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.
Abstract:
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.
Although each cocycle for a action of the integers is
specified by the sequence of Birkhoff sums of a function,
it is relatively difficult to specify cocycles for the actions of
multidimensional groups such as $\Bbb Z^2$.
We'll see that if $(X,T)$ is a transitive action of the finitely
generated (countable) group $\Gamma$ by homeomorphism of the polish space $X$,
and $\Bbb B$ is a separable Banach space, there is a cocycle
$F:\Gamma\times X \to\Bbb B$
with each $x\mapsto F(g,x)$ bounded and continuous
so that the skew product action $(X x \Bbb B,S)$ is transitive where
Abstract:
The Chabauty method is a remarkable tool which employs p-adic analitic methods (in particular Colman integration.) To study rational points on curves. However the method can be applied only when the genus of the curve in question is larger than its Mordell-Weil rank. Kim developed a sophisticated "nonableian" generalisation.
We shall present the classical methid, and give an approachable introduction to Kim's method.
I'm basically going to follow http://math.mit.edu/nt/old/stage_s18.html
Abstract: I will discuss applications of algebraic results to combinatorics, focusing in particular on Lefschetz theorem, Decomposition theorem and Hodge Riemann relations. Secondly, I will discuss proving these results combinatorially, using a technique by McMullen and extended by de Cataldo and Migliorini. Finally, I will discuss Lefschetz type theorems beyond positivity.
Recommended prerequisites: basic commutative algebra
Abstract:
The Chabauty method is a remarkable tool which employs p-adic analitic methods (in particular Colman integration.) To study rational points on curves. However the method can be applied only when the genus of the curve in question is larger than its Mordell-Weil rank. Kim developed a sophisticated "nonableian" generalisation.
We shall present the classical methid, and give an approachable introduction to Kim's method.
I'm basically going to follow http://math.mit.edu/nt/old/stage_s18.html
Abstract:
The Chabauty method is a remarkable tool which employs p-adic analitic methods (in particular Colman integration.) To study rational points on curves. However the method can be applied only when the genus of the curve in question is larger than its Mordell-Weil rank. Kim developed a sophisticated "nonableian" generalisation.
We shall present the classical methid, and give an approachable introduction to Kim's method.
I'm basically going to follow http://math.mit.edu/nt/old/stage_s18.html
Abstract: I will discuss applications of algebraic results to combinatorics, focusing in particular on Lefschetz theorem, Decomposition theorem and Hodge Riemann relations. Secondly, I will discuss proving these results combinatorially, using a technique by McMullen and extended by de Cataldo and Migliorini. Finally, I will discuss Lefschetz type theorems beyond positivity.
Recommended prerequisites: basic commutative algebra
Abstract:
The Chabauty method is a remarkable tool which employs p-adic analitic methods (in particular Colman integration.) To study rational points on curves. However the method can be applied only when the genus of the curve in question is larger than its Mordell-Weil rank. Kim developed a sophisticated "nonableian" generalisation.
We shall present the classical methid, and give an approachable introduction to Kim's method.
I'm basically going to follow http://math.mit.edu/nt/old/stage_s18.html
Abstract:
The Chabauty method is a remarkable tool which employs p-adic analitic methods (in particular Colman integration.) To study rational points on curves. However the method can be applied only when the genus of the curve in question is larger than its Mordell-Weil rank. Kim developed a sophisticated "nonableian" generalisation.
We shall present the classical methid, and give an approachable introduction to Kim's method.
I'm basically going to follow http://math.mit.edu/nt/old/stage_s18.html
