**CUNY Graduate Center**

**Room 6417**

**Fridays 10:00am-12:00pm**

**Organized by Victoria Gitman and Corey Switzer**

**Calendar**

**May 10**

**Kameryn Williams**,
University of Hawai‘i at Mānoa

**TBA**

**Abstract**

**April 26**

**Seminar cancelled**

Spring Break

**April 19**

**Seminar cancelled**

Spring Break

**April 12**

**Jonas Reitz**,
CUNY

**TBA**

**Abstract**

**March 29**

**Grigor Sargsyan**,
Rutgers University

**TBA**

**Abstract**

**March 22**

**Special time: 12:30-2pm**

**Arthur Apter**,
CUNY

**TBA**

**Abstract**

**March 15**

**Stefan Mesken**,
University of Münster

**TBA**

**Abstract**

**March 8**

**Brent Cody**,
Virginia Commonwealth University

**TBA**

**Abstract**

**March 1**

**Chris Lambie-Hanson**,
Virginia Commonwealth University

**TBA**

**Abstract**

**February 22**

**Joseph Van Name**,
CUNY

**Lower bounds on the cardinalities of quotient algebras of elementary embeddings**

**Abstract**

From non-trivial elementary embeddings $j_{1}\dots j_{r}:V_{\lambda}\rightarrow V_{\lambda}$, we obtain a sequence of polynomials $(p_{n}(x_{1},\dots,x_{r}))_{n\in\omega}$ that satisfies the infinite product $$\prod_{k=0}^{\infty}p_{k}(x_{1},\dots,x_{r})=\frac{1}{1-(x_{1}+\dots+x_{r})}.$$ From this infinite product, we deduce lower bounds of the cardinality of $|\langle j_{1},...,j_{r}\rangle/\equiv^{\alpha}|$ using analysis and analytic number theoretic techniques. Computer calculations that search for Laver-like algebras give some empirical evidence that these lower bounds cannot be greatly improved.

**February 15**

**Dan Saattrup Nielsen**,
University of Bristol

**Level-by-level virtual large cardinals**

**Abstract**

A virtual large cardinal is (usually) the critical point of a generic elementary embedding from a rank-initial segment of the universe into a transitive $M\subset V$, as introduced by Gitman and Schindler (2018). A notable feature is that all virtual large cardinals are consistent with $V=L$, and they've proven useful in characterising several properties in descriptive set theory. We'll work with the virtually $\theta$-measurable, $\theta$-strong and $\theta$-supercompact cardinals, where the $\theta$ in particular indicates that the generic embeddings have $H_\theta^V$ as domain, and investigate how these level-by-level virtual large cardinals relate both to each other and to the existence of winning strategies in certain games. This is work in progress and joint with Philipp Schlicht.

**February 8**

**Sean Cox**,
Virginia Commonwealth University

**Martin's Maximum and the Diagonal Reflection Principle**

**Abstract**

Several years ago I introduced the Diagonal Reflection Principle (DRP), a maximal form of simultaneous stationary reflection. Roughly, DRP asserts that for all regular $\theta \ge \omega_2$, there are stationarily many sets $W$ of size $\omega_1$ such that every stationary element of $W$ reflects to $W$ (i.e. if $S \subset [\theta]^\omega$ is stationary and $S \in W$, then $S \cap [W \cap \theta]^\omega$ is stationary). In that paper I showed that that $\text{MM}^{+\omega_1}$--even just the '$+\omega_1$' version of the forcing axiom for $\sigma$-closed forcings--implies DRP. In this talk I will prove that MM, even its technical strengthening $\text{MM}^{+\omega}$, does NOT imply DRP, contrary to my initial expectations. This is joint work with Hiroshi Sakai.

**February 1**

**Gunter Fuchs**,
CUNY

**Bounded forcing axioms and the preservation of wide Aronszajn trees**

**Abstract**

I will talk about connections between three properties of a forcing class $\Gamma$: the most well-known one is the bounded forcing axiom at $(\omega_1,\lambda)$ for this class, introduced by Goldstern and Shelah for proper forcing. The second property is a two cardinal version of $\Sigma^1_1$ absoluteness, at $(\omega_1,\lambda)$, and the third is the preservation of Aronszajn trees of height $\omega_1$ and width $\lambda$. Under the assumption that $\lambda=\lambda^\omega$ and forcings in $\Gamma$ don't add countable subsets of $\lambda$, I will show that these properties are equivalent. The class of forcing notions satisfying Jensen's property of subcompleteness is a canonical class that does not add reals and it is a corollary of the above result that the bounded forcing axiom for subcomplete forcing at $(\omega_1,2^\omega)$ is equivalent to the preservation of Aronszajn trees of height $\omega_1$ and width $2^\omega$. This generalizes a prior result, obtained jointly with Kaethe Minden. I will discuss some further results, as well as some open questions, on the preservation of Aronszajn trees of height $\omega_1$ and other widths.

**Previous Semesters**