Department of

# Mathematics

Seminar Calendar
for Graduate Student Homotopy Theory Seminar events the year of Thursday, October 10, 2019.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
    September 2019          October 2019          November 2019
Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
1  2  3  4  5  6  7          1  2  3  4  5                   1  2
8  9 10 11 12 13 14    6  7  8  9 10 11 12    3  4  5  6  7  8  9
15 16 17 18 19 20 21   13 14 15 16 17 18 19   10 11 12 13 14 15 16
22 23 24 25 26 27 28   20 21 22 23 24 25 26   17 18 19 20 21 22 23
29 30                  27 28 29 30 31         24 25 26 27 28 29 30



Monday, January 14, 2019

3:00 pm in 343 Altgeld Hall,Monday, January 14, 2019

#### Organizational Meeting

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

Monday, April 1, 2019

3:00 pm in 343 Altgeld Hall,Monday, April 1, 2019

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

Abstract: In this talk I will discuss how (higher) operads help us encode monoidal structures in (higher) categories. I will also discuss how to generalize this to parametrized settings and hope to convey the usefulness of such formalism in equivariant and motivic homotopy theories.

Monday, April 8, 2019

3:00 pm in 343 Altgeld Hall,Monday, April 8, 2019

#### Mapping space spectral sequences

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

Abstract: The classical story of obstruction theory for computing maps into a space Y involves lifting maps up the Postnikov tower of Y. In this talk, I will introduce a form of this obstruction theory for computing maps between highly structured objects in homotopy theory. Along the way, we will see why Quillen cohomologies show up in homotopy theory, take derived categories of derived categories, and take multiplicative Postnikov towers of nonconnective ring spectra.

Monday, April 15, 2019

3:00 pm in 343 Altgeld Hall,Monday, April 15, 2019

#### Group Theory for Homotopy Theorists

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

Abstract: In this expository talk, we'll introduce a model structure on the category of groups and demonstrate how to effectively study groups using this model. This model has the technical advantage of avoiding the overly abstract definition of a group via sets with binary operation. It also allows for clean definitions of colimits and free objects. If time permits, we'll discuss monoidal structures for a certain localization of this model structure. This is based on a short article by Krause-Nikolaus.

Monday, April 22, 2019

3:00 pm in 343 Altgeld Hall,Monday, April 22, 2019

#### Complex structures on Real vector bundles

###### Abhra Abir Kundu (UIUC Math)

Abstract: In this talk, I will state the first and the second obstruction to having a stable complex structure on a real vector bundle. I will then show how one can go from stable complex structure to complex structure. And, if time permits, I will try to sketch how the second obstruction can be expressed as a secondary cohomology operation.

Monday, April 29, 2019

3:00 pm in 343 Altgeld Hall,Monday, April 29, 2019

#### Crystalline period map

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

Abstract: In Chromatic homotopy theory, one tries to understand the homotopy groups of spheres using the height filtration on formal group laws. This way at each height we get a spectral sequence whose $E_2$ term is the group cohomology of the Morava stabilizer group with coefficients in the Lubin-Tate ring. In this talk, I hope to explain a conceptual way to figure out the action of the Morava Stabilizer group on the Lubin-Tate ring.

Monday, August 26, 2019

3:00 pm in 441 Altgeld Hall,Monday, August 26, 2019

#### Organizational Meeting

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

Monday, September 9, 2019

3:00 pm in 441 Altgeld Hall,Monday, September 9, 2019

#### Algebraic theories and homotopy theory

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

Abstract: In this talk, I will motivate and introduce algebraic theories as a category-theoretic approach to finite product theories. I will then talk about a well-behaved notion of an infinitary algebraic theory, and the introduction of homotopy-theoretic structure, which can be used to define notions of homology and cohomology for the models of an algebraic theory. This is the first of two talks; the second will use these ideas to produce applications in stable homotopy theory.

Monday, September 16, 2019

3:00 pm in 441 Altgeld Hall,Monday, September 16, 2019

#### Modeling higher algebra with product-and-loop theories

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

Abstract: In this talk, I will introduce the extra homotopical properties of a (suitably infinitary) algebraic theory that make it suitable for modeling spectral, or otherwise higher, algebra, rather than merely derived forms of ordinary algebra. To illustrate the utility of this viewpoint, I will indicate some of the computational tools that can be constructed and understood from this perspective. Time permitting, I will discuss some applications to chromatic homotopy theory.

Monday, September 23, 2019

3:00 pm in 441 Altgeld Hall,Monday, September 23, 2019

#### Splitting $BP\langle 1\rangle \wedge BP\langle 1\rangle$ at odd primes

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

Abstract: The Adams Spectral Sequence is a tool for approximating $\pi_{*}X$, where $X$ is a connective spectrum. If $E$ is a ring spectrum satisfying certain properties, then we can define an $E$-based Adams spectral sequence converging to $\pi_{*}\hat{X}$, where $\hat{X}$ is the $E$-completion of $X$. When $E_{*}E$ is flat over $E_{*}$, the $E^{2}$-page of the spectral sequence can be described as $Ext_{E_{*}E}(E_{*}, E_{*}X)$. But if $E_{*}E$ is not flat over $E_{*}$, then there is no such description. Instead, we must study $E\wedge E$ to understand the spectral sequence. The Brown-Peterson spectra $BP\langle n \rangle$ are an example of such spectra. One approach is to split the product $E \wedge E$ into more manageable pieces. When $n=1$, we can construct a splitting $BP\langle 1 \rangle \wedge BP\langle 1 \rangle$ as $\vee_{k=0}^{\infty}\Sigma^{2k(p-1)} BP\langle 1 \rangle 1 \wedge B(k)$, where $B(k)$ is the $k^{th}$ integral Brown-Gitler spectrum. We give a sketch of Kane's construction of this splitting for odd primes.

Monday, September 30, 2019

3:00 pm in 441 Altgeld Hall,Monday, September 30, 2019

#### The Recognition Principle for Infinite Loop Spaces

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

Abstract: In this expository talk, I'd like to discuss the infinite loop space recognition principle. In particular, I'd like to examine Boardman-Vogt's infinite loop space machine from a modern view point.

Monday, October 7, 2019

3:00 pm in 441 Altgeld Hall,Monday, October 7, 2019

#### Motivating Higher Toposes: Geometric Characteristic Classes

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

Abstract: This will be part one of two talks aimed at motivating higher topos theory from physics. In this talk we will give a brief history of classical obstructions for manifolds, then suddenly find ourselves naturally requiring tools from higher topos theory. In the end, we shall see how working with simplicial sheaves on Manifolds allows us to define (but more importantly compute) geometric characteristic classes.

Monday, October 14, 2019

3:00 pm in 441 Altgeld Hall,Monday, October 14, 2019

#### Motivating Higher Toposes: Higher Bundle Theory

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

Abstract: In this (self-contained) talk, I will begin with a quick recap of the motivation for higher bundle theory from the first talk. I will then say a few words about Toposes, and proceed to spend the majority of the talk attempting to develop a general theory of higher bundles. Along the way, we will see how the necessary properties for this development (almost) force higher topos structure. (Technical details will be sacrificed for intuitive clarity. No particular model of higher categories will be imposed.)