Past Logic Seminar

17 June 2021
11:30
Abstract

I will discuss compressible types and relate them to uniform definability of types over finite sets (UDTFS), to uniformity of honest definitions and to the construction of compressible models in the context of (local) NIP. All notions will be defined during the talk.
Joint with Martin Bays and Pierre Simon.

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4 June 2021
11:30
Kobi Peterzil
Abstract

(Joint with Y. Halevi and A. Hasson.) We consider two kinds of expansions of a valued field $K$:

(1) A $T$-convex expansion of real closed field, for $T$ a polynomially bounded o-minimal expansion of $K$.

(2) A $P$-minimal field $K$ in which definable functions are PW differentiable.

We prove that any interpretable infinite field $F$ in $K$ is definably isomorphic to a finite extension of either $K$ or, in case (1), its residue field $k$. The method we use bypasses general elimination of imaginaries and is based on analysis of one dimensional quotients of the form $I=K/E$ inside $F$ and their connection to one of 4 possible sorts: $K$, $k$ (in case (1)), the value group, or the quotient of $K$ by its valuation ring. The last two cases turn out to be impossible and in the first two cases we use local differentiability to embed $F$ into the matrix ring over $K$ (or $k$).

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27 May 2021
11:30
Zou Tingxiang
Abstract

The Elekes-Szabó's theorem says very roughly that if a complex irreducible subvariety V of X*Y*Z has ''too many'' intersection with cartesian products of finite sets, then V is in correspondence with the graph of multiplication of an algebraic group G. It was noticed by Breuillard and Wang that the algebraic group G must be abelian. There is a constraint for the finite sets witnessing ''many'' intersections with V, namely a condition called in general position, which plays a key role in forcing the group to be abelian.  In this talk, I will present a result which shows that in the case of the graph of complex algebraic groups, with a weaker general position assumption, nilpotent groups will appear. More precisely, for a connected complex algebraic group G the following are equivalent:

1. The graph of G has ''many'' intersections with finite sets in weak general position;

2. G is nilpotent;

3. The ultrapower of G has a pseudofinite coarse approixmate subgroup in weak general position.

Surprisingly, the proof of the direction from 2 to 3 invokes some form of generic Mordell-Lang theorem for commutative complex algebraic groups.

This is joint work with Martin Bays and Jan Dobrowolski.

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20 May 2021
11:30
Abstract
This is joint work with Itay Kaplan and Saharon Shelah.
Given a graph $(G,E)$, its chromatic number is the smallest cardinal $\kappa$ of a legal coloring of the vertices. We will mainly concentrate on the following strong form of Taylor's conjecture:
If $G$ is an infinite graph with chromatic number$\geq \aleph_1$ then it contains all finite subgraphs of $Sh_n(\omega)$ for some $n$, where $Sh_n(\omega)$ is the $n$-shift graph (which we will introduce).

 
The conjecture was disproved by Hajnal-Komjath. However, we will sketch a proof for a variant of this conjecture for $\omega$-stable\superstable\stable graphs. The proof uses a generalization of  Ehrenfeucht-Mostowski models, which we will (hopefully) introduce.

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13 May 2021
16:30
Abstract

We survey some of the applications of generalized indiscernible sequences, both in model theory and in structural Ramsey theory.  Given structures $A$ and $B$, a semi-retraction is a pair of  quantifier-free type respecting maps $f: A \rightarrow B$ and $g: B \rightarrow A$ such that $g \circ f: A \rightarrow A$ is quantifier-free type preserving, i.e. an embedding.  In the case that $A$ and $B$ are locally finite ordered structures, if $A$ is a semi-retraction of $B$ and the age of $B$ has the Ramsey property, then the age of $A$ has the Ramsey property.

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5 May 2021
17:00
Sebastian Eterović
Abstract

There are many open conjectures about the algebraic behaviour of transcendental functions in arithmetic geometry, one of which is the Existential Closedness problem. In this talk I will review recent developments made on this question: the cases where we have unconditional existence of solutions, the conditional existence of generic solutions (depending on the conjecture of periods and Zilber-Pink), and even a few cases of unconditional existence of generic solutions. Many of the results I will mention are joint work with (different subsets of) Vahagn Aslanyan, Jonathan Kibry, Sebastián Herrero, and Roy Zhao. 

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11 March 2021
11:30
Boris Zilber
Abstract

I will discuss the multi-sorted structure of analytic covers H -> Y(N), where H is the upper half-plane and Y(N) are the N-level modular curves, all N, in a certain language, weaker than the language applied by Adam Harris and Chris Daw.  We define a certain locally modular reduct of the structure which is called "pure" structure - an extension of the structure of special subvarieties.  
The problem of non-elementary categorical axiomatisation for this structure is closely related to the theory of "canonical models for Shimura curves", in particular, the description of Gal_Q action on the CM-points of the Y(N). This problem for the case of curves is basically solved (J.Milne) and allows the beautiful interpretation in our setting:  the abstract automorphisms of the pure structure on CM-points are exactly the automorphisms induced by Gal_Q.  Using this fact and earlier theorem of Daw and Harris we prove categoricity of a natural axiomatisation of the pseudo-analytic structure.
If time permits I will also discuss a problem which naturally extends the above:  a categoricity statement for the structure of unramified analytic covers H -> X, where X runs over all smooth curves over a given number field.  

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4 March 2021
11:30
Raf Cluckers
Abstract

Point counting on definable sets in non-archimedean settings has many faces. For sets living in Q_p^n, one can count actual rational points of bounded height, but for sets in C((t))^n, one rather "counts" the polynomials in t of bounded degree. What if the latter is of infinite cardinality? We treat three settings, each with completely different behaviour for point counting : 1) the setting of subanalytic sets, where we show finiteness of point counting but growth can be aribitrarily fast with the degree in t ; 2) the setting of Pfaffian sets, which is new in the non-archimedean world and for which we show an analogue of Wilkie's conjecture in all dimensions; 3) the Hensel minimal setting, which is most general and where finiteness starts to fail, even for definable transcendental curves! In this infinite case, one bounds the dimension rather than the (infinite) cardinality. This represents joint work with Binyamini, Novikov, with Halupczok, Rideau, Vermeulen, and separate work by Cantoral-Farfan, Nguyen, Vermeulen.

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25 February 2021
17:00
Abstract

Zilber's Restricted Trichotomy Conjecture predicts that every sufficiently rich strongly minimal structure which can be interpreted from an algebraically closed field K, must itself interpret K. Progress toward this conjecture began in 1993 with the work of Rabinovich, and recently Hasson and Sustretov gave a full proof for structures with universe of dimension 1. In this talk I will discuss a partial result in characteristic zero for universes of dimension greater than 1: namely, the conjecture holds in this case under certain geometric restrictions on definable sets. Time permitting, I will discuss how this result implies the full conjecture for expansions of abelian varieties.

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3 December 2020
09:00
John Welliaveetil
Abstract

In 2010, Hrushovski--Loeser studied the homotopy type of the Berkovich analytification of a quasi-projective variety over a valued field. In this talk, we explore the extent to which some of their results might hold in a relative setting. More precisely, given a morphism of quasi-projective varieties over a valued field, we ask if we might construct deformation retractions of the analytifications of the source and target which are compatible with the analytification of the morphism and whose images are finite simplicial complexes. 

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