Department of

Mathematics


Seminar Calendar
for Graduate Student Homotopy Theory Seminar events the year of Friday, October 22, 2021.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
    September 2021          October 2021          November 2021    
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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.

Friday, October 8, 2021

3:00 pm in 243 Altgeld Hall,Friday, October 8, 2021

Simplicial Localizations and How to Find Them

Doron Grossman-Naples (UIUC Math)

Abstract: Abstractly defining \(\infty\)-categorical localization is easy, but explicitly constructing it is hard. Following a series of papers by Dwyer and Kan, I will describe the construction known as the hammock localization and use it to obtain a clearer picture of some important \(\infty\)-categories.

Friday, October 15, 2021

3:00 pm in 243 Altgeld Hall,Friday, October 15, 2021

Cellular homotopy type theory: why and how?

Samuel Hsu (UIUC Math)

Abstract: In this talk we will give some motivations for cellular homotopy type theory, by first looking at its expected semantics and discussing some applications and related literature. We will then go over a bit of basic machinery that goes into the semantics. Despite the title, we won't actually spend much time on syntax, instead focusing on connections to other work in homotopy theory, algebraic topology, and higher category theory, at least when n=1 i.e. the simplicial case. No prior knowledge of the topics is assumed.

Friday, October 22, 2021

3:00 pm in 243 Altgeld Hall,Friday, October 22, 2021

An introduction to equivariant motivic homotopy theory

Timmy Feng (UIUC Math)

Abstract: Equivariant motivic homotopy theory is the homotopy theory of motivic spaces and spectra with algebraic group actions. Following [Hoy17], I will introduce both the unstable and stable equivariant motivic homotopy ∞-category. I will also talk about the so called Ambidexterity Theorem which identifies the left and right adjoint of the pullback functor f* up to a suspension. Then I will show how to prove the Atiyah Duality by using the Ambidexterity and some other properties of these functors. [Hoy17] Marc Hoyois, The six operations in equivariant motivic homotopy theory.

Friday, October 29, 2021

3:00 pm in 243 Altgeld Hall,Friday, October 29, 2021

Connections between motivic and classical homotopy theory

Johnson Tan (UIUC Math)

Abstract: In the first part of this talk we will motivate the construction of the stable motivic homotopy category over the complex numbers through a non-standard construction of the stable homotopy category. After some comparisons between the motivic and classical theory, we will introduce the Chow t-structure and explain how it relates to some chromatic theory.

Friday, November 12, 2021

3:00 pm in 243 Altgeld Hall,Friday, November 12, 2021

∞-categorical Kummer theory

Abhra Kundu (UIUC Math)

Abstract: Kummer theory provides a way of recognizing the isomorphism classes of (certain) Galois extensions of fields containing enough roots of unity. Running a similar machinery for general classical commutative rings leads to the classification of (certain) Galois extensions of the ring which involves the Picard spectrum of the ring. Schlank, et al. have designed a version of this theory that works for nice presentable additive symmetric monoidal ∞-categories which involves the Picard spectrum of this category. I will go over their version of the theory in this talk.

Friday, November 19, 2021

3:00 pm in 243 Altgeld Hall,Friday, November 19, 2021

Introduction to the Stolz-Teichner Program

Connor Grady (UIUC Math)

Abstract: The Stolz-Teichner program is a far-reaching research program that aims to connect QFT to the cohomology of manifolds. Importantly for homotopy theorists, it is expected to provide a cochain model for TMF. In this talk I will sketch some of the broad strokes of the program and describe some of the partial results currently known.

Friday, December 3, 2021

3:00 pm in 243 Altgeld Hall,Friday, December 3, 2021

Towards Splitting $BP<2> ⋀ BP<2>$ at Odd Primes

Liz Tatum (UIUC Math)

Abstract: In the 1980s, Mahowald and Kane used Brown-Gitler spectra to construct splittings of $bo ⋀ bo$ and $l ⋀ l$. These splittings helped make it feasible to do computations using the $bo$- and $l$-based Adams spectral sequences. I will discuss progress towards an analogous splitting for $BP<2> ⋀ BP<2>$ at odd primes.