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Abstract. In his famous paper, "An Unsolvable Problem of Elementary Number Theory," Alonzo Church (1936) identified the intuitive notion of effective calculability with the mathematically precise notion of recursiveness. This proposal, known as Church's Thesis, has been widely accepted. In this talk I consider two early arguments against it. Pepis criticized Church's Thesis already in 1937 in his dissertation and in a letter written to Church. I will display recently found documents showing Pepis's stance. The main focus of the talk will be László Kalmár's famous "An Argument Against the Plausibility of Church's Thesis" from 1959. As this paper is quite short, my aim will be to present Kalmár's argument and to fill in missing details based on his general philosophical thoughts on mathematics and his writings published on these issues in Hungarian.

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Abstract. It became clear through work of Borovik and Cherlin in 2008 that the classification theory for groups of finite Morley rank (fMr) is sufficiently developed that general questions about permutation groups of fMr can be settled even though the classification itself remains incomplete. This is a particularly salient direction since the study of uncountably categorical theories is intertwined with binding groups of fMr (acting on realizations of types).

Borovik and Cherlin posed several motivating problems, many of which are centered around the notion of ``generic $n$-transitivity''. In this talk, we will discuss the status and implications of their conjecture that the only transitive and generically $(n+2)$-transitive group of fMr acting on a set of rank $n$ is $operatorname{PGL}_{n+1}$ acting naturally on projective $n$-space. Among other things, we will highlight a general approach to constructing a projective geometry in this context, and we will also illustrate how this conjecture is intertwined with minimal fMr representations of the finite symmetric groups. The talk will begin with a brief overview of the fMr landscape---knowledge of the advanced theory of groups of fMr will be not be required.

Abstract. We will talk about some partial results and methods which point in the direction of more definable counterexamples of the continuum hypothesis. The goal is to try to identify a canonical generic extension of L(R) in which Theta^L(R)>aleph_3, the axiom of choice holds and whose theory is set-forcing invariant in the presence of large cardinals. From the point of view of Woodin's Omega-logic, the idea behind this work is to identify axioms which mirror forcing axioms and which settle Sigma_2 truth of initial segments of V but under which there are definable counterexamples to CH. We will first review the context and motivation of this investigation and previous results.

Abstract. The orbit equivalence relation of any continuous action of a Polish group $G$ has trivial Borel complexity if $G$ is compact. Similarly it is a theorem of A.S. Kechris that whenever a locally--compact Polish group acts continuously on a Polish space, the orbit equivalence relation of the action is essentially countable ---that is, Borel reducible to the orbit equivalence relation of an action of a countable group. A.S. Kechris asked for ``inverses'' to these results: (1) does every non--compact Polish group admit a continuous action that is not concretely classifiable? (2) does every non--locally--compact Polish group admit a continuous action that is not essentially countable?

Question (1) was positively answered by S. Solecki in a paper where he also answered question (2) for the special case where $G$ is the additive group of a separable Banach space. It is also resolved for the case where $G$ is an Abelian pro--countable group, by results of M. Malicki. In this talk I will present recent work on this problem, including a new dynamical proof of question (1) and a positive solution of question (2) in the case of non--Archimedean Polish groups.

This is in joint work with J. Zielinski.

Abstract. The main object of study of ergodic theory are the measure-preserving actions of (countable) groups on probability spaces. I will discuss a formalization of this setup in the framework of continuous logic and explain how some important notions studied in ergodic theory have a natural model-theoretic interpretation. This allows for some quick proofs of known results as well as a new rigidity theorem for strongly ergodic, distal actions. This is joint work with Tomas Ibarlucia.

Abstract. NIP theories are the first class of the hierarchy of n-dependent structures. The random n-hypergraph is the canonical object which is n-dependent but not (n-1)-dependent. Thus the hierarchy is strict. But one might ask if there are any algebraic objects (groups, rings, fields) which are strictly n-dependent for every n? We will start by introducing the n-dependent hierarchy and present all known results on n-dependent groups and fields.

Abstract. The property of "flatness" of a pregeometry (matroid) is best known in model theory as the device with which Hrushovski showed that his example refuting Zilber's conjecture does not interpret an infinite group. I will dedicate the first part of this talk to explaining what flatness is, how it should be thought of, and how closely it relates to hypergraphs and Hrushovski's construction method. In the second part, I will conjecture that the family of flat pregeometries associated to strongly minimal sets is model theoretically nice, and share some intermediate results.

Abstract. In this talk I will review some fundamental concepts in computability theory and show how they can be applied to analyze the relative complexity of real-valued functions. I will review some old results that come from this analysis, in particular, the continuous degrees of Miller. Then I will introduce some new hierarchies that results from joint work with Downey and Westrick. One of these new hierarchies refines the Bourgain rank for Baire class 1 functions and allows a new way to extend this rank to all Baire measurable functions.

Abstract. For $L$-structures $B$, $C$ we use the notation ${C choose B}$ to denote the set of all substructures of $C$ isomorphic to $B$. We say that a countable, locally finite structure $I$ ordered by a relation $<$ has RP (the Ramsey property) if for all $A_0, B_0 in textrm{age}(I)$ and integers $k geq 1$ there is some $C_0 in textrm{age}(I)$ such that

$C_0 rightarrow (B_0)^{A_0}_k$. In other words, for all functions $c: {C_0 choose A_0} rightarrow k$ there is some $B' subseteq C_0$, $B' cong B_0$ such that $c$ restricted to ${B' choose A_0}$ is a constant function.

We will approach the question of when RP transfers from one countable structure to another, where these structures are in possibly different languages. We will look at universal algebraic and model theoretic criteria.

Abstract. I will present a few basic applications of model theory in theoretical computer science, e.g. in verification, databases, and algorithms. I will also discuss some initial ideas employing (ideas from) stability theory to solve algorithmic problems concerning graphs.

Abstract. Shelah's work on saturation spectra, Hrushovski on PAC structures, and Cherlin-Hrushovski on quasi-finite structures gave the initial impetus for the development of simple theories. A general theory, which unified and explained these different lines of research, was developed by Kim and Pillay using the notion of non-forking independence, which in turn spawned a remarkably rich line of model-theoretic research. In my talk, we will describe a parallel theory for the broader class of NSOP1 theories centered around the notion of Kim-independence and the applications that this theory made possible. We will survey results in a series of papers joint with Artem Chernikov, Itay Kaplan, Alex Kruckman, and Saharon Shelah (though not all at once).

Abstract. For $L$-structures $B$, $C$ we use the notation ${C choose B}$ to denote the set of all substructures of $C$ isomorphic to $B$. We say that a countable, locally finite structure $I$ ordered by a relation $<$ has RP (the Ramsey property) if for all $A_0, B_0$ in age$(I)$ and integers $k geq 1$ there is some $C_0$ in age$(I)$ such that $C_0 rightarrow (B_0)^{A_0}_k$. In other words, for all functions $c: {C_0 choose A_0} rightarrow k$ there is some $B' subseteq C_0$, $B' cong B_0$ such that $c$ restricted to ${B' choose A_0}$ is a constant function.

We will approach the question of when RP transfers from one countable structure to another, where these structures are in possibly different languages. We will look at universal algebraic and model theoretic criteria.

Abstract. Let $T_{k+1, k}$ denote the theory of the k-ary, k+1-clique free random hypergraph, for k >= 3. Malliaris and Shelah have famously proven that $T_{k+1, k}$ is not below $T_{k'+1, k'}$ in Keisler's order, whenever k+1 < k'; hence, Keisler's order has infinitely many classes. I have since improved the combinatorics to obtain the same result whenever k < k', and I obtain model-theoretic upper and lower bounds for the relevant dividing lines detected by Keisler's order. These bounds correspond to various kinds of k-dimensional amalgamation properties.

The combinatorics involved is rather technical; however, the model-theoretic upper and lower bounds are not. I aim to introduce and motivate them; in particular, we will explore a connection between generalized amalgamation properties and the chromatic numbers of hypergraphs of partial types. It is open if the various k-dimensional amalgamation properties we introduce are equivalent.

Abstract. This talk discusses a new combinatorial proof of the existence of expander graphs, which can be carried out in the bounded arithmetic theory VNC$^1$ corresponding to alternating linear time. As an application, we prove that the monotone propositional sequent calculus polynomially simulates the full propositional sequent calculus. Prior to this, only a quasipolynomial simulation was known. Joint work with Valentine Kabanets, Antonina Kolokolova, and Michal Koucky.

Abstract. Lachlan's problem is asking whether any countable theory $T$ with $1<I(omega, T)$=the no. of countable models of $T<omega$ must have the strict order property. It is not fully answered yet, but Lachlan himself in 70s proved that such $T$ is non-superstable. The result is extended by me in 90s to non-supersimple $T$. Now I extend it further in NSOP$_1$ theory context.

Theorem: If $T$ is NSOP$_1$ with nonforking existence, then $1<I(omega, T)<omega$ implies that there must be a finite tuple whose own preweight is $omega$.

The proof of the theorem relies on recent exciting developments on NSOP$_1$ theories initiated by I. Kaplan and N. Ramsey for models, and continued for sets in a joint work with J. Dobrowolski and N. Ramsey.

Abstract. The extent to which ordinal definable sets can capture essential information about the universe V has been extensively studied in the last few years.

One main line of study in this vein has been initiated by Shelah, who proved that for every singular strong limit cardinal k of uncountable cofinality, there exists a single subset x of k such that the ordinal definable class HOD(x) (consisting of sets which are hereditarily definable in ordinals and x) contains all other subsets of k. The theorem does not address cardinals k of countable cofinality and it is natural to ask whether Shelah's approximation result can be extended to all strong limit singular cardinals.

In 2016, James Cummings, Sy Friedman, Menachem Magidor, Assaf Rinot, and Dima Sinapova, have established the failure of a version of Shelah's theorem for cardinals k of countable cofinality. The purpose of the talk is to discuss the consistency strength of this result and described new consistency bounds. This is a joint work with Moti Gitik, Itay Neeman, and Spencer Unger.

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Abstract. (This is joint work with Martin Pizarro). We prove the equationality of the theory of pairs of algbraically closed fields by exhibiting a nice set of equations. We do no know if these equations have the DCC outside characteristic 0.

Abstract. I will give an overview of the many developments in the descriptive set theory of maximal almost disjoint ("mad") families (and their cousins, maximal eventually different families and maximal cofinitary groups) that have happened in the last 5 years.

Abstract. We explore how generalisations of Kleene's theory of recursion in type 2 objects (which can be used to characterise complete $Pi^1_1$ sets and open determinacy) can be lifted to $Sigma^0_3$-Determinacy. The generalisation requires the use of so-called infinite time Turing machines, and the levels of the Gödel constructible hierarchy needed to see that such machines models produce an output are, perhaps surprisingly, intimately connected with those needed to prove the existence of such strategies: the generalised halting problem has the same informational content as a listing of the $Sigma^0_3 $-games won by Player I. The subsystem of analysis needed for this work is $Pi^1_3$-$mathsf{CA_0}$.

Abstract. In 1932 von Neumann proposed classifying the statistical behavior of diffeomorphisms of compact manifolds. In modern language this means classifying up to conjugacy by measure preserving transformations. Von Neumann's question resulted in an enormous literature, with partial solutions involving invariants such as entropy, spectrum and several other properties.

In this talk we show that the general problem is impossible in the sense that no countable protocol, using arbitrary countable resources can solve it. Thus concrete methods such as spectral theory or entropy have no chance of succeeding-any classification must involve a some tool such as the uncountable axiom of choice.

Precisely, we show: {(S,T): S and T are conjugate ergodic diffeomorphisms of the 2-torus} is NOT a Borel set in the C-infinity topology on pairs of diffeomorphism.

For dessert, we give an example of a recursive diffeomorphism T such that the statement "T is isomorphic to its inverse" is independent of ZFC (and an S such that S being isomorphic to its inverse is independent of "ZFC + there is a supercompact cardinal" and so forth).

This is joint work with Benjy Weiss and Hans Gaebler.

Abstract. Fons van Engelen used the description of Wadge degrees of Borel sets to analyze Borel homogeneous spaces. I will explain the first steps we have made with Andrea Medini and Sandra Müller towards the generalization of van Engelen's results in the projective hierarchy.

Abstract. This talk looks at the relationship between three foundational systems: Goedel's Constructible Universe of Sets, the naive conception of set found in consistent fragments of Frege's Grundgesetze, and the intensional logic of Church's Logic of Sense and Denotation. One basic result shows how to use the constructible sets to build models of fragments of Frege's Grundgesetze from which one can recover these very constructible sets using Frege's definition of membership. This result also allows us to solve the related consistency problem and joint consistency problems for abstraction principles with limited amounts of comprehension. Another basic aim of this talk is to show how to "factor" this result via a consistent fragment of Church's Logic of Sense and Denotation: so one may use the constructible sets to build models of Church's Logic of Sense and Denotation, from which one may then define models of the consistent fragments of Frege's Grundgesetze. This talk is based on [1]-[3].

[1] S. Walsh. Fragments of Frege's Grundgesetze and Goedel's constructible universe. The Journal of Symbolic Logic, 81(2):605-628, 2016.

[2] S. Walsh. Predicativity, the Russell-Myhill paradox, and Church's intensional logic. The Journal of Philosophical Logic, 45(3):277-326, 2016.

[3] S. Walsh. The strength of abstraction with predicative comprehension. The Bulletin of Symbolic Logic, 22(1):105-120, 2016.

Abstract. Given a countable family G of analytic hypergraphs on a Polish space X, one can form the sigma-ideal generated by Borel sets which are anticliques in at least one hypergraph in the family G. One can also form the quotient forcing P(G) of Borel sets modulo the sigma ideal. It turns out that in large classes of cases, the quotient forcing is proper, and simple combinatorial properties of the hypergraph family G translate into forcing properties of the quotient poset P(G). Various dichotomies lead to partial classification of posets obtained from restrictive classes of hypergraphs. The machinery allows for simple computation of the iteration and product ideal in terms of the hypergraphs, which in turn leads to overwhelmingly simple proofs of various preservation theorems.

Abstract. $H$-fields are ordered differential fields which serve as an abstract generalization of both Hardy fields (ordered differential fields of germs of real-valued functions at $+\infty$) and transseries (ordered valued differential fields such as $\mathbb{T}$ and $\mathbb{T}_{\log}$). A Liouville closure of an $H$-field $K$ is a minimal real-closed $H$-field extension of $K$ that is closed under integration and exponential integration. In 2002, Lou van den Dries and Matthias Aschenbrenner proved that every $H$-field $K$ has exactly one, or exactly two, Liouville closures, up to isomorphism over $K$. Recently (in arxiv.org/abs/1608.00997), I was able to determine the precise dividing line of this dichotomy. It involves a technical property of $H$-fields called $\lambda$-freeness. In this talk, I will review the 2002 result of van den Dries and Aschenbrenner and discuss my recent contribution.

Abstract. We consider a 1903 theorem of G. H. Hardy and a more recent question of Peter Nyikos regarding its converse. We answer that question by producing a forcing model of (ZF + there exists an uncountable wellorderable set of reals + omega_1 is regular + there does not exist a ladder system).

Abstract. Classification problems occur in all areas of mathematics. Descriptive set theory provides methods to assign complexity to such problems. Using a technique developed by Hjorth, Kechris and Sofronidis proved, for example, that the problem of classifying all unitary operators $\mathcal{U}(\mathcal{H})$ of an infinite dimensional Hilbert space up to unitary equivalence $\simeq_U$ is strictly more difficult than classifying graph structures with domain $\mathbb{N}$ up to isomorphism. We present a game--theoretic approach to anti--classification results for orbit equivalence relations and use this development to reorganize conceptually the proof of Hjorth's turbulence theorem. We also introduce a dynamical criterion for showing that an orbit equivalence relation is not Borel reducible to the orbit equivalence relation induced by a CLI group action; that is, a group which admits a complete left invariant metric (recall that, by a result of Hjorth and Solecki, solvable groups are CLI). We deduce that $\simeq_U$ is not classifiable by CLI group actions.

This is a joint work with Martino Lupini.

Abstract. I will present results on expansions of the additive group of integers which satisfy certain model theoretic notions of tameness, including stability, simplicity, and dp-minimality. The focus will be on how these model theoretic notions control the combinatorial and number theoretic properties of definable sets.

Abstract. Valuations yield a way of understanding (an expansion of) a field in terms of two, often simpler, associated structures, namely its residue field and value group. Here, we consider real closed valued fields, i.e. real closed fields with a valuation determined by a convex subring. We discuss how the types in such a structure are determined by the residue field and value group, and what this implies for notions of independence in this structure.

This is joint work with C. F. Ealy and D. Haskell.

Abstract. We describe some recent work concerning the hereditarily ordinal definable sets in models M of the Axiom of Determinacy. Under a natural iterability hypothesis, HOD^M admits a fine structural analysis, and hence, for example, satisfies the GCH. Underlying this fine structural analysis is a general comparison theorem for iteration strategies.

Abstract. We will discuss Woodin's AD-conjecture, which gives a deep relationship between very large cardinals and determined sets of reals. In particular we will show that the AD-conjecture holds for the axiom I0 and that there are many interesting consequences of this fact. We will also discuss variations of the AD-conjecture and their consequences, including generic absoluteness properties for I0.

Abstract. Analytic sets enjoy a classical representation theorem based on wellfounded relations. I will explain a similar representation theorem for Pi^1_2 sets due to Marcone. This can be used to answer negatively the primary outstanding question from (Kechris, Solecki and Todorcevic; 1999): the shift graph is not minimal among the graphs of Borel functions which have infinite Borel chromatic number.

Abstract. An operator space is a norm closed linear subspace of the Banach space B(H) of bounded linear operators on a Hilbert space. For reasons that will be explained in this talk, operator spaces are the noncommutative analogs of Banach spaces. A fundamental result of Junge and Pisier shows that there are many more operator spaces than there are Banach spaces in a way to be made precise in the talk. I will explain the model-theoretic content of their result. Most of my contributions mentioned in this talk represent joint work with Thomas Sinclair.

Abstract. A hereditary graph property is a class of finite graphs closed under isomorphism and induced subgraphs. Given a hereditary graph property H, the speed of H is the function which sends n to the number of distinct elements in H with underlying set 1,..., n. There are many wonderful results from combinatorics concerning what functions can occur as speeds of hereditary graph properties. These results show there are discrete "jumps" in the possible speeds of hereditary graph properties. In this talk we use VC_n-dimension, a generalization of VC-dimension, to extend one of these results to the setting of arbitrary finite relational languages. In particular, we show that bounded VC_n-dimension characterizes the gap between the fastest and penultimate speeds.

Abstract. This talk will review the $\Sigma^0_1$ notions of algorithmic information and dimension and survey very recent applications of these to classical (non-algorithmic) questions in fractal geometry. These applications include N. Lutz and D. Stull's strengthened lower bounds on the dimensions of generalized Furstenberg sets and N. Lutz's extension of the fractal intersection formulas in Euclidean space from Borel sets to arbitrary sets.

Abstract. We investigate various aspects of compactness of omega_1 under ZF + DC (the Axiom of Dependent Choice). We say that omega_1 is X-supercompact if there is a normal, fine, countably complete nonprincipal measure on powerset_{omega_1}(X) (in the sense of Solovay). We say omega_1 is X-strongly compact if there is a fine, countably complete nonprincipal measure on powerset_{omega_1}(X). A long-standing open question in set theory asks whether (under ZFC) "supercompactness" can be equiconsistent with "strong compactness. We ask the same question under ZF+DC. More specifically, we discuss whether the theories "omega_1 is X-supercompact" and "omega_1 is X-strongly compact" can be equiconsistent for various X. The global question is still open but we show that the local version of the question is false for various X. We also discuss various results in constructing and analyzing canonical models of AD^+ + omega_1 is X-supercompact.

Abstract. I will present a logician-friendly overview of the classification problem for group actions on C*-algebras from the perspective of Borel complexity theory, including joint works with Gardella, Kerr, and Phillips.

Abstract. The core question of this talk is: Given a group G and an arbitrary subgroup H which is abelian, nilpotent or solvable, can one find a definable envelope of H, that is a definable subgroup of G containing H with the same or similar algebraic properties. In the past decades there has been remarkable progress on groups fulfilling model theoretic properties as well as on groups satisfying certain chain conditions on centralizers (Mc-groups) which will ensure the existence of definable envelopes. We briefly present the work in stable, simple, and NIP theories as well as Mc-groups. In the end of this talk we generalize these results to groups which merely satisfies the same chain condition on centralizers as groups definable in simple theories as well as NTP2 theories.

Abstract. Let K(x) denote the prefix-free Kolmogorov complexity of a finite bit string x. A string is incompressible (random) if the value of K(x) is at least its length minus a constant. An infinite bit sequence (real) is random in the sense of Martin-Löf (ML) iff each of its initial segments is incompressible (for the same constant). In the first part of the talk we characterise randomness of a real in terms of effective analysis and ergodic theory.

The K-trivial bit sequences are antirandom in the sense that the initial segment complexity in terms of K grows as slowly as possible. Since 2002, many alternative characterisations of this class have been found: properties such as low for ML-randomness, and basis for ML-randomness, which state in one way or the other that the sequence is close to computable.

Initially the class looked quite amorphous. More recently, internal structure has been found. Bienvenu, Greenberg, Kucera, N., and Turetsky (JEMS 2016) showed that there is a ``smart" K-trivial set, in the sense that any ML-random computing it must compute all K-trivials. Greenberg, Miller and N. (submitted) showed that there is a dense hierarchy of subclasses. Even more recent results with Turetsky combine the two approaches using cost functions.

Abstract. The study of differential fields using model theory has a long and rich history. The theory of differentially closed fields of characteristic 0, $DCF_0$, has been studied intensively and has played an important role in the development of geometric stability theory. Furthermore, many new results in number theory and differential Galois theory have been obtained using the model theoretic approach to differential algebra.

Nevertheless, only very recently has the techniques from geometric stability been used to study well-known differential equations. In this talk we highlight some of the contributions of logic, via model theory, to the classification and study of the classical Painlev\'e and Schwarzian equations.

Abstract. The pseudoarc is the generic compact, connected, metric space. It can be represented as a canonical quotient of the pre-pseudoarc, a projective Fra{\" i}ss{\' e} limit. I will present a homogeneity result for tuples of points in the pre-pseudoarc. The proof of this result uses tools coming from combinatorics and logic. I will deduce from it the topological homogeneity for tuples of points in the pseudoarc. I will finish with speculations on what the ultimate homogeneity result for the pseudoarc may be and present results obtained in this direction.

The talk will continue the theme of my last year Logic Colloquium talk, but it will be self-contained. The talk will be based on joint work with Todor Tsankov.

Abstract. Let R be a binary relation on a set X. A subset A of X is R-discrete if no two distinct elements of A are R-related.

Discrete sets arise in many context: If R is a graph, a discrete set is often called an independent set; or if R os the relation of having a mutual extension in some partial ordering, then a discrete set would be called an antichain.

In this talk I will discuss the existence of maximal discrete sets for an analytic binary relations on a Polish spaces. If V=L we can always obtain a \Delta^1_2 maximal discrete set for a given \Sigma^1_1 relation, but there are easy examples that show that this is not the case if we add a Cohen or random real to L. In this talk, I will show that we _can_ add a Sacks or Miller real to L and there will still be a \Delta^1_2 maximal discrete set for any Sigma^1_1 relation. I will then discuss an application of this theorem to a question concerning the existence of definable sets of orthogonal Borel probability measures.

The results discussed in the talk are joint with David Schrittesser.

Abstract. Approximate groups are generalizations of groups where one relaxes the condition of closure under the group operation to hold "an appreciable fraction of the time" rather than always. Somewhat surprisingly, there is a rich structure theory for approximate groups, starting in additive combinatorics in the 50s. In this talk I will define approximate groups, give some of the intuitions around them, and describe how ideas from model theory led to a breakthrough in the case of finite approximate groups. Then I will describe my own work extending these results to continuous approximate groups, where a couple new ideas are needed, especially from Lie theory.

Abstract. The early 2000's saw the description by Haskell, Hrushovksi and Macpherson of the interpretable sets in algebraically closed valued fields as higher dimensional equivalents of balls --- more precisely, they proved elimination of imaginaries in the emph{geometric language}. These same years also saw a growing interest in the model theory of valued fields with operators. Most of the questions that were solved for these structure revolve around quantifier elimination and tameness results. But, in the light of Haskell, Hrushovksi and Macpherson's result, it is also tempting to try to classify interpretable sets in these structures.

In this talk, I will treat two of the most tractable examples: Scanlon's existentially closed valued fields with a contractive derivation and separably closed valued fields of finite imperfection degree. In particular, I will show how the elimination of imaginaries in these structures relates to computing the canonical basis of definable types and how the independence property (or rather its absence) can play a role in controlling those canonical bases.

Abstract. The Categoricity Conjecture has been central in the development of Model Theory for Non-Elementary Classes. I will describe this role, and partial results and then will focus on the role of "abstract compactness properties" (tameness and type shortness being the most famous) and their connections to large cardinals (recent results of Boney and Unger). Finally, I will explore a couple of new directions stemming from these questions.

Abstract. We say that a class of finite structures for a finite first- order signature is r-compressible for an unbounded function $r \colon \mathbb{N} \rightarrow \mathbb{N}^+$ if each structure G in the class has a first-order description of size at most O(r(|G|)). We show that the class of finite simple groups is log- compressible, and the class of all finite groups is log3 -compressible. The result relies on the classification of finite simple groups, the bi-interpretability of the twisted Ree groups with finite difference fields, the existence of profinite presentations with few relators for finite groups, and group cohomology.

This is joint work with A. Nies.

Abstract. Most set theorists tend to think of Gödel-Bernays class theory, hereafter GB, as a minor variant of ZF-set theory, perhaps because of the perspicuity of the model-theoretic conservativity proof of GB over ZF. My talk will discuss the following old and new results about GB that seriously challenge this conception:

(1) [Mostowski 1950] The unprovability of the scheme of induction over natural numbers in GB.

(2) [Pudlák, 1985] The superexponential speed-up of GB over ZF.

(3) [E, 2004] The conservativity of GB + ``the class of ordinals is weakly compact'' over the extension of ZFC obtained by adding the scheme whose instances are statements of the form ``there is an n-reflective n-Mahlo cardinal'', where n ranges over natural numbers in the meta-theory.

(4) [E and Hamkins, 2016] The veracity of the statements ``the class of ordinal is not weakly compact" and ``there is a global well-ordering of sets iff the class of ordinals carries a diamond sequence (in the sense of Jensen)" in canonical models of GB, i.e., models of GB whose classes are precisely the parametrically definable subsets of a model of ZFC.

Abstract. The constructible universe L is built by series of stages where each stage is the set of (first order) definable subsets of the previous stage. L is a very nice inner model, but it misses many canonical objects, like $0^\sharp$.

One possible attempt to define rich class of inner models is by imitating the construction of L but replacing "first order definability" by definability by stronger logics. A classical theorem by Myhill and Scott claims that that if we use second order logic, we get HOD - the class of sets hereditarily ordinal definable. HOD is not very canonical, it depends very much on the universe of Set Theory from which we start.

This talk will present some joint work with J. Kennedy and J. Vaananen, in which we study the inner models that are constructed by using logics that are stronger than first order logic, but weaker than full second order logic. One interesting case is the logic of the quantifier $Q x, y \Phi(x, y)$ which means: "The formula $\Phi(x, y)$ defines a linear order which has cofinality $\omega$. The model we get is rather canonical, in the presence of large cardinals, and contains many canonical definable objects.

Another interesting case is the case of stationary logic studied by Barwise and Kaufmann and Makkai.

Abstract. The Mitchell order is a partial ordering on normal ultrafilters which plays an important role in the theory of inner models. Let U, W be two normal ultrafilters in a transitive model of set theory V, U < W if U appears in the ultrapower of V by W.

How complicated can the Mitchell ordering be? Mitchell proved it is a well-founded poset and its possible structure was previously studied by Mitchell, Baldwin, Cummings, Witzany, Friedman and Magidor. We will discuss the possible structure of the Mitchell order and explain how to construct models with various Mitchell order structures using forcing methods, inner model theory, and some purely combinatorial ideas about a certain families of posets.

Abstract. In their celebrated approximate solution of Artin's conjecture on homogeneous forms over the $p$-adic numbers, Ax and Kochen showed that the first-order theory of a henselian valued field of residue characteristic zero is determined by the theories of its residue field and of its value group. Over the years, this principle (also identified by Ershov) has been greatly refined and extended to include much more complicated structures on valued fields (for example, by allowing analytic functions), to permit relative quantifier elimination theorems beyond the original relative completeness theorem, and even to permit a deep structural analysis of the types from the valued field sort in terms of the much simpler types in the residue field and value group.

We shall discuss limits of the Ax-Kochen-Ershov principle in two senses: in the original sense used in the 1960s of considering the situation as $p \to \infty$, but then also how for some natural structures on valued fields, we cannot expect this principle to hold in the form we have come to expect.

Abstract. Let {a_n} be a combinatorial sequence counting the number of certain combinatorial objects, e.g. permutations, trees, graphs, etc. How easy is it to compute this sequence? Are there sequences that are hard to compute? More specifically, are the sequences P-recursive and ADE? (these will be defined in the talk) I will try to answer these questions and discuss how computability theory naturally arises in this context. The main result is our recent disproof of the Noonan-Zeilberger conjecture.

This talk is aimed at a general audience. Joint work with Scott Garrabrant.

Abstract. Lindenbaum Maximalization Theorem (LMT) is a fundamental theorem of modern logic. It is also known as Lindenbaum lemma, since it is a fundamental step for the proof of the completeness theorem. The abstract version of LMT is equivalent to the axiom of choice, as proved by Dzik. In this talk I will explain the relations between LMT, compactness, the axiom of choice and the completeness theorem. I will also present another version of LMT, known as Lindenbaum-Asser theorem (LAT), which is stronger and more interesting in view of applications to many different systems of logic, in particular those with a weak negation or without negation.