Past Logic Seminar

4 February 2016
16:00
Damian Rössler
Abstract

We shall describe a new proof of the Mordell-Lang conjecture in positive characteristic, in the situation where the variety under scrutiny is a smooth subvariety of an abelian variety. Our proof is based on the theory of semistable sheaves in positive characteristic, in particular on Langer's theorem that the Harder-Narasimhan filtration of sheaves becomes strongly semistable after a finite number of iterations of Frobenius pull-backs. Our proof produces a numerical upper-bound for the degree of the finite morphism from an isotrivial variety appearing in the statement of the Mordell-Lang conjecture. This upper-bound is given in terms of the Frobenius-stabilised slopes of the cotangent bundle of the variety.

28 January 2016
17:30
Abstract

I will report on joint work with Arno Fehm in which we apply
our previous `existential transfer' results to the problem of
determining which fields admit diophantine nontrivial henselian
valuation rings and ideals. Using our characterization we are able to
re-derive all the results in the literature. Also, I will explain a
connection with Pop's large fields.

3 December 2015
17:30
Franziska Jahnke
Abstract

Abstract: (Joint work with Sylvy Anscombe) We consider four properties 
of a field K related to the existence of (definable) henselian 
valuations on K and on elementarily equivalent fields and study the 
implications between them. Surprisingly, the full pictures look very 
different in equicharacteristic and mixed characteristic.

26 November 2015
11:00
Benedikt Loewe
Abstract

 If you fix a class of models and a construction method that allows you to construct a new model in that class from an old model in that class, you can consider the Kripke frame generated from any given model by iterating that construction method and define the modal logic of that Kripke frame.  We shall give a general definition of these modal logics in the fully abstract setting and then apply these ideas in a number of cases.  Of particular interest is the case where we consider the class of models of ZFC with the construction method of forcing:  in this case, we are looking at the so-called "generic multiverse".

19 November 2015
17:30
Raf Cluckers
Abstract

In the real, p-adic and motivic settings, we will present recent results on oscillatory integrals. In the reals, they are related to subanalytic functions and their Fourier transforms. In the p-adic and motivic case, there are furthermore transfer principles and applications in the Langlands program. This is joint work with Comte, Gordon, Halupczok, Loeser, Miller, Rolin, and Servi, in various combinations. 
 

5 November 2015
17:30
Abstract

We consider the decision problem of determining whether an exponential
polynomial has a real zero.  This is motivated by reachability questions
for continuous-time linear dynamical systems, where exponential
polynomials naturally arise as solutions of linear differential equations.

The decidability of the Zero Problem is open in general and our results
concern restricted versions.  We show decidability of a bounded
variant---asking for a zero in a given bounded interval---subject to
Schanuel's conjecture.  In the unbounded case, we obtain partial
decidability results, using Baker's Theorem on linear forms in logarithms
as a key tool.  We show also that decidability of the Zero Problem in full
generality would entail powerful new effectiveness results concerning
Diophantine approximation of algebraic numbers.

This is joint work with Ventsislav Chonev and Joel Ouaknine.

29 October 2015
17:30
Abstract

In contrast to the Artin-Schreier Theorem, its p-adic analog(s) involve infinite Galois theory, e.g., the absolute Galois group of p-adic fields.  We plan to give a characterization of p-adic p-Henselian valuations in an essentially finite way. This relates to the Z/p metabelian form of the birational p-adic Grothendieck section conjecture.

22 October 2015
17:30
Abstract

In the course of work with Jamshid Derakhshan on definability in adele rings, we came upon various problems about definability and model completeness for possibly infinite dimensional algebraic extensions of p-adic fields (sometimes involving uniformity across p). In some cases these problems had been closely approached in the literature but never  explicitly considered.I will explain what we have proved, and try to bring out many big gaps in our understanding of these matters. This  seems appropriate just over 50 years after the breakthroughs of Ax-Kochen and Ershov.

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