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.

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.

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.

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.

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.

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.

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.

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.

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.

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.