The Painlevé equations are nonlinear 2nd order ODE and come in six families P1-P6, where P1 consists of the single equation y''=6y^2+t, and P2-P6 come with some complex parameters. They were discovered strictly for mathematical considerations at the beginning of the 20th century but have arisen in a variety of important physical applications including for example random matrix theory and general relativity. In this talk I will explain how one can use model theory to study the structure of the sets of solutions of these equations. Indeed, after recalling the basics of the theory of differentially closed fields of characteristic zero, I will explain how the trichotomy theorem, a powerful result proved by Hrushovski and Sokolovic in the early 90's, plays a fundamental role in this work.

We give an exposition of results in recent papers (of Cluckers-Derakhshan-Leenknegt-Macintyre, and joint work of the speaker with Fehm and Koenigsmann) about the definability of valuation rings in valued fields by existential formulas and by universal formulas, using few parameters. We mainly work in power series fields and use parameters from the embedded residue field.

The (right) Ziegler spectrum, Zg_R, of a ring R is a topological space attached to the (right) module category of R. This space encodes model theoretic (and algebraic) information about the module category. A valuation ring is a ring whose ideals are ordered by inclusion. A Prüfer ring is a ring whose localisations at maximal ideals are valuation rings. The value monoid of a commutative ring is its monoid of principal ideals. For a valuation domain, this is just the positive cone of the value group. I will explain how to construct the Ziegler spectrum of a Prüfer ring from its value monoid alone and how this perspective allows us to transfer soberness for Ziegler spectra of Prüfer domains to the non-domain case.

Over the past year we have become interested in extending probability functions to Second Order languages. While Second Order languages allow for more expressive strength, it also comes with the dangers of Incompleteness that is inherent in Second Order Logic. In this talk, I will present our approach to get around the Incompleteness problem. Furthermore I will introduce a rational principle for this Second Order version of Inductive Logic and give some results concerning this principle.

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The value ring used in Kontsevich's theory of motivic integration is closely associated to the Grothendieck ring of varieties. I will describe the construction of the Grothendieck ring of varieties over an algebraically closed field and discuss two questions about its structure posed by Gromov and Larsen-Lunts. The aim of this talk is to show the equivalence of these two questions and that the underlying group of the Grothendieck ring is free abelian.

The Pila-Wilkie theorem gives a bound on the number of rational points of bounded height that lie on the transcendental part of a set definable in an o-minimal expansion of the real field. The bound is optimal, but there is hope that it could be substantially improved in some particular cases. I'll describe one method for improving it in the case that the definable set is a curve, and the various results obtained by this method.

This is joint work with Angus Macintyre. I will state quantifier elimination theorems for the ring of adeles of a number field and its consequences for definable sets. I will then talk about measures of definable sets, relations to works by Weil, Tate, Langlands, Birch-Swinnerton Dyer, and Kontsevich-Zagier around Tamagawa numbers and L-functions, a conjectural picture, and partial results and several questions and challenges.

It is well known that quasianalytic systems of real functions can generate polynomially bounded o-minimal expansions of the real field. Through a few examples, we study several algebraic properties of such systems, including recent results on Weierstrass preparation.

In his work on Dulac’s problem, Ilyashenko uses a quasianalytic class of functions that is a group under composition, but not closed under addition or multiplication. When trying to extend Ilyashenko’s ideas to understand certain cases of Hilbert’s 16th problem, it seems desirable to be able to define corresponding quasianalytic classes in several variables that are also closed under various algebraic operations, such as addition, multiplication, blow-ups, etc. One possible way to achieve this requires us to first extend the one-variable class into a quasianalytic algebra whose functions have unique asymptotic expansions based on monomials definable in R_{an,exp}. I will explain some of the difficulties that arise in constructing such an algebra and how far (or close) we are to obtaining it. (This is joint work with Tobias Kaiser.)

I compare Zilber's fields with the complex field , both terms of behaviour of classical exponential polynomials and in terms of complexity of models of the respective theories.

The Principle of Indifference says that, in the absence of any evidence, we ought to adopt the uniform distribution over the space of possibilities. Why? I survey some existing justifications of the principle, due to Jaynes, Paris and Vencovská, and Williamson. Then I offer a new justification that appeals to the notion of accuracy along with decision-theoretic principles that counsel risk aversion. At the end, I ask how the Principle of Indifference interacts with calibration norms.

Probabilistic reasoning following the principle of maximum entropy is a viable and convenient alternative to Bayesian networks, relieving the user from providing complete (local) probabilistic information and observing rigorous conditional independence assumptions. This talk presents a novel approach to performing computational MaxEnt reasoning that makes use of symbolic computations instead of graph-based techniques. Given a probabilistic knowledge base, the MaxEnt optimization problem is encoded into a system of polynomial equations, and then Groebner basis theory is applied to find MaxEnt inferences as solutions of this system. Moreover, I will also present an approach to apply the MaxEnt methodology to knowledge bases built over a fragment of first-order probabilistic logic.

In recent times topological dynamics has been rediscovered as a useful tool for model theoretical observations. The seminal works of Newelski and Pillay are leading the way to new methods to expand stability-like results outside the stable context with the creation of a “tame” or “definable” topological dynamics. In this talk I will give an introduction to topological dynamics, show how it is used in model theory and give an overview of the results obtained so far.

Spectrum Exchangeability, Sx, is an irrelevance principle of Pure Inductive Logic, an extension of atom exchangeability to polyadic languages. We investigate the theory of Spectrum Exchangeability: for a fixed language L the set of sentences of L which must be assigned probability 1 by every probability function satisfying Sx. We find that the theory of Sx is equal to the theory of finite structures for L, and it emerges that Sx is inconsistent with the principle of Super-Regularity (Universal Certainty).

When trying to prove that a given real number is irrational it helps to have an irrationality criterion to appeal to. These usually take the form of inequalities that a sequence of rational numbers' numerators and denominators all have to satisfy for their limit to be irrational. Viggo Brun asked whether it was possible to find irrationality criteria for numbers given by an infinite series with rational summands, where the criterion took the form of an inequality on the summands themselves, not their numerators and denominators. In this talk we'll see what happens if the criteria are general first order formula in some structure, and in particular in the case of o-minimal structures.

The well-known duality of classical algebraic geometry between affine varieties and their co-ordinate rings has a perfect analogue in the theory of commutative C^*-algebras, which can be seen by the Gel'fand-Naimark theorem as the algebras of continuous complex-valued functions on a locally compact Hausdorff space. In modern geometry and physics one deals with much more complex generalisations of co-ordinate algebras, such as schemes, stacks and non-commutative C^*-algebras, where a geometric counterpart is no longer readily available and in many cases is believed impossible. We will discuss a model-theoretic project which challenges this point of view.

The ENT model of belief was introduced by Alena Vencovská and myself in a paper in AI in 1993. Since then two other papers have appeared which provide some further possible support for this model. In my talk I shall sketch the original construction and these subsequent observations.

Given a polynomial f in Z[x], where x is an n-tuple of variables, one can ask how many solutions the equation f(x) = 0 has in the ring Z/mZ. In the 80s, Denef-Igusa-Meuser discovered a (rather mysterious) relation between the number of solutions in Z/p^rZ where p is a fixed prime and r varies. A (seemingly) completely different problem consists in describing the singularities of the variety {x in C^n | f(x) = 0}. I will present "t-stratifications", which are certain definable partitions in valued fields, and explain how they are useful for both of the above problems.

In the foundations of probabilistic reasoning, there exist a variety of quite separate justifications for the Maximum Entropy Inference Process (ME), the oldest and conceptually the simplest of which is a "counting of possible worlds" argument (Jaynes, Paris, Vencovska), which is in reality a "logical" reinterpretation of the 19th century "balls-in-boxes" statistical argument of classical thermodynamics. A curious aspect here is that there is a certain ambiguity about exactly what might be meant by a possible world, but since at a technical level all obvious modelling of the notion of possible worlds lead to exactly the same inference process, no one appears to have paid too much attention to this. It turns out however that as soon as one tries to generalise ME to the case when the probabilistic information comes from more than one agent, the exact interpretation of what is meant by a possible world becomes of crucial importance. A finer logical analysis along these lines yields an interesting connection to the normalised geometric mean pooling operator, LogOp, familiar to decision theorists, suggests a possible multi-agent (or "social") inference process, and in turn perhaps also casts a new light on the relationship between entropy and the foundations of single-agent probabilistic reasoning. The philosophical approach which underlies the above is that of intersubjective probability. Consider a fixed finite propositional language, and the set of all possible probability functions on that language. Each agent, from some fixed finite set of agents, is assumed to have declared all the probabilistic constraints ("beliefs") that she has on her possible probabilistic belief function, and in each case these constraints are assumed to be consistent and to arise from an objective state of information (or lack of information) of that particular agent * . The probabilistic constraints of different agents arising in this manner may not however be collectively consistent, owing to the different states of information of the agents. The task of the independent observer (or chairman) is to generate by some objectively justifiable procedure a single probabilistic belief function which represents the collective probabilistic information provided by all the agents. A method of doing this is called a social inference process, (SIP). Several SIP's generalising ME have been studied (e.g. Kern-Isberner, Wilmers, Adamcik), but of these the one which appears to have the nicest axiomatic properties so far is the Social Entropy Process (SEP). However while SEP has some plausible justification in information theoretic terms it does not appear to be generated by any obvious possible worlds argument of the kind above, despite the fact that it generalises LogOp as well as ME. It also fails to satisfy a natural generalisation of the independence axiom satisfied by ME. It is for these reasons that I believe it necessary to return to the more basic foundational problematic above in search of the optimal social inference process. The reason for the title will emerge in the talk. * The constraints are assumed to have certain reasonable mathematical properties: they generate a closed convex set of probability functions.