Department of

Mathematics


Seminar Calendar
for Model Theory and Descriptive Set Theory Seminar events the year of Monday, October 26, 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

What generic automorphisms of the random poset look like

Dakota Thor Ihli (UIUC Math)

Abstract: The Fraïssé limit of the class of finite posets, also called the random poset, admits generic automorphisms — that is, its automorphism group admits a comeagre conjugacy class. This result, due to D. Kuske and J. Truss, was proven without explicitly describing the automorphisms in question. Here we give a new, concrete description of the generic automorphisms, and we discuss the tools-and-tricks involved.

Friday, November 6, 2020

4:00 pm in Zoom (email ruiyuan at illinois for info),Friday, November 6, 2020

The fractal properties of real sets defined by Büchi automata

Christian Schulz (UIUC Math)

Abstract: Büchi automata are a natural counterpart to finite automata that accept infinite strings instead of finite strings as input. We consider the k-automatic sets, which are subsets of $[0, 1]^n$ consisting of all tuples with a base-k expansion recognized by a given Büchi automaton. These sets often exhibit fractal behavior; for instance, the Cantor set and Sierpinski carpet are both 3-automatic. In this talk, we give methods for measuring various fractal properties of a k-automatic set, including box-counting dimension and Hausdorff dimension and measure, by examining the structure of its Büchi automaton. This is joint work with Alexi Block Gorman and Philipp Hieronymi.

Friday, November 13, 2020

4:00 pm in Zoom (email ruiyuan at illinois for info),Friday, November 13, 2020

GraphSAT -- a novel decision problem combining SAT and graph theory

Vaibhav Karve (UIUC Math)

Abstract: Graph theorists often care about forbidden-graph characterizations of graph properties. For example, Kuratowski's theorem says that a graph is planar iff it does not "contain" K5 or K3,3. We replace "planar" by a new graph property inspired by Boolean Satisfiability (SAT) and look for similar forbidden graphs in the hope that it will tell us something about SAT. We will also talk about various ways to translate "containment" to GraphSAT and as a result we will encounter a host of other decision problems on the way. We will show a full forbidden-graph characterization of 2-GraphSAT and partial results for 3-GraphSAT. This talk requires no prerequisites other than familiarity with complexity classes P, NP, NP-complete etc. This is joint work with Anil Hirani (UIUC Math)

Friday, November 20, 2020

4:00 pm in Zoom (email ruiyuan at illinois for info),Friday, November 20, 2020

A universal separable homogeneous Banach lattice

Mary Angelica Tursi (UIUC Math)

Abstract: A metric structure $\mathfrak{M}$ is considered approximately ultra-homogeneous if for any finitely generated structure $A =\langle a_1,\dotsc,a_n\rangle$, embeddings $f,g:A \rightarrow \mathfrak{M}$, and $\varepsilon > 0$, there exists an automorphism $\phi$ on $\mathfrak{M}$ such that $d(\phi \circ f (a_i), g(a_i) ) < \varepsilon $. We show that there exists a separable approximately ultra-homogeneous Banach lattice $\mathfrak{BL}$ that is isometrically universal for separable Banach lattices by proving that finitely generated Banach lattices form a metric Fraïssé class. If time permits, we will also explore some interesting properties about $\mathfrak{BL}$.