Generalized logic is a language for expressing properties of a class of a structures in a given signnature. In its most abstract version the only requirement is the invariance of the satisfaction relation under isomorphism. A natural generalization is a the logic as a language for expressing properties of elements of structures.

If \(\mathcal{L}\) is a generalized logic, then one can naturally define a natural class of embeddings between structures respecting the properties described by \(\mathcal{L}\). ("\(\mathcal{L}\) elementary embeddings".) Regularity properties of abstract logics, like compactness of Skolem Lowenheim properties are connected with reflection properties in Set theory. e.g. Vopenka principle is equivalent to the statement that every abstract logic has a Skolem-Lowenheim property.

The concept of category can be considered as an abstraction of the concept of a class of structures with the appropriate class of mappings ("morphisms"). In this talk we shall consider the problem of extending the concept of generalized logic to abstract categories. We shall discuss the extent in which reflection principles like Vopenka's principle can be extended to the new context. We shall try to make the talk accessible, even for a listener who knows very little about category theory.

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Assume AD\(_{\mathbb R}\) and suppose there is no iteration strategy for a model with a superstrong cardinal. We will introduce the following two conjectures,

1. The Direct Limit Independence and
2. Bounding.

We will then show how 1 and 2 above imply Generation, which is the statement that every set of reals is Wadge reducible to an iteration strategy of some fine-structural, perhaps hybrid, countable model.

We will then show how to prove both conjectures assuming there is no iterable countable structure with a Woodin cardinal that is a limit of Woodin cardinals.

As a consequence, we obtain that Generation holds below a Woodin cardinal that is a limit of Woodin cardinals. We will also explain a method of attacking both 1 and 2.

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Measure equivalence is an equivalence relation on countable groups introduced by Gromov as a measure theoretic counterpart to the goemetric notion of quasi-isometry. In the first part of this talk I will give a brief introduction to measure equivalence and its relationship to countable Borel equivalence relations. I will then discuss some joint work with Lewis Bowen in which we show that the class B, of groups which satisfy the conclusion of Popa's Cocycle Superrigidity Theorem for Bernoulli shifts, is invariant under measure equivalence. As a consequence we show that any nonamenable lattice in a product of noncompact locally compact groups must belong to the class B. This also has implications for entropy: we introduce a new kind of entropy called weak Pinsker entropy, and show that equivalence relations generated by free measure preserving actions of groups in the class B completely "remember" the weak Pinsker entropy of the action.

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For a long time it has been known that enumeration operators on the the powerset of the integers form a model of the \(\lambda\)-calculus. More recently, the speaker realized that well-known methods allow for the adjunction of random variables to the model. Also other well-known ideas can expend the basic model into a model for Martin-Löf type theory. Some recent work combines the two approaches by invoking Boolean-valued models. The talk will focus on asking how to give this natural modeling interesting applications.

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In 1985, Baldwin and Shelah proved a dichotomy among stable theories. Either \(T\) is monadically stable, i.e., every expansion of every model by unary predicates remains stable, or else some model \(M\) has a monadic expansion that codes graphs.

Here, we note a further dichotomy among monadically stable theories. A subset \(Y\subseteq\lambda^k\) is mutually algebraic if there is some natural number \(m\) so that every \(\alpha\in\lambda\) is contained in at most \(m\) elements of \(Y\). [In particular, every subset of \(\lambda^1\) is mutually algebraic.] A structure with universe \(\lambda\) is mutually algebraic if every definable set is a boolean combination of mutually algebraic sets. It follows that every mutually algebraic expansion of every mutually algebraic structure remains mutually algebraic.

We give many characterizations of mutually algebraic structures and prove that if \(T\) is not mutually algebraic, then for every non-monadically definable, mutually algebraic \(Y\subseteq\lambda^k\), \(T\) has a model \(M\) with universe \(\lambda\) whose expansion \((M,Y)\) is not monadically stable (hence has a monadic expansion that codes graphs).

This is joint work with Sam Braunfeld.

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There is a familiar uniform effective procedure for coding directed graphs in undirected graphs. Moreover, the input structures are defined in the output structures (uniformly) in a way that guarantees uniform effective decoding. Harrison-Trainor, Melnikov, R. Miller, and Montalban defined a very general notion of effective interpretation. We observe that there is a graph \(G\) such that for all linear orderings \(L\), some copy of \(L\) does not compute a copy of \(G\). It follows that \(G\) is not effectively interpreted in \(L\). Similarly, there is a graph that is not interpreted in any linear ordering using computable \(\Sigma_2\) formulas. Every graph can be interpreted in a linear ordering using computable \(\Sigma_3\) formulas. Friedman and Stanley described a uniform effective coding \(L\) of directed graphs in linear orderings. We show that there do not exist \(L_{\omega_1\omega}\)-formulas that, for all graphs \(G\), interpret \(G\) in \(L(G)\). We suspect that Friedman and Stanley did the best they could, in the sense that for any uniform effective coding \(\Phi\) of graphs in linear orderings, there do not exist formulas that, for all graphs \(G\), interpret \(G\) in \(\Phi(G)\). This is joint work with Alexandra Soskova and Stefan Vatev.

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I will discuss joint work with Su Gao, Steve Jackson, and Ed Krohne that proves that \({\mathbb Z}^2\) has Borel chromatic number 3. Specifically, given any standard Borel space \(X\) and a free Borel action of \({\mathbb Z}^2\) on \(X\), one constructs a Borel graph on \(X\) by laying down a copy of the canonical (unmarked) Cayley graph of \({\mathbb Z}^2\) on each \({\mathbb Z}^2\)-orbit. We prove that this graph always admits a graph-theoretic 3-coloring that is Borel.

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Years ago when I arrived at Cornell, I started thinking about Thompson's group in the spirit of branching out, learning a new subject, and talking to new colleagues about their mathematics. In the years since, I've come to realize that set-theoretic techniques are in fact fruitful in studying problems about Thompson's group. This talk will give survey some results, conjectures, and problem in this spirit.

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Dating back to Birkhoff, pointwise ergodic theorems for probability measure preserving (pmp) actions of countable groups are bridges between the global condition of ergodicity (measure-transitivity) and the local combinatorics of the actions. Each such action induces a Borel equivalence relation with countable classes and the study of these equivalence relations is a flourishing subject in modern descriptive set theory. Such an equivalence relation can also be viewed as the connectedness relation of a locally countable Borel graph. These strong connections between equivalence relations, group actions, and graphs create an extremely fruitful interplay between descriptive set theory, ergodic theory, measured group theory, probability theory, and descriptive graph combinatorics. I will discuss how descriptive set theoretic thinking combined with combinatorial and measure theoretic arguments yields a pointwise ergodic theorem for quasi-pmp locally countable graphs. This theorem is a general random version of pointwise ergodic theorems for group actions and is provably the best possible pointwise ergodic result for some of these actions.

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The final synthesis of AD\(^+\) and fine structure will yield many theorems. For example, it will yield the theorem that assuming AD\(^+\) and \(V = L({\mathcal P}({\mathbb R}))\), GCH holds in HOD.

AD\(^+\) is a technical refinement of the axiom AD and the context is that \(V = L({\mathcal P}({\mathbb R}))\), together with DC for reals and ZF. It is known that AD\(^+\) is not a strengthening of AD in terms of consistency. For example, AD\(^+\) must hold if \(V = L({\mathbb R})\) and AD holds. Further, assuming ZF + DC, and that all real games are determined, then AD\(^+\) must hold in \(L({\mathcal P}({\mathbb R}))\).
AD\(^+\) is simply a structural refinement of AD and it is conjectured that AD implies AD\(^+\).

We give two recent examples where one can prove the predicted theorems now, without appealing to that final synthesis. The first example is that assuming AD\(^+\), then there can be at most one Woodin cardinal in HOD below \(\Theta_0\) where \(\Theta_0\) is the supremum of the lengths of the prewellorderings of the reals which are ordinal definable.

To set the context, it is a theorem of AD that \(\Theta_0\) is a Woodin cardinal in HOD. It is also theorem of AD\(^+\) that \(\Theta_0\) is the least Woodin cardinal in HOD\(_x\) for a Turing cone of \(x\). This strongly motivates the conjecture that assuming AD\(^+\), \(\Theta_0\) is the least Woodin cardinal in HOD, and this is actually one of the predicted theorems. While we cannot prove this predicted theorem, we can prove the indicated close approximation. The main application is that if \(V = \)Ultimate \(L\) then the \(\Omega\) Conjecture must hold.

We note that there are many examples from the theory of AD illustrating that the "lightface" version of a "boldface" theorem can be significantly more difficult. To illustrate, assuming AD, CH must hold in HOD\(_x\) for a Turing cone of \(x\). In fact, assuming AD\(^+\) and \(V = L({\mathcal P}({\mathbb R}))\), CH must hold in HOD but that is more subtle since without assuming \(V = L({\mathcal P}({\mathbb R}))\), CH can fail in HOD.

The second example solves a question about just \(L({\mathbb R})\) (where the synthesis is known) and which involves just countable unions of projective sets. But this solution uses methods completely outside the usual methods of descriptive set theory and also gives a general AD\(^+\) theorem.

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Cantor's early (1883) Grundlagen einer allgemeinen Mannigfaltigkeitslehre is badly organized and has significant errors and omissions. Nevertheless it is a great work, rich in both mathematical and philosophical content, and its concepts are in some ways superior to Cantor's later ones.

I will discuss a few interrelated topics of the Grundlagen: (1) Cantor's distinction between ordinary infinity and absolute infinity; (2) his quasi-axiomatic, iterative account of ordinal numbers and number classes; (3) the role that a version of the Replacement Axiom plays in this account; (4) differences between Cantor's Grundlagen notion of absolute infinity and his later notion of inconsistent multiplicities.

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A mouse pair is a pair \((P,\Sigma)\) such that \(P\) is a premouse and \(\Sigma\) is an iteration strategy for \(P\) having a certain condensation property. The basic theorems of inner model theory (e.g. the Comparison Lemma and the Dodd-Jensen Lemma) are best stated as theorems about mouse pairs.

One type of mouse pair can be used to analyze the \({\mbox{HOD}}\)s of models of \({{\sf AD}_{\mathbb R}}\), leading to

Theorem Assume \({{\sf AD}_{\mathbb R}} + {\sf HPC}\); then \({\mbox{HOD}} \models {\sf GCH}\).

Here \({\sf HPC}\) stands for "hod pair capturing", a natural assumption concerning the existence of mouse pairs.

An analysis of optimal Suslin representations for mouse pairs leads to

Theorem Assume \({{\sf AD}_{\mathbb R}} + {\sf HPC}\); then the following are equivalent:

1. \(\delta\) is Woodin in \({\mbox{HOD}}\), and a cutpoint of the extender sequence of \({\mbox{HOD}}\),
2. \(\delta = \theta_0\), or \(\delta = \theta_{\alpha+1}\) for some \(\alpha\).

Here the \(\theta_\alpha\)'s are the Solovay sequence, that is, \(\theta_0\) is the sup of the lengths of the ordinal definable prewellorderings of \(\mathbb{R}\), \(\theta_{\alpha+1}\) is the sup of the lengths of prewellorderings of \(\mathbb{R}\) ordinal definable from some set of Wadge rank \(\theta_\alpha\), and \(\theta_\lambda = \sup_{\alpha < \lambda}\theta_\alpha\) for \(\lambda\) a limit.

Grigor Sargsyan introduced a refinement of the Solovay sequence in which ordinal definability from sets of reals is replaced by ordinal definablity from countable sequences of ordinals. Calling these ordinals \(\eta_\alpha\), we have

Theorem Assume \({{\sf AD}_{\mathbb R}} + {\sf HPC}\); then the following are equivalent

1. \(\delta\) is a successor Woodin in \({\mbox{HOD}}\),
2. \(\delta = \eta_0\), or \(\delta = \eta_{\alpha+1}\) for some \(\alpha\).

This theorem was conjectured by Sargsyan.

Finally, we have the following conjecture

Conjecture Assume \({{\sf AD}_{\mathbb R}} + {\sf HPC}\); then the following are equivalent

1. \(\kappa\) is a Suslin cardinal,
2. \(\kappa\) is a cardinal of \(V\), and a cutpoint of the extender sequence of \({\mbox{HOD}}\).

We can prove (2)\(\Rightarrow\)(1). With Jackson and Sargsyan, we have shown that (1)\(\Rightarrow\)(2) holds whenever \(\kappa\) is a limit of Suslin cardinals, or the next Suslin after a limit of Suslin cardinals.

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I will introduce the notion of amenable theory as a natural counterpart of the notion of definably amenable group. Roughly speaking, amenability means that there are invariant (under the action of the group of automporphism of a sufficiently saturated model), Borel, probability measures on various types spaces. I will discuss several equivalent definitions and give some examples. Then I will discuss the theorem which says that each amenable theory is G-compact. This is a part of my recent paper with Udi Hrushovski and Anand Pillay.

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We will discuss aspects of Diophantine Approximation which are motivated and informed by Recursion Theoretic considerations.

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