For which pairs of linear orders $A$ and $B$ are the sums $A + B$ and $B + A$ isomorphic? In the 1930s, Tarski conjectured that $A + B = B + A$ if and only if

(i.) There is an order $C$ and natural numbers $n$ and $m$ such that $A = nC$ and $B = mC$, or

(ii.) There is an order $M$ such that $B = \omega A + M + \omega^* A$, or  

(iii.) There is an order $N$ such that $A = \omega B + N + \omega^* B$. 

Notably, these conditions on $A$ and $B$ are “arithmetic” in the sense that they are expressed in terms of finitary and $\omega$-ary sums of linear orders. 

Tarski proved his conjecture over the class of scattered linear orders, but Lindenbaum was able to produce a non-scattered counterexample. Building on Lindenbaum’s work, Aronszajn found a structural characterization of all additively commuting pairs of linear orders. 

Aronszajn’s characterization is somewhat complicated: in modern language it can be described in terms of orbit equivalence relations of groups of translations on $\mathbb{R}$. Tarski lamented that Aronszajn’s result could not be formulated arithmetically — that is, purely in terms of sums — and the line of work was abandoned. 

Building on our recent work on sums of linear orders, Eric Paul and I showed that there is an arithmetic condition equivalent to commutativity for linear orders. And in fact, the condition is a natural extension of the one appearing in Tarski’s original conjecture. In this talk, I will state our result, outline the proof, and discuss some related problems. 

The Halpern-Läuchli theorem was first introduced for its use in Halpern and Lévy's proof of BPI in the Cohen model. Since then, several other theorems establish emergent connections between variants of the Halpern-Läuchli Theorem and BPI in certain symmetric extensions. In this talk, we develop the forcing perspective given by Harrington's proof of the Halpern-Läuchli Theorem. By doing so, we will more clearly identify a connection between variants of the Halpern-Läuchli Theorem and the existence of certain filters in symmetric extensions. Using tools from the study of BPI in symmetric extensions, we use this connection to give simple positive and negative proofs of new variants of the Halpern-Läuchli Theorem.

For an infinite cardinal κ, the ultrafilter number u_κ is the smallest size of a family that generates a uniform ultrafilter on κ. We say that u_κ is bounded if u_κ<2^κ.
The case κ=ω is well understood, with the consistency of a bounded ultrafilter number at ω first noted in the early 1970s by Kunen. The method used in Kunen’s construction does not generalize to the case κ=ω_1, and Kunen asked whether u_{ω_1}<2^{ω_1} is even consistent; this remains wide open.
The consistency of u_κ<2^κ for general uncountable κ has been studied extensively in the last decade. In this series of talks, we will survey some recent progress towards bounding the ultrafilter number at successor values of κ, with emphasis on a key theorem of Raghavan and Shelah. We will also discuss some crucial limitations to applying this theorem for a bounded ultrafilter number at any of the ω_n’s.

We survey recent work on the Sealing phenomenon. Woodin shows that various forms of Sealing hold in a generic extension of the universe of sets in which there is a supercompact cardinal and a proper class of Woodin cardinals. Sargsyan and I study Sealing in hod mice and compute the exact consistency strength of Sealing. I will give background and state precisely these results. I will also discuss the impact of Sealing on the universe of sets and the inner model program.

I will introduce the notion of “Ramsey partition regularity,” a generalization of partition regularity involving infinite configurations.  This notion is characterized in terms of certain ultrafilters related to tensor products, and called Ramsey witnesses.  We use the properties of this characterization in the nonstandard context of hypernatural numbers to determine whether various patterns involving polynomials and exponentials are Ramsey partition regular.

(Joint work with L. Luperi Baglini, M. Mamino, R. Mennuni, and M. Ragosta.)