Generic derivations on o-minimal structures Antongiulio Fornasiero
A derivation on a field K is a map d from K to K such that d(x + y) = d(x) + d(y) and d(x y) = x d(y) + d(x) y.
Given an o-minimal structure M in a language L, we introduce the notion L-derivation, i.e derivation compatible with L. For example, if M is the field of reals with exponentiation, then we further require that the derivation d satisfies d(exp x) = exp(x) d(x).
This talk is a survey on results concerning the Teichmuller space of negatively curved Riemannian metrics on M. It is defined as the quotient space of the space of all negatively curved Riemannian metrics on M modulo the space of all isotopies of M that are homotopic to the identity. This space was shown to have highly non-trivial homotopy when M is real hyperbolic by Tom Farrell and Pedro Ontaneda in 2009.
Title: Chang's Conjecture (joint with Monroe Eskew)
Abstract:
I will review some consistency results related to Chang's Conjecture (CC).
First I will discuss some classical results of deriving instances of CC from huge cardinals and the new results for getting instances of CC from supercompact cardinals, and present some open problems.
Then, I will review the consistency proof of some versions of the Global Chang's Conjecture - which is the consistency of the occurrence many instances of CC simultaneously.
We address the semistable reduction conjecture of Abramovich and Karu: we prove that every surjective morphism of complex projective varieties can be modified to a semistable one. The key ingredient is a combinatorial result on triangulating lattice Cayley polytopes. Joint work with Karim Adiprasito and Michael Temkin.
The lecture consists of two parts: first 30 minutes an algebra-geometric introduction by Michael Temkin, and then a one hour talk by Gaku Liu about the key combinatorial result.
The purpose of this talk is to survey several results from Hjorth's theory of turbulent polish group actions.
We will start by discussing certain classification problems associated with Borel equivalence relations, and present the notions of Borel reductions and smooth relations, and the E_0 dichotomy theorem of Harrington-Kechris-Louveau.
For $\kappa < \lambda$ infinite cardinals let us consider the following generalization of the Lowenheim-Skolem theorem: "For every algebra with countably many operations over $\lambda^+$ there is a sub-algebra with order type exactly $\kappa^+$".
We will discuss the consistency and inconsistency of some global versions of this statement and present some open questions.
Model theory and geometry of fields with automorphism
I will review some of the model-theoretic geometry of difference varieties, and some open problems. A difference variety is defined by polynomial equations with an additional operator $\si$ interpreted as a field automorphism.
Abstract: The goal of this (and the next) talk is to introduce automorphic L-functions
for GL(n) and other split groups, and to discuss some of their properties and some conjectures.
Key words: L-functions, Langlands dual group, modular forms
Abstract: The starting point of the geometric approach to the theory of automorphic forms over function fields is a beautiful observation of Weil asserting that there is a natural bijection between the two-sided quotient GL(n,F)\GL(n,A)/GL(n,O) and the set of isomorphism classes rank n vector bundles on a curve. The goal of my talk will be to explain this result and to give some applications.
Key words: adeles and ideles in the function field case, algebraic curves, line and vector bundles on curves, Picard group, Riemann-Roch theorem.
Last week we discussed what does it means for a functor to be a "sheaf" in the etale topology.
Our goal now will be to complete the definition of algebraic stacks and to give examples.
Key words: algebraic stacks, faithfully flat morphisms, faithfully flat descent, moduli spaces of vector bundles
on curves.
The main goal of this talk will be to define algebraic stacks and to give examples.
Our main example will be moduli "space" of vector bundles on a smooth projective curve.
Key words: groupoids, Grothendieck topologies, etale and smooth morphisms of schemes, G-torsors,
algebraic stacks.
Having defined the standard automorphic L-function for GL(n) in the first talk, we now proceed to the definition of L-functions for general split groups and representations of the Langlands dual group
(which will be discussed as well). I then want to discuss some results and conjectures regarding these
L-functions.
Key words: L-functions, Langlands dual group, modular forms
Abstract: The goal of this talk will be to explain what are algebraic stacks and why they naturally appear.
If time permits, we will start discussing our main example of moduli spaces of vector bundles on a smooth projective curve.
Key words: groupoids, Grothendieck topologies, etale and smooth morphisms of schemes, algebraic stacks.
