# Algebraic K-theory, motivic cohomology and motivic homotopy theory

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Workshop
23rd March 2020 to 27th March 2020
Organisers:
Aravind Asok University of Southern California
Roy Joshua Ohio State University
Marc Levine Universität Duisburg-Essen
Marco Schlichting University of Warwick

## Workshop Theme:

Algebraic $K$-theory, motivic cohomology and motivic homotopy theory

Motivic cohomology has its roots going back to theories of algebraic cycles, including such basic topics as enumerative geometry, intersection theory and theories of adequate equivalence relations such as rational, algebraic and numerical equivalence. Algebraic $K$-theory, beginning with the study of algebraic vector bundles, is a more recent development. Through Grothendieck's introduction of the theory of Chern classes of algebraic vector bundles and his Riemann-Roch theorem, the two subjects quite quickly became closely intertwined. At the same time, Grothendieck's introduction of theories of motives of smooth projective varieties formalized the link between algebraic cycles and cohomology theories. The search for an extension of a theory of motives to understand cohomology of non-proper varieties, by Beilinson, Bloch, Deligne, Lichtenbaum and many others, eventually led to Voevodsky's development of a triangulated category of (mixed) motives in the 1990s. Parallel to this, ideas for a corresponding homotopy theory were made precise in the work of Morel and Voevodsky, which opened the field of motivic homotopy theory. In this setting, motivic cohomology and algebraic $K$-theory (for regular schemes) become merely one of the universe of possible motivic cohomology theories represented by objects in the motivic stable homotopy category.

Voevodsky used the triangulated category of motives and motivic homotopy theory in an essential way in his proofs of the Milnor conjecture and his contribution to the proof of the Bloch-Kato conjecture.  After these stunning successes, the field has had a remarkable further development.

At the same time, the motivic approach has a serious, built-in limitation: it can only analyze "$\mathbb{A}^1$-homotopy invariant" phenomena. A generalization of motivic cohomology through the theory of higher Chow groups with modulus seeks to develop a theory that captures interesting non-$\mathbb{A}^1$-invariant phenomena. Parallel to this, the algebraic $K$-theory of non-regular schemes is rarely $\mathbb{A}^1$-homotopy invariant and theories which exploit this fact, such as topological Hochschild homology and topological cyclic homology, have seen major developments in the past few years.

This workshop will cover all these areas of recent developments in algebraic $K$-theory, motivic cohomology and motivic homotopy theory. We will examine a number of topics:

• Advances in the computation of the basic "structure constants" of motivic homotopy theory, that is, the stable  and unstable motivic homotopy groups of spheres, with applications to questions of splitting of vector bundles
• Using motivic techniques for computing classical stable homotopy groups of spheres.
• Analyzing motivic spectra and recognizing motivic infinite loop spaces through the theory of framed cobordism.
• Applying motivic methods to problems in enumerative geometry over non-algebraically closed fields.
• Analyzing properties of topological Hochschild homology and topological cyclic homology by various methods and using these theories to understand the $K$-theory of non-reduced schemes, using these theories to understand problems in the deformation theory of $K$-theory and connecting these theories to higher Chow groups with modulus.
• Constructing and analyzing higher structures in motivic homotopy theory, such as multiplicative norms.

### Deadline for applications: 29th December 2019

KAH programme participants DO NOT need to apply, programme participants with visit dates during KAHW02 will automatically be added to the attendee list.

Please note members of Cambridge University are welcome to turn up and sign in as a non-registered attendee on the day(s) during the workshop and attend the lecture(s). Please note that we cannot provide you with any support including name badge, meals or accommodation.

In addition to visiting the INI, there are multiple ways in which you can participate remotely.

Apply now

#### Fees

##### Registration Only
• Registration Package: £200
• Student Registration Package: £150

The Registration Package includes admission to all seminars, lunches and refreshments on the days that lectures take place (Monday - Friday), wine reception and formal dinner, but does not include other meals or accommodation.

##### Registration and Accommodation
• Accommodation Package: £568

The Accommodation Package includes a registration fee, bed and breakfast accommodation at Churchill College from the evening of Sunday to breakfast on Saturday, together with lunches and refreshments during the days that lectures take place (Monday - Friday). The formal dinner is also included, but no other evening meals.

##### Formal Dinner Only
• Formal Dinner: £50

Participants on the Accommodation Package or Registration Package, including organisers and speakers, are automatically included in this event. For all remaining participants who would like to attend, such as programme participants, the above charge will apply.

#### Accommodation

Accommodation in single study bedrooms with shared facilities and breakfast are provided at Churchill College,

#### Meals

##### Lunch

Lunch timings and location will be confirmed with timetable.

##### Evening Meal

Participants are free to make their own arrangements for dinner.

##### Formal Dinner

The Formal Dinner location and date is to be confirmed. Participants on the Accommodation Package or Registration Package, including organisers and speakers, are automatically included in this event.