The electronic Computational Homotopy Theory research community organizes occasional minicourses on topics of current interest.

Contact Dan Isaksen (Wayne State University) for additional information.

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See below for the schedule of previous minicourses, in reverse chronological order.

**15, 17, and 22 September 2020**

3-hour online mini-course, offered on consecutive Tuesdays and Thursdays at 11:30am Eastern time.

Speaker: Clover May, UCLA

Title: Equivariant cohomology

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Abstract: 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.

**12, 14, 19, and 21 May 2020
**

4-hour online mini-course, offered on consecutive Tuesdays and Thursdays at 11:30am in Detroit (Eastern Time).

Speaker: Gijs Heuts, Utrecht University

Title: Unstable homotopy groups

First talk video: Rational homotopy theory

First talk notes: Rational homotopy theory

Second talk video: The EHP sequence

Second talk notes: The EHP sequence

Third talk video: The Goodwillie tower

Third talk notes: The Goodwillie tower

Fourth talk video: Periodicity

Fourth talk notes: Periodicity

Abstract: 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.

**12, 14, 19 November 2019**

3-hour online mini-course, offered on consecutive Tuesdays and Thursdays at 11:30am in Detroit (Eastern Time).

Speaker: Oliver Röndigs, Universität Osnabrück

Title: The motivic slice spectral sequence

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Abstract: 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.

**14, 16, 21, 23 May 2019**

4-hour online mini-course, offered on consecutive Tuesdays and Thursdays at 11:30am in Detroit (Eastern Time).

Mini-course information, including presentation slides and references

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Speaker: Agnes Beaudry, University of Colorado

Title: An introduction to chromatic homotopy theory

Abstract: 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.

**16, 18, 23, 25 October 2018
**

This is a special event. The ECHT community is experimenting with a 4-hour online mini-course, offered on consecutive Tuesdays and Thursdays at 11:30am in Detroit (Eastern Time).

Speaker: Akhil Mathew, University of Chicago

Title: Topological Hochschild homology and its applications

First talk video (Beware that the video quality in the later part of this talk is poor.)

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Third talk (Due to technical difficulties, this file is audio only.)

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Abstract: 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.