Abstract. By the Glimm-Effros dichotomy, every non-smooth countable Borel equivalence relation contains a copy of the non-smooth hyperfinite Borel equivalence relation. We will discuss how local rigidity properties of the usual action of SL_2(Z) on R^2 can be used to rule out analogous results for the class of non-measure-hyperfinite countable Borel equivalence relations. I hope to make the talk accessible to a broad audience.

Abstract. We investigate the density requirement on a given normed space X that guarantees the existence of an infinite unconditional basis sequence of elements of $X.$ This is a joint work with J. Lopez-Abad.

Abstract. In a paper from 1965, Effros asked if the conjugacy relation on unitary representations of a countable discrete group (or, more generally, a locally compact second countable group) on a separable Hilbert space is a Borel equivalence relation. In this talk, I will show that the answer is yes. This is joint work with Greg Hjorth, completed in December 2010.

Abstract. The Hrushovski amalgamation method was originally built to generate theories with unexpected geometric properties. I will present how the Hrushovski amalgamation construction can be used to construct theories with strange recursion theoretic properties as well, answering problems in recursive model theory. In particular, we will focus on characterizing the degrees of strongly minimal and countably categorical theories all of whose countable models are recursive.

Abstract. The notion of a nonstandard hull of a normed space has proven very useful in applications of nonstandard analysis to Banach space theory. Given a normed space E with nonstandard extension E*, the nonstandard hull of E is the normed space of "finite'' elements of E* quotiented by the subspace of "infinitesimal" elements of E*. If E happened to be a Banach-Lie algebra, that is, a Banach space equipped with a continuous Lie bracket, then the nonstandard hull of E is naturally a Banach-Lie algebra. Pestov used this construction to produce nonstandard hulls of Banach-Lie groups, which he then used to derive a very useful local criterion for the enlargeability of Banach-Lie algebras. In this talk, I will explain all of the relevant nonstandard analysis and Lie theory to understand the aforementioned construction and results of Pestov. Then I will discuss my own extension of Pestov's results to the wider class of locally exponential Lie groups and algebras.

Abstract. This talk will suggest that a certain "view", readily available from basic ingredients of mathematical logic, bears a resemblance with views found respectively in the philosophers Parmenides, Hegel, and Heidegger. And this talk will attempt to indicate the possibility that, while staying inside mathematical logic, this view can be elaborated to maintain such a resemblance with a philosophical view found in common in the works ofthese philosophers.

Abstract. We will discuss the question, "What first order statements follow from the existence of an infinite random sequence by effective means?" The answer depends on the degree of randomness in the infinite source. In one case, it isolates a mysterious subtheory of arithmetic strictly between $\Sigma_1$-Induction and $\Sigma_2$-Bounding.

Abstract. The NP-complete problem of Satisfiability is one of the central decision problems studied computational complexity. In this talk, we discuss alternation trading proofs that give lower bounds for sublinear space algorithms for Satisfiability. Ryan Williams proved that any deterministic algorithm for Satisfiability that uses space n^{0{1}} must have run time of at least n^c for c equal to 2\cos(\pi/7), and this result applies to many other NP complete problems. This talk discusses this result, plus how to prove that this value for c is the best that can be obtained with currently known techniques. We also present similar results for Satisfiability algorithms that use space n^e for constant values e<1. (Joint work with R. Williams.)

Abstract. The statistical behavior of deterministic dynamical systems can be completely random. The ergodic theorem tells us that the statistical behavior of a system carrying an invariant ergodic measure is captured by the isomorphism class of that measure. Motivated by this type of phenomenon, in 1932 von Neumann proposed the program of classifying the ergodic measure preserving systems. This talk discuss applications of the tools of descriptive set theory to show that this program is impossible. In earlier joint work with Weiss, it was shown that the isomorphism relation is "turbulent" and hence no generic class can be reduced to an S-infinity action. With Rudolph and Weiss it was shown that the isomorphism relation is a complete analytic set. But what about concrete systems: C-infinity measure preserving diffeomorphisms of the torus? Recent work with Weiss extends these results to such transformations. Finally a very recent result shows that the "graph-isomorphism problem" can be reduced to isomorphism of measure preserving diffeomorphisms. As a corollary the isomorphism relation is complete for S-infinity actions.

Abstract. Reverse Mathematics is a program in foundations of mathematics initiated by Friedman ([1, 2]) and developed extensively by Simpson ([4]). Its aim is to determine which minimal axioms prove theorems of ordinary mathematics. Nonstandard Analysis plays an important role in this program ([3, 5]). We consider Reverse Mathematics where equality is replaced by the predicate $\approx$, i.e. equality up to infinitesimals from Nonstandard Analysis. A `copy' of Reverse Mathematics for WKL$_0$ and ACA$_0$ is obtained in a weak system of Nonstandard Analysis. We consider possible connections with constructive analysis and recursion theory. Our results also have implications for the Philosophy of Science. In particular, we show how the very nature of Mathematics in Physics implies that real numbers are not needed in Physics (cf. the famous Quine-Putnam indispensability argument).

[1] Harvey Friedman, Some systems of second order arithmetic and their use, Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1, Canad. Math. Congress, Montreal, Que., 1975, 235-242.

[2] , Systems of second order arithmetic with restricted induction, I & II (Abstracts), Journal of Symbolic Logic 41 (1976), 557-559.

[3] H. Jerome Keisler, Nonstandard arithmetic and reverse mathematics, Bull. Symbolic Logic 12 (2006), no. 1, 100-125.

[4] Stephen G. Simpson, Subsystems of second order arithmetic, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1999.

[5] Kazuyuki Tanaka, The self-embedding theorem of WKL0 and a non-standard method, Ann. Pure Appl. Logic 84 (1997), no. 1, 41-49.

Abstract. Many different foundational theories have been developed and studied by set theorists. Remarkably, no divergence at the level of statements about "concrete" objects (for example, natural numbers) has emerged, and indeed, these theories are linearly ordered by the inclusion order on their sets of arithmetic consequences. The place of a theory in this order is known as its consistency strength.

Our methods for comparing the consistency strengths of set theories center on three intertwined canonical hierarchies of sets. In this talk we shall describe those three hierarchies in very general terms. We shall focus on the philosophical significance of the results and open questions concerning them.

Abstract. We study fragments of first-order logic and of least fixed point logic that allow only unary negation: negation of formulas with at most one free variable. These logics generalize many interesting known formalisms, including modal logic and the mu-calculus, as well as conjunctive queries and monatic Datalog. We show that satisfiability and finite satisfiability are decidable for both fragments, and we pinpoint the complexity of of satisfiability, finite satisfiability, and model checking. We also show that the unary negation fragment of first-order logic is model-theoretically very well behaved. In particular, it enjoys Craig interpolation and the Beth property. Based on joint work with Luc Segoufin (STACS 2011).

Abstract. Iterated forcing constructions are a crucial tool in consistency proofs, and their use typically requires preservation theorems showing that the iteration does not destroy cardinals. General preservation theorems exist for several classes of forcing posets, most notably the classes of c.c.c. posets, proper posets, and semi-proper posets. Each of these theorems, used with a universal iteration, leads to a consistency proof for a forcing axiom stating that the universe is to some extent saturated under forcing with posets in the class. These axioms, most notably Martin's Axiom (c.c.c. posets), the Proper Forcing Axiom, and the Semi-Proper Forcing Axiom, are in turn used as center points for many consistency proofs.

While c.c.c. posets preserve all cardinals, proper posets need only preserve $\omega_1$. This limits the level of saturation possible in the Proper Forcing Axiom. When the axiom was first introduced, some 30 years ago, it was expected that there would be a parallel of the axiom and the iteration theorem for proper posets, that works for an analogous class of posets preserving more cardinals, for example preserving both $\omega_1$ and $\omega_2$. But the naive attempt to obtain such an analog fails, and no analog had been obtained.

In the talk I will show how to obtain an analog of the Proper Forcing Axiom for posets that preserve both $\omega_1$ and $\omega_2$. The analog requires a new iteration method, that combines standard iterations with uses of models as side conditions.

Abstract. I will describe an algorithm that determines whether or not a given system of equations and inequations over a torsion-free word-hyperbolic group has infinitely many conjugacy classes of solutions. One application is to enumerate `immutable' sub-groups.

Abstract. (I will define and explain "stronly minimal," "modular," "recursive spectrum," and all other frightening words.) The recursive spectrum of a fixed strongly minimal theory T is the set of integers n such that the model of T of dimension n admits a recursive presentation. How complicated can this set be? Some interesting examples can be constructed by coding complicatedness into an infinite language, or by using Hrushovski constructions. We are working towards showing that the recursive spectrum of a strongly minimal locally modular theory in a finite language must be nothing, everything, or only the prime model. The title describes the current state of this joint work with Uri Andrews.

Abstract. A distinction is standardly drawn between so-called "structural" axioms (those that define a type of mathematical structure, like a group or a field) and "foundational" axioms (those like PA or ZFC that characterize some pre-existing structure). However, some mathematicians with structuralist sympathies deny that there is such a distinction and assimilate all axioms to the "structural" type. In this talk, I will consider Godel's Completeness Theorem and show that its standard interpretation depends on a non-mathematical thesis parallel to the Church-Turing thesis in computability theory. Consideration of this thesis shows that the type of structuralist view mentioned above is untenable, since it suggests that any consistent system is equally good, but says that there is no fact of the matter about whether certain systems are consistent.

Abstract. The (constructive Hausdorff) dimension of a point x in Euclidean space is the algorithmic information density of x. Roughly speaking, this is the least real number dim(x) such that r·dim(x) bits suffice to specify x on a general-purpose computer with arbitrarily high precisions 2-r. This talk will survey the development of this notion, its geometric meaning, and its role in the analysis of several classes of fractals. The results surveyed are the work of many investigators.

Abstract. In the case of continuous actions of the infinite symmetric group, the Scott analysis provides a sharp analysis of the Borel Complexity of orbits. This talk will compare the relatively well understood situation of the symmetric group with the still poorly understood situation for general Polish groups.

Abstract. The lecture describes some model theoretic transfer properties on $\omega_4$ that can be proved to be between a huge cardinal and a 2-huge cardinal in consistency strength

Abstract. I will discuss the extension of stability-theoretic ideas to a broader context but with measures replacing types.

Abstract. Computable model theorists have analysed many classical theorems on linear orders. A standard (and important) example is that although any infinite linear order must contain either an infinite increasing chain or an infinite decreasing chain, there is a computable linear order with no computable suborder of type omega or omega*. We will show that this theorem fails in the automatic structures setting. This pronounced difference between computable model theory and automatic model theory is the departure point for further questions about categoricity, dimension, homoegeneity, and complexity of substructures. Differences between the computable and automatic context with respect to each of these will be illustrated with recent results about automatic linear orders. This is joint work with Asher Kach.

Abstract. See: http://www.math.ucla.edu/~hbe/joel.pdf

Abstract. Forcing Axioms are axioms added to the usual axioms of Set Theory with the intuitive motivation of ``A Set that can be imagined to exist, does exist''.

These axioms, like Martin's Axiom (MA), the Proper Forcing Axiom (PFA), Martin's Maximum (MM) and other variations, turned out to be very successful in deciding many problems that are otherwise independent of Set Theory.

The combinatorial principle square, which was introduced by Jensen, seems to hold in almost all of the canonical inner models for large cardinals constructed so far. There is a class of weakenings of the square principle , and the weakest ``weak square'' that holds in a given model seems to be an indication of the model's deviation from a canonical inner model.

One of the interesting global consequences of forcing axioms stronger than Martin's Axiom is that square fails everywhere. The talk will survey the known results about the interaction of different forcing axioms like PFA and MM with weak square principles. This study may be the first step to the determination of the consistency strength of these axioms.

Abstract. The talk articulates a minimal, formalist framework for the representation of mathematical reasoning. We develop basic arithmetic in an elementary theory of concatenation, simulating induction and primitive recursion without taking the standard natural numbers for granted. The concatenation theory is proved to be mutually interpretable with the well-known arithmetical theory $Q$.

Abstract. One of the main tools that model theory offers for the analysis of structures is different abstract notions of "smallness" (such as forking). The goal of my talk is to describe the behavior of some of these notions in the context of generically stable types in dependent theories (theories without the independence property).

First I will discuss the concept of generic stability, which allows to carry many techniques from stability theory over to theories without the independence property (such as certain valued fields). Then I will talk about the notions of forking and thorn-forking, and their meaning in this context. In particular, I will discuss the recent result (joint with D. Garcia and A. Onshuus) that all the "reasonable" notions of "smallness" coincide for realizations of generically stable

Abstract. The ancient Euclidean algorithm computes the greatest common divisor $\gcd(m,n)$ of two natural numbers by iterating the remainder operation. It requires no more than $2\log(\min(m,n))$ extractions of remainders to compute $\gcd(m,n)$ (for $m>n>1$), and so to also check whether $m$ and $n$ are coprime. It is not known whether it is optimal, either for the computation of $\gcd(m,n)$ or for deciding coprimeness.

In this lecture I will develop an axiomatic approach to the theory of algorithms in the style of abstract model theory, which makes it possible to make precise and (in some cases) derive non-trivial lower bounds for problems like the computation of $\gcd(x,y)$ or deciding coprimeness. These axiomatically derived lower bounds apply to many natural complexity measures, all known computation models and (arguably) to all algorithms from specified primitives. The method will be illustrated with some specific results on problems in arithmetic like those above which are joint with Lou van den Dries.

Abstract. We are going to consider the space of free, measure preserving, ergodic actions of a countable group on a standard probability space. This space is of high importance in functional analysis. We will consider equivalence relations on this space - in particular, the equivalence relation given by orbit equivalence - and discuss the Borel complexity of the equivalence relation. Two group actions are orbit equivalent if these is a measurable way of almost everywhere identifying the orbits given by the actions. Descriptive set theoretic aspects that are of interest will be introduced as well as recent results concerning these group actions.

Abstract. The differential field of Laurent series over the reals is not closed under exponentiation, but has a canonical extension to an exp-closed differential field $T$ of ``transseries.'' I will describe $T$ and mention some of the intriguing conjectures about its model-theoretic behavior, which are analogous to the well-known model-theoretic facts abot the real field. This is a survey of joint work with Matthhias Aschenbrenner and Joris van der Hoeven

Abstract. It is by now a well-known theorem of Martin, Steel and Woodin that if there is a measurable cardinal above omega-many Woodin cardinals then AD holds in $L(\mathbb{R})$. Since this theorem was proved, many theorems of a similar nature have appeared in set theory. For instance, Steel recently showed that PFA, the proper forcing axiom, implies that AD holds in $L(\mathbb{R})$. Many results that followed Martin-Steel-Woodin theorem, including Steel's aforementioned result used a substantial amount of inner model theory to prove that all sets in a certain "canonical" collection of sets of reals are determined. Canonical can probably be taken to be Universally Baire though it is a conjecture whether the actual definition that will be used in the talk is equivalent to Universal Baireness. At any rate, letting $\Gamma$ be this collection of canonical sets of reals, many of the theorems that have been proven after the Martin-Steel-Woodin result are about the size of $\Gamma$, and in particular, the earlier results just show that under certain hypotheses all sets of reals in $L(\mathbb{R})$ are in $\Gamma$. Descriptive inner model theory then can be defined as the the study of the properties of $\Gamma$ in various mathematically rich extensions of ZFC, using methods from inner model theory. Inner model theory enters into the study of the properties of $\Gamma$ through a conjecture known as the Mouse Set Conjecture, which allows one to translate questions about sets of reals into questions about iteration strategies for a certain kind of mice known as hod mice. In this talk, we will explain all this terminology and will outline some of the more recent theorems, including the following.

Theorem (S.) If there is a Woodin limit of Woodins then there is $\Gamma^*$ which is a subset of Gamma and such that and $L(\Gamma^*, \mathbb{R})$ satisfies $AD_{\mathbb{R}}$ + $\theta$ is regular.

We will also explain some applications in calibrating lower bounds of consistency strength and in the inner model problem.

Abstract. We start by briefly describing Shoenfield's functional interpretation of Peano arithmetic. Tao's finitization of the infinite convergence principle can be seen as an instance of this interpretation together with a simple majorizability argument (uniformity result). In 1962, Spector gave a remarkable characterization of the provably recursive functionals of full second-order arithmetic (a.k.a. analysis). We explain what is bar-recursion and outline a direct (i.e., not via intuitionistic logic) proof of Spector's result. We note that bar-recursive functionals are majorizable and, hence, that majorizability arguments extend to analysis. The bounded functional interpretation was introduced in 2005 by Oliva and the present author. In its version for classical arithmetic, it is an interpretation which "injects" some non set-theoretic uniformities into Peano arithmetic. Due to its Soundness Theorem, the interpretation extracts correct uniform bounds of AE-theorems (Proof Mining). Together with Engrácia, we recently showed that the bounded functional interpretation also extends to analysis.

Abstract. The most common version of Hilbert's Fifth Problem (H5) asks whether every locally euclidean topological group (i.e. a topological group whose identity has an open neighborhood homeomorphic to some R^n) can be given the structure of a real analytic manifold so that the group multiplication and inversion become real analytic maps; briefly, whether every locally euclidean topological group is a Lie group. A positive solution to the H5 was given by Gleason, Montgomery, and Zippin in the 1950s. Shortly after, Jacoby claimed to have proven that every locally euclidean "local group" was locally isomorphic to a Lie group. Roughly speaking, a local group is a hausdorff space which has a partial group-like operation defined on it which is continuous. However, Jacoby's proof was discovered to be flawed in the '90s, leaving some important theorems whose truth rests on this local version of the H5 on shaky foundation. By modifying techniques used by Hirschfeld in a proof of the H5 using nonstandard analysis, I was able to give a correct proof of the local H5. In this talk, I will try and sketch a part of the proof of the local H5. No background knowledge in Lie theory or nonstandard analysis will be necessary and all relevant terms will be defined.

Abstract. We classified the class of locally finite quantifier eliminable graphs in the language of distance predicates. This is a joint work with Shawn Hedman.

Abstract. Schema mappings are high-level specifications, typically expressed in some logical formalism, that describe the relationship between two database schemas. Schema mappings constitute the essential building blocks in formalizing the main data inter-operability tasks, including data exchange and data integration. The aim of this talk is to present an overview of recent developments in the study of schema mappings with emphasis on the interaction of this area of research with logic and computation.

Abstract. Standard abstract model theory has mainly looked at extensions of first-order logic, taking its expressive power for encoding structure for granted. And the classical Lindström Theorem analyzes what makes this base system tick at a suitable abstraction level. But small can be beautiful, too. In particular, modal languages are natural descriptions of processes and information that can be viewed as fragments of first-order logic with a nice balance between expressive power and (decidable) complexity. These, too, can be captured via Lindström theorems, but we need new proofs. Lindström's original argument breaks down at some stage, because not enough expressive power is left to encode partial isomorphisms or bisimulations with the right extension properties. We present an alternative proof, its consequences for preservation and interpolation theorems, and its upward generalizations to larger fragments. We also discuss how far Lindström's original proof extends 'downward' into fragments, as well as new open problems at the fault line between the two methods. Finally, we raise some questions about fixed-point extensions of these weaker languages: an area where so far, Lindström theorems have been famously missing. What does this mean? References J. van Benthem, 2007, A New Modal Lindström Theorem, "Logica Universalis" 1, 125-138. J. van Benthem, B. ten Cate & J. Väänännen, 2007, Lindström Theorems for Fragments of First-order Logic, Proceedings of LICS 2007, 280-292. To appear in "Logical Methods for Computer Science".

Abstract. Computer scientists have created over the past thirty years new revolutionary notions of mathematical proofs. The introduction of randomness created proofs that allow for a probability of error. Zero Knowledge Proofs enable a demonstration of a mathematical truth without revealing any information beyond the claimed truth. We shall also present a new practically efficient method for ZKPs and describe applications to business transactions such as the secure and secrecy preserving conduct of auctions. The talk will be self contained and readily accessible.

Abstract. I will discuss various aspects of the work that led to a proof that Hilbert's Tenth Problem (Diophantine Equations) is unsolvable. The unsolvability result is a consequence of the equivalence between two notions, one from logic/computability theory, the other, from number theory. Interesting and curious applications of this equivalence will be discussed including a universal polynomial equation, a prime representing function, and Diophantine form of famous problems.

Abstract. Logic started as a branch of philosophy, going back to Greeks in the classical period. Computers are relatively young, dating back to the middle of the 20th century. This talk tells the story of logic begat computers, tracing the path from Aristotle to the Pentium. This is a story full of both intellectual drama, as well as real-life drama, with most of the characters dying young, miserably, or both.

Abstract. It's natural to formulate theories of real vector spaces using a 2-sorted first-order language with a sort for the scalars and a sort for the vectors. Introduction of coordinates reduces the theory of a vector space of a specific finite dimension to the first-order theory of the real numbers, known to be decidable since Tarski. Experience in the actual formalization of mathematics motivates an investigation into decision problems for various more general 2-sorted first-order theories of vector spaces, with or without an inner product, norm, assumption of completeness, or restriction on dimension.

Solovay, Arthan and the speaker have carried out a systematic study of decidability for such theories. The theories of real vector spaces, inner product spaces, and Hilbert spaces turn out to be decidable and to admit quantifier elimination in a slightly expanded language. However, similar theories of normed spaces, Banach spaces and metric spaces are not even arithmetical: the theory of 2-dimensional Banach spaces, for example, has the same many-one degree as the set of truths of second-order arithmetic. However, by restricting quantifier alternations, we can arrive at some decidable fragments of these theories, a fact that has proved useful in mechanizing proofs.

Abstract. This talk describes work with Francois Dorais on the (strong) Choquet game. We are interested in spaces where the second player has a stationary winning strategy, which only considers the previous move of the first player, or has a convergent strategy, which causes the result of any play to contain a single point. We show these properties are closely related to the structure of bases for the space. In particular, the second player has a stationary strategy on every second-countable T_1 Choquet space.

Abstract. Some old questions about Borel functions, in particular their generation by filter limits and generalizations of the Jayne-Rodgers result, have been studied by various people, including Debs, Saint Raymond, Motto Ross, and Semmes. I will discuss the results obtained so far and some possible conjectures.

Abstract. Computing the Mahlo rank of a large cardinal provides perspective on the cardinal. In particular it provides a "conservative" quantitative method for evaluating whether the existence of the cardinal should be accepted as an axiom. The $\Pi_1^1$-indescribable cardinals are a fundamental example. Results related to this problem will be given, including a lower bound on the rank.

Abstract. Conceptual structuralism is a view of the nature of mathematics according to which mathematics emerges from humanly constructed, intersubjectively established, basic structural conceptions. Some of these are so clear that every problem posed in the language of the structure conceived can be said to have a definite truth value, whether or not that can be determined. Other conceptions may contain an inherently vague aspect, so that truth is only partial in them and not every problem can be asserted to have a definite truth value. In particular, this is behind my view that the Continuum Hypothesis is not a definite mathematical problem.

Abstract. I will present some consistency results about the following combinatorial properties at small regular cardinals: stationary set reflection, existence of nonregular ultrafilters, and variants of Chang's Conjecture.

Abstract. In the last decade there has been a renewed interest in the Furstenberg correspondence, a technique for relating problems in additive combinatorics. We discuss how methods from mathematical logic apply to this correspondence and can be used to expand the class of problems the correspondence applies to.

Abstract. The idea of the model theory of a valued field being controlled by the residue field and value group has been a common thread since the theorem of Ax and Kochen, and recently has found a new level of sophistication in work of Haskell, Hrushovski, and Macpherson. We give an overview, and then describe an ongoing attempt to consolidate this through a general analysis of characteristic-0 henselian fields relative to the leading terms. The talk should be broadly accessible.

Abstract. I will describe analysis of minimal stable wide types over Banach spaces, which in particular leads to the solution of Henson's conjecture on categorical classes of Banach spaces. The proofs are quite elementary, and I will define all the basic concepts, so this talk will require virtually no background in model theory (or logic in general).

Abstract. Over the last 30 years, a great variety of dichotomy theorems concerning Borel structures in Polish spaces have been discovered. Despite the fact that these results only deal with classical notions, many of their proofs require somewhat advanced ideas from logic, e.g., recursion theory and forcing. We will discuss an approach to giving classical proofs of these theorems which utilizes ideas from graph theory.

Abstract. Many logical systems today describe intelligent interacting agents over time. This talk will survey various approaches to modeling rational interaction and the questions that arise when comparing them.

Abstract. The Higman-Neumann-Neumann Embedding Theorem states that every countable group $G$ can be embedded into a 2-generator group $K$. In this talk, using the theory of Borel equivalence relations, I will explain why there does not exist an explicit choice of the group $K$ which has the property that isomorphic groups $G$ are assigned isomorphic groups $K$.

Abstract. Formalizations of arithmetic ordinarily proceed along one of the three avenues: (1) as a first-order theory in which numbers are taken as primitive and their properties laid down by means of particular extra-logical axioms; (2) as a second-order theory in the manner of Frege and Russell, in which numbers are naturally identified with classes of (extensions of) equinumerous concepts, and their properties derived from such a characterization; or (3) as a first-order theory embedded within a set-theoretic apparatus, in the manner of Zermelo or von Neumann, selecting particular representatives for the equivalence classes (with respect to equinumerosity). None of these options is completely satisfying in every respect. Option (1) leaves the nature of number unexplained, and option (3) does not seem general enough, in that the connection of numbers with their cardinal properties (i.e., ultimately, with the act of counting) is lost. Option (2) is perhaps the most conceptually satisfying, in that the characterization of numbers as equivalence classes is well-motivated and completely general. Unfortunately, option (2) also takes us into the hostile landscape of second-order logic.

In this talk we show how to overcome these shortcomings. After looking at abstraction principles in general, and with the aid of a non-standard (but still first-order) cardinality quantifier and an extra-logical operator representing numerical abstraction, we propose a formalization of arithmetic at the first order, in which numbers are abstracta of the equinumerosity relation, their properties derived from those of the cardinality quantifier and the abstraction operator.

Abstract. The notion of forking independence, introduced by Shelah, turned out to be the main tool in the study of stable and, more generally, simple theories. This notion generalizes classical notions of independence, e.g.~linear independence in vector spaces and algebraic independence in algebraically closed fields. The powerfulness of forking independence in stable and simple theories comes from the fact that it satisfies the long list of nice properties (such as symmetry or transitivity), which actually determine this notion.

The main motivation for what I am going to discuss is to apply various ideas from geometric stability theory to purely topological (or descriptive set theoretic) objects. The point is that we can use a certain notion of independence defined in a purely topological way, even if we do not work in a first-order context.

I will discuss topological notions of profinite structure and ${m}$-independence introduced by Newelski in the late 90?s. Examples, conjectures and some structural theorems about profinite groups regarded as profinite structures, proved by Newelski and Wagner, will be given. Then I will concentrate on a more general notion of Polish structure and non-meager independence that I have introduced and been investigating for about two years.

A Polish structure is a pair ${(X,G)}$ where ${G}$ is a Polish group acting (faithfully) on a set ${X}$ so that the stabilizers of all singletons are closed. Profinite structures are particular cases of Polish structures for ${X}$ and ${G}$ profinite and the action of ${G}$ on ${X}$ continuous. Polish ${G}$-spaces and, more generally, Borel ${G}$-spaces are also examples of Polish structures. We say that a Polish structure ${(X,G)}$ is small if for every ${n \in \omega}$ there are countably many orbits on ${X^n}$ under ${G}$.

Newelski has introduced the topological notion of ${m}$-independence in profinite structures which, under the assumption of smallness, satisfies nice properties and allows us to prove counterparts of some deep results from stability theory. However, it is not easy to find many explicit examples of small profinite structures (e.g.~I have proved that abelian profinite groups of finite exponent being inverse limits of countable systems are such examples). In contrast, there are many examples of small Polish structures. A simple non-profinite example is the unit circle with the full group of homeomorphisms. More complicated examples are Hilbert cube and the pseudo-arc with the full group of homeomorphisms.

I will discuss a purely topological notion of independence, called non-meager independence, that satisfies some nice properties (e.g.~symmetry, transitivity, existence of independent extensions) in small Polish structures, and so allows us to introduce basic stability-theoretic concepts and to prove fundamental results about them (e.g.~Lascar inequalities). In profinite structures this notion of independence coincides with ${m}$-independence introduced by Newelski. If time permits, I will also discuss some examples and results describing the structure of small compact ${G}$-groups, i.e., small Polish structures ${(X,G)}$ where ${X}$ is a compact group and ${G}$ acts continuously on ${X}$ as a group of automorphisms.

Abstract. I will talk about a combinatorial coloring property for countable groups that is equivalent to the existence of compact free subflows of the Bernoulli flow. We will show that every countable group has this coloring property. This is joint work with Steve Jackson and Brandon Seward.

Abstract. Epistemic logic is a modal logic that expresses knowledge and belief of agents. We obtain dynamic epistemic logic (DEL) by adding to epistemic logic expressions that assert that certain actions result in certain new beliefs. In this talk, I will discuss one of several languages that add temporal logic to DEL, focusing on the addition of a previous time operator to DEL.

Abstract. Motivated by a conjecture of Florian Pop, we show that for each finitely generated field K, there is a sentence in the language of rings so that for any finitely generated field L, the sentence is true in L if and only if L is isomorphic to K. In this talk, we will outline some of the key steps involved in connecting algebra and logic and then discuss some related questions about commutative rings.

Abstract. Formal language theory likes to deal with languages as sets of strings. But clearly, languages are sets of signs, which in turn are (at least) pairs consisting of a string and some meaning. A mathematical theory of such objects is so far missing. In this talk I shall outline the beginnings of such a theory, and give some results as well as open problems. For example, we shall show that it is possible to define a notion of strong generative capacity that does not assume structure to begin with and yet is provably different from weak generative capacity normally studied in formal language theory.

Abstract. A Maharam algebra is a complete Boolean algebra that carries a strictly positive continuous submeasure. In 1947, Maharam asked if every such algebra is in fact a measure algebra. This problem, which was also known as the Control Measure Problem, was finally solved by Talagrand in 2005 who provided a counterexample. In this talk, we will survey some recent work on characterizations and combinatorial properties of Maharam algebras and discuss some open questions.

Abstract. Since the fourteenth century and certainly not before then, mankind has demonstrated a universal preference for simplicity. This principle, known as Ockham's Razor, still manifests itself today: humans love short computer programs. Not surprisingly, we have a short description for the set of shortest programs:

MIN = {e : for no j < e is phi_j = phi_e}. I will survey the history of this set, from Cold War Russia to modern-day Chicago.

The only thing better than a Turing machine which tells you whether or not your program code is the shortest possible would be a machine that does this correctly. But if your friend can do this correctly, then she is not a Turing machine. Even an oracle machine which completely knows the halting set will be clueless when it comes to shortest programs. More precisely, the set of shortest programs is Turing equivalent to 0'', the halting set of the halting set.

Recently, Stephan and Teutsch have characterized the canonical Turing degrees 0', 0'', 0''', ... in terms of shortest programs. This should be viewed as a generalization of the above theorem by Meyer. Our characterization gives a partial solution to the most tantalizing problem in all of computability theory, Schaefer's so-called MIN* problem. The problem is this. Replace "=" in the definition for MIN by "=*" (equal except for a finite set), and call the new set MIN*. Schaefer himself showed that MIN* join 0' was Turing equivalent to 0''', leaving open the question whether or not the halting set itself might be computable from MIN*. Our answer is that this reduction holds for all Kolmogorov numberings. For Gödel numberings in general I'm still unsure.

Abstract. Let k be a positive integer. A function f : [N]^k --> N is called regressive iff f(u) < min(u) whenever min(u) > 0. A subset B of N is min-homogeneous for f iff f(u) = f(v) whenever u and v are in [B]^k and 0 < min(u) = min(v). Kanamori and McAloon showed that any regressive function admits an infinite min-homogeneous set, but this is not provable in Peano Arithmetic. For k = 2, this translates into the corresponding regressive Ramsey numbers having Ackermannian rate of growth. We present a combinatorial proof of this result that considerably improves the previous bounds.