Department of

# Mathematics

Seminar Calendar
for Graduate Student Homotopy Theory Seminar events the year of Friday, November 9, 2018.

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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.
     October 2018          November 2018          December 2018
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1  2  3  4  5  6                1  2  3                      1
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30 31


Monday, January 22, 2018

2:00 pm in 345 Altgeld Hall,Monday, January 22, 2018

#### Segal's Infinite Loop Space Machine

###### Brian Shin (UIUC Math)

Abstract: Infinite loop spaces arise as the spaces underlying connective spectra. The construction and analysis of such spaces has been an important part of homotopy theory. In this talk I will introduce Segal's infinite loop space machine, a device used to construct infinite loop spaces from algebraic data. Using this machine, I will prove a theorem of Barratt-Priddy-Quillen. Finally, I will indicate how these ideas are being put to use in motivic homotopy theory.

Monday, January 29, 2018

2:00 pm in 345 Altgeld Hall,Monday, January 29, 2018

#### The Bezoutian of a Rational Function

###### Daniel Carmody   [email] (UIUC Math)

Abstract: The computation of the pointed endomorphisms of $\mathbb P^1$ gives a first approximation to the zeroth stable homotopy group of the motivic sphere. In this introductory talk, following Cazanave, I’ll give some examples of a basic construction which associates a bilinear form to a rational function $\mathbb P^1 \rightarrow \mathbb P^1$. This will hint at a reason for the appearance of Grothendieck-Witt groups in stable motivic homotopy theory.

Monday, February 12, 2018

3:00 pm in Altgeld Hall 345,Monday, February 12, 2018

#### Thom Spectra

###### William Balderrama (UIUC Math)

Abstract: Thom spectra are suspension spectra with a twist, in the same way fiber bundles are products with a twist. Following this idea, I will introduce these objects and talk about some of the structure they can support.

Monday, February 19, 2018

3:00 pm in Altgeld Hall 345,Monday, February 19, 2018

#### Why should homotopy theorists care about homotopy type theory?

###### Nima Rasekh (UIUC Math)

Abstract: There is this new branch of mathematics known as homotopy type theory, which combines concepts from logic, computer science and homotopy theory. While it is clearly not necessary to know homotopy type theory to learn homotopy theory there are certain aspects that can be beneficial. The goal of this talk is to point to some of those aspects of homotopy type theory that can be beneficial to our understanding of homotopy theory.

Monday, March 5, 2018

3:00 pm in Engineering Hall 106B1,Monday, March 5, 2018

#### Multiplicative Norms

###### Tsutomu Okano (UIUC Math)

Abstract: I will introduce the idea of multiplicative norms in equivariant and motivic homotopy theories. This is a necessary structure behind the theory of genuine rings, which are supposed to be like E_\infty algebras with additional multiplication structure. For example, in equivariant orthogonal spectra, norms help us establish a model structure on commutative rings that plays well with restriction along the inclusion of a subgroup. In motivic homotopy theory, there is a formulation of norms in terms of infinity-categorical Grothendieck construction. I will show that although this is a nice way to package the story, it is difficult to work with in terms of computations.

Monday, March 12, 2018

3:00 pm in 345 Altgeld Hall,Monday, March 12, 2018

#### The geometry of the moduli stack of formal groups

###### Ningchuan Zhang (UIUC Math)

Abstract: The geometry of $\mathcal{M}_{fg}$, the moduli stack of formal groups, reflects many important concepts and results in chromatic homotopy theory. In this talk, I’ll first give the basic definition of stacks. After that, I’ll define $\mathcal{M}_{fg}$ and talk about its geometry: sheaves, substacks and Lubin-Tate deformation theory. I’ll also explain how the geometry of $\mathcal{M}_{fg}$ is related to chromatic homotopy theory.

Monday, April 9, 2018

3:00 pm in 345 Altgeld Hall,Monday, April 9, 2018

#### Unexpecteded Grassmannians in the Scanning Area

###### Cameron Rudd (UIUC Math)

Abstract: This talk will be a brief overview of a geometric corner of homotopy theory from the perspective of someone who doesn't know any homotopy theory. I will discuss spaces of embeddings, the scanning map and theorems of Barratt-Priddy-Quillen and Madsen-Weiss. Title is result of joint work with Hadrian Quan.

Monday, April 16, 2018

3:00 pm in Altgeld Hall 345,Monday, April 16, 2018

#### Tale of an Exotic Sphere

###### Venkata Sai Narayana Bavisetty (UIUC Math)

Abstract: I will start out by explaining how trying to classify manifolds naturally leads to the discovery of exotic spheres and then construct an exotic sphere in dimension 7.

Monday, April 23, 2018

3:00 pm in Altgeld Hall 345,Monday, April 23, 2018

#### Equivariant Homotopy Theory of Finite Spaces and Sylow Theorems

###### Joseph Rennie (UIUC Math)

Abstract: Finite topological spaces serve as great pedagogical tools, and not just because they are a source of counter examples. In this talk, I will go from zero to Sylow in about forty minutes, covering the essentials of the theory of finite spaces along the way. On our way to this unsurprising result are some rather surprising ones. In the remaining ten minutes, I will discuss a conjecture in Sylow Theory by Quillen.

Monday, September 17, 2018

3:00 pm in 345 Altgeld Hall,Monday, September 17, 2018

#### Trichotomies in Cohomology and Integrable Systems

###### Matej Penciak (UIUC Math)

Abstract: In integrable systems, it is often the case that solutions can be divided into 3 classes: rational, trigonometric, and elliptic. This trichotomy is also apparent in the 3 classes of cohomology theories admitting chern classes: (singular) cohomology, K-theory, and elliptic cohomology. In this talk I will describe work by Ginzburg, Kapranov, and Vasserot (and subsequently, many others) synthesizing these two phenomena.

Monday, October 8, 2018

3:00 pm in 345 Altgeld Hall,Monday, October 8, 2018

#### Descent and the Adams Spectral Sequence

###### Brian Shin (UIUC Math)

Abstract: This talk will be an exposition on Hess's framework for monadic descent. We will start with a discussion on monads. We will then move towards an understanding of some of the homotopical aspects of the theory. We will conclude with a discussion of the spectral sequence that naturally arises, and in particular demonstrate that the celebrated Adams and Adams-Novikov Spectral Sequences are instance of our general framework.

Monday, October 15, 2018

3:00 pm in 345 Altgeld Hall,Monday, October 15, 2018

#### Congruence and Group Cohomology

###### Ningchuan Zhang (UIUC Math)

Abstract: In this talk, I’ll explain the relation between congruence and (continuous) group cohomology of $\mathbb{Z}_p^\times$-representations in invertible $\mathbb{Z}_p$-modules. The first half of the talk will focus on explicit computations of the two sides (including the $p=2$ case). In the second half, the connection between congruence and group cohomology will be built using the chromatic resolution (Cousin complex) of the $\mathbb{Z}_p^\times$-representations. The discussion here also applies to open subgroups of $\mathbb{Z}_p^\times$.

Monday, October 22, 2018

3:00 pm in 345 Altgeld Hall,Monday, October 22, 2018

#### Power operations in complex $K$-theory

###### William Balderrama (UIUC Math)

Abstract: The complex $K$-theory of a finite-dimensional space $X$ is a ring that can be defined in terms of the structure of complex vector bundles over $X$. Certain operations on vector bundles, such as taking exterior powers, enrich the $K$-theory of $X$ with further structure, and this extra structure has historically added to the applicability of complex $K$-theory. In this talk, I will explain how these operations can be obtained from certain equivariant refinements of the $n$'th tensor power operations, relating these operations to the representation theory of symmetric groups.

Monday, October 29, 2018

3:00 pm in 345 Altgeld Hall,Monday, October 29, 2018

#### Multiplicative norms in equivariant and motivic homotopy theory

###### Tsutomu Okano (UIUC Math)

Abstract: The equivariant multiplicative norm construction has proven to be useful in the work of Hill Hopkins Ravenel. It can also be used to formulate genuine equivariant commutative ring spectra. I will give an overview of these ideas and discuss their connection with motivic homotopy theory.

Monday, November 5, 2018

3:00 pm in 345 Altgeld Hall,Monday, November 5, 2018

#### Witt groups and Real K-theory

###### Daniel Carmody (UIUC Math)

Abstract: Swan's theorem gives an equivalence of categories between vector bundles on a manifold and finitely-generated projective modules over the ring of continuous functions on the manifold. I'll discuss a variant of this theorem for vector bundles on a manifold with the additional data of a non-degenerate symmetric bilinear form.

Monday, November 12, 2018

3:00 pm in 345 Altgeld Hall,Monday, November 12, 2018

#### Modalities and Blakers-Massey

###### Joseph Rennie (UIUC Math)

Abstract: In this talk, I’ll introduce modality from a logical standpoint, then explore the interpretation within the logic of spaces provided by recent Type Theories. I’ll present the recent generalization of Blakers-Massey theorems to arbitrary higher toposes as well as some applications to homotopy theory.

Monday, December 3, 2018

3:00 pm in 345 Altgeld Hall,Monday, December 3, 2018

#### Whitehead’s Theorem for Modules Over the Steenrod Algebra

###### Liz Tatum (UIUC Math)

Abstract: In this talk, I’ll discuss a stable category of modules over the Steenrod algebra. This category has many parallels to the homotopy category of spaces, with Margolis homology playing the role of homotopy groups. For example, this category includes a version of Whitehead’s Theorem. I will present this theorem along with some related results and applications.