Conference in honor of Saharon Shelah’s 80th birthday
July 14 and 15, 2025, at TU Wien, Austria
No satisfaction guaranteed.
Schedule including some abstracts
| Monday July 14: Model Theory Day | ||
| 08:30–09:10 | Registration (at seminar room) | |
| 09:20–09:50 | Boris Zilber | TBA |
| 10:00–10:30 | Itay Kaplan | Existence over a predicate Joint project with Martin Bays and Pierre Simon. Suppose that L contains a predicate symbol P and that T is some complete theory such that P is stably embedded. We give a sufficient condition that implies existence: any model of the induced theory T^P is the P part of a model T and give some examples where existence fails. |
| 10:30–11:10 | Coffee break (at lecture hall) | |
| 11:10–11:40 | Artem Chernikov | Externally definable groups A set is externally definable in a structure M if it is given by the intersection with M of a set definable (with parameters) in some elementary extension of M. If all types over M are definable (for example, if the theory of M is stable), then all externally definable sets are already definable. This fails beyond stability, e.g. in linear orders (take a cut of some irrational number over the rationals) or in the generic Rado graph (where all subsets of a model are externally definable). An important theorem of Shelah shows that the former case is much better behaved than the latter: the expansion of an NIP structure M by all externally definable sets M^Sh eliminates quantifiers, and remains NIP. We discuss externally definable groups, and show that all "definably compact" groups externally definable in an NIP structure are already definably isomorphic to groups interpretable in it. Our proof relies on honest definitions and a group chunk result reconstructing a hyper-definable group from its multiplication given generically with respect to a translation invariant definable Keisler measure. |
| 11:50–12:20 | Chris Laskowski | Equivalents of NOTOP Shelah introduced the notions of NDOP and NOTOP in his celebrated proof of the `Main Gap' for countable theories. We survey the history and state many new equivalents of NOTOP within the context of countable, superstable theories. We prove that for any such theory, NOTOP implies that an arbitrary model $N$ is atomic over an independent tree of countable, elementary substructures. In particular, $N$ is determined up to back-and-forth equivalence by the tree. This is joint work with Danielle Ulrich. |
| (break) | ||
| 13:40–14:10 | John Baldwin | When is an $\aleph_1$-categorical $L_{\omega_1,\omega}$-sentence $\omega$-stable? Morley's proof that an $\aleph_1$ categorical first order theory $T$ is $\omega$-stable depends on the existence of a models of $T$ with cardinality up to $\beth_{\omega_1}$ A first order theory is categorical in $\aleph_1$ iff it is $\omega$-stable and has no two cardinal models; this characterization is easily seen to be absolute. Already in \cite{Sh88} Shelah had given an example of an $L_{\omega_1,\omega}(Q)$ sentence that was categorical under MA but not under $2^{\aleph_0}<2^{\aleph_1}$. Whether there is a similar example for $L_{\omega_1,\omega}$ remains open. I review a series of works with Laskowski and Shelah addressing this issue. We reformulated the problem as the study of atomic models of first order theories. In \cite{BLSmanymod}, we introduced the notion of pseudo-algebraicity and then of a pseudo-minimal set as an analog for strong minimality and prove: If $T$ is a countable first order theory has an atomic model and fewer than $2^{\aleph_1}$ models in $\aleph_1$ then every non-pseudo-algebraic formula is implied by a pseudo-minimal formula. In \cite{BLlarge} we show a pseudo-minimal sentence has a model in the continuum. In \cite{BLSwhendoes} we showed that a sentence of $L_{\omega_1,\omega}$ that is categorical in $\aleph_0$ and $\aleph_1$ and has model in $\beth_1^+$ is syntactically $\omega$-stable. This reduces the Hanf number from $\beth_{\omega_1}$ to $\beth_1^+$. We note that the sentence need not be Galois $\omega$-stable. |
| 14:20–14:35 | Gregory Cherlin | Laudatio |
| 14:40–15:10 | Saharon Shelah | |
| 15:10–15:50 | Coffee break (at lecture hall) | |
| 15:50–16:20 | Pierre Simon | On expansions of linear orders and NIP I will present an elementary question on expansions of linear orders and explain how it would help in developing classification theory for NIP structures. |
| 16:30–17:00 | Thomas Scanlon | Tilting as a bi-interpretation We propose a model theoretic interpretation of the theorems about the equivalence between mixed characteristic perfectoid spaces and their tilts. More specifically, we show that the tilting construction may be seen as half of a bi-interpretation in the sense of continuous logic and that fundamental results in the theory of perfectoids fall out of general properties of preservation under bi-interpretation. We close with an open problem on the classification of untilts up to isomorphism insprired by constructions of Kedlaya and Temkin. This is a report on joint work with Silvain Rideau-Kikuchi and Pierre Simon. |
| 17:10–17:40 | Maryanthe Malliaris | TBA |
| 18:00–21:00 | Conference dinner (invitation only) | |
| Tuesday July 15: Set Theory Day | ||
| 09:20–09:50 | Mohammad Golshani | Questions that haunt my nights, while Shelah might answer by dawn In a conversation between Saharon Shelah and Alexander Soifer, Shelah asked, ""Why do people attend conferences?"" Soifer responded, ""To show their latest results, to learn about the achievements of others, and to socialize."" Shelah disagreed, saying, ""None of this makes any sense. People should attend conferences in order to solve together problems they could not solve on their own."" In this talk, I will present some of my favorite problems—some that I’ve thought about, and others that I’d like to explore in the future (which means I may know very little about some of them!). |
| 10:00–10:30 | Gianluca Paolini | Classifications problems in abelian group theory I will survey my work with Saharon on abelian groups, in various different directions. I will focus mainly on our solution to the problem of Borel completeness of torsion-free abelian groups and other anti-classification results stemming from those techniques (completeness of various rigidity conditions on abelian groups). We will then explore various results on Polish abelian groups, including non-admissibility of Polish topologies for certain abelian groups and the existence of many $\aleph_1$-free non-archimedean Polish abelian groups. |
| 10:30–11:10 | Coffee break (at lecture hall) | |
| 11:10–11:25 | Tomek Bartoszynski | Laudatio |
| 11:30–12:00 | Saharon Shelah | |
| (break) | ||
| 13:20–13:50 | Assaf Rinot | Marginalia to [Sh:365] Solovay famously proved that every stationary subset of a regular uncountable cardinal kappa may be decomposed into kappa many stationary sets. Classical variations of which are due to Ulam and Hajnal (with respect to maximally-complete ideals) and due to Shelah (with respect to club-guessing ideals). Here, we study a general case that captures and extends the above results in addition to results of Chang, Kunen-Prikry, and Eisworth. Our main result is a sought-after extension of [Sh:365, Claim 3.3] that yields an optimal extension of [Sh:365, Conclusion 4.8(2)]. Specifically, we get that at every inaccessible admitting a stationary set non-reflecting at inaccessibles, the square-bracket Ramsey relation fails with the maximal number of colors. The breakthrough here is obtained by replacing the classical `least’ function associated with ideals by a two-dimensional `last’ function associated with walks on ordinals. This is joint work with Tanmay Inamdar. |
| 14:00–14:30 | Moti Gitik | TBA |
| 14:40–15:10 | Stevo Todorcevic | TBA |
| 15:10–15:50 | Coffee break (at lecture hall) | |
| 15:50–16:20 | Jerome Keisler | Probability theory with continuous model theory We give an efficient way to apply continuous model theory to study random variables and stochastic processes. |
| 16:30–17:00 | Justin Moore | The second to last word on the consistency of the five element basis for uncountable linear orders We prove (in ZFC) that if $A$ is an Aronszajn line, there is a proper forcing extension in which $A$ has a Countryman suborder. As a corollary, if $L$ is an uncountable linear order, $X$ is any set of reals of cardinality $\aleph_1$ and $C$ is any Countryman line, there is a proper forcing extension in which $L$ contains an isomorphic copy of one of the following linear orders: $\omega_1$, $\omega_1^*$, $X$, $C$, or $C^*$. In particular if there is an inaccessible cardinal, then there is a generic extension in which there is a five element basis for the uncountable linear orders. This is joint work with John Krueger. |
| 17:10–17:40 | Matt Foreman | TBA |
Organizers: Martin Goldstern and Jakob Kellner