Department of

# Mathematics

Seminar Calendar
for Topology events the year of Thursday, July 2, 2020.

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events for the
events containing

More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
      June 2020              July 2020             August 2020
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30 31


Friday, January 24, 2020

4:00 pm in 141 Altgeld Hall,Friday, January 24, 2020

#### Organizational Meeting

###### Nachiketa Adhikari (UIUC)

Abstract: We will draft a schedule of the seminar talks this semester. Please join us and sign up if you want to speak (you don't have to decide on a topic or abstract now). As usual, there will be cookies. All are welcome!

Friday, January 31, 2020

4:00 pm in 141 Altgeld Hall,Friday, January 31, 2020

#### Introduction to Orbifolds

###### Brannon Basilio (UIUC)

Abstract: We can generalize the notion of a manifold to include singularities; thus we can define a new object called orbifolds. In this talk, we will give an introduction to the notion of orbifolds, including examples, covering orbifold, Euler number of an orbifold, and the classification theorem of 2-orbifolds.

Tuesday, February 4, 2020

3:00 pm in 243 Altgeld Hall,Tuesday, February 4, 2020

#### Moduli spaces of Lagrangians in symplectic topology and mirror symmetry

###### James Pascaleff (UIUC)

Abstract: Moduli spaces of Lagrangians (as objects in Fukaya categories), and the geometry on such moduli spaces, may be used to understand problems in symplectic topology and mirror symmetry. In this talk, I will introduce these ideas and give an example showing how to use symplectic topology to solve a problem about Laurent polynomials (based on joint work with Dmitry Tonkonog).

Friday, February 7, 2020

4:00 pm in 141 Altgeld Hall,Friday, February 7, 2020

#### A probabilistic approach to quantizing Yang-Mills theory

###### Kesav Krishnan (UIUC)

Abstract: I will introduce the problem of rigorously quantizing Yang Mills Theory, and how probability theory can be used to that end. If time permits, I will talk about the discrete gauge-string duality as introduced by Sourav Chatterjee

Monday, February 10, 2020

3:00 pm in 243 Altgeld Hall,Monday, February 10, 2020

#### The moduli space of objects in the Fukaya category

###### James Pascaleff (Illinois)

Abstract: In this talk I will survey some tools from derived algebraic geometry that, when applied to Fukaya categories have applications to symplectic topology and mirror symmetry. (Note: this talk is connected with the talk I gave last week in the algebraic geometry seminar, but will be self-contained and largely disjoint.)

Tuesday, February 11, 2020

11:00 am in 243 Altgeld Hall,Tuesday, February 11, 2020

#### Vanishing and Realizability

###### Shane Clark (University of Kentucky)

Abstract: The Reidemeister trace of an endomorphism of a CW complex is a lower bound for the number of fixed points (up to homotopy) of that endomorphism. For an endomorphism $f$, the Reidemeister trace of $f^n$ is a lower bound for the number of fixed points of $f^n$, however it can be a far from an optimal lower bound. One method of addressing this discrepancy constructs an equivariant map, the n^{th} Fuller trace $f$, which carries information about the periodic points of a map $f$. However, we must ask how much information is retained by this equivariant construction? In this talk we show that the n^{th} Fuller trace of $f$ is a complete invariant for describing a minimum set of periodic points for maps of tori.

Friday, February 14, 2020

4:00 pm in 141 Altgeld Hall,Friday, February 14, 2020

#### Bounds on volumes of mapping tori

###### Heejoung Kim (UIUC)

Abstract: For a surface $S$ and a homeomorphism $f: S\to S$, the mapping torus of $S$ by $f$ is defined by $M_f=(S\times [0,1])/((x,0)\sim (f(x), 1))$. In particular, for a closed surface $S$ of genus at least 2 and a pseudo-Anosov element $f$ of the mapping class group of $S$, $M_f$ is a hyperbolic manifold. Brock provided bounds of the hyperbolic volume of $M_f$ from a hyperbolic structure on $M_f$ by using its Weil-Petersson metric. And then Agol gave a sharp upper bound for the volume in terms of the translation distance on the pants graph $P(S)$ which is associated with pants decomposition on $S$. In this talk, we will discuss mapping class groups and Agol's proof on the upper bound.

Friday, February 21, 2020

4:00 pm in 141 Altgeld Hall,Friday, February 21, 2020

#### Unifying Galois Theories with Categorification

###### Robert (Joseph) Rennie (UIUC)

Abstract: Since its inception nearly two centuries ago, what we call "Galois Theory" (say in an undergraduate algebra course) has led to many analogous results, and thus attained the status of a sort of metatheorem. In Galois' case, this concept was applied to fields, yielding an equivalence between some lattice of field extensions and a lattice of subgroups of a corresponding "galois group" ... under certain conditions. Later on, the same concept was shown to be present in Topology, with extensions being replaced by their dual notion of covering spaces, and the galois group being replaced by the fundamental group... again, under certain conditions. Even later, Galois' results for fields were generalized to arbitrary rings, introducing new associated data along the way. In this talk, we explore the process of formally unifying all of these "Galois Theories" into one Galois Principle, with the aim of developing an intuition for identifying some of its infinite use-cases in the wilds of Math (e.g. Algebra, Topology, and Logic). Along the way, I aim to discuss explicitly and to motivate categorification to the working mathematician using the results of this talk as concrete examples.

Tuesday, February 25, 2020

11:00 am in 243 Altgeld Hall,Tuesday, February 25, 2020

#### \'Etale K-theory

###### Akhil Mathew (U Chicago)

Abstract: I will explain some general structural results about algebraic K-theory and its \'etale sheafification, in particular its approximation by Selmer K-theory. This is based on some recent advances in topological cyclic homology. Joint work with Dustin Clausen.

Friday, February 28, 2020

4:00 pm in 141 Altgeld Hall,Friday, February 28, 2020

#### Arnold-Liouville Theorem

###### Jungsoo (Ben) Park (UIUC)

Abstract: This talk will be an introduction to fundamental concepts of symplectic geometry. Furthermore, we will delve into the proof of Arnold-Liouville theorem: https://en.wikipedia.org/wiki/Liouville–Arnold_theorem.

Monday, March 2, 2020

4:00 pm in 245 Altgeld Hall,Monday, March 2, 2020

#### Linear Analysis on Singular Spaces

Abstract: As an undergraduate, one may be introduced to the 3 classic linear differential equations: Laplace’s equation, the heat equation, and the wave equation. Simply trying to solve these equations in different coordinate systems leads to a zoo of different solutions; such variation reflects the strong connection between the geometry of a space, and the behavior of solutions to these PDE on that space. Passing from Euclidean space to more general manifolds, these three equations can be studied whenever our manifold is equipped with the geometric structure of a Riemannian metric. In this talk I will highlight a few of the many surprising theorems exhibiting this connection between the geometry and topology of a manifold and the behavior of solutions to the Laplace, heat, and wave equation. Time permitting, I’ll highlight recent joint work with Pierre Albin of some new phenomena on certain singular spaces.

Tuesday, March 3, 2020

11:00 am in 243 Altgeld Hall,Tuesday, March 3, 2020

#### Approximating higher algebra by derived algebra

###### William Balderrama

Abstract: Obstruction theories and spectral sequences from the basic computational tools for accessing complex homotopical structure by means of pure algebra. Many of these are constructed via a careful examination of the relevant notion of free resolution''; the difficulties in their construction are then in maintaining sufficient control over these resolutions, as well as in the identification of the relevant obstruction groups. I will describe a general conceptual framework for producing these tools, based on a higher categorical variation on the notion of an algebraic theory, which is easily applicable to a wide variety of situations and provides a direct bridge between homotopical structure and algebraic structure.

Thursday, March 5, 2020

4:00 pm in 245 Altgeld Hall,Thursday, March 5, 2020

#### Plane Trees and Algebraic Numbers

###### George Shabat (Russian State University for the Humanities and Independent University of Moscow)

Abstract: The main part of the talk will be devoted to an elementary version of the deep relations between the combinatorial topology and the arithmetic geometry. Namely, an object defined over the field of algebraic numbers, a polynomial with algebraic coefficients and only two finite critical values, will be associated to an arbitrary plane tree. Some applications of this construction will be presented, including polynomial Pell equations and quasi-elliptic integrals (going back to N.-H. Abel). The relations with finite groups and Galois theory will be outlined. At the end of the talk the possible generalizations will be discussed, including the dessins d'enfants theory initiated by Grothendieck.

Tuesday, March 10, 2020

11:00 am in 243 Altgeld Hall,Tuesday, March 10, 2020

#### $C_2$-equivariant homotopy groups of spheres

###### Mark Behrens (Notre Dame Math)

Abstract: I will explain how $RO(C_2)$-graded $C_2$-equivariant homotopy groups of spheres can be deduced from non-equivariant stable homotopy groups of stunted projective spaces, and the computation of Mahowald invariants.

Friday, March 13, 2020

4:00 pm in 141 Altgeld Hall,Friday, March 13, 2020

#### Poincare duality for singular spaces

###### Gayana Jayasinghe (UIUC)

Abstract: Poincare duality of manifolds is a classical theorem which can be phrased in terms of the homology and cohomology groups of manifolds. However, when we look at singular spaces, this fails to hold for the usual homology and cohomology groups. In the setting of a certain class of singular spaces know as topological pseudomanifolds, which include orbifolds, algebraic varieties, moduli spaces and many other natural objects, one can extend these groups in order to recover some form of Poincare duality. I'll present how this was achieved by Goresky and MacPherson with their Intersection homology, and by Cheeger using L^2 cohomology and explain how they are related to each other, in similar spirit to the equivalence in the smooth setting. I'll only assume a basic knowledge of homology and cohomology.

Tuesday, March 24, 2020

11:00 am in 243 Altgeld Hall,Tuesday, March 24, 2020

#### CANCELLED

###### Christina Osborne (OSU Math)

Tuesday, April 7, 2020

11:00 am in 243 Altgeld Hall,Tuesday, April 7, 2020

#### To Be Announced

###### Ben Antieau (UIC )

Wednesday, May 13, 2020

3:30 pm in https://illinois.zoom.us/j/806582029 (email Anush Tserunyan for password),Wednesday, May 13, 2020

#### Introduction to 𝓁2-Betti numbers for groups, part 3: 𝓁2-homology

###### Ruiyuan (Ronnie) Chen (UIUC Math)

Abstract: We will first finish our discussion of ordinary group homology by giving an alternative definition via equivariant chain complexes, completely bypassing topology. We will then discuss $\ell^2$-homology, defined by replacing (chain) homotopy quotients by $\ell^2$-completion to Hilbert $\Gamma$-modules. The $\ell^2$-Betti numbers are the von Neumann dimensions of the resulting homology Hilbert $\Gamma$-modules.