eCHT Minicourses

The telescopic approach to understanding the stable homotopy category involves breaking up and reconstructing the sphere spectrum into pieces that are built by telescopes, which are periodic spectra obtained from finite spectra by inverting a self map. Although the homotopy groups of these telescopes remain difficult to access at the moment, certain algebraic K-theory spectra give powerful invariants to detect information about them. The goal of this course is to explain some computational techniques for understanding the homology of telescopes with respect to algebraic K-theory spectra, and how features of the telescopic stable homotopy categories are reflected in these computations, such as the failure of the telescope conjecture.

This minicourse will begin with an introduction to global algebra through the lens of the representation ring. We will discuss restriction maps, transfer maps, character theory, power operations, Adams operations, and symmetric powers and how these structures fit together. This may take the first two lectures. Then we will describe the analogous structures for Burnside rings and Morava E-theory. Finally, we will end with a look at partition functors — which are algebraic structures built to study power operations.

In these lectures, we will present our work on computing differentials in the Adams spectral sequence, culminating in a solution to the Last Kervaire Invariant Problem in dimension 126. Specifically, we demonstrate that the element h6^2 survives, proving the existence of framed manifolds with Kervaire invariant one in dimension 126. There will be two talks each day, with a short break.

Talk I: Zhouli Xu will provide an overview of classical and modern techniques for computing differentials in the Adams spectral sequence. (Slides)

Talk II: Guozhen Wang will introduce two new methods we developed: the Generalized Leibniz Rule and the Generalized Mahowald Trick.

Talk III: Weinan Lin will demonstrate how to use his computer programs to do these computations, including running illustrative examples. (Slides)

Talk IV part 1, part 2: In the first half, Weinan Lin will highlight some important differentials whose proofs were found by computer programs. In the second half, Zhouli Xu will discuss the final ad hoc arguments needed to address the case in dimension 126. (Slides)

The notion of Segal space has been useful for modeling up-to-homotopy topological categories, and can be described via the so-called Segal maps. Two different approaches have led to the same generalization, known as 2-Segal spaces: while Dyckerhoff and Kapranov sought to generalize the Segal maps from a geometric point of view, Galvez-Carrillo, Kock and Tonks were motivated by making homotopy-theoretic versions of constructions in combinatorics. A key example of a 2-Segal space is the output of Waldhausen’s S-construction when applied to an exact category, and 2-Segal spaces satisfying certain finiteness assumptions provide a unifying treatment to Hall algebra constructions. In the first talk, we will give definitions and basic examples, and in subsequent talks we’ll introduce these two main themes in the subject, as well as some of the other applications of 2-Segal spaces that are currently being investigated.

The course will give the mathematical background of the recently settled (by Burklund, Hahn, Levy, and Schlank) telescope conjecture in stable homotopy theory. A “trailer” for the course will be the 30-minute introduction I plan to give in a special edition of the Princeton algebraic topology seminar on October 28. There will be some overlap with the eCHT course on chromatic homotopy theory given by Agnès Beaudry in May 2019. I will discuss Morava K-theory, Bousfield localization, the Hopkins-Smith periodicity theorem, formal group laws, the Morava stabilizer group and the chromatic filtration.

The slice filtration has become a powerful tool in equivariant stable homotopy theory over the past decade. In this minicourse, we will discuss the equivariant slice filtration and associated spectral sequence, beginning with the case of G=C_2. Along the way, we will discuss Mackey functors and Bredon homology. The goal of the minicourse is for participants to become comfortable with the techniques that are commonly involved in determining a slice tower and computing a slice spectral sequence. The course will be aimed at graduate students.

One of the central problems of stable homotopy theory is the computation of the stable homotopy groups. The most effective tool for computing the (2-primary) stable homotopy groups is the Adams spectral sequence. We will discuss how to set up and compute this spectral sequence, including the determination of differentials and the resolution of hidden extensions. The goal is for participants to become familiar with the skills required for carrying out the computation. We will discuss a variety of contributing ingredients, including the May spectral sequence, Massey products and Toda brackets, deformations of classical stable homotopy theory (i.e., motivic and synthetic stable homotopy theory), and machine-assisted computation.

This minicourse of four lectures is an introduction to a useful tool in an algebraic topologist’s arsenal: homological stability. Our goal is to explain what homological stability is, give you the tools to prove your own homological stability results, and provide an overview of the state-of-the-art. We will start by working out in detail homological stability for symmetric groups, originally due to Nakaoka. This example displays all the features of the “usual” homological stability argument, as formalised by Randal-Williams and Wahl. After we explain their argument and its input, we end by surveying more refined stability phenomena. These lectures are intended to be accessible to graduate students with a background in algebraic topology.

Equivariant homotopy theory has close connections to classical, chromatic, and motivic homotopy theory. Perhaps the most notable application is the use of equivariant computations in the solution to the Kervaire invariant one problem by Hill, Hopkins, and Ravenel. More recently, Pitsch, Ricka, and Scherer have completely characterized conjugation spaces using RO(G)-graded cohomology. Despite a growing body of work, many equivariant analogues of classical computations remain unknown.

This course will begin with a brief introduction to equivariant homotopy theory and RO(G)-graded cohomology. We will then turn to computations, focusing mainly on C_2, the cyclic group of order two. Even for C_2, the cohomology of a point is an infinite-dimensional non-Noetherian ring. While this complicates computations, we will learn some tools and techniques that simplify the calculations for a number of C_2-spaces. We will apply these techniques in several low-dimensional and accessible examples, including real projective space and the torus. This course is aimed at graduate students in homotopy theory.

This will be a four lecture crash course on calculational methods in unstable homotopy theory, containing a mixture of classical and some more modern material. There is a wide range of tools available with which to study the structure of the homotopy groups of spheres (and other spaces). In this course I will focus on the following: rational homotopy theory, the EHP sequence, the Goodwillie tower, and v_n-periodicity in unstable homotopy groups. I hope to explain how each of these informs our understanding of unstable homotopy theory and how they can be applied to calculations. These lectures are intended to be accessible to graduate students.

Motivic or A^1-homotopy theory provides a friendly environment for the construction of a good spectral sequence converging to Quillen’s algebraic K-groups of a smooth variety. The aim of this lecture series is to explain Voevodsky’s approach to this construction, which proceeds by introducing a specific “slice” filtration on the motivic stable homotopy category of a scheme. This filtration will be applied to a few P^1-spectra, among them the P^1-spectra representing hermitian K-theory and algebraic K-theory. A major example will be the slice filtration on the motivic sphere spectrum. Its analysis provides a partial proof of Morel’s theorem, which identifies the zeroth stable homotopy groups of motivic spheres with Milnor-Witt K-theory, as well as a computation of the first stable homotopy groups of motivic spheres.

At the center of homotopy theory is the classical problem of understanding the stable homotopy groups of spheres. Despite its simple definition, this object is extremely intricate; there is no hope of computing it completely. It hides beauty and pattern behind a veil of complexity.

Chromatic homotopy theory gives us a way to see through this veil by using the algebraic geometry of formal groups to organize theory and computations. The simplest case is via K-theory: Bott periodicity gives rise to a repeating family of elements in the stable homotopy groups of spheres via the image of the J homomorphism. It is an insight of Morava that there are higher analogues of K-theory and that they should give rise to higher periodicity in the stable homotopy groups of spheres.

In the 1980s, Ravenel and Hopkins made a series of conjectures describing this connection, most of which were proved in the 1980s and 1990s by Devinatz, Hopkins, Smith and Ravenel. Two of these problems remain open: the chromatic splitting conjecture and the telescope conjecture. The ultimate goal of the course will be to motivate and state these two famous problems.

The course will include a quick reminder of spectra and a brief introduction to complex orientations and localizations. We will discuss periodicity and the chromatic filtration, leading to a statement of the two conjectures. We will also discuss the higher K-theories and the role they play in modern computations. Many topics will only be touched briefly, as my intention is to provide a roadmap of the field to non-experts.

I will assume the knowledge of an advanced course in algebraic topology, and some familiarity with the category of spectra, K-theory, the Adams operations and cobordism.

Topological Hochschild homology (THH) is an invariant of rings, a variant of ordinary Hochschild homology when the base ring is replaced with the sphere spectrum and all relevant constructions are carried out in spectra rather than chain complexes. The construction THH contains more information than Hochschild homology, and it acquires a rich additional structure called a cyclotomic spectrum. The resulting construction of topological cyclic homology (TC) has been used in many fundamental calculations of algebraic K-theory. More recently, THH has been used in the work of Bhatt, Morrow, and Scholze to build p-adic cohomology theories, which provide one-parameter deformations of algebraic de Rham cohomology in mixed characteristic.

In these talks, I will give an introduction to topological Hochschild homology and to the work of Bhatt-Morrow-Scholze.