Department of

# Mathematics

Seminar Calendar
for Graduate Student Homotopy Theory Seminar events the year of Monday, October 26, 2020.

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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 27, 2020

3:00 pm in Altgeld Hall 441,Monday, January 27, 2020

#### Organizational meeting

###### William Balderrama (Illinois Math)

Monday, February 3, 2020

3:00 pm in 441 Altgeld Hall,Monday, February 3, 2020

#### Exotic elements in Picard groups

###### Ningchuan Zhang (Illinois Math)

Abstract: In this talk, I will discuss the subgroup of exotic elements in the $K(h)$-local Picard groups. We will first show this subgroup is zero when $p\gg h$ and then focus on the $(h,p)=(1,2)$ and $(2,3)$ cases.

Monday, February 10, 2020

3:00 pm in 441 Altgeld Hall,Monday, February 10, 2020

#### Exotic elements in Picard groups (part 2)

###### Ningchuan Zhang (Illinois Math)

Abstract: In this talk, I will discuss the subgroup of exotic elements in the $K(h)$-local Picard groups. We will first show this subgroup is zero when $p\gg h$ and then focus on the $(h,p)=(1,2)$ and $(2,3)$ cases.

Monday, February 17, 2020

3:00 pm in 441 Altgeld Hall,Monday, February 17, 2020

#### A geometric perspective on the foundations of modern homotopy theory

###### Brian Shin (Illinois Math)

Abstract: Homotopy theorists have always been interested in studying spaces. However, the meaning of the word space'' has evolved over the years. Whereas one used to say space to mean a topological space, it seems the modern stance is to view a space as an $\infty$-groupoid. In this expository talk, I would like to connect the modern stance back to geometry. In particular, I will demonstrate how the $\infty$-category of spaces can be built out of the category of manifolds. As an application, we will use this connection to give a geometric perspective on infinite loop space theory.

Monday, February 24, 2020

3:00 pm in 441 Altgeld Hall,Monday, February 24, 2020

#### An introduction to motivic homotopy theory

###### Brian Shin (Illinois Math)

Abstract: Motivic homotopy is often thought of as the homotopy theory of algebraic varieties. In this expository talk, we'll see exactly what that means. In particular, we'll see how the construction of the category of motivic spaces is a direct algebro-geometric analog of that of the category of spaces. More interestingly, we'll also see how the analogy breaks down.

Monday, March 2, 2020

3:00 pm in 441 Altgeld Hall,Monday, March 2, 2020

#### Relations between Spectral Sequences

###### Liz Tatum (Illinois Math)

Abstract: Consider a ring spectrum E and a spectrum X. The E-based Adams Spectral Sequence is a tool for approximating the homotopy groups $\pi_{*}X$. Depending on the choice of ring spectrum E, the Adams spectral sequence might be easier to compute, but might give a weaker approximation to $\pi_{*}X$. One could ask “If A, B are two different ring spectra, what can an A-based Adams spectral sequence tells us about a B-based Adams spectral sequence”? In the paper “On Relations Between Adams Spectral Sequences, With an Application to the Stable Homotopy of a Moore Space”, Miller proves a theorem addressing this question. In this talk, I’ll introduce some of the tools Miller uses to formulate and prove this theorem, and outline the previously mentioned application.

Monday, September 21, 2020

3:00 pm in Zoom,Monday, September 21, 2020

#### Straightening and unstraightening

###### Antonio Ruiz (UIUC)

Abstract: For any quasicategory $X$ with objects $w,x$, $\textrm{Hom}(w,x)$ is a Kan complex. Given any edge, $f: x \rightarrow y$ in $X$, we can construct a functor $f_{*}: \textrm{Hom}(w,x) \rightarrow \textrm{Hom}(w,y)$ but this construction is such that for a pair of composable edges, $f: x \rightarrow y$ and $g: y \rightarrow z$, the lifts $g_{*}f_{*}$ and $(gf)_{*}$ are not necessarily equal, only homotopic. Consequently, $\textrm{Hom}(w,-)$ does not induce a functor from $X$ into the category of Kan complexes. We will show how we can `straighten' a homotopy coherent functor (i.e. a left fibration) such as $\textrm{Hom}(w,-)$ into an actual functor between infinity categories. Please email vb8 at illinois dot edu for the zoom details.

Monday, October 12, 2020

3:00 pm in Zoom,Monday, October 12, 2020

#### Picard groups in homotopy theory

###### Sai (UIUC)

Abstract: A very classical invariant of a commutative ring $R$ is the Picard group. In this talk I will introduce some "interesting" Picard groups in homotopy theory and some techniques to compute them. Please email vb8 at illinois dot edu for the zoom details.

Monday, October 19, 2020

3:00 pm in Zoom,Monday, October 19, 2020

#### An Introduction to Algebraic $K$-Theory

###### Brian Shin (UIUC)

Abstract: In this expository talk I'd like to give a brief introduction to algebraic $K$-theory. We'll start with an short historical overview. We'll then go into one of the many successful definitions for higher algebraic $K$-theory, that given by Waldhausen. If time permits, I'll give a slightly more conceptual perspective on Waldhausen $K$-theory. Please email vb8 at illinois dot edu for the zoom details.

Monday, October 26, 2020

3:00 pm in Zoom,Monday, October 26, 2020

#### Introduction to simplicial sets

###### Haoyuan Li (UIUC)

Abstract: This is an introduction to simplicial sets. I will first talk about the definition of simplicial sets and also some examples of simplicial sets. Then I will introduce the Kan complex and some of its properties. If time permits, I will talk a little bit about the homotopy groups of Kan complex. Please email vb8 at illinois dot edu for the zoom details.

Monday, November 2, 2020

3:00 pm in Zoom,Monday, November 2, 2020

#### Unfolding Orbifold Theory

###### Joseph Rennie (UIUC)

Abstract: Orbifolds, since their inception in the 1950's have taken on many different mathematical definitions despite their simple underlying concept. This talk will start with the conceptual definition and end with a definition based on equivariant sheaves, all without making any unmotivated changes to the definitions along the way. Please email vb8 at illinois dot edu for the zoom details.

Monday, November 9, 2020

3:00 pm in Zoom,Monday, November 9, 2020

#### Equivariant formal group laws

###### Tsutomu Okano (UIUC)

Abstract: Formal group laws arise in homotopy theory through complex oriented cohomology theories. They are particularly interesting for finite characteristic and led to the creation of many “designer spectra" in stable homotopy theory. In this talk I will introduce equivariant formal group laws for abelian compact Lie groups and highlight some known structural results. I will also talk about some key differences and difficulties compared to the non-equivariant setting. Please email vb8 at illinois dot edu for the zoom details.

Monday, November 16, 2020

3:00 pm in Zoom,Monday, November 16, 2020

#### Brown-Gitler Spectra and Their Applications

###### Elizabeth Tatum (UIUC)

Abstract: Brown-Gitler spectra are a class of spectra whose cohomology form certain submodules of the Steenrod algebra. These spectra were originally constructed to study immersions of manifolds, but they have turned out to have many other applications in homotopy theory. In this talk, we’ll define Brown-Gitler spectra and discuss some applications. In particular, we’ll focus on their use in investigating $bo$- and Adams $l$-summand cooperations. Please email vb8 at illinois dot edu for the zoom details.