y in X are E-related.