Motivic homotopy theory

On this page are open problems and resources in various topics in motivic homotopy theory. The resources in motivic homotopy theory that I have found useful tend to be papers.

Open Problems

Here are some open problems in motivic homotopy theory. If you have one to contribute, or no longer feel that one of these problems is open, let me know!

1. Show that Voevodsky’s Steenrod algebra of power operations is indeed the algebra of all bistable operations in mod-p motivic cohomology over any base scheme.
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   • A nice discussion is given here
2. (Hopkins-Morel) Show that the $\text{H}\mathbb{Z}$ is equivalent to the quotient of $\text{MGL}$ by the generators of the Lazard ring.
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   • A nice discussion is given here
   • See Section 1.2 of this paper for some interesting consequences
3. Better understand the relationship between cellular motivic spectra and synthetic spectra.
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   • Bachmann-Burklund-Xu do this for a large swath of base fields using Galois reconstruction
   • Recent work of Fabio Tanania offers an exciting and surprising new direction
4. (Motivic Hopkins-Mahowald Theorem) Is the motivic Eilenberg-MacLane spectrum $\text{H}\mathbb{F}_p$ a Thom spectrum?
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   • The classical and many equivariant analogues of this are true (for the state of the art equivariant version, see Ishan Levy's paper)
   • This would better our understanding of the motivic Dyer-Lashof algebra
   • An interesting take is given on this is given in Dundas-Hill-Ormsby-Østvær

Personal Favorite Resources

• Lecture notes from PCMI 2024
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  This webpage has a great selection of notes accompanying the many mini-courses from PCMI 2024. Touches on a whole bunch of topics. Particular notes I enjoy are Burt Totaro's notes on classifying spaces, Kirsten Wickelgren's notes on Weil conjectures, Frederic Deglise's notes on characteristic classes, and Sabrina Pauli's notes on enumerative geometry.
• Talbot 2023: Computations in Stable Motivic Homotopy Theory
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  All Talbot notes are good, and these ones especially have many gems.
• Motivic Stable Homotopy Groups, Dan Isaksen and Paul Arne Østvær
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  Just filled to the brim with everything I've ever wanted to know about stable homotopy groups. Ends with a nice set of open problems.
• Algebraic K-theory from the Viewpoint of Motivic Homotopy Theory, Tom Bachmann
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  Hands on, infinity-categorical approach to the stable motivic homtoopy category. Does the slice spectral sequence. Low bar for entry!

Topics

In Progress

• Algebraic K-theory

• Algebraic Vector Bundles

• Betti and Etale Realization
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  Realization functors typically take some form of motivic spectra to some form of classical spectra.

  • The real Betti realization of motivic Thom spectra and of very effective Hermitian K-theory, Julie Bannwart. Great resource collecting and clarfying semi-folkore results about Real realization.

• Bockstein Spectral Sequences

• Cobordism

• Equivariant aspects

• Enumerative Geometry and the A1 degree

• Etale motivic homotopy theory

• Finite Field Computations

• Hermitian K-theory
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  Hermitian K-theory plays a very important role in stable motivic homotopy theory. It is often useful to consider the Hermitian K-theory spectrum as a motivic analogue of real topological K-theory spectrum.

  • kq-resolutions I, Dominic Culver and J.D. Quigley. Inspired by Mahowald's work on the bo-resolution, this paper initiates the study of the C-motivic kq-based Adams spectral sequence. Determines the v1-periodic elements in the C-motivic stable stems, discusses the motivic telescope conjecture, and uses synthetic/filtered technology to construct a connective image of J spectrum..
  • On the ring of cooperations for real Hermitian K-theory, Jackson Morris. I initiated the study of the kq-resolution over the real numbers, computing the cooperations algebra and proving a splitting result for symplectic K-theory.
  • Rings of cooperations for Hermitian K-theory over finite fields, Jackson Morris. I initiated ths study of the kq-resolution over finite fields. Much of the story here is controlled by the integral motivic cohomology of the base field.
  • On the very effective Hermitian K-theory, Alexey Ananyevskiy, Oliver Röndigs, and Paul Arne Østvær. Constructs a useful connective variant of Hermitian K-theory, akin to the connective variant of real topological K-theory.
  • A motivic spectrum representing hermitian K-theory, Baptiste Calmès, Yonatan Harpaz, and Denis Nardin. Uses the machinery of Poincare-infinity categories to construct a hermitian K-theory spectrum without concern of invertibility of 2, recovers expected fiber sequences.
  • Hermitian K-theory and Milnor-Witt motivic cohomology of Z, Håkon Kolderup, Oliver Röndigs, and Paul Arne Østvær. Uses the newly defined very-effective K-theory specturm in SH(Z) and makes fundamental computations with the slices spectral sequence.
  • Slices of hermitian K-theory and Milnor's conjecture on quadratic forms, Oliver Röndigs and Paul Arne Østvær. Computes the slices of Hermitian K-theory, makes computations with the slices spectral sequence.
  • Cellularity of hermitian K-theory and Witt theory, Oliver Röndigs, Markus Spitzweck, and Paul Arne Østvær. short paper proving cellularity.
  • The motivic Hopf map solves the homotopy limit problem for K-theory, Oliver Röndigs, Markus Spitzweck, and Paul Arne Østvær. Over fields of characteristic different from 2 with finite vcd, the motivic Hopf map gives an equivalence between the eta-completion of KQ and the C2 homotopy fixed points of KGL. Remarkably, this is also shown to hold for the connective variants, unlike in the topological case.
  • Hermitian K-theory, Dedekind ζ-functions, and quadratic forms over rings of integers in number fields, Jonas Irgens Kylling, Oliver Röndigs, Paul Arne Østvær. Makes computations of the Hermitian K-theory for rings fo integers in number fields.
  • The multiplicative structure on the graded slices of Hermitian K-theory and Witt-Theory, Oliver Röndigs and Paul Arne Østvær. Computes the graded slices as a ring in relation to the Steenrod algebra.
  • The generalized slices of Hermitian K-theory, Tom Bachmann. Computes the very-effective slices of Hermitian K-theory, displaying a real Bott periodic pattern.
  • The homotopy limit problem for Hermitian K-theory, equivariant motivic homotopy theory and motivic Real cobordism, Po Hu, Igor Kriz, and Kyle Ormsby. Sets up foundations in equivariant motivic homotopy to show that over all fields of characteristic 0 with finite vcd, the 2-completion of KQ is equivalent to the homotopy fixed points for the C2-action on KGL.
  • The homotopy limit problem and the cellular Picard group of Hermitian K-theory, Drew Heard. Solves the homotopy limit problem over qcqs base schemes using descent. As a neat application, computes the C-motivic Picard group of cellular modules over hermitian K-theory.

• Infinite Loop Spaces

• Logarithmic Structures

• Motivic Adams Spectral Sequence

• Motivic Adams-Novikov Spectral Sequence

• Motivic Cohomology

• Motivic Hochschild Homology

• Motivic Mahowald Invariant

• Nilpotence

• Non A1-invariant

• Normed Motivic Spectra

• Orientations

• Periodicity

• Picard Groups

• Power Operations

• Rational Computations

• Slices

• Stable Motivic Homotopy Theory Foundations

• Stable Motivic Homotopy Groups of Spheres

• Steenrod Algebra

• Synthetic Spectra

• Syntomic Cohomology

• Tensor-Triangular Geometry

• Twisted K-theory

• Unstable Motivic Homotopy Theory Foundations

• Witt K-theory