Forthcoming events in this series


Thu, 21 Nov 2019

11:30 - 12:30
C4

On NIP formulas in groups

Gabriel Conant
(Cambridge)
Abstract

I will present joint work with A. Pillay on the theory of NIP formulas in arbitrary groups, which exhibit a local formulation of the notion of finitely satisfiable generics (as defined by Hrushovski, Peterzil, and Pillay). This setting generalizes ``local stable group theory" (i.e., the study of stable formulas in groups) and also the case of arbitrary NIP formulas in pseudofinite groups. Time permitting, I will mention an application of these results in additive combinatorics.

Thu, 07 Nov 2019
11:30
C4

Functional Modular Zilber-Pink with Derivatives

Vahagn Aslanyan
(UEA)
Abstract

I will present Pila's Modular Zilber-Pink with Derivatives (MZPD) conjecture, which is a Zilber-Pink type statement for the j-function and its derivatives, and discuss some weak and functional/differential analogues. In particular, I will define special varieties in each setting and explain the relationship between them. I will then show how one can prove the aforementioned weak/functional/differential MZPD statements using the Ax-Schanuel theorem for the j-function and its derivatives and some basic complex analytic geometry. Note that I gave a similar talk in Oxford last year (where I discussed a differential MZPD conjecture and proved it assuming an Existential Closedness conjecture for j), but this talk is going to be significantly different from that one (the approach presented in this talk will be mostly complex analytic rather than differential algebraic, and the results will be unconditional).

Thu, 31 Oct 2019
11:30
C4

Constructing geometries

Kobi Kremnitzer
(Oxford)
Abstract

In this talk I will explain a category theoretic perspective on geometry.  Starting with a category of local objects (of and algebraic nature), and a (Grothendieck) 
topology on it, one can define global objects such as schemes and stacks. Examples of this  approach are algebraic, analytic, differential geometries and also more exotic geometries  such as analytic and differential geometry over the integers and analytic geometry over  the field with one element. In this approach the notion of a point is not primary but is  derived from the local to global structure. The Zariski and Huber spectra are recovered  in this way, and we also get new spectra which might be of interest in model theory.

Thu, 06 Jun 2019
11:30
C4

The (non-uniform) Hrushovski-Lang-Weil estimates

Shuddhodan Kadattur Vasudevan
(Hebrew University Jerusalem Israel)
Abstract

In 1996 using techniques from model theory and intersection theory, Hrushovski obtained a generalisation of the Lang-Weil estimates. Subsequently the estimates have found applications in group theory, algebraic dynamics and algebraic geometry. We shall discuss a geometric proof of the non-uniform version of these estimates.

Thu, 23 May 2019
11:30
C4

Parameterization

Alex Wilkie
((Oxford University))
Abstract

I will give an introduction to the theory of definable parameterization of definable sets in the o-minimal context and its application to diophantine problems. I will then go on to discuss uniformity issues with particular reference to the subanalytic case. This is joint work with Jonathan Pila and Raf Cluckers

Thu, 16 May 2019
11:30
C4

An Imaginary Ax-Kochen-Ershov principle

Silvain Rideau
(CNRS / Institut de Mathématiques de Jussieu-Paris Rive Gauche)
Further Information

 (work in progress with Martin Hils)

Abstract

In the spirit of the Ax-Kochen-Ershov principle, one could conjecture that the imaginaries in equicharacteristic zero Henselian fields can be entirely classified in terms of the Haskell-Hrushovski-Macpherson geometric imaginaries, residue field imaginaries and value group imaginaries. However, the situation is more complicated than that. My goal in this talk will be to present what we believe to be an optimal conjecture and give elements of a proof.

Tue, 14 May 2019
11:30
C4

TBA

Anand Pillay
(University of Notre Dame)
Thu, 02 May 2019
11:30

CANCELLED

Shuddhodan Kadattur Vasudevan
Further Information

The talk will be rescheduled to another time.  

Thu, 07 Mar 2019
17:00
L5

Proving Lower Bounds on the Sizes of Proofs and Computations

Rahul Santhanam
(Oxford)
Abstract

The well known (and notoriously hard) P vs NP problem asks whether every Boolean function with polynomial-size proofs is also computable in
polynomial time.

The standard approach to the P vs NP problem is via circuit complexity. For progressively richer classes of Boolean circuits (networks of AND, OR and NOT
logic gates), one wishes to show super-polynomial lower bounds on the sizes of circuits (as a function of the size of the input) computing some Boolean
function known to be in NP, such as the Satisfiability problem.

However, there is a more logic-oriented approach initiated by Cook and Reckhow, going through proof complexity rather than circuit complexity. For
progressively richer proof systems, one wishes to show super-polynomial lower bounds on the sizes of proofs (as a function of the size of the tautology) of
some sequence of propositional tautologies.

I will give a brief overview on known results along these two directions, and on their limitations. Somewhat surprisingly, similar techniques have been found
to be useful for these seemingly different approaches. I will say something about known connections between the approaches, and pose the question of
whether there are deeper connections.

Finally, I will discuss how the perspective of proof complexity can be used to formalize the difficulty of proving lower bounds on the sizes of computations
(or of proofs).

 

Tue, 26 Feb 2019
16:00
L1

Geometric model theory in separably closed valued fields

Martin Hils
(University of Muenster)
Further Information

joint work with Moshe Kamensky and Silvain Rideau

Abstract

Let $p$ be a fixed prime number and let $SCVF_p$ be the theory of separably closed non-trivially valued fields of
characteristic $p$. In the talk, we will see that, in many ways, the step from $ACVF_{p,p}$ to $SCVF_p$ is not more
complicated than the one from $ACF_p$ to $SCF_p$.

At a basic level, this is true for quantifier elimination (Delon), for which it suffices to add parametrized $p$-coordinate
functions to any of the usual languages for valued fields. It follows that all completions are NIP.

At a more sophisticated level, in finite degree of imperfection, when a $p$-basis is named or when one just works with
Hasse derivations, the imaginaries of $SCVF_p$ are not more complicated than the ones in $ACVF_{p,p}$, i.e., they are
classified by the geometric sorts of Haskell-Hrushovski-Macpherson. The latter is proved using prolongations. One may
also use these to characterize the stable part and the stably dominated types in $SCVF_p$, and to show metastability.

Thu, 21 Feb 2019
17:00
L5

Actions of automorphism groups of omega-categorical structures on compact spaces

David Evans
(Imperial College, London)
Abstract

If G is a topological group, a G-flow X is a non-empty, compact, Hausdorff space on which G acts continuously; it is minimal if all G-orbits are dense. By a theorem of Ellis, there is a (unique) minimal G-flow M(G) which is universal: there is a continuous G-map to every other G-flow. 

Here, we will be interested in the case where G = Aut(K) for some structure K, usually omega-categorical. Work of Kechris, Pestov and Todorcevic and others gives conditions on K under which structural Ramsey Theory (due to Nesetril - Rodl and others) can be used to compute M(G). 

In the first part of the talk I will give a description of the above theory and when it applies (the 'tame case'). In the second part, I will describe joint work with J. Hubicka and J. Nesetril which shows that the omega-categorical structures constructed in the late 1980's by Hrushovski as counterexamples to Lachlan's conjecture are not tame and moreover, minimal flows of their automorphism groups have rather different properties to those in the tame case. 

Thu, 14 Feb 2019
17:00
L5

A Dichotomy for Some Elementarily Generated Modal Logics

Stanislav Kikot
(Oxford)
Abstract

 The talk is about the normal modal logics of elementary classes defined by first-order formulas of the form
 'for all x_0 there exist x_1, ..., x_n phi(x_0, x_1, ... x_n)' with phi being a conjunction of binary atoms.
 I'll show that many properties of these logics, such as finite axiomatisability,
 elementarity,  axiomatisability by a set of canonical formulas or by a single generalised Sahlqvist formula,
 together with modal definability of the initial formula, either simultaneously hold or simultaneously do not hold.
 

Thu, 07 Feb 2019
17:00
L5

Intermediate models of ZF

Asaf Karagila
(Norwich)
Abstract

Starting with a countable transitive model of V=L, we show that by 
adding a single Cohen real, c, most intermediate models do no satisfy choice. In 
fact, most intermediate models to L[c] are not even definable.

The key part of the proof is the Bristol model, which is intermediate to L[c], 
but is not constructible from a set. We will give a broad explanation of the 
construction of the Bristol model within the constraints of time.

Thu, 31 Jan 2019
17:00
L5

Z + PROVI

A.R.D. Mathias
(Université de la Réunion)
Abstract

Here Z is Zermelo’s set theory of 1908, as later formulated: full separation, but no replacement or collection among its axioms. PROVI was presented in lectures in Cambridge in 2010 and later published with improvements by Nathan Bowler, and is, I claim, the weakest subsystem of ZF to support a recognisable theory of set forcing: PROV is PROVI shorn of its axiom of infinity. The provident sets are the transitive non-empty models of PROV. The talk will begin with a presentation of PROV, and then discuss more recent applications and problems: in particular an answer in the system Z + PROV to a question posed by Eugene Wesley in 1972 will be sketched, and two proofs (fallacious, I hope) of 0 = 1 will be given, one using my slim models of Z and the other applying the Spector–Gandy theorem to certain models of PROVI. These “proofs”, when re-interpreted, supply some arguments of Reverse Mathematics.

Thu, 24 Jan 2019
11:00
L6

Kim-independence in NSOP1 theories

Itay Kaplan
(Hebrew University)
Abstract

NSOP1 is a class of first order theories containing simple theories, which contains many natural examples that somehow slip-out of the simple context.

As in simple theories, NSOP1 theories admit a natural notion of independence dubbed Kim-independence, which generalizes non-forking in simple theories and satisfies many of its properties.

In this talk I will explain all these notions, and in particular talk about recent progress (joint with Nick Ramsey) in the study of Kim-independence, showing transitivity and several consequences.

 

Tue, 22 Jan 2019
16:00
L5

EPPA and RAMSEY

Jaroslav Nesetril
(Charles University, Prague)
Abstract

We survey recent research related to the Extension Property of Partial Isomorhisms (EPPA, also known as Hrushovski property) and, perhaps surprisingly, relate it to structural Ramsey theory.   This is based on a joint work with David Evans, Jan Hubicka and Matej Konecny.
 

Thu, 17 Jan 2019
11:00
L6

Philosophical implications of the paradigm shift in model theory

John Baldwin
(University of Illinois at Chicago)
Abstract



Traditionally, logic was thought of as `principles of right reason'. Early twentieth century philosophy of mathematics focused on the problem of a general foundation for all mathematics. In contrast, the last 70 years have seen model theory develop as the study and comparison of formal theories for studying specific areas of mathematics. While this shift began in work of Tarski, Robinson, Henkin, Vaught, and Morley, the decisive step came with Shelah's stability theory. After this paradigm shift there is a systematic search for a short set of syntactic conditions which divide first order theories into disjoint classes such that models of different theories in the same class have similar mathematical properties. This classification of theories makes more precise the idea of a `tame structure'. Thus, logic (specifically model theory) becomes a tool for organizing and doing mathematics with consequences for combinatorics, diophantine geometry, differential equations and other fields. I will present an account of the last 70 years in model theory that illustrates this shift. This reports material in my recent book published by Cambridge: Formalization without Foundationalism: Model Theory and the Philosophy of Mathematical Practice.

Tue, 15 Jan 2019
16:00
L5

On strongly minimal Steiner systems Zilber’s Conjecture, Universal Algebra, and Combinatorics

John Baldwin
(University of Illinois at Chicago)
Abstract

With Gianluca Paolini (in preparation), we constructed, using a variant on the Hrushovski dimension function, for every k ≥ 3, 2^µ families of strongly minimal Steiner k systems. We study the mathematical properties of these counterexamples to Zilber’s trichotomy conjecture rather than thinking of them as merely exotic examples. In particular the long study of finite Steiner systems in reflected in results that depend on the block size k. A quasigroup is a structure with a binary operation such that for each equation xy = z the values of two of the variables determines a unique value for the third. The new Steiner 3-systems are bi-interpretable with strongly minimal Steiner quasigroups. For k > 3, we show the pure k-Steiner systems have ‘essentially unary definable closure’ and do not interpret a quasigroup. But we show that for q a prime power the Steiner q systems can be interpreted into specific sorts of quasigroups, block algebras. We extend the notion of an (a, b)-cycle graph arising in the study of finite and infinite Stein triple systems (e.g Cameron-Webb) by introducing what we call the (a, b)-path graph of a block algebra. We exhibit theories of strongly minimal block algebras where all (a, b)-paths are infinite and others in which all are finite only in the prime model. We show how to obtain combinatorial properties (e.g. 2-transitivity) by the either varying the basic collection of finite partial Steiner systems or modifying the µ function which ensures strong minimality

Sat, 05 Jan 2019
16:15

TBA

Rahul Santhanam
(Oxford)
Tue, 20 Nov 2018
16:00
L5

Definably simple groups in valued fields

Dugald Macpherson
(Leeds)
Abstract

I will discuss joint work with Gismatullin, Halupczok, and Simonetta on the following problem: given a henselian valued field of characteristic 0, possibly equipped with analytic structure (in the sense stemming originally from Denef and van den Dries), describe the possibilities for a definable group G in the valued field sort which is definably almost simple, that is, has no proper infinite definable normal subgroups. We also have results for an algebraically closed valued field K in characteristic p, but assuming also that the group is a definable subgroup of GL(n, K).

Tue, 13 Nov 2018
16:00
L5

Projective geometries arising from Elekes-Szabó problems

Martin Bays
(Muenster)
Abstract

I will explain how complex varieties which have asymptotically large intersections with finite grids can be seen to correspond to projective geometries, exploiting ideas of Hrushovski. I will describe how this leads to a precise characterisation of such varieties. Time permitting, I will discuss consequences for generalised sum-product estimates and connections to diophantine problems. This is joint work with Emmanuel Breuillard.

Tue, 06 Nov 2018
16:00
L5

Standard conjectures in model theory, and categoricity of comparison isomorphisms

Misha Gavrilovich
(Higher School of Economics)
Abstract


abstract:

In my talk I shall try to explain the following speculation and present some
evidence in the form of "correlations" between categoricity conjectures in
model theory and motivic conjectures in algebraic geometry.

Transfinite induction constructions developed in model theory are by now
sufficiently developed to be used to build analogues of objects in algebraic
geometry constructed with a choice of topology, such as a singular cohomology theory,
the Hodge decomposition, and fundamental groups of complex algebraic varieties.
Moreover, these algebraic geometric objects are often conjectured to satisfy
homogeneity or freeness properties which are true for objects constructed by
transfinite induction.


An example of this is Hrushovski fusion used to build Zilber pseudoexponentiation,
i.e. a group homomorphism  $ex:C^+ \to C^*$ which satisfies Schanuel conjecture,
a transcendence property analogous to Grothendieck conjecture on periods.


I shall also present a precise conjecture on "uniqueness" of Q-forms (comparison isomorphisms)
of complex etale cohomology, and will try to explain its relation to conjectures on l-adic
Galois representations coming from the theory of motivic Galois group.
 

Tue, 30 Oct 2018
16:00
L5

On a question of Babai and Sós, a nonstandard approach.

Daniel Palacin
(Freiburg)
Abstract

In 1985, Babai and Sós asked whether there exists a constant c>0 such that every finite group of order n has a product-free set of size at least cn, where a product-free set of a group is a subset that does not contain three elements x,y and z  satisfying xy=z. Gowers showed that the answer is no in the early 2000s, by linking the existence of product-free sets of large density to the existence of low dimensional unitary representations.

In this talk, I will provide an answer to the aforementioned question by model theoretic means. Furthermore, I will relate some of Gowers' results to the existence of nontrivial definable compactifications of nonstandard finite groups.
 

Tue, 23 Oct 2018
16:00
L5

Decidability of continuous theories of operator expansions of finite dimensional Hilbert spaces

Alexander Ivanov
(Wroclaw)
Abstract

 
We study continuous theories of classes of finite dimensional Hilbert spaces expanded by 
a finite family (of a fixed size) of unitary operators. 
Infinite dimensional models of these theories are called pseudo finite dimensional dynamical Hilbert spaces. 
Our main results connect decidability questions of these theories with the topic of approximations of groups by metric groups.