Department of

Mathematics


Seminar Calendar
for Model Theory and Descriptive Set Theory Seminar events the year of Sunday, October 18, 2020.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
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Friday, September 25, 2020

4:00 pm in Zoom (email ruiyuan at illinois for info),Friday, September 25, 2020

Characterizing companionability for expansions of o-minimal theories by a dense, proper subgroup

Alexi Block Gorman (UIUC Math)

Abstract: Recent works in model theory have established natural and broad criteria concerning the existence of model companions and the preservation of certain neostability properties when passing to the model companion. In this talk, we restrict our attention to the o-minimal setting. By doing so, we can isolate the sort of necessary and sufficient condition that can be elusive in more general settings. The central result is a full characterization for when the expansion of a complete o-minimal theory by a unary predicate that picks out a dense, divisible subgroup has a model companion. We will discuss examples both in which the predicate is an additive subgroup, and in which it is a mutliplicative subgroup. The o-minimal setting allows us to provide a full and geometric characterization for companionability, with a particularly elegant dividing line when the group operation is multiplication. We conclude with a brief discussion of neostability properties, and give examples that illustrate the lack of preservation for properties such as strong, NIP, and NTP2, though there are also examples for which some or all three of those properties hold.

Friday, October 2, 2020

4:00 pm in Zoom (email ruiyuan at illinois for info),Friday, October 2, 2020

T-convex T-differential fields and their immediate extensions (Part 1)

Elliot Kaplan (UIUC Math)

Abstract: Let T be an o-minimal theory extending the theory of ordered fields. A T-convex T-differential field is a model of T equipped with a T-convex valuation ring and a continuous T-derivation. This week and next week, I will discuss some of my recent work on immediate extensions of T-convex T-differential fields. This week will be focused on background (what a T-convex valuation ring is, what a T-derivation is, what immediate extensions are) and on examples of T-convex T-differential fields.

Friday, October 9, 2020

4:00 pm in Zoom (email ruiyuan at illinois for info),Friday, October 9, 2020

T-convex T-differential fields and their immediate extensions (Part 2)

Elliot Kaplan (UIUC Math)

Abstract: Continuing from last week, I will discuss T-derivations and strict extensions, define "eventual smallness", and sketch a proof of the main result: if T is polynomially bounded, then any T-convex T-differential field has an immediate strict extension which is spherically complete.

Friday, October 23, 2020

4:00 pm in Zoom (email ruiyuan at illinois for info),Friday, October 23, 2020

On the Pila-Wilkie Theorem

Neer Bhardwaj (UIUC Math)

Abstract: We prove Pila and Wilkie’s Counting theorem, following the original paper, but exploit cell decomposition more thoroughly to simplify the deduction from its main ingredients. Our approach in particular completely avoids ‘regular’ or $C^1$ smooth points, and related technology; which also allows simplifications around Pila’s ‘block family’ refinement of the result.

Friday, October 30, 2020

4:00 pm in Zoom (email ruiyuan at illinois for info),Friday, October 30, 2020

TBA

Dakota Thor Ihli (UIUC Math)