Motivic homotopy theory seminar

Motivic homotopy theory learning seminar 2022

This seminar aims to cover the basic language and results of motivic homotopy theory, and go over applications depending on the interests of the participants. Some topics that may be touched on include Thom spaces, algebraic K-theory, motivic cohomology, and depending on interest, also algebraic cobordism, enumerative geometry, Milnor-Witt groups, the slice filtration, and six functor formalisms

We are only scheduling a few talks in advance while we go over the basics.

Here are some resources:

Introductory:
- Algebraic K-theory from the viewpoint of motivic homotopy theory (includes material on slice filtration, motivic cohomology, higher Chow groups)
- Unstable motivic homotopy theory (includes material on enumerative geometry)
- A primer for unstable motivic homotopy theory
- Théorie homotopique des S-schémas
- Motivic homotopy types of projective curves
- An introductory course on motivic homotopy theory and rational mixed motives
- A1-Homotopy Theory from an (∞,1)-Categorical Viewpoint (notes)
Classical reference (may have some mistakes?): A1-homotopy theory of schemes , other version
A1-algebraic topology over a field (Levine), also this
Stable A1-connectivity: Stable A1-connectivity over a base
From algebraic cobordism to motivic cohomology ("A proof of the Hopkins–Morel isomorphism in motivic homotopy theory.")
The Hilbert scheme of infinite affine space and algebraic K-theory
Milnor-Witt Motives
Motivic cohomology:
- A commutative P^1-spectrum representing motivic cohomology over Dedekind domains
- Motivic cohomology over Dedekind rings (Thomas Geisser)
- Techniques of localization in the theory of algebraic cycles (Marc Levine)
- Modules over motivic cohomology (Oliver Röndigs, Paul Arne Østvær)
- Lecture Notes on Motivic Cohomology (Mazza, Voevodsky, Weibel)
- An overview of motivic cohomology (Peter Haine)
- Motivic cohomology and Steenrod operations (lecture by Hoyois)
- Algebraic cycle complexes (slides by Levine)
- The homotopy coniveau tower
- Voevodsky's slice conjectures via Hilbert schemes
Motivic cohomology, K-theory and topological cyclic homology
On the motivic spectra representing algebraic cobordism and algebraic K-theory
Descente par éclatements en K-théorie invariante par homotopie (cdh descent for KH)
Motivic infinite loop spaces:
- Motivic infinite loop spaces
- Notes on motivic infinite loop space theory
- On the infinite loop spaces of algebraic cobordism and the motivic sphere
Modules over algebraic cobordism
Cancellation theorem for motivic spaces with finite flat transfers (contains discussion of modules over K-theory)
Milnor excision for motivic spectra
Framed correspondences:
- Introduction to framed correspondences
- Framed transfers and motivic fundamental classes
- The localization theorem for framed motivic spaces
Six functor formalisms
- Voevodsky's for constructible categories of coefficients (for algebraic spaces)
- Generalized cohomology theories for algebraic stacks
- The universal six functor formalism
- An introduction to six-functor formalisms
- The six operations in equivariant motivic homotopy theory
- Ayoub's thesis
Cdh descent in equivariant homotopy K-theory
Étale motives
Nice survey: Vector bundles on algebraic varieties
Nice survey: A^1-homotopy theory and contractible varieties: a survey
Affine representability results in A1-homotopy theory: vector bundles , principal bundles and homogeneous spaces , finite fields and complements
On modules over motivic ring spectra
Fundamental classes in motivic homotopy theory
Norms in motivic homotopy theory
Triangulated categories of mixed motives
A1-contractibility of Koras-Russel threefolds
Motivic homotopy theory in derived algebraic geometry
A quadratic refinement of the Grothendieck-Lefschetz-Verdier trace formula
A cohomological classification of vector bundles on smooth affine threefolds
Non A1-invariant:
- Motivic spectra and universality of K-theory

If you would like to participate, contact Roy Magen.
We meet on Fridays 1:30pm-3:00pm in room 307.

Talks

Date	Speaker	Topic	Main resource(s)
September 30	Roy Magen	Organizational meeting
October 7	Emily Saunders	Quick overview of ∞-categories, commutative monoids, K-theory	Section 1 and maybe 2 of Algebraic K-theory from the viewpoint of motivic homotopy theory
October 14	Morena Porzio	Basics of the unstable motivic homotopy category	Section 2 of Algebraic K-theory from the viewpoint of motivic homotopy theory and
October 21	Andres Fernandez Herrero	On the A1-equivalences between Hilb_d(A^infinity), FFlat_d, Vect_{d - 1}, Gr_{d - 1}(A^infinity)	Sections 2 and 4 of The Hilbert scheme of infinite affine space and algebraic K-theory
October 28	Andres Fernandez Herrero	On the A1-equivalences between Hilb_d(A^infinity), FFlat_d, Vect_{d - 1}, Gr_{d - 1}(A^infinity)	Sections 2 and 4 of The Hilbert scheme of infinite affine space and algebraic K-theory
November 4	Patrick Lei	On the A1-invertibility after group completion of the map from FFlat to Vect, proof of lemma 4.1	Section 3 and lemma 4.1 of The Hilbert scheme of infinite affine space and algebraic K-theory
November 11	Ben Church	Homotopy groups, purity, stability results for Hilbert schemes	Section 8 of The Hilbert scheme of infinite affine space and algebraic K-theory
December 2	Roy Magen, Ben Church	Stability results for Hilbert schemes	Section 8 of The Hilbert scheme of infinite affine space and algebraic K-theory
December 9	Roy Magen	More discussion of A1-homotopy groups, purity
		Classification of vector bundles
		Stable homotopy, motivic spectra
		Motivic cohomology
		Framed transfers
		...