Spring 2023 Talks

Day	Speaker	Title and Abstract
01/27	Sam	A gentle introduction to ambidexterity via examples In this talk we will motivate ambidexterity starting from some basic constructions in algebraic topology and finite group representations, with a focus on examples from a class of topological quantum field theories. No knowledge of physics is assumed. Along the way, we will discuss what ambidexterity even is, and how a toy construction behooves us to consider it. Time permitting, we’ll look at (∞,n)-categorical (extended TQFT’s) and higher chromatic analogues. Hardly any chromatic homotopy theory will be required however. In fact, having seen Morava K-theory in some context before is probably sufficient. A related but different example will be reserved for the next talk.
02/03	Sam	Ambidexterity via examples again This talk will motivate and define ambidexterity in low dimensions using a twisted version of Dijkgraaf–Witten theory over the complex numbers, noting how the twists make the underlying structure more readily apparent. Giving a simple (inductive) class of examples, we will see why even after just a few steps it will be natural to ask for ambidexterity in settings more exotic than complex vector spaces. Finally, we’ll go over some historical context and results. We likely won’t say much about the proofs since they are rather technical aside from some very broad steps, but if there’s enough time left, we may say a few words about insights gained from the proofs. No recollection of the previous talk (01/27) is assumed.
02/10	Yigal	Proving Bott periodicity for homotopy theorists Over the past 64 years, many proofs of the Bott periodicity theorem have appeared. Initially, such proofs required tools outside the domain of homotopy theory. Bott’s 1959 proof made heavy use of Morse theory, and Atiyah-Singer’s 1969 proof described the spaces in terms of Fredholm operators. In 1975, McDuff suggested that the Atiyah-Singer proof could be modified to “remove from it all the analysis”. This project was pursued by Aguilar-Prieto in 1999, and completed by Behrens in 2002 and 2004. In this talk, I will present Behrens’s proof, with the aim of convincing the disheartened homotopy theorist that they do in fact have the technology to prove their own theorem.
02/17	Sam	Transchromatic characters In the context of local systems on finite groups (and their higher analogues), it makes sense to consider higher (categorical and chromatic) representation theory. Interesting phenomena such as those intermediating between complete reducibility and unipotence show up, familiar from the difference between finite representation theory in characteristic 0 and characteristic p. It is then natural to wonder whether an analogous character theory exists. The answer is yes in many regards, but it is quite fascinating because we may additionally refine it by chromatic height and that it brings together several structures of interest in their own right. In this talk we will introduce the basic ingredients and context for transchromatic characters. Time permitting, we will look at three crucial properties which make this construction tick in a way that is amenable to further extension. No familiarity with classical character theory is assumed; we will recall some at the start. Some passing familiarity with Lubin–Tate spectra is recommended, however.
02/24	Anthony	Monads: My Best Friend The monad is a categorical descendant of the monoid. Also called the triple, it is derived from Godement’s standard resolution, and the modern monad is the standard resolution’s nearest living relative. Monads were the first category-theoretic tool domesticated by computer scientists over 15 years ago before the development of Rust. Due to their long association with adjunctions, monads have expanded to a large number of mathematical applications and gained the ability to describe a wide range of topological-like concepts. In this talk I will share an introduction to the theory of monads, emphasizing concrete definitions rather than the typical “monoidal object in an endofunctor category’’ approach. This will lead us to several examples of monads in topology and cohomology, as well as some interesting parallels to computer science. After sharing these examples, I will describe one key application of monads to homotopy theory: the bar construction.
03/03	CANCELLED
03/10	Doron	Towards a Chromatic Langlands Program Originally developed as a tool for computing the stable homotopy groups of spheres in topology, chromatic homotopy theory has proven to be highly interdisciplinary, possessing seemingly fundamental connections to number theory and the mathematics of high-energy physics. On the arithmetic side, it can be interpreted as a Brave New class field theory, with geometric models like tmf acting as a spectral version of arithmetic geometry. On the physical side, these complex-oriented cohomology theories act as receivers of “index maps” central to both concrete computations and geometry in quantum field theory. In this talk, I will propose a chromatic Langlands program that unifies notions of ramification, globalization, and equivariance appearing in these three fields. The approach taken involves modular and global equivariance for topological automorphic forms, cyclotomic trace as a description of geometric and algebraic ramification, and chromatic redshift; and, as suggested by the name, should be thought of as a spectral version of the Langlands program. Broadly speaking, the program aims to give a unified description of transchromatic geometry, and is expected to produce new computational tools in stable homotopy theory. Applications outside of topology include a rigorous interpretation of Witten’s equivariant index theory and arithmetic geometry over the field with one element.
03/24	NU CONFERENCE
03/31	Johnson	THH and cyclotomic structure In this talk we will be looking at the construction of the topological Hochschild homology THH(R) where R is a ring spectrum. In the situation where R has the structure of a commutative ring spectra then there is an equivalent description of THH(R) where certain structure becomes more easily constructed and we will explore what this provides for us. Time permitted variants of THH will be discussed.
04/07	Timmy	Algebraic cobordism and algebraic K theory In the 70’s, Snaith defined algebraic cobordism theory by using Quillen’s algebraic K theory space. Later in 08, Gepner and Snaith considered algebraic cobordism as a motivic spectrum and showed the motivic Conner-Floyd theorem in their paper. However, just like algebraic K theory is not A¹-invariant, there should be a non A¹-invariant version of algebraic cobordism. The relation between this non A¹-invariant version and the usual A¹-invariant version of algebraic cobordism should be analogous to the relation between algebraic (Thomason-Trobough) K theory and Weibel’s homotopy K theory. In this talk, we will start by introducing the non A¹-invariant version of algebraic cobordism based on the non A¹-invariant theory developed by Toni Annala, Marc Hoyois and Ryomei Iwasa. Then we will see how Algebraic Conner-Floyd isomorphism work in this context.
04/14	CANCELLED
04/21	Heyi	A construction for the ℂ-motivic topological modular forms spectrum I will talk about the work of Gheorghe, Isaksen, Krause and Ricka where they construct a cellular stable -motivic homotopy theory at p=2 without assuming algebro-geometric input via filtered spectra, which comes with a convenient set-up for the -motivic Adams-Novikov spectral sequence. In this context, the classical construction for tmf generalizes to a -motivic version whose cohomology is the quotient A//A(2) motivically as expected.
04/28	Langwen	The Kervaire invariant A surprising discovery in differential topology is that homeomorphic manifolds can admit different smooth structures; moreover, it is often possible to classify exotic smooth structures by homotopy theory. The work of Kervaire and Milnor (1963) determines the exotic smooth structures on spheres of dimension ≥5 up to knowing the stable homotopy groups of spheres and the “Kervaire invariant”. Hill, Hopkins and Ravenel (2016) recently determined this invariant in all dimensions except 126 using homotopy theoretic methods. I will give an introduction to the Kervaire invariant problem with a focus on the homotopy theoretic technique applied to solve it, namely equivariant stable homotopy theory and the equivariant slice spectral sequence.
05/04	Gabby	Dieudonne module and the classification of formal groups A Dieudonne module is a module over the ring of Witt vectors over some field k with two endomorphisms Frobenius F and Verschiebung V that satisfy certain relations. The category of formal groups over a perfect field k is equivalent to the category of finitely generated Dieudonné module under some mild condition, so the classification of formal groups reduces to the classification of such modules. In this talk, I will introduce the construction of Dieudonné modules and compute some Dieudonné modules associated with various formal groups.

Note that 05/04 is a Thursday, which is reading day. The time of day and location are still the same however.