Fall 2022 Talks

Day	Speaker	Title
09/02	Sai	Picard groups in homotopy theory Picard groups have been classically studied in commutative algebra, algebraic number theory and algebraic geometry. These can be suitably generalised to define Picard groups of 𝔼 ͚ -ring spectra. However, computing these groups is not very easy. In this talk, I will introduce two main methods to compute Picard groups of 𝔼 ͚ -ring spectra.
09/09	Zach	“Real” K-Theory, Bott Periodicity, and the Equivariant Slice Filtration The equivariant slice filtration and spectral sequence was one of the key tools used in the Hill, Hopkins, Ravenel resolution of the nonexistence of elements of Kervaire invariant 1. In this talk, I wish to motivate and introduce the slice filtration, by looking at the context where (a specific case of it) was first studied by Dan Dugger. So called “Real” K−theory is a C₂−equivariant spectrum, mixing information about complex K−theory with orthogonal K−theory. There exists an RO(C₂)−graded analog of Bott periodicity that implies both complex and orthogonal Bott periodicity as corollaries. However this periodicity is not detected by the equivariant Postnikov filtration. The slice filtration is then presented as an alternative to the Postnikov filtration which does in fact detect this periodicity.
09/16	Johnson	From algebraic K-theory to THH and cyclotomic spectra The algebraic K-theory of a commutative ring is an important invariant with applications in algebra, number theory, and topology. Compared to its topological counterpart, computations for algebraic K-theory are much more subtle as it lacks many nice properties of the former. In this talk we will give an overview of a method of studying algebraic K-theory by looking at computable approximations such as Hochschild homology and its variants.
09/23	Yigal	Power operations in K-theory Cohomology theories are invariants of spaces that are relatively computable while retaining much of the structure of a space. When a ring structure is present, some topological information is detected via the multiplication operation alone. Power operations are additional algebraic structures that a cohomology theory might have, and they have a particularly neat description in K-theory through an equivariant perspective. After a brief overview of cohomology operations and power operations in general, I will introduce equivariant K-theory, and use it to describe the power operations in K-theory. If there is time, I will try to address relationships with operations in other cohomology theories, as well as the relationship between power operations and spectra.
09/30	Sam	Barwick and Schommer-Pries’s Unicity Axioms We sketch out some of the ideas behind Barwick and Schommer-Pries’s unicity axioms for the ∞-category of (∞,n)-categories for 1<n<∞. In this talk, we will try to give some motivations including some examples beyond the cobordism hypothesis. Next, we will try to emphasize the more geometric, algebraic, or homotopical aspects, in particular invoking topological intuition when possible, and try to keep the combinatorics to a minimum (or really gloss over a large amount of content in general). No knowledge of higher category theory should be required so long as one is willing to take some basic ∞-categorical results for granted. Time permitting, we may discuss an extension of the unicity axioms to d-Cartesian fibrations, how to get a description of the ∞-equipment of (∞,n)-categories, and something resembling an internal language of the latter.
10/07	Anthony	Homotopical Ideas in First-Order Logic The interaction between logic and homotopy theory is dominated by higher category theory and type theories. However, recent developments in traditional first-order logic are being interpreted in an increasingly homotopical paradigm, leading to new Stone-duality-esque theorems by Makkai, Butz, Moerdijk, Lurie, and others. These results show that—while type theories have an internal homotopy theory—there exists an external homotopy theory of first-order theories. This talk is a survey of the history of categorical logic and its applications for homotopy theory, as well as a discussion on why these homotopical results are showing up in the first place.
10/14	Alex	An Introduction to Twisted K-theory Twisted ordinary cohomology arises naturally when one attempts to construct a notion of Poincare Duality for nonoriented manifolds. One might wonder how much this situation generalizes to other cohomology theories. Twisted K-theory is one such generalization. It should be thought of as K-theory with a more refined notion of degree. We will go over the basic definitions of Twisted K-theory and show how it naturally arises when attempting to produce pushforward maps.
10/21	Langwen	An overview of Lubin-Tate spectra In their 1966 paper, Lubin and Tate studied the deformation theory of 1-dimensional commutative formal groups. They showed that this moduli problem is representable by a certain ring with an associated “universal deformation” formal group law. It turns out that this formal group law is Landweber exact and the associated spectrum has a unique E ͚ multiplication, due to a theorem of Goerss, Hopkins and Miller. I will introduce these spectra and briefly discuss their central role in the local structure of chromatic homotopy theory.
10/28	Doron	Chromatic Homotopy: What, Why, and How Despite its success in computational and abstract applications, the budding field of chromatic homotopy theory remains somewhat inaccessible to newcomers. In contrast to some other areas of homotopy theory (stable, simplicial, rational, etc.), it can be difficult to say what exactly chromatic homotopy is about. This is because its central concept is not one idea, but rather the relationship between three different ones. In this talk, I will describe the structural, computational, and geometric aspects of chromatic homotopy, and how they relate to one another. The goal is that the next time attendees encounter a reference to the “chromatic point of view”, they have the context to understand what that means. This talk should be accessible to anyone familiar with stable homotopy theory and algebraic geometry.
11/04	Timmy	The motivic Adams isomorphism for finite group actions Frank Adams proves the following surprising statement in 1984: For a finite CW complex X with a nice finite group action, the orbit spectrum and the fixed point spectrum of Σ ᪲ X are isomorphic. Lewis, May and Steinberg then generalized Adams’ result to arbitrary spectra in 86. In this talk, I will discuss a motivic analogue of the above results, following the work of Gepner and Heller. The proof is dependent on the existence of the ‘six functors formalism’ but is more than that. I will roughly explain how the proof works and show you some applications of the motivic Adams isomorphism.
11/11	CANCELLED
11/18	Heyi	Basic constructions for the 1-category theory of ∞-categories I will start with my interpretation of one major motivation for homotopy theorists to work with ∞-categories and introduce a nowadays favorite model, i.e., quasi-categories, for these. Then I will discuss the first properties of the nerve and homotopy category construction and justify the viewpoint of 1-categories as ∞-categories. I will end with the definition of an (∞)-space in this framework and give the easier direction of its identification with a Kan complex. This talk is (almost entirely) based on the first two chapters of Charles’ lecture notes for the higher cats course he offers every ~three semesters and the speaker hopes that her talk serves as a proper advertisement for its upcoming installment!
12/02	Abhra	Descent theory In this talk, I will discuss descent in the ∞-categorical context. I will try to motivate the theory of descent and state some basic properties of descent. Along the way we will also look at some examples and applications of this theory. I will try to make the case for descent being an important tool not just for understanding complicated ∞-categories using relatively simpler ones but also as a tool for getting extra information out of particular kinds of co-simplicial rings which is not visible through Dold-Kan correspondence. Some prior knowledge of ∞-categories will be helpful but will not be assumed. I will try to give as “down to earth” description as I can of the things I will be talking about.