**Lecture courses:**

*Ulrich Kohlenbach (Darmstadt)* "Proof Mining"

This course gives a survey on the methods and some recent applications of the `proof mining' program. This program, which has its historical roots in G. Kreisel's pioneering work on `unwinding proofs' dating back to the 50s, is concerned with the extraction of effective uniform bounds from ineffective proofs in mathematics using techniques from proof theory. Often the uniformity of the bounds also implies new qualitative results. This is particularly true in the areas of nonlinear analysis, fixed point theory and ergodic theory on which we will focus.

*Kobi Peterzil (Haifa)* "Complex Analysis in o-minimal Structures"

The aim of this mini course is to present an interaction between model theory of o-minimal structures (here always expansion of a real closed field which we denote by R) and analysis over the algebraic closure K of R (since K is extension of degree 2, we identify it with the R-plane). While R might be non-Archimedean and thus not locally compact and possibly totally disconnected, restricting the investigation to sets and functions which are definable in o-minimal expansions of R allows us to achieve two goals:

1. Develop an analytic theory parallel ot classical complex analysis for functions definable in o-minimal expansions of R. As we will note, because every germ of complex holomorphic function is definable in the o-minimal structure R_an, the theory we develop, when specialized to this structure, must agree with the classical theory.

2. Using the strong topological restrictions which o-minimality puts on definable sets, we prove theorems which are substentially stronger than classicsl results, for classically holomorphic functions and analytic sets which are definable in o-minimal structures. Since the basic tools of complex analysis, such as power series and integration, are not available in this context one has to replace them with topological tools based on Whyburn's Topological Analysis.

*Rough Plan: *

a. Toplogical fields and differentiability.

b. The winding number and properties of definable, differentiable 1-variable functions.

c. A model theoretic application: Zilber's Trichotomy conjecture and definability in o-minimal structures vs. algebraically closed fields.

d. Functions of several variables and closure properties of definable analytic sets.

All work described here is joint with Sergei Starchenko and appears in a sequence of papers going back to 2001.

Suggested surveys (joint with S. Starchenko)

1. "Complex like" analysis, Proceedings of the RAAG Summer school, Lisbon 2003".

2. "Tame complex analysis and o-minimality", Proceedings of the ICM, Hyderabad 2010.

"Pre-requisites": An undergraduate course in complex analysis will probably help as well as familiarity with very basics properties of o-minimal structures.

*Jochen Koenigsmann (Oxford)* "Hilbert's 10th Problem"

Hilbert's 10th problem asks for an algorithm that decides on input a polynomial with integer coefficients (in several variables) whether or not the polyonomial has a zero in integers. The course will give a full proof of Matijasevich's Theorem that the problem is unsolvable and will discuss the analogous problem for rings other than the integers, notably the rationals, for which the problem is still open.

*Andreas Doering (Oxford)* "Quantum Logic and Connections to Physics"

*Lecture 1.* Standard Quantum Logic Quantum logic started with Birkhoff and von Neumann's seminal 1936 paper, in which they set out to "discover what logical structure one may hope to find in physical theories that, like quantum mechanics, do not conform to classical logic". We will present some rough outline of basic quantum theory, show why classical logic is not sufficient, and what Birkhoff and von Neumann suggested as logical structure instead. Key aspects like non-distributivity, existence of an orthocomplement, relation to projective geometry, etc. will be discussed and visualised where possible. We will consider the important theorems by Wigner, Gleason and Kochen-Specker, and some more recent developments like effect algebras will be mentioned. Some conceptual problems of quantum logic will be discussed.

*Lecture 2.* Topos-based Logic for Quantum Systems In the second lecture, the main aspects of a new form of logic for quantum systems based on generalised state spaces in the form of presheaves will be presented. This leads to an intuitionistic, multi-valued form of quantum logic in which contextuality plays a central role. It will be indicated how the internal logic of (presheaf) topoi and local set theory come into play. Moreover, we will show that co-Heyting algebras, paraconsistent logic and bi-Heyting algebras show up naturally in this formalism. We will put emphasis on conceptual aspects and will mention some open questions.

**Contributed Talks:**

*Franziska Jahnke (Oxford) *"Definable Henselian Valuations"

In this talk, we give a short introduction to definable henselian valuations and give a Galois-theoretic criterion for a henselian valued field to carry a definable henselian valuation in the language of rings.

*Jizhan Hong (McMaster)* "Definable Non-divisible Henselian Valuations"

On a valued field K with valuation ring V, if V is Henselian and the value group is non-divisible and regular, then V is 0-definable over K in the language of rings. As a part of a recently submitted paper by the speaker, this in particular generalizes a Lemma of Koenigmann. The proof is very elementary.

*Zaniar Ghadernezhad (Münster)** *"On the automorphism group of ab-initio generic structures"

In this talk, we show that the automorphism group of ab-initio generic structures with rational coefficients is a (boundedly) simple group. The proof is based on the machinery (stationary independence) that used by Tent and Ziegler in the paper about the isometry group of Urysohn space and an interesting unpublished note of Evans modifying their method.

*William Anscombe (Oxford) "*F-Definability in power series fields F((t))"

*Kristian Strommen (Oxford) *"Introduction to the Section Conjecture"

Grothendieck's 'anabelian geometry' is the study of how much information the etale fundamental group of a scheme knows about the scheme itself. The celebrated Section Conjecture is a specific instance of this, which states that for hyperbolic curves, the rational points are in bijection with the sections of a certain exact sequence. We give a brief introduction to these ideas, including a birational analogue, where definite results have been proven by Koenigsmann using the model theory of p-adics.

*Sebastian Müller (Prague*) "Cuts in Models of Weak Arithmetic with an Application to the Complexity of Proofs"

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*Bernhard Elsner (Oxford)* "A Geometric Notion of Smoothness"

*Mohsen Khani (Manchester) *"The field of reals with predicates for powers of 2 and real algebraic numbers"

*David Bradley-Williams (Leeds) *"A family of unfamiliar Jordan groups"

*Artem Chernikov (Lyon)* "On the number of Dedekind cuts and two-cardinal models of NIP theories"

For a cardinal k, we let ded(k) be the supremum of the number of Dedekind cuts a linear order of size k may have. While it is always true that k < ded(k) and that ded(k) is bounded by 2^k, everything else seems to depend on the model of ZFC one is working with. We establish some new equalities and inequalities. Model-theoretic importance of ded(k) comes from the fact that it describes the size of type-spaces of dependent theories (an important class of first-order theories containing both stable and o-minimal theories, but also e.g. algebraically closed valued fields and p-adics). We give some applications to the two-cardinal model transfer. (Joint work with Saharon Shelah)

*Vincenzo Mantove (Pisa)* "A pseudoexponentiation-like structure on the algebraic numbers"

When constructing pseudoexponentiation, Zilber introduced a notion of “exponential-algebraic closure” for exponential fields, stating that certain systems of exponential-polynomial equations must have solutions. It is possible to construct an exponential-algebraic closed field whose elements are just the algebraic numbers, and such that the kernel of the exponential function is cyclic. In some ways, it still resembles complex and pseudo exponentiation, but on the other hand any statement of Schanuel type is false. The problem of finding an exponential field of this kind is mostly an arithmetic one, and its solution is obtained by controlling the likely intersections inside the product of copies of the additive and of the multiplicative group.

*Pablo Cubides Kovacsics (Paris) * "Locally constant functions in C-minimal structures"

We present some results on the behaviour of definable locally constant functions in dense C-minimal structures having a canonical tree with infinite branching at each node. If time allows, we show as corollaries known results in algebraically closed valued fields that can be then derived for their C-minimal expansions.

*Samuel Volkweis Leite (Konstanz) *"Divisibilities and an Axiomatization of the Ring of Continuous Functions from a Compact Hausdorff Space to the Field of p-adic Numbers"

I will talk about divisibilities as being the pre-concept of valuations when treated axiomatically. As an application, I will use them to give an axiomatization of a certain class of rings of continuous functions having p-adic values.