Department of

# Mathematics

Seminar Calendar
for Graduate Student Homotopy Theory Seminar events the year of Tuesday, March 13, 2018.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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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.