Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, February 19, 2019.

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events for the
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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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Tuesday, February 19, 2019

11:00 am in 345 Altgeld Hall ,Tuesday, February 19, 2019

#### G-equivariant factorization algebras

###### Laura Wells (Notre Dame Math)

Abstract: Factorization algebras are a mathematical tool used to encode the data of the observables of a field theory. There are various notions of factorization algebra: one can define a factorization algebra on the open subsets of some fixed manifold; or alternatively, one can define a factorization algebra on the site of all manifolds of a given dimension with specified geometric structure. In this talk I will outline a comparison between two such notions: G-equivariant factorization algebras on a fixed model space M and factorization algebras on the site of all manifolds quipped with a (G, M)-structure (given by an atlas of charts in M and transition maps in G). I will introduce the definitions of these two concepts and then sketch the proof of their equivalence as (\infy,1)-categories.

1:00 pm in 345 Altgeld Hall,Tuesday, February 19, 2019

#### Realizations of countable Borel equivalence relations

###### Forte Shinko (Caltech)

Abstract: By a classical result of Feldman and Moore, it is known that every countable Borel equivalence relation can be realized as the orbit equivalence relation of a continuous action of a countable group on a Polish space. However, if we impose further conditions, such as requiring the action to be minimal, then it is no longer clear if such a realization exists. We will detail the progress on characterizing when realizations exist under various conditions, including a complete description in the hyperfinite case. This is joint work with Alexander Kechris.

2:00 pm in 243 Altgeld Hall,Tuesday, February 19, 2019

#### Small Doublings in Abelian Groups of Prime Power Torsion

###### Souktik Roy (Illinois Math)

Abstract: Let $A$ be a subset of $G$, where $G$ is a finite abelian group of torsion $r$. It was conjectured by Ruzsa that if $|A+A|\leq K|A|$, then $A$ is contained in a coset of $G$ of size at most $r^{CK}|A|$ for some constant $C$. The case $r=2$ received considerable attention in a sequence of papers, and was resolved by Green and Tao. Recently, Even-Zohar and Lovett settled the case when $r$ is a prime. In joint work with Yifan Jing (UIUC), we confirm the conjecture when $r$ is a power of prime.

3:00 pm in 243 Altgeld Hall,Tuesday, February 19, 2019

#### Symplectic Springer theory

###### Kevin McGerty (University of Oxford and UIUC)

Abstract: One of the classical results of geometric representation theory is Springer's realization of representations of a Weyl group in the cohomology of the vanishing locus of nilpotent vector fields on the associated flag variety. A rich strain of current research focuses on attempting to extend aspects of Lie theory to the more general context of conical symplectic resolutions''. We will discuss, based on the discovery of Markman and Namikawa that such varieties have a natural analogue of a Weyl group, to what extent one can build an analogue of Springer's theory in this context, recovering for example a construction of Weyl group actions on the cohomology of quiver varieties, first discovered by Nakajima, which unlike previous construction does not require painful explicit verification of the braid relation.

4:00 pm in 243 Altgeld Hall,Tuesday, February 19, 2019

#### Learnability Can Be Undecidable

###### Jacob Trauger (University of Illinois at Urbana–Champaign)

Abstract: This seminar will be on the paper by Shai Ben-David et al, NATURE Mach. Intel. vol 1, Jan 2019, pp 44–48. The author's abstract reads: "The mathematical foundations of machine learning play a key role in the development of the field. They improve our understanding and provide tools for designing new learning paradigms. The advantages of mathematics, however, sometimes come with a cost. Gödel and Cohen showed, in a nutshell, that not everything is provable. Here we show that machine learning shares this fate. We describe simple scenarios where learnability cannot be proved nor refuted using the standard axioms of mathematics. Our proof is based on the fact the continuum hypothesis cannot be proved nor refuted. We show that, in some cases, a solution to the ‘estimating the maximum’ problem is equivalent to the continuum hypothesis. The main idea is to prove an equivalence between learnability and compression."

4:00 pm in 245 Altgeld Hall,Tuesday, February 19, 2019

#### Altgeld-Illini Renovation/Building Project Feedback Session

Abstract: This feedback session will focus on recent renovations done at other math departments. Specific questions that will be discussed at the meeting are:

a. What other math departments have been built or redone in the last 20 years?
b. What is your general impression of each of these spaces?
c. What specific features of particular places are worth copying?