Spring 2021 Talks

Day	Speaker	Title
02/01	Heyi	Thom spectra via rigid spaces The classical definition of the space GL₁(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 GL₁(R) as an A ͚ -space. If time permits, we will also briefly look at their E ͚ -version.
02/08	William	Spectral sequences and deformations of homotopy theories I 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.
02/15	William	Spectral sequences and deformations of homotopy theories II
02/22	Brian	Structure of the Motivic Stable Homotopy Category In classical homotopy theory, a first step in understanding the stable homotopy category is understanding the zeroth stable stem π₀ 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.
03/01	Joseph	Generalized Gauge Theory: Where Logic Meets Homotopy Theory 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.
03/15	Tsutomu	Equivariant motivic orientations 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.
03/22	Timmy	An introduction to Milnor conjecture 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.
03/29	Liz	The Telescope Conjecture 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.
04/05	Sam	Equivariant BPQ and Bicategorical Enrichment 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.
04/19	Abhra	Topological Modular forms with level structure Goerss-Hopkins-Miller theorem gives us a way of extracting homotopic information hidden inside the Moduli Stack of Elliptic curves by constructing an étale 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 étale 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 étale 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.