Oberseminar Algebra und Geometrie

Winter term 2025/26: Definable Groups in o-Minimal Structures

Chaired by I. Halupczok, H. Kammeyer and B. Klopsch.

All talks take place on Fridays at 14:30 in 25.22.03.73.

If you want get announcements about the seminar, please get in touch with I. Halupczok so that your address is added to the mailing list.

Infos für Studierende

Das Oberseminar richtet sich an alle, die einen Einblick in aktuelle Forschung erhalten möchten, ist aber tendenziell eher für fortgeschrittene Studierende geeignet (ab Master). Besonders empfohlen wird die Teilnahme an Oberseminaren, wenn Sie sich vorstellen können zu promovieren. Wenn Sie intessiert sind, können Sie sich einfach (ohne Anmeldung) ins Seminar reinsetzen - gerne auch nur zu einzelnen Vorträgen, die Sie interessieren.

Die Vorträge in diesem Oberseminar sind auf englisch. Üblicherweise nehmen an Oberseminaren auch viele Doktorand:innen, Postdocs und Professor:innen teil. Wahrscheinlich werden Sie nicht alles verstehen; das passiert aber auch den fortgeschritteneren Teilnehmer:innen, und auch, wenn man nicht alles verstanden hat, hat man doch am Ende oft einen interessanten Einblick in ein neues Thema erhalten.

Aims and Content

tba

Schedule

(The dates of the talks are temporary and might still change.)

17.10.:tba: Tameness and semi-algebraic sets
Present the material from [Cos, §1.1-§1.2], i.e.: Give some intuition about tame vs. non-tame geometry; explain in which sense semi-algebraic sets in $\mathbb{R}^n$ provide a tame geometry; in particular state the Tarski-Seidenberg Theorem; mention that all this also works in other real closed fields than $\mathbb{R}$. (If necessary, recall what a real closed field is.)
Other sources: [Mar, p.1], [Dri, p.1 to middle of p.2], [BCR, ch. 1, 2]
24.10.:tba: O-Minimality
Present the material from [Cos, §1.3 up to ex. 1.11], i.e.: define structures expanding real closed fields; define o-minimality; prove (or mention) some basic facts about them (1.6 - 1.11); mention some examples of o-minmimal structures from [Mar, §3], at least $\mathcal{S}_{\mathrm{an}}$ and $\mathcal{S}_{\mathrm{exp}}$ (they are defined on p. 353).
Other sources: [Mar, §2], [Dri, §1.2-§1.3]
31.10.:tba: Basic results in o-minimal structures
Present the material from [Cos, §1.3, Prop 1.12 onwards] and [Dri, §1.3, (3.3)-(3.7)]; most importantly: following [Cos], introduce the notion of first order formula and prove Thm 1.13. (You can use the proof of [Cos, 1.12] as a motivation for [Cos, 1.13]: it becomes a lot easier using [1.13]; see [Dri, §1.3, (3.4)].) Concerning definable connectedness [Dri, (3.5)], not all of [Dri, Lemma (3.6)] needs to be stated, but as a motivation, at least state (1), and mention also why classical connectedness fails.
7.11.:tba: The Monotonicity Theorem and the Finiteness Lemma
Present the material from [Dri, §3.1], but skip the (long and complicated) proofs of the Monotonicity Theorem ((1.3)-(1.5)) and of the Finiteness Lemma (p.47-48); instead: state the Monotonicity Theorem (1.2); prove Corollaries (1.6) 1 and 2 (about existence of limits and minima/maxima); state the Finiteness Lemma, but instead of the version (1.7), state the stronger version (2.13) on p. 53; if time permits, state consequence (1.8).
If you want to give a taste of the proof of the Monotonicity Theorem, you can prove one of its ingredients, namely (1.3) Lemma 1. (The proof is in (1.5).)
Other source: [Cos, §2.1 and Thm 2.9]
14.11.:tba: Cell Decomposition and Piecewise Continuity
Present the material from [Cos, §2.2] and [Dri, §3.2] (those are more or less the same), but omit the big proofs. More concretely: define Cells and (Cylindrical Definable) Cell Decompositions [Cos, Defn 2.4) = [Dri, (2.3), (2.4), (2.10)]; state and partially prove some basic facts about cells, most importantly [Dri, (2.5)] $\approx$ [Cos, 2.6], [Dri, (2.7)], [Dri, (2.9)]; state the Cell Decomposition Theorem [Cos, 2.10] and Piecewise Continuity [Cos, 2.12] (see also [Dri, (2.11)]); maybe give some easy application like [Cos, 2.11], [Dri, (2.18)]. Maybe mention the (essentially trivial) $n=1$ case of the proofs of [Cos, 2.10+2.12].
21.11.:tba: Dimension
Present the material from [Cos, §3.3], i.e.: mention the ad-hoc definition of dimension of a cell given above [Cos, 2.5]; give the "proper" definition [Cos, 3.14]; mention [Cos, 2.5] and deduce that for cells, we have proper dim $\ge$ naive dim. (Maybe make [Cos, 2.5] plausbible by proving it for intervals.) State [Cos, Lemma 3.15] (the "key technical result about dimension"); deduce nice properties of dimension: [Cos, 3.16, 3.17, 3.18]; for 3.18, introduce the notion of a "definable family", which is given above [Cos, 3.11]. Also state [Cos, 3.22] (but omit the proof). If time permits, you can also say something about the proof of [Cos, Lemma 3.15].
Other source: [Dri, §4.1]
28.11.:tba: Generic elements
This talk should probably be given by someone with a decent background in model theory. Goals of this talk: introduce the notion of dimension $\dim(a/A)$ of a tuple over a parameter set [Pil, Defn. 1.1]; introduce the notion of generic elements from [Pil, below Defn. 1.3]; convince the audience that we can pretend that generic elements exist. Here is a possible plan for such a talk: Introduce the notion of $A$-definability, for parameter sets $A$; define a tuple $b$ to be generic in $X$ over $A$ (for $X$ $A$-definable) if $b$ is not contained in a lower-dimensional $A$-definable set. Maybe define $\operatorname{dcl}(A)$. (This might be useful later in your talk.) Prove that in $\RR$ with e.g. the pure ring structure (or any other o-minimal structure with countably many definable sets), generic elements exist: In intervals, this is for cardinality reasons; in general, this can be obtained using cell decomposition. To obtain generics in general, mumble something about elementary extensions and/or ultraproducts. (You can give a quick intro to that, but only if this does not take away too much time. Note: It seems to me that we *only* need a sufficiently big model, but no real saturation.) Mention and maybe prove some basic properties of $\dim(b/A)$, like [Pil, 1.2], but use your definition of $\dim(b/A)$. Note that the properties of dimensions of definable sets induce properties of $\dim(b/A)$. Also show that if $b_1$ and $b_2$ are interdefinable over $A$, then $\dim(b_1/A) = \dim(b_2/A)$.
Other sources: your favourite model theory books; ask Immi
5.12.:tba: Largeness, definable groups and definable manifolds
Introduce the notion of a definable group [Pil, first paragraph of §0]. State [Pil, 1.8]. (Note: An relation $\sim$ on a definable set $X$ is called definable if the set $\{(a,b) \in X \times X \mid a \sim b\}$ is definable.) Introduce the notion of "definable manifold" from [Pil, Rem. 1.10]. Define largeness [Pil, 1.11] and prove the basic properties [Pil, 1.12, 1.13] and the first equality of [Pil, 1.14]. Present [Pil, 2.1-2.4] (the main goal being 2.4). State [Pil, 2.5] (the main goal of Pillay's paper) and relate it to [Pil, 1.10]. (Don't say anything about the proof of [Pil, 2.5]; that's the next talk.)
12.12.:tba: Definable groups are topological groups
Present as much as seems reasonable from the proof of [Pil, 2.5]. Instead of trying to rush through everything, rather focus on some "example arguments" where one can see the main ideas. You could for example skip many technical details related to getting the group operation continuous.
19.12.:tba: Definable groups behave like Lie groups
Present (some of) the material from [Pil, 2.6-2.16]. Feel free to choose what seems interesting to you.
9.1.:tba: Pillay's Conjecture
Present the material of [Pet, §2], but only talk about o-minimal structures that expand real closed fields. Probably, you should be vague concerning the notion of the real closed field $\mathcal{R}$ being $\kappa$-saturated: just say that it is much bigger than $\mathbb{R}$ (and in particular contains infinitesimal elements). And let's pretend that we can take $\kappa = |\mathbb{R}|$, so that one can define ``type-definable'' to mean ``intersection of less than $|\mathbb{R}|$ many definable sets''. Spending some time on the examples on p. 6 is probably nice.
16.1.:tba: Existence of $G^{00}$
Sketch the proof of parts (1) and (2) of Pillay's Conjecture (as stated in [Pet, p.5]). Present the material of [Pet, §4]. I suggest explaining the statement of [Pet, Thm 4.1], not saying much about its proof, but explain how that Theorem implies the above (1) and (2).
23.1.:tba: The dimension statement
Sketch the proof of part (3) of Pillay's Conjecture (as stated in [Pet, p.5]). I suggest following the sketch given in [Pet, §3].

References:

[Cos] Coste, M.: An Introduction to o-Minimal Geometry
[Dri] van den Dries, L.: Tame Topology and o-Minimal Structures
[Mar] Marker, D.: Review of Tame topology and o-minimal structures
[Pet] Peterzil, Y.: Pillay's Conjecture and its Solution-a Survey
[Pil] Pillay, A.: On Groups and Fields Definable in o-Minimal Structures