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
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: 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: 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: 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: 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: 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
