Department of

# Mathematics

Seminar Calendar
for Graduate Student Homotopy Theory Seminar events the year of Tuesday, August 3, 2021.

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events for the
events containing

More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
      July 2021             August 2021           September 2021
Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
1  2  3    1  2  3  4  5  6  7             1  2  3  4
4  5  6  7  8  9 10    8  9 10 11 12 13 14    5  6  7  8  9 10 11
11 12 13 14 15 16 17   15 16 17 18 19 20 21   12 13 14 15 16 17 18
18 19 20 21 22 23 24   22 23 24 25 26 27 28   19 20 21 22 23 24 25
25 26 27 28 29 30 31   29 30 31               26 27 28 29 30



Monday, January 25, 2021

3:00 pm in Zoom,Monday, January 25, 2021

#### Organisational meeting

###### Sai (UIUC)

Abstract: This is the organisational meeting for the graduate student homotopy theory seminar. Please email vb8 at illinois dot edu for the zoom details.

Monday, February 1, 2021

3:00 pm in Zoom,Monday, February 1, 2021

#### Thom spectra via rigid spaces

###### Heyi Zhu (UIUC)

Abstract: The classical definition of the space $GL_1(R)$ of units, given a ring spectrum $R$, does not play well with more modern models of spectra. In this talk, we will introduce Ando-Blumberg-Gepner-Hopkins-Rezk's Thom spectra functor, which builds upon a construction of $GL_1(R)$ as an $A_{\infty}$-space. If time permits, we will also briefly look at their $E_{\infty}$-version. Please email vb8 at illinois dot edu for the zoom details.

Monday, February 8, 2021

3:00 pm in Zoom,Monday, February 8, 2021

#### Spectral sequences and deformations of homotopy theories I

###### William Balderrama (UIUC)

Abstract: One of the oldest problems in stable homotopy theory is simple: just compute the stable homotopy groups of spheres. This turns out to be difficult, and a complete answer may never be known, but the computations continue. A recent technique being applied to great success can be summarized as: don't just compute the stable homotopy groups of the sphere spectrum, also compute the stable homotopy groups of other sphere spectra. These other sphere spectra include, for example, those arising in motivic and equivariant contexts, and can be thought of as deformations of the classical sphere spectrum. The homotopy theories they live in can then be thought of as deformations of classical stable homotopy theory. There are a few methods for building these deformations; the goal of this sequence of two talks is to describe the concrete approach to this deformation story using filtered objects. Please email vb8 at illinois dot edu for the zoom details.

Monday, February 15, 2021

3:00 pm in Zoom,Monday, February 15, 2021

#### Spectral sequences and deformations of homotopy theories II

###### William Balderrama (UIUC)

Abstract: One of the oldest problems in stable homotopy theory is simple: just compute the stable homotopy groups of spheres. This turns out to be difficult, and a complete answer may never be known, but the computations continue. A recent technique being applied to great success can be summarized as: don't just compute the stable homotopy groups of the sphere spectrum, also compute the stable homotopy groups of other sphere spectra. These other sphere spectra include, for example, those arising in motivic and equivariant contexts, and can be thought of as deformations of the classical sphere spectrum. The homotopy theories they live in can then be thought of as deformations of classical stable homotopy theory. There are a few methods for building these deformations; the goal of this sequence of two talks is to describe the concrete approach to this deformation story using filtered objects. Please email vb8 at illinois dot edu for the zoom details.

Monday, February 22, 2021

3:00 pm in Zoom,Monday, February 22, 2021

#### Structure of the Motivic Stable Homotopy Category

###### Brian Shin (UIUC)

Abstract: In classical homotopy theory, a first step in understanding the stable homotopy category is understanding the zeroth stable stem $\pi_0 \mathbf{S}$. The fact that this is the ring of integers leads to the idea that we may be able to study things "one prime at a time". In this expository talk, I'll talk about the analogous story in the setting of motivic homotopy theory. After reviewing basics of the motivic story, we'll see how knowledge of the motivic zeroth stable stem can be used to better understand the motivic stable homotopy category. Please email vb8 at illinois dot edu for the zoom details.

Monday, March 1, 2021

3:00 pm in Zoom,Monday, March 1, 2021

#### Generalized Gauge Theory: Where Logic Meets Homotopy Theory

###### Joseph Rennie (UIUC)

Abstract: Higher categories admit a notion of internal groupoids which Nikolaus et. al have shown yield a nice theory of principle bundles in any higher topos. An example of the practical use of this can be seen work of Freed-Hopkins where they define a higher topos of “generalized spaces” which then admits a universal bundle with connection. In an attempt to extend the results of Nikolaus to more kinds of categories, we inevitably end up working with the same kinds of structures as logicians. Namely, with pretoposes and logical functors, as opposed to the more homotopy theoretic grothendieck toposes and geometric morphisms. The goal of this talk will be to demystify this deep connection between model theory (in the logician’s sense) and homotopy theory. This talk will mostly operate at a conceptual level to more insightfully navigate the fact that many results that we want don’t yet have analogs proven in the higher-categorical setting, and the fact that the lower setting doesn’t quite have as nice of a picture. I assume no background in logic, and a vague awareness of the use (conceptually) of toposes in homotopy theory. Please email vb8 at illinois dot edu for the zoom details.

Monday, March 15, 2021

3:00 pm in Zoom,Monday, March 15, 2021

#### Equivariant motivic orientations

###### Tsutomu Okano (UIUC)

Abstract: For a finite abelian group A, I will introduce the notion of oriented spectra in A-equivariant motivic homotopy theory. Orientation yields a theory of Chern classes which can be used to compute the cohomology of Grassmannians. As an application, we obtain the equivariant motivic analogue of the Snaith theorem. Please email vb8 at illinois dot edu for the zoom details.

Monday, March 22, 2021

3:00 pm in Zoom,Monday, March 22, 2021

#### An introduction to Milnor conjecture

###### Timmy Feng (UIUC)

Abstract: Milnor conjecture (1970 by J.Milnor) states that the Milnor K-theory (mod 2) and the Galois/etale cohomology of a field (char not 2) in mod 2 coefficient are equivalent. In 1996, V.Voevodsky proved Milnor conjecture by using new theories and techniques including motivic cohomology, splitting varieties and cohomology operations. Bloch-Kato conjecture, which generalizes Milnor conjecture to mod l coefficients, was also proved in the following years. In the talk, I will start from the Milnor K-theory and its relation with the quadratic forms. I will also introduce the motivic cohomology and the higher dimensional analogues of Hilbert’s Theorem 90. Then, if time allowed, I’ll talk about the strategy of Voevodsky’s proof on mod 2 Milnor conjecture. Please email vb8 at illinois dot edu for the zoom details.

Monday, March 29, 2021

3:00 pm in Zoom,Monday, March 29, 2021

#### The Telescope Conjecture

###### Liz Tatum (UIUC)

Abstract: In his 1984 paper “Localization with Respect to Certain Periodic Homotopy Theories”, Ravenel made seven major conjectures about homotopy theory. While the rest of these conjectures were quickly proven and are an important part of the framework for chromatic homotopy theory, the telescope conjecture remains open. Roughly, the telescope conjecture claims that: “finite localization and smashing localization in the stable homotopy category are the same”. In this talk, we’ll discuss localization in the stable homotopy category and various ways to state the telescope conjecture. Time permitting, we’ll briefly discuss a generalization of this conjecture to other categories. Please email vb8 at illinois dot edu for the zoom details.

Monday, April 5, 2021

3:00 pm in Zoom,Monday, April 5, 2021

#### Equivariant BPQ and Bicategorical Enrichment

###### Samuel Hsu (UIUC)

Abstract: Following the work of Guillou, May, Merling, and Osorno, we give a (very) broad overview of the (2-)algebraic input that goes into their proof of the multiplicative equivariant Barratt-Priddy-Quillen theorem. Although it is not explicitly invoked, an underlying point we wish to make is the presence of bicategorical enrichment over the 2-category of categories internal to G-spaces when G is a finite group, where bicategorical enrichment is meant in the sense of e.g. Garner--Shulman, Franco, or Lack. This also opens up a pathway to concepts like enriched analogues of bicategorical concepts, less celebrated structures such as double multi or poly categories, and other devices which are related to the usual celebrities in formal category theory, which we might discuss existing or hoped applications for, time permitting. This talk is intended to be accessible with hardly any knowledge of homotopy theory. Please email vb8 at illinois dot edu for the zoom details.

Monday, April 19, 2021

3:00 pm in Zoom,Monday, April 19, 2021

#### Topological Modular forms with level structure

###### Abhra Kundu (UIUC)

Abstract: Goerss-Hopkins-Miller theorem gives us a way of extracting homotopic information hidden inside the Moduli Stack of Elliptic curves by constructing an ́etale presheaf of E∞-ring spectra. Evaluating the presheaf on some particular Modular curves produces TMF with level structure. This presheaf is not only defined on ́etale sites of the Moduli Stack of Elliptic curves but also on Moduli Stack of generalized Elliptic curves but unfortunately the modular curves in this case are no longer ́etale over this stack. So, the presheaf can no longer be evaluated on these modular curves. But, it turns out that by refining the topology on this stack one can define a presheaf which not only produces the universal object, Tmf , but also produces a functorial family of objects, Tmf with level structures, which are the analogs of TMF with level structures. In this talk I will state this result and try to explain this refinement. Please email vb8 at illinois dot edu for the zoom details.

Friday, October 1, 2021

3:00 pm in 243 Altgeld Hall,Friday, October 1, 2021

#### On the Lichtenbaum-Quillen conjectures in algebraic K theory

###### Likun Xie (UIUC Math)

Abstract: Starting with some motivations and brief expositions on algebraic K theory, I’ll introduce some early important computations of algebraic K theory, including computations of K theory of finite fields and of rings of integers for which I will briefly outline the proofs. Then we’ll move on to K theory with finite coefficients of separably closed fields. With the motivation of recovering some information of K theory of an arbitrary field from its separable closure, we introduce a few versions of the Lichtenbaum-Quillen conjectures as descent spectrum sequences of etale Cohomology groups. If time permits, I’ll mention relation to motivic Cohomology that a key tool is some “motivic-to-K-theory” spectral sequence.