Logic Seminar and Colloquium
Organizers:
Peter Koellner,
and W. Hugh Woodin.
To be added to the mailing list, please contact
logic@fas (dot harvard dot edu).
For seminars and colloquia in previous years, see the archived schedules.
The Logic Colloquium meets roughly once a month and involves a major figure in logic, speaking on a broad area of general interest. Unless otherwise noted, the talks meet at 2 Arrow Street in Room 420 from 4:006:00 pm.
Fall 2016 Logic Seminar and Colloquium Schedule

Thursday, April 6, 2017.
Tim Button
(Cambridge): "Internal categoricity for ScottPotter set theory"
Abstract.
• Many mathematicians and philosophers think that mathematics is about 'structure'. Many would also explicate this notion of 'structure' via model theory. But the Compactness and LöwenheimSkolem theorems lead to famous philosophical problems for this view; they threaten that we cannot talk about any particular 'structure'.
In this talk, we show how 'structure' might be explicated without any model theory, and indeed without any kind of semantic ascent. The approach involves making use of internal categoricity. The target theorem is the internal categoricity for secondorder ScottPotter set theory. (Unless I am instructed by the audience otherwise, I will not presuppose familiarity with the notions of internal categoricity or with ScottPotter set theories.)
The ScottPotter set theory we use is very weak. Indeed, it is consistent with the claim "there are exactly 2^n sets", for any natural number n. But it is internally categorical, in a sense that we can prove the following in deductive pure secondorder logic: if I have a setlikeproperty and membershiplikerelation, and you have a setlikeproperty and membershiplikerelation, then a secondorder function maps isomorphically between our properties and relations.
En route to this Theorem, we explain some internal quasicategoricity results. We also explain how internal categoricity leads to intolerance. Roughly put: if I have a setlikeproperty and membershiplikerelation, and you have a setlikeproperty and membershiplikerelation, then it is deductively inconsistent for "our sets" to behave differently. This suggests an argument for the determinacy of the continuum hypothesis  and any other settheoretic claim you like  which does not invoke the full semantics for secondorder logic (or, indeed, any semantics).
• 46 pm, Room 420, 2 Arrow Street.

Thursday, October 20, 2016.
Joel David Hamkins
(CUNY): "Recent advances in settheoretic geology"
Abstract.
• Settheoretic geology is the study of the settheoretic universe V in the context of all its ground models and those of its forcing extensions. For example, a bedrock of the universe is a minimal ground model of it and the mantle is the intersection of all grounds. In this talk, I shall explain some recent advances, including especially the breakthrough result of Toshimichi Usuba, who proved the strong downward directed grounds hypothesis: for any setindexed family of grounds, there is a deeper common ground below them all. This settles a large number of formerly open questions in settheoretic geology, while also leading to new questions. It follows, for example, that the mantle is a model of ZFC and provably the largest forcinginvariant definable class. Strong downward directedness has also led to an unexpected connection between large cardinals and forcing: if there is a hyperhuge cardinal κ, then the universe indeed has a bedrock and all grounds use only κsmall forcing. (Post questions and commentary on my blog at http://jdh.hamkins.org/recentadvancesinsettheoreticgeologyharvardlogiccolloquiumoctober2016/.)
• 46 pm, Room 420, 2 Arrow Street.

Thursday, November 3, 2016.
Theodore A. Slaman
(UC Berkeley): "Recursion Theory and Diophantine Approximation"
Abstract.
• Recursion Theory deals with the definability of sets, especially sets of natural numbers or equivalently real numbers. Diophantine Approximation deals with the approximation of real numbers by rational numbers, which can be viewed as a number theoretic form of definability. We will discuss connections between these areas.
• 46 pm, Room 420, 2 Arrow Street.

Thursday, November 17, 2016.
Haim Horowitz
(Hebrew University of Jerusalem): "On the nonexistence and definability of mad families"
Abstract.
• By an old result of Mathias, there are no mad families in the Solovay model constructed by the Levy collapse of a Mahlo cardinal. By a recent result of Törnquist, the same is true in the classical model of Solovay as well. In a recent paper, we show that ZF+DC+"there are no mad families" is actually equiconsistent with ZFC. I'll present the ideas behind the proof in the first part of the talk. In the second part of the talk, I'll discuss the definability of maximal eventually different families and maximal cofinitary groups. In sharp contrast with mad families, it turns out that Borel MED families and MCGs can be constructed in ZF. Finally, I'll present a general problem in Borel combinatorics whose solution should explain the above difference between mad and maximal eventually different families, and I'll show how large cardinals must be involved in such a solution. This is joint work with Saharon Shelah.
• 46 pm, Room 408, 2 Arrow Street.
Spring 2016 Logic Seminar and Colloquium Schedule

Wednesday, February 3, 2016.
Ilijas Farah
(York University): "Set theory and representations of $C^*$algebras"
Abstract.
• In the 1960s Glimm proved a dichotomy result for representations of locally compact groups and $C^*$algebras. This result was later refined by Effros and in 1990 Harrington, Kechris and Louveau proved the "Glimm —Effros Dichotomy," a landmark result in descriptive set theory and abstract classification theory. I shall discuss the complexity of the space of (irreducible) representations of a $C^*$algebra from the settheoretic vantage point, both in separable and nonseparable case. Some compelling problems remain open in the latter case.
• 46 pm, Room 420, 2 Arrow Street.

Wednesday, February 24, 2016.
John Baldwin
(University of Illinois at Chicago): "The divorce of set theory and first order model theory"
Abstract.
• Around 1970 there seemed to be inevitable ties between axiomatic set theory and even first order model theory. The advent of stability theory erased this impression, by showing that while cardinality is deeply entangled with first order model theory, the niceties of cardinal arithmetic are not. We describe the types of entanglement as oracular (consistency inspires a ZFC proof), transitory (the hypothesis is eliminable), and full (there is an equivalence with an independent set theoretic proposition). In the last case, set theoretic pluralism entails model theoretic pluralism.
• 46 pm, Room 420, 2 Arrow Street.

Wednesday, April 20, 2016. (Note: The colloquium will be held from 24 pm.)
Scott Cramer
(Rutgers): "Woodin's ADconjecture for I0"
Abstract.
• We will discuss Woodin's ADconjecture, which gives a deep relationship between very large cardinals and determined sets of reals. In particular we will show that the ADconjecture holds for the axiom I0 and that there are many interesting consequences of this fact. We will also discuss variations of the ADconjecture and their consequences, including generic absoluteness properties for I0.
• 24 pm, Room 420, 2 Arrow Street.

Thursday, April 28, 2016. (Note: The colloquium will be held on a Thursday.)
Andrew Marks
(UCLA): "Martin measure and strong ergodicity"
Abstract.
• Assuming the axiom of determinacy, Martin has shown that any set of Turing degrees either contains a Turing cone, or is disjoint from a Turing cone. Thus, the collection of sets of Turing degrees which contain a cone forms a countably complete measure which is called Martin's measure. It is an open problem to characterize the additivity of Martin measure, in the general sense of for what (not necessarily wellorderable) sets X can the Turing degrees be partitioned into X many Martin measure 0 sets. We will describe an approach to this problem which is based on the ergodictheoretic ideas of cocycle superrigidity and strong ergodicity. This leads to some new problems in recursion theory (related to the study of never continuously random reals) and descriptive set theory (in Borel combinatorics). This is joint work with Adam Day.
• 46 pm, Room 420, 2 Arrow Street.
Fall 2015 Logic Seminar and Colloquium Schedule

Wednesday, November 11, 2015.
Sebastien Vasey
(Carnegie Mellon University): "Shelah's eventual categoricity conjecture in universal classes"
Abstract.
• Abstract elementary classes (AECs) are an axiomatic framework encompassing classes of models of an $L_{\lambda, \omega}$ sentence, as well as numerous algebraic examples. They were introduced by Saharon Shelah in the mid seventies. One of Shelah's goals was to study generalizations of Morley's categoricity theorem to the infinitary setup. Among several variations, Shelah conjectured the following eventual version: An AEC categorical in a highenough cardinal is categorical on a tail of cardinals.
In this talk, we will prove the conjecture for universal classes. It is an interesting type of AEC introduced by Shelah in a milestone 1987 paper [Sh:300] (the work was done in 1985). They correspond approximately to classes of models of a universal $L_{\lambda, \omega}$ sentence. The proof of the conjecture proceeds by first observing that any universal class satisfies tameness: a locality property isolated by Grossberg and VanDieren which says that orbital types are determined by their small restrictions. Next, several structural properties are derived from categoricity: the class has amalgamation on a tail and in fact admits a wellbehaved forkinglike independence relation. Finally, a definition of a unidimensionalitylike property (due to Shelah) is shown to follow from categoricity in a single cardinal and imply categoricity on a tail of cardinals. The argument generalizes to tame AECS which have primes over sets of the form $Ma$.
• 4:30pm talk and subsequent reception in the Logic Center, Room 420, 2 Arrow Street.

Wednesday, December 9, 2015.
Monica VanDieren
(Robert Morris University): To be announced.
Abstract.
• To be announced.
• 4:30pm talk and subsequent reception in the Logic Center, Room 420, 2 Arrow Street.
Spring 2015 Logic Seminar and Colloquium Schedule

Monday, February 23, 2015.
Will Boney
(University of Illinois at Chicago): "Tameness in Abstract Elementary Classes"
Abstract.
• Tameness is a locality property of Galois types in AECs. Since its isolation by Grossberg and VanDieren 10 years ago, it has been used to prove new results (upward categoricity transfer, stability transfer) and replace settheoretic hypotheses (existence of independence notions). In this talk, we will outline the basic definitions, summarize some key results, and discuss some open questions related to tameness.
• 4:30pm talk and subsequent reception in the Logic Center, Room 420, 2 Arrow Street.
Fall 2014 Logic Seminar and Colloquium Schedule

Wednesday, September 24, 2014.
Gil Sagi (LudwigMaximiliansUniversität, München): "What is a Fixed Term?"
Abstract.
• In standard modeltheoretic semantics, logical terms are said to be fixed in the system while nonlogical terms remain variable. Much effort has been devoted to characterizing logical terms, those terms that should be fixed, but little has been said on their role in logical systems: on what fixing them precisely amounts to. My proposal is that when a term is considered logical in a system, what gets fixed is its intension rather than its extension. I provide a rigorous way of spelling out this idea. Further, I show that under certain natural assumptions, some paradigmatic examples of nonlogical terms cannot be fixed in a standard system: they require more structure than such a system affords. We thus obtain a precondition for logical terms. I then propose a graded account of logicality: the less structure a term requires, the more logical it is. Finally, I relate this idea to invariance criteria for logical terms. Invariance criteria can be used as a tool in determining how much structure a term needs in order to be fixed. Thus, rather than settling on one criterion for logicality, I use invariance conditions as a measure for logicality.
• 4:30pm talk and subsequent reception in the Logic Center, Room 408, 2 Arrow Street. 
Wednesday, October 29, 2014.
Joan Bagaria (Universitat de Barcelona):
"Reflection phenomena in the set theoretic universe"
Abstract.
• The phenomenon of reflection occurs at different layers of the settheoretic universe V, and strengthened forms of reflection associated to each layer give rise to natural additional axioms for set theory of different sorts. In this talk we shall review some strong forms of reflection pertaining to different layers such as the class of ordinals, some inner models, V itself, and even the ideal extensions of V provided by the forcing method, while exposing some intriguing connections among the axioms that arise in each case.
• 4:305:30pm talk and 5:306:30pm reception in the Logic Center, Room 408, 2 Arrow Street. 
Wednesday, November 19, 2014.
Maryanthe
Malliaris (University of Chicago): "Comparing the complexity of unstable theories"
Abstract.
• In 1967 Keisler posed the problem of Keisler’s order, a suggested program for comparing the complexity of classes of mathematical structures using an asymptotic (ultrapower) point of view. The talk will be about recent results in this area, due to Malliaris and to Malliaris and Shelah, which advance this program by developing a sort of fine structure theory for pseudofinite behavior in model theory. In particular, the focus will be on simple theories, a key modeltheoretic class which includes the random graph and pseudofinite fields.
• 4:30pm talk and subsequent reception in the Logic Center, Room 420, 2 Arrow Street.
Spring 2014 Logic Colloquium Schedule

Wednesday, February 19, 2014.
Grigor Sargsyan (Rutgers University): "Covering, core model induction and hod mice"
Abstract.• One of the main themes in set theory is determining whether large cardinals are needed to establish the consistency of various theories such as the Proper Forcing Axiom (PFA) or failure of the square principle $\square$. Covering principles are principles that allow one to establish that large cardinals are indeed indispensable. In this talk, we will introduce basic notions from inner model theory and work our way towards a new covering principle, Covering With Derived Models, that can be used to derive strength from PFA or from failure of $\square$.
• 45pm talk and 56pm reception in the Logic Center, Room 420, 2 Arrow Street. 
Wednesday, March 12, 2014.
Paul Larson (Miami University): "Forcing axioms in $\mathbb{P}_{\mathrm{max}}$ extensions"
Slides.
Abstract.• Martin's Maximum, the maximal forcing axiom, was introduced in the 1980's by Foreman, Magidor and Shelah, who proved its consistency relative to a supercompact cardinal. In the 1990's, Woodin introduced $\mathbb{P}_{\mathrm{max}}$, a method for forcing over models of determinacy, and used it produce a new consistency proof for MM($\mathfrak{c}$), the restriction of Martin's Maximum to partial orders of cardinality the continuum. Later results of Sargsyan showed that the determinacy hypothesis used in Woodin's result has consistency strength below a Woodin limit of Woodin cardinals. We will discuss a recent attempt to extend Woodin's theorem to partial orders of cardinality $\mathfrak{c}^{+}$. In particular, we will show that in the $\mathbb{P}_{\mathrm{max}}$ extension of a suitable determinacy model, certain $\square$ principles fail at both $\omega_2$ and $\omega_3$, significantly reducing the consistency strength upper bound for such a result. Part of the talk will focus on models of determinacy. This is joint work with Caicedo, Sargsyan, Schindler, Steel and Zeman.
• 45pm talk and 56pm reception in the Logic Center, Room 420, 2 Arrow Street. 
Wednesday, March 26, 2014.
Ralf Schindler (University of Münster): "Martin's Maximum with an asterisk"
Abstract.• There are two prominent axioms of set theory which both yield that the continuum be of size $\aleph_2$, namely Martin's Maximum (MM) and Woodin's (*). In recent years, Jensen's Lforcing has been exploited to verify that many combinatorial consequences of (*) also follow from MM, but the relationship between MM and (*) remains a mystery. We propose and discuss a natural amalgamation of MM and (*), "Martin's Maximum with an asterisk."
• 45pm talk and 56pm reception in the Logic Center, Room 420, 2 Arrow Street. 
Wednesday, April 16, 2014.
Ilijas Farah (York University): "Logic and operator algebras"
Slides.
Paper.
Abstract.• 'Operator algebras' are subalgebras of the algebra of bounded operators on a complex Hilbert space. They are usually in addition assumed to be selfadjoint and normclosed (C*algebras) or closed in weak operator topology (von Neumann algebras). Connections between logic and operator algebras in the past century were few and sparse. This has changed radically in the recent years. Several diverse tools from logic were successfully applied to solve longstanding problems in operator algebras and new, purely logical, tools had to be developed to this effect. I will survey some of the most recent developments.
• 45pm talk and 56pm reception in the Logic Center, Room 420, 2 Arrow Street.
Fall 2013 Logic Seminar and Colloquium Schedule

Tuesday, October 8, 2013. Logic Colloquium: Rahim Moosa (University of Waterloo): "The Canonical Base Property and the Zilber Dichotomy Revisited". Slides. Abstract.
• Abstract: The truth of the Zilber dichotomy in several firstorder theories of fields with additional operators was behind Hrushovski's dramatic application of model theory to diophantine geometry in the nineties. About ten years ago, abstracting from a theorem in complexanalytic geometry, Pillay introduced a new modeltheoretic condition now called the canonical base property (CBP). This condition provides a direct proof of the Zilber dichotomy in various contexts, and has other strong geometric consequences. What seems to be required in establishing the CBP in any given situation is an appropriate notion of "jet space". This talk will be a largely expository introduction to, and overview of, the subject.
• 4:30pm in Science Center 507. 
Tuesday, October 15, 2013. Paul Christiano (University of California, Berkeley): "Probabilistic metamathematics and the definability of truth".
Abstract.• Abstract: No model $M$ of a sufficiently expressive theory can contain a truth predicate $T$ such that for all $S$, we have $M \models T(``S")$ if and only if $M \models S$. I'll consider the setting of probabilistic logic, and show that there are probability distributions over models which contain an "objective probability function" $P$ such that $M \models a < P(``S") < b \ $ almost surely whenever $a < P(M \models S) < b\ $. This demonstrates that a probabilistic analog of a truth predicate is possible as long as we allow infinitesimal imprecision. I'll argue that this result significantly undercuts the philosophical significance of Tarski's undefinability theorem, and show how the techniques involved might be applied more broadly to resolve obstructions due to selfreference.
• 4:30pm in Science Center 507. 
Tuesday, October 22, 2013. Cameron Donnay Hill (Wesleyan University):
"On Filters in Fraïssé Classes"
Abstract.• I will make several observations about a certain very natural filter on the age of a Fraïssé limit. In particular, I will show that several possible properties of classes of finite structures — e.g. zeroone laws and the (algebraic) finite submodel property — are characterized by the existence of certain extensions of this filter. Time permitting, I will also sketch out some applications in recovering geometry/dimension theory in Ramsey classes and ZeroOne classes.
• 4:30pm in Science Center 507. 
Tuesday, October 29, 2013. Spencer Breiner (Carnegie Mellon University):
"A scheme construction for logic and model theory".
Abstract.• Although contemporary model theory has been called "algebraic geometry minus fields" [Hodges 07], the formal methods of the two fields are radically different. In this talk I will present a theory of "logical schemes," geometric entities which relate to firstorder logical theories in much the same way that algebraic schemes relate to commutative rings.
Recall that the affine scheme associated with a commutative ring $R$ consists of two components: a topological space $\textrm{Spec}(R)$ (the spectrum) and a sheaf of rings $\mathcal{O}_R$ (the structure sheaf). Moreover, the scheme satisfies two important properties: its stalks are local rings and its global sections are isomorphic to $R$. In this work we replace $R$ by a firstorder logical theory $\mathbb{T}$, regarded as a generalized algebraic structure, and then associate the theory with a topological spectrum $\textrm{Spec}(\mathbb{T})$ and a sheaf of theories $\mathcal{O}_\mathbb{T}$ which satisfy similar local/global properties.
The spectrum of $\mathbb{T}$ is a topological groupoid built from the semantics of the theory (i.e., $\mathbb{T}$models and isomorphisms); this is based on a construction of Butz and Moerdijk [Butz 91]. The structure sheaf $\mathcal{O}_\mathbb{T}$ is then assembled from the Henkin extensions of $\mathbb{T}$; the stalk of $\mathcal{O}_\mathbb{T}$ over a $\mathbb{T}$model $M$ is the "elementary diagram" of $M$. These stalks satisfy a locality property corresponding to the logical existence and disjunction properties of the Henkin theories. These local theories inherit an action by the groupoid of isomorphisms and, up to a conservative extension $\mathbb{T}\subseteq\mathbb{T}^\mathrm{eq}$, the isomorphismstable sections of $\mathcal{O}_\mathbb{T}$ recover the original theory. If time allows, I will discuss some constructions from algebraic geometry which generalize to logical schemes.
• 5:00pm in Science Center 232. 
Monday, November 11, 2013. Isaac Goldbring (University of Illinois at Chicago):
"A survey of the model theory of tracial von Neumann algebras".
Slides.
Abstract.• Von Neumann algebras are certain algebras of bounded operators on Hilbert spaces. In this talk I will survey some of the model theoretic results about (tracial) von Neumann algebras, focusing mainly on (in)stability, quantifiercomplexity, and decidability. No prior knowledge of von Neumann algebras will be necessary. Some of the work presented is joint with Ilijas Farah, Bradd Hart, David Sherman, and Thomas Sinclair.
• 4:30pm in Science Center 507. 
Monday, November 18, 2013. Logic Colloquium: Alexander S. Kechris (Caltech):
"Topological dynamics and ergodic theory of automorphism groups of countable structures".
Slides.
Abstract.• I will discuss some aspects of the topological dynamics and ergodic theory of automorphism groups of countable firstorder structures and their connections with logic, finite combinatorics and probability theory. This is joint work with Omer Angel and Russell Lyons.
• 5:30pm in Science Center 507. 
Tuesday, November 26, 2013. Brian Wynne (Bard College at Simon's Rock):
"Upper products of existentially closed Abelian $\ell$groups".
Abstract.• I will start with some background on existentially closed Abelian $\ell$groups, and then discuss my work on upper products, a construction due to Ball, Conrad, and Darnel that generalizes the direct product, the lexicographic product, and the wreath product. In particular, I will present conditions under which the upper product of two existentially closed Abelian $\ell$groups is existentially closed, and some examples.
• 4:30pm in Science Center 507. 
Tuesday, December 3, 2013. Lynn Scow (Vassar College): "$\mathcal{I}$indexed indiscernible sets and trees".
Slides.
Abstract.• Fix any structure $\mathcal{I}$ on an underlying set $I$. An $\mathcal{I}$indexed indiscernible set is a set of parameters $A = \{a_i : i \in I\}$ where the $a_i$ are samelength finite tuples from some structure $M$ and $A$ satisfies a certain homogeneity condition.
In this talk, I will discuss examples of trees $\mathcal{I}$ for which $\mathcal{I}$indexed indiscernible sets are particularly wellbehaved. In particular, we will look at the structure $\mathcal{I}_0 = (\omega^{<\omega},\unlhd,<_{\mathrm{lex}},\wedge)$ where $\unlhd$ is the partial order on the tree, $\wedge$ is the meet in this order, and $<_{\mathrm{lex}}$ is the lexicographical order. By a dictionary theorem that I will present in this talk, known results about indiscernibles from model theory yield alternate proofs that certain classes of finite trees are Ramsey.
• 4:30pm in Science Center 507. 
Friday, December 6, 2013. Sam Sanders (Ghent University):
"Higherorder Reverse Mathematics: Where existence meets computation via infinitesimals".
Abstract.• Classically, the existence of an object tells us very little about how to construct said object. We consider a nonstandard version of Ulrich Kohlenbach's higherorder Reverse Mathematics in which there is a very elegant and direct correspondence between, on one hand, the existence of a functional computing an object and, on the other hand, the classical existence of this object with the same standard and nonstandard properties. We discuss how these results –potentially– contribute to the programs of finitistic and predicativist mathematics.
• 4:30pm in Science Center 507.