Sandra Müller

Österreichische Studienstiftung, Wien - Unbegreifliches verstehen: eine Reise zu großen Unendlichkeiten und den Grenzen der Mathematik

2026-04-09T00:00:00+02:00

I am invited to give a talk at an information event of the Österreichische Studienstiftung and TU Wien on April 10, 2026. This event will take place at the Festsaal of TU Wien from 4 pm to 8 pm.

Unbegreifliches verstehen: eine Reise zu großen Unendlichkeiten und den Grenzen der Mathematik

Building Materials Seminar, TU Wien - The boundaries of mathematics: infinite games and very large sets

2026-04-08T00:00:00+02:00

I am invited to give a talk at the seminar of the Research Unit Building Materials of TU Wien on April 9, 2026.

The boundaries of mathematics: infinite games and very large sets

Logic Colloquium, University of Vienna - Sealing the universe: How far can inner model theory reach?

2026-03-17T00:00:00+01:00

I am invited to give a talk in the Logic Colloquium of the University of Vienna on March 19, 2026.

Sealing the universe: How far can inner model theory reach?

The inner model program is one of the central pillars of modern research in set theory. Grounded in Gödel’s work, it aims at reaching stronger and stronger canonical inner models for large cardinals. These inner models provide a controlled environment to study the structural behaviour of large cardinals and can be regarded as witnesses for their consistency. But the program is currently undergoing a revolution: Established methods such as Woodin’s Core Model Induction provably fail even below a Woodin limit of Woodin cardinals. This is grounded in a principle called Sealing. In this talk we will outline the impact Sealing has on the inner model program as well as scenarios for overcoming the obstacles introduced by Sealing for a non-expert audience.

(with T. Gappo) Long games just beyond fixed countable length

2026-01-28T00:00:00+01:00

Submitted. PDF. arXiv. Bibtex.

 @ARTICLE{GaMu26,
   AUTHOR = {T. Gappo and S. Müller},
   TITLE = {Long games just beyond fixed countable length},
   JOURNAL = {},
   VOLUME = {},
   ISSUE = {},
   YEAR = {2026},
   PAGES = {},
   EPRINT = {2601.03749},
   DOI = {}}

We introduce a new type of game on natural numbers of variable countable length, which can be regarded as a diagonalization of all games of fixed countable length on natural numbers. Building on previous work by Trang and Woodin, we show that analytic determinacy of the game is equivalent to the existence of a sharp for a canonical inner model with a limit of Woodin cardinals $\lambda$ such that the order type of Woodin cardinals below $\lambda$ is $\lambda$.

6th Austrian Day of Women in Mathematics - The boundaries of mathematics: infinite games and very large sets

2026-01-26T00:00:00+01:00

I am invited to give a plenary talk at the 6th Austrian Day of Women in Mathematics on February 27, 2026.

The boundaries of mathematics: infinite games and very large sets

The independence phenomenon is a central theme in set theory and more generally in mathematics. Starting from the Continuum Problem, there are numerous statements in various areas of mathematics that can neither be proven nor disproven in the usual axiomatic framework given by the Zermelo Fraenkel Axioms. The most promising approach to attack this issue is by studying extensions of the usual axiomatic framework, their connections, and their impact on previously independent statements. Two famous examples in set theory are the hierarchies given by the determinacy of infinite two-player games and large cardinals. The deep connection between these two hierarchies forms the backbone of inner model theory. We outline this connection and discuss recent progress as well as important open questions in the area.

The consistency strength of determinacy when all sets are universally Baire

2025-12-19T00:00:00+01:00

Advances in Mathematics. Volume 481, December 2025. 110548.
DOI: 10.1016/j.aim.2025.110548. PDF. arXiv. Bibtex.

 @ARTICLE{Mu21,
   AUTHOR= {S. Müller},
   TITLE= {The consistency strength of determinacy when all sets are universally {B}aire},
   JOURNAL = {Advances in Mathematics},
   VOLUME = {481},
   ISSUE = {},
   YEAR = {2025},
   PAGES = {110548},
   EPRINT ={2106.04244},
   DOI = {10.1016/j.aim.2025.110548}}

The large cardinal strength of the Axiom of Determinacy when enhanced with the hypothesis that all sets of reals are universally Baire is known to be much stronger than the Axiom of Determinacy itself. In fact, Sargsyan conjectured it to be as strong as the existence of a cardinal that is both a limit of Woodin cardinals and a limit of strong cardinals. Larson, Sargsyan and Wilson showed that this would be optimal via a generalization of Woodin’s derived model construction. We introduce a new translation procedure for hybrid mice extending work of Steel, Zhu and Sargsyan and use it to prove Sargsyan’s conjecture.

Náboj Junior Competition, Vienna - Eine spielerische Reise zu großen Unendlichkeiten an der Grenze der Mathematik

2025-12-08T00:00:00+01:00

I was invited to give a talk with the title Eine spielerische Reise zu großen Unendlichkeiten an der Grenze der Mathematik at the Náboj Junior Competition in Vienna, taking place November 21, 2025.

University of Münster - Translating strategy mice

2025-12-08T00:00:00+01:00

I was invited to give a talk at the University of Münster on December 1, 2025.

Translating strategy mice

We discuss a procedure to translate rigidly layered strategy mice with omega Woodin cardinals into mice with a limit of Woodin and strong cardinals. We will also outline obstacles when attempting to generalize this procedure to strategy mice with more large cardinals or least branch strategy mice. If time permits, we will also sketch related work in progress, joint with Dominik Adolf and Benjamin Siskind, on translating complexity in derived models into large cardinals in a mouse.

Berkeley Inner Model Theory Conference - Translating between large cardinals and iteration strategies

2025-12-05T00:00:00+01:00

I was invited to give a talk on June 6, 2025, as well as a tutorial from June 30 to July 3,2025 at the Berkeley Inner Model Theory Conference.

Talk: Translating between large cardinals and iteration strategies

My PhD thesis under Ralf’s supervision focused on mice with finitely many Woodin cardinals, the projective complexity of their iteration strategies and models of levels of projective determinacy. Since then, I have been interested to connect inner models with large cardinals and models of determinacy at various levels. In this talk I will outline some classical results as well as new developments in this area.

Tutorial. Derived models from genericity iterations

Woodin’s derived model theorem as well as Steel’s stationary-tower-free proof of it are powerful tools to construct models of determinacy. We outline Steel’s proof of the stationary-tower-free version using genericity iterations as well as iterability. Moreover, we sketch recent extensions to models of determinacy with non-trivial sets of sets of reals, obtained jointly with Sargsyan, as well as applications of this method, obtained jointly with Koschat and Sargsyan.

Lecture notes are available on the conference website.

(with G. Sargsyan) Towards a generic absoluteness theorem for Chang models

2025-11-30T00:00:00+01:00

Advances in Mathematics. Volume 476, August 2025. 110357.
DOI: 10.1016/j.aim.2025.110357. PDF. arXiv. Bibtex.

 @ARTICLE{MS_GenAbsCM,
   AUTHOR= {S. Müller and G. Sargsyan},
   TITLE= {Towards a generic absoluteness theorem for Chang models},
   JOURNAL={Advances in Mathematics},
   VOLUME = {476},
   ISSUE = {},
   YEAR = {2025},
   PAGES = {110357},
   EPRINT = {2304.07623},
   DOI = {10.1016/j.aim.2025.110357}}

Let $\Gamma^\infty$ be the set of all universally Baire sets of reals. Inspired by earlier work of Sargsyan and Trang, we introduce a new technique for establishing generic absoluteness results for models containing $\Gamma^\infty$.

Our main technical tool is an iteration that realizes $\Gamma^\infty$ as the sets of reals in a derived model of some iterate of $V$. We show, from a supercompact cardinal $\kappa$ and a proper class of Woodin cardinals, that whenever $g \subseteq \operatorname{Col}(\omega, 2^{2^\kappa})$ is $V$-generic and $h$ is $V[g]$-generic for some poset $\mathbb{P}\in V[g]$, there is an elementary embedding $j: V\rightarrow M$ such that $j(\kappa)=\omega_1^{V[g*h]}$ and $L(\Gamma^\infty, \mathbb{R})$ as computed in $V[g*h]$ is a derived model of $M$ at $j(\kappa)$.

As a corollary we obtain that $\mathsf{Sealing}$ holds in $V[g]$, which was previously demonstrated by Woodin using the stationary tower forcing. Also, using a theorem of Woodin, we conclude that the derived model of $V$ at $\kappa$ satisfies $\operatorname{AD}_{\mathbb{R}}+``\Theta$ is a regular cardinal”.

Inspired by core model induction, we introduce the definable powerset $\mathcal{A}^\infty$ of $\Gamma^\infty$ and use our derived model representation mentioned above to show that the theory of $L(\mathcal{A}^\infty)$ cannot be changed by forcing. Working in a different direction, we also show that the theory of $L(\Gamma^\infty, \mathbb{R})[\mathcal{C}]$, where $\mathcal{C}$ is the club filter on $\wp_{\omega_1}(\Gamma^\infty)$, cannot be changed by forcing. Proving the two aforementioned results is the first step towards showing that the theory of $L(Ord^\omega, \Gamma^\infty, \mathbb{R})([\mu_\alpha: \alpha \in Ord])$, where $\mu_\alpha$ is the club filter on $\wp_{\omega_1}(\alpha)$, cannot be changed by forcing.