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2018

Squares, ascent paths, and chain conditions. (JMM, AMS-ASL Special Session on Set Theory, Logic, and Ramsey Theory, San Diego, January 2018.)
(Slides)
Using a variety of square principles, we obtain results on the consistency strengths of the non-existence of kappa-Aronszajn trees with narrow ascent paths and of the infinite productivity of strong kappa-chain conditions. In particular, we show that, if kappa is an uncountable regular cardinal that is not weakly compact in L, then:
1. for every lambda < kappa, there is a kappa-Aronszajn tree with a lambda-ascent path;
2. there is a kappa-Knaster poset P such that P^omega does not have the kappa-chain condition;
3. there is a kappa-Knaster poset that is not kappa-stationarily layered.
This answers questions of Cox and Lücke and consists of joint work with Philipp Lücke.

2017

Reflections on graph coloring. (MAMLS Logic Friday, CUNY Graduate Center, October 2017.)
(Slides)
In 1951, de Bruijn and Erdős published a compactness theorem for graphs with finite chromatic number, proving that, if G is a graph, k is a natural number, and all finite subgraphs of G have chromatic number at most k, then G has chromatic number at most k. Since then, infinitary generalizations of this theorem, for the chromatic number as well as the coloring number of graphs, have attracted much attention. In this talk, we will briefly review some of the historical highlights in this area and then present some new work. These results show that the coloring number can exhibit only a limited amount of incompactness, while large amounts of incompactness for the chromatic number are implied by relatively weak hypotheses. This indicates that the coloring number and chromatic number behave quite differently with respect tocompactness and illustrates the difficulty involved in obtaining infinitary analogues of the de Bruijn-Erdős result at infinite, accessible cardinals. This is joint work with Assaf Rinot.

A forcing axiom deciding the generalized Souslin Hypothesis. (Carnegie Mellon University Logic Seminar, October 2017.)
Given a regular, uncountable cardinal kappa, it is often desirable to be able to construct objects of size kappa^+ using approximations of size less than kappa. Historically, such constructions have often been carried out with the help of a (kappa, 1)-morass and/or a diamond(kappa)-sequence. We present a framework for carrying out such constructions using diamond(kappa) and a weakening of Jensen's principle square_kappa. Our framework takes the form of a forcing axiom, SDFA(P_kappa). We show that SDFA(P_kappa) follows from the conjunction of diamond(kappa) and our weakening of square_kappa and, if kappa is the successor of an uncountable cardinal, that SDFA(P_kappa) is in fact equivalent to this conjunction. We also show that, for an infinite cardinal lambda, SDFA(P_{lambda^+}) implies the existence of a lambda^+-complete lambda^{++}-Souslin tree. This implies that, if lambda is an uncountable cardinal, 2^lambda = lambda^+, and Souslin's Hypothesis holds at lambda^{++}, then lambda^{++} is a Mahlo cardinal in L, improving upon an old result of Shelah and Stanley. This is joint work with Assaf Rinot.

A forcing axiom deciding the generalized Souslin Hypothesis. (Miami University Logic Seminar, September 2017.)
Given a regular, uncountable cardinal kappa, we isolate a forcing axiom, SDFA(mathcal{P}_kappa), that provides a framework for constructing objects of size kappa^+ using approximations of size <\kappa. We show that, for uncountable successor $\kappa$, SDFA(mathcal{P}_kappa) is equivalent to the conjunction of 2^{ < kappa} = kappa and square^B_kappa, which is a weakening of Jensen's square_kappa. In our main application, we show that, for an infinite cardinal lambda, SDFA(mathcal{P}_{lambda^+}) implies the existence of a lambda^+-closed lambda^{++}-Souslin tree. This yields a corollary stating that, if lambda is an uncountable cardinal, 2^lambda = lambda^+, and Souslin's Hypothesis holds at lambda^{++}, then lambda^{++} is Mahlo in L, improving upon a result of Shelah and Stanley. This is joint work with Assaf Rinot.

Constructions from squares and diamonds. (6th European Set Theory Conference, Budapest, July 2017.)
(Slides)
In 1982, Shelah and Stanley proved that, if kappa is a regular, infinite cardinal, 2^kappa = kappa^+, and there is a (kappa^+, 1)-morass, then there is a kappa^{++}-super-Souslin tree, which is a type of normal kappa^{++}-tree that necessarily has a \kappa^{++}-Souslin subtree and continues to do so in any outer model in which kappa^{++} is preserved and no new subsets of kappa are present. This result establishes a lower bound of an inaccessible cardinal for the consistency strength of the conjunction of 2^kappa = kappa^+ and Souslin's Hypothesis at kappa^{++}. In this talk, we will present a method for constructing objects of size lambda^+ from square_lambda + diamond_lambda, where lambda is a regular, uncountable cardinal. As an application, we will use square_{kappa^+} + diamond_{kappa^+} to construct a kappa^{++}-super-Souslin tree. For uncountable kappa, this increases Shelah and Stanley's lower bound from an inaccessible cardinal to a Mahlo cardinal. This is joint work with Assaf Rinot.

Constructions from square and diamond, with an application to super-Souslin trees. (Oberseminar mathematische Logik, University of Bonn, May 2017.)
In 1982, Shelah and Stanley proved that, if kappa is a regular, infinite cardinal, 2^kappa = kappa^+, and there is a (kappa^+, 1)-morass, then there is a kappa^{++}-super-Souslin tree, which is a type of normal kappa^{++}-tree that necessarily has a \kappa^{++}-Souslin subtree and continues to do so in any outer model in which kappa^{++} is preserved and no new subsets of kappa are present. This result establishes a lower bound of an inaccessible cardinal for the consistency strength of the conjunction of 2^kappa = kappa^+ and Souslin's Hypothesis at kappa^{++}. In this talk, we will present a method for constructing objects of size lambda^+ from square_lambda + diamond_lambda, where lambda is a regular, uncountable cardinal. As an application, we will use square_{kappa^+} + diamond_{kappa^+} to construct a kappa^{++}-super-Souslin tree. For uncountable kappa, this increases Shelah and Stanley's lower bound from an inaccessible cardinal to a Mahlo cardinal. This is joint work with Assaf Rinot.

Reflections on the coloring and chromatic numbers. (University of Helsinki Logic Seminar, April 2017.)
Compactness phenomena play a central role in modern set theory, and the investigation of compactness and incompactness for the coloring and chromatic numbers of graphs has been a thriving area of research since the mid-20th century, when De Bruijn and Erdős published their compactness theorem for finite chromatic numbers. In this talk, we will briefly review some of the highlights in this area and then present new results indicating, firstly, that the coloring number can only exhibit a limited amount of incompactness, and, secondly, that large amounts of incompactness for the chromatic number are compatible with strong compactness statements, including compactness for the coloring number. This indicates that the chromatic and coloring numbers behave quite differently with respect to compactness. This is joint work with Assaf Rinot.

Partition relations and generalized scattered orders. (Bar-Ilan University Set Theory Seminar, March 2017.)
The class of scattered linear orders, isolated by Hausdorff, plays a prominent role in the study of general linear orders. In 2006, Dzamonja and Thompson introduced classes of orders generalizing the class of scattered orders. For a regular cardinal kappa, they defined the classes of kappa-scattered and weakly kappa-scattered linear orders. For kappa = omega, these two classes coincide and are equal to the classical class of scattered orders. For larger values of kappa, though, the two classes are provably different. In this talk, we will investigate properties of these generalized scattered orders with respect to partition relations, in particular the extent to which the classes of kappa-scattered or weakly kappa-scattered linear orders of size kappa are closed under partition relations of the form tau -> (phi, n) for all natural numbers n. Along the way, we will prove a generalization of the Milner-Rado paradox and look at some results regarding ordinal partition relations. This is joint work with Thilo Weinert.

Trees with ascent paths. (Hebrew University of Jerusalem Logic Seminar, March 2017.)
The notion of an ascent path through a tree, isolated by Laver, is a generalization of the notion of a cofinal branch and, in many cases, the existence of an ascent path through a tree provides a concrete obstruction to the tree being special. We will discuss some recent results regarding ascent paths through kappa-trees, where kappa > omega_1 is a regular cardinal. We will discuss the consistency of the existence or non-existence of a special mu^+-tree with a cf(mu)-ascent path, where mu is a singular cardinal. We will also discuss the consistency of the statement, "There are omega_2-Aronszajn trees but every omega_2-tree contains an omega-ascent path." We will connect these topics with various square principles and with results about the productivity of chain conditions.

Pseudo-Prikry sequences. (Arctic Set Theory 3. Kilpisjärvi, Finland, January 2017.)
(Slides)

Reflections on the coloring and chromatic numbers. (Hebrew University of Jerusalem Logic Seminar, January 2017.)
Compactness phenomena play a central role in modern set theory, and the investigation of compactness and incompactness for the coloring and chromatic numbers of graphs has been a thriving area of research since the mid-20th century, when De Bruijn and Erdős published their compactness theorem for finite chromatic numbers. In this talk, we will briefly review some of the highlights in this area and then present new results indicating, firstly, that the coloring number can only exhibit a limit amount of incompactness, and, secondly, that large amounts of incompactness for the chromatic number are compatible with strong compactness statements, including compactness for the coloring number. This is joint work with Assaf Rinot.

2016

Squares, stationary reflection, and incompactness. (Young Researcher's Seminar Week. Centre de Recerca Matemàtica, Barcelona, November 2016.)
The study of the tension between compactness and incompactness has played a central role in modern set theory. In this talk, we will consider the interplay between various compactness and incompactness principles, focusing in particular on square principles, stationary reflection, and Aronszajn trees. We will begin by discussing joint work with Yair Hayut regarding the extent to which the existence of weak square sequences is compatible with simultaneous stationary reflection, establishing a tight connection between these two notions. We then discuss applications of this work to higher Souslin trees and incompactness for the chromatic number of graphs.

Square sequences and simultaneous stationary reflection. (SETTOP 2016. Fruska Gora, Serbia, June 2016.)
(Slides)
(Joint work with Yair Hayut) There has been much work done in set theory investigating the tension between compactness phenomena, such as stationary reflection, and incompactness phenomena, such as Jensen's square principle. It is a folklore result that, for a cardinal mu, square_mu implies a strong failure of stationary reflection at mu^+, while, for a regular cardinal kappa, Todorcevic's square principle square(kappa) implies the failure of simultaneous stationary reflection for pairs of stationary subsets of kappa. We obtain results about the extent to which weakenings of square(kappa) are compatible with simultaneous stationary reflection at kappa.

Robust reflection principles. (ASL Winter Meeting, Seattle, January 2016.)
(Slides)
It is a common motif in set theory that, if kappa is a large cardinal (Mahlo, weakly compact, measurable, supercompact, etc.), then kappa satisfies certain interesting reflection principles. In addition, because most large cardinal notions are preserved under small forcing extensions, i.e. forcing extensions by posets with cardinality less than kappa, these reflection principles, when they hold at large cardinals, are also robust under small forcing. It has been a fruitful line of research to consider the extent to which these reflection principles can hold at smaller cardinals. However, these principles can in general fail to be preserved by small forcing when they hold at small cardinals. Focusing on stationary reflection and the tree property, we discuss situations in which reflection principles can fail to be robust under small forcing, introduce natural strengthenings which are implied by large cardinals and which are in all cases robust under small forcing, and consider the extent to which these strenthenings can hold at small cardinals.

2015

Robust reflection principles. (Cornell University Logic Seminar, September 2015.)
Large cardinals are useful in set theory in part because they imply certain reflection properties. For example, if kappa is a weakly compact cardinal, then kappa satisfies the tree property and every stationary subset of kapp reflects. An interesting direction of research involves investigating the extent to which these reflection properties can hold at smaller cardinals. By results of Levy and Solovay, for most large cardinal notions (inaccessible, weakly compact, measurable, etc.), if kappa is such a large cardinal in V and P is a forcing poset of size less than kappa, then kappa remains large after forcing with P. Therefore, reflection properties of a cardinal kappa that are implied by kappa being a particular large cardinal are themselves indestructible by small forcing when kappa is such a large cardinal. However, these reflection properties may no longer be indestructible at kappa if kappa is a small cardinal. For example, it is consistent that aleph_{omega+1} has the tree property but there is a forcing P of size omega_1 such that aleph_{omega+1} fails to have the tree property after forcing with P. We will begin by reviewing the relevant large cardinal and reflection notions and will then consider strengthenings of certain reflection properties that are always indestructible under small forcing, focusing in particular on stationary reflection and the tree property. We will look at the extent to which these reflection properties can hold at small cardinals and the extent to which they are in fact stronger than the weaker principles from which they are derived.

Coloring classes and the Hanf number for amalgamation. (Ben Gurion University Logic, Set Theory, and Topology Seminar, March 2015.)
The amalgamation property is a topic of fundamental importance in model theory and is still imperfectly understood. In the 1980s, Grossberg asked a question, which remains open, about the existence of a Hanf number for amalgamation in abstract elementary classes. In particular, Grossberg conjectured that the Hanf number for amalgamation for classes given by an L_{omega_1, omega} sentence is beth_{omega_1}. We introduce a new collection of abstract elementary classes, called coloring classes, and use them to give a partial answer to Grossberg's question, significantly improving upon work of Baldwin, Kolesnikov, and Shelah. Analysis of these coloring classes leads to some purely combinatorial questions that are of interest in their own right. This is joint work with Alexei Kolesnikov.

Patterns of stationary reflection. (Winter School in Abstract Analysis, Set Theory and Topology Section. Hejnice, Czech Republic. February 2015.)
(Slides)
If kappa is a regular cardinal, alpha < kappa has uncountable cofinality, and S is a stationary subset of kappa, we say S reflects at alpha if S intersect alpha is stationary in alpha. S reflects if there is alpha < kappa such that S reflects at alpha. Questions regarding the extent of stationary reflection have been extensively studied and are intimately related to a number of topics concerning large cardinals, combinatorial set theory, and cardinal arithmetic. Eisworth, motivated in part by his work on square-bracket partition relations, asked whether it must be the case that if lambda is a singular cardinal and every stationary subset of lambda^+ reflects, then every stationary subset of lambda^+ reflects at ordinals of arbitrarily high cofinality below lambda. We will answer this in the negative and go on to consider variants of Eisworth's question. Along the way, we will explore some connections between stationary reflection and the combinatorial notion of approachability.

2014

Coloring classes and amalgamation. (Ariel University Seminar on Algorithms, Combinatorics, Graph Theory and Algebra, December 2014.)
The amalgamation property for a class of models is the assertion that, under certain conditions, two models in the class can be realized as sub-models of a single larger model in the class. The amalgamation property for first-order logic is a useful consequence of the compactness theorem, but it may fail for generalizations of first-order logic. In the 1980s, Grossberg made a conjecture regarding the extent of amalgamation for models of sentences in infinitary logics. We introduce a class of combinatorial structures called well-colorings, prove results about the existence of well-colorings of certain cardinalities, and use these results to show the optimality of Grossberg's conjecture, improving upon results of Baldwin, Kolesnikov, and Shelah. This is joint work with Alexei Kolesnikov.

Bounded stationary reflection. (Hebrew University of Jerusalem Students' Set Theory Seminar, December 2014.)
Eisworth, motivated in part by his work on square-bracket partition relations, asked whether it must be the case that if mu is a singular cardinal and every stationary subset of mu^+ reflects, then every stationary subset of mu^+ reflects at an ordinal of arbitrarily high cofinality below mu. We will give some background motivation, sketch a proof answering this question in the negative, and discuss various variations on Eisworth's question.

Well-colorings and the Hanf number for amalgamation. (Bar-Ilan University Infinite Combinatorics Seminar, November 2014.)
The amalgamation property is a topic of fundamental interest in model theory and is still imperfectly understood. In the 1980s, Grossberg asked a question, which remains open to this day, about the existence of a Hanf number for amalgamation in abstract elementary classes. We introduce a new class of structures, called well-colorings, and use them to give a partial answer to Grossberg's question, significantly improving upon previous work of Baldwin, Kolesnikov, and Shelah. We shall start the talk by briefly discussing the relevant model-theoretic definitions and will then give proofs of the main results, which are entirely set-theoretic and combinatorial in nature and of interest in their own right. This is joint work with Alexei Kolesnikov.

Bounded stationary reflection. (Graduate Student Conference in Logic, April 2014.)
(Slides)
We will discuss the effects of square-bracket partition relations on stationary reflection at the successors of singular cardinals. We will then sketch a proof of the result that, relative to large cardinals, it is consistent that there is a singular cardinal mu such that every stationary subset of mu^+ reflects but that there is a stationary subset of mu^+ that does not reflect at ordinals of arbitrarily high cofinality below mu. This answers a question of Todd Eisworth and is joint work with James Cummings.

Jónsson cardinals, partition relations, and stationary reflection, part I. (CMU Logic Seminar, February 2014.)
I will give an introduction to Jónsson cardinals and related square bracket partition relations. We will prove some of the basic facts about Jónsson cardinals, focusing in particular on the important open question of whether the successor of a singular cardinal can be Jónsson. This will involve a discussion of the connections between Jónsson cardinals and stationary reflection, which will lead into a recent result of Cummings and myself.

Jónsson cardinals, partition relations, and stationary reflection, parts II and III. (CMU Logic Seminar, February/March 2014.)
I will present a proof that, relative to large cardinal assumptions, it is consistent that there is a singular cardinal mu such that every stationary subset of mu^+ reflects but that there is a stationary subset of mu^+ that does not reflect at ordinals of arbitrarily high cofinality. This answers a question of Eisworth motivated by the study of Jónsson cardinals and square-bracket partition relations and is joint work with James Cummings.

The transfinite subway and closure properties of uncountable cardinals. (CMU Graduate Student Seminar, January 2014.)
It is a little-known fact that there is a subway line with uncountably many stops connecting the Hilbert Hotel to its nearest airport. In this talk, we will analyze the behavior and efficiency of this subway line. In the process, we will develop some of the combinatorial theory of uncountable cardinals and, time permitting, construct every Borel subset of the real numbers.

2013

The wonderful world of singular cardinals. (CMU Graduate Student Seminar, September 2013.)
The cardinal numbers divide into two very different classes: regular cardinals and singular cardinals. Cardinal arithmetic involving regular cardinals is in a certain sense completely understood and subject only to a few simple rules. Cardinal arithmetic involving singular cardinals, on the other hand, is a rich, fascinating, and still imperfectly-understood subject, yielding surprises to this day. In this talk, I will introduce some of the basic facts about singular cardinals and survey some of the most important recent developments in cardinal arithmetic, taking short detours through Heidelberg in 1904 and Moscow in the 1920s.

Intermediate square principles. (New York Graduate Student Logic Conference, April 2013.)
(Slides)
Since their introduction by Jensen, square principles have been an important and well-studied example of combinatorial incompactness in set theory, with connections to many areas of the field, including large cardinals, inner model theory, and PCF theory. In this talk, we present some naturally arising square principles intermediate between the classical notions of square_kappa and square(kappa^+), where kappa is an uncountable regular cardinal, and provide a detailed picture of the implications and non-implications between these principles.

2012

Covering matrices and squares, parts I and II. (CMU Logic Seminar, October 2012.)
Covering matrices were introduced by Matteo Viale in his proof that the Singular Cardinals Hypothesis follows from the Proper Forcing Axiom. In particular, he showed that PFA implies that certain covering matrices exhibit strong covering and reflection properties. In this series of talks, I will construct counterexamples to these covering and reflection properties and investigate their relationships with square principles. This will lead to an examination of a variety of square principles intermediate between square_kappa and square(kappa^+). In the first lecture, I will introduce the notion of a covering matrix and present results about the existence of certain types of kappa-covering matrices for kappa^+. We will show that the existence of transitive, normal, uniform kappa-covering matrices for kappa^+ follows from square_{kappa, < kappa} (but not from weak square). In the second lecture, we will show that the converse fails by constructing a model in which there is a transitive, normal, uniform kappa-covering matrix for kappa^+ but in which square_{kappa, < kappa} fails. If time permits, we will begin a discussion of Todorcevic's rho-functions on square sequences and their use in constructing covering matrices.

The chromatic number of the plane: an unfinished conversation in two acts. (CMU Graduate Student Seminar, August 2012.)
In this talk, I will introduce the fascinating and still-open problem of determining the chromatic number of the unit-distance graph on the plane and discuss some progress that has been made toward its solution. I will present the best known upper and lower bounds for the chromatic number of the plane and some of its variants and also some recent work suggesting that the solution may depend on the set-theoretic axioms assumed. I will also present two delightful proofs of the De Bruijn-Erdos Compactness Theorem, of considerable interest in its own right.

2011

The continuum hypothesis, the axiom of choice, and Lebesgue measurability. (CMU Graduate Student Seminar, October 2011.)
(Slides)

The Continuum Hypothesis has fascinated mathematicians ever since it was advanced by Georg Cantor in the late 19th century, and work inspired by CH has revolutionized set theory on multiple occasions. In this talk, I will review the history of the Continuum Hypothesis and some of the major results arising out of its study. In the process, I will briefly discuss Chris Freiling's Axiom of Symmetry, present a model of set theory, due to Solovay, in which every set of reals is Lebesgue measurable, and prove Cantor's theorem by playing games.

2010

A mathematical investigation of juggling. (CMU Graduate Student Seminar, November 2010.)
In the 1980s, a handful of jugglers developed a mathematical notation for juggling that changed the art forever. This street of influence went two ways, though, as mathematicians considering the intricacies of juggling began proving theorems in a surprising variety of fields. In this talk, I will develop a simple mathematical model of juggling and investigate its combinatorial and group-theoretic aspects. This will culminate in a theorem, which has led to original work in the combinatorics of symmetric groups, counting the number of possible juggling patterns.