Department of

# Mathematics

Seminar Calendar
for Graduate Student Homotopy Theory Seminar events the year of Wednesday, September 16, 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.