GENERAL FAVORITES
PCMI 2024: Motivic Homotopy Theory, lecture notes https://www.ias.edu/pcmi/pcmi-2024-gss-lecture-notes-and-problem-sets
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 enumerativ geometry
Computations in Stable Motivic Homotopy Theory, Talbot notes https://drive.google.com/file/d/1VFIAfv6F3a3gRD973dRR00lnNXVg012c/view
I really love these notes.
Here are some open problems and conjectures in the field. If you think that they are no longer open or think I've missed something, let me know!
For all fields F, the Steenrod algebra of power operations as described by Voevodsky is equivalent to the algebra of all bistable operations in motivic cohomology.
This conjecture is discussed in https://arxiv.org/pdf/2506.05585
(Hopkins--Morel) For all fields F, the quotient map MGL/(x1, x2, ...) -> HZ is an equivalence, where xi are generators for the Lazard ring.
As discussed in https://arxiv.org/pdf/2506.05585, this conjecture is implied by Conjecture 1
Section 1.2 of https://arxiv.org/pdf/2111.02320 contains some interesting consequences if the conjecture is true
For all fields F, describe the cellular F-motivic stable stems in terms of synthetic spectra and arithmetic of the base field.
In other words, extend the techniques of https://arxiv.org/pdf/2503.12060. This would be particularly interesting for F=R.
Is the Eilenberg-MacLane spectrum HFp a Thom spectrum?
This is deeply related to power operations and the Dyer--Lashof algebra.
An interesting discussion of this problem and its relation with MHH is in https://arxiv.org/pdf/2204.00441
Textbooks/Course notes
Online writings/Talk Notes/Minicourse notes
Algebraic K-theory from the viewpoint of motivic homotopy theory, Tom Bachmann https://www.mathematik.uni-muenchen.de/~bachmann/KGL-minicourse.pdf
A great infinity categorical introduction to motivic homotopy theory. Nice application to the K-theory of fields in the last section.
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The real Betti realization of motivic Thom spectra and of very effective Hermitian K-theory, Julie Bannwart https://www.arxiv.org/abs/2505.07297
Wonderful paper collecting semi-folklore, or at least scattered, facts about Real Betti realization. Also discusses multiplicative structures.
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Ext and the motivic Steenrod algebra over R, Mike Hill https://arxiv.org/abs/0904.1998
Develops the rho-Bockstein spectral sequence as a way to lift C-motivic computations over the Steenrod algebra to R-motivic computations over the Steenrod algebra. Development is brisk, and most of the paper deals with explicit computations.
Low-dimensional Milnor--Witt stems over R, Dan Dugger and Dan Isaksen https://arxiv.org/abs/1502.01007
Section 3 contains a robust description of the rho-Bockstein spectral sequence, which is thoroughly employed through the paper.
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Motivic invariants of p-adic fields, Kyle Ormsby https://arxiv.org/abs/1002.5007
Uses the motivic Adams spectral sequence to compute the homotopy of BPGL and its truncations over p-adic fields for p not 2.
Motivic Brown--Peterson invariants of the rationals, Kyle Ormsby and Paul Arne Ostvaer https://arxiv.org/abs/1208.5007
Uses the motivic Adams spectral sequence and a local-to-global principle to compute the homotopy of BPGL and its truncations over the rationals and the 2-adics. Reintroduces some theorems on algebraic K-theory in a new light.
Splittings of truncated motivic Brown--Peterson cooperations algebras, Jackson Morris, Sarah Petersen, and Liz Tatum https://arxiv.org/abs/2509.19542
Produces a spectrum level splitting of the BPGL<1>-cooperations algebra and computes the E1-page of the BPGL<1>-based motivic Adams spectral sequence over C, R, and Fq at all primes. Also shows that all differentials in question are determined by integral motivic cohomology of base field.
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On the relation of special linear algebraic cobordism to Witt groups, Alexey Ananyevskiy https://arxiv.org/abs/1212.5780
Proves a Conner--Floyd style isomorphism for expressing Witt theory in terms of MSL, after inverting eta.
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Motivic explorations in enumerative geometry, Sabrina Pauli https://www.ias.edu/sites/default/files/PCMI_lecture_notes-5.pdf
First two sections are a great example driven introduction to the A1 degree as a tool for solving enumerative problems. Second two sections contain interesting connections with tropical geometry.
Papers
The evolution of enumerative geometry: A narrative from classical problems to enriched invariants, Thomas Brazelton and Candace Bethea https://tbrazel.github.io/papers/PCMI_Conference_Proceedings.pdf
Wonderful survey article written by two wonderful people.
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kq-resolutions I, Dominic Culver and J.D. Quigley https://arxiv.org/abs/1905.11952
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 https://arxiv.org/abs/2506.16672
Initiates the study of the R-motivic kq-resolution. Determines the E1-page in terms of Ext of Brown--Gitler comodules, and proves a splitting result for effective symplectic K-theory ksp.
Rings of cooperations for Hermitian K-theory over finite fields, Jackson Morris https://arxiv.org/abs/2509.02786
Initiates the study of the Fq-motivic kq-resolution, where Fq is a finite field of characteristic different from 2. Determines the E1-page in terms of Ext of Brown--Gitler comodules, and shows that all differentials in the Adams spectral sequence computing the cooperations are determined by integral motivic cohomology of Fq.
On very effective hermitian K-theory, Alexey Ananyevskiy, Oliver Rondigs, and Paul Arne Ostvaer https://arxiv.org/abs/1712.01349
Argues that kq, the very effective cover of Hermitian K-theory KQ, is a good connective cover.
Textbook accounts
Online writings/Talk notes/Minicourse notes
Notes on motivic infinite loop space theory, Tom Bachmann and Elden Elmanto https://arxiv.org/abs/1912.06530
Exanded set of talk notes going over the main theorems of motivic infinite loop space theory.
Papers
Textbook accounts/Lecture notes/Memoirs
Logarithmic motivic homotopy theory, Federico Binda, Doosung Park, Paul Arne Ostvaer https://arxiv.org/abs/2303.02729
AMS Memoir which contructs the logarithmic stable motivic homotopy category. LogTHH and logTC is representable here.
Online writings
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On the logarithmic slice filtration, Federico Binda, Doosung Park, Paul Arne Ostvaer https://arxiv.org/pdf/2403.03056
Develops a motivic logarithmic slice filtration, shows it agrees with previous known slice filtrations on log theories
On the log motivic stable homotopy groups, Doosung Park https://arxiv.org/pdf/2308.07683
Shows that for a field admitting resolution of singularities, the log stable stems are the same as the motivic stable stems
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A1-invariant motivic cohomology of schemes, Tom Bachmann, Elden Elmanto, and Matthew Morrow https://arxiv.org/abs/2508.09915
A masterwork realizing Voevodsky's vision over any qcqs base scheme.
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Hochschild homology of mod-p motivic cohomology over algebraically closed fields, Bjorn Dundas, Mike Hill, Kyle Ormsby, and Paul Arne Ostvaer https://arxiv.org/abs/2204.00441
Constructs the spectrum MHH and computes MHH(Fp). Shows a difference between the classical case, and includes interesting remarks relating MHH with motivic Thom spectra.
A motivic Greenlees spectral sequence towards motivic Hochschild homology, Federico Ernesto Mocchetti https://arxiv.org/abs/2408.00338
Constructs a very general spectral sequence, which is then employed to compute the tau-inverted MHH of finite fields.
Modulo τp−1 motivic Hochschild homology of modulo p motivic cohomology, Federico Ernesto Moccheti https://arxiv.org/abs/2409.18540
Uses the motivic Greenlees spectral sequence to compute MHH of finite fields modulo powers of tau.
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The motivic Mahowald invariant, J.D. Quigley https://arxiv.org/abs/1801.06035
Introduces the motivic Mahowald invariant. Computes the Mahowald invariant of the motivic analogue of Mahowald's eta_j classes, makes many Tate fixed point computations to exhibit blueshift.
Real motivic and C2-equivariant Mahowald invariants, J.D. Quigley https://arxiv.org/abs/1904.12996
Makes Mahowald invariant computations in R-motivic and C2-equivariant homotopy theory.
Motivic Mahowald invariants over general base fields, J.D. Quigley https://arxiv.org/abs/1905.03902
Makes Mahowald invariant computations over general base fields. Also discusses v1-periodicity.
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Nilpotence in normed MGL-modules, Tom Bachmann nd Jeremy Hahn https://arxiv.org/abs/1906.01306
Proves that if E is a normed motivic spectrum which is HZ-acyclic, then it is MGL-acyclic.
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Algebraic cobordism and a Conner--Floyd isomorphism for algebraic K-theory, Toni Annala, Marc Hoyois, and Ryomei Iwasa https://arxiv.org/pdf/2303.02051
Proves a Conner--Floyd isomorphism for algebraic K-theory in terms of MGL over any qcqs dervied scheme. Also proves a Snaith theorem for PMGL.
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Norms in motivic homotopy theory, Tom Bachmann and Marc Hoyois https://arxiv.org/abs/1711.03061
Introduces normed motivic ring spectra, a stronger notion than E\infty.
Textbook accounts
Online writings/Minicourse notes
Characteristic classes in motivic homotopy theory, Frederic Deglise https://www.ias.edu/sites/default/files/deglise4.pdf
Thorough exploration of GL-orientations and their connections with formal group laws, before pivoting to the quadratic story.
Papers
SL-oriented cohomology theories, Alexey Ananyevskiy https://arxiv.org/abs/1901.01597
Establishes a criterion for a motivic spectrum to admit an SLc-orientation, and proves a Thom isomorphism for SLc-bundles in SL-oriented cohomology.
Witt sheaves and the η-inverted sphere spectrum, Alexey Ananyevskiy, Marc Levine, and Ivan Panin https://arxiv.org/abs/1504.04860
Establishes a version of Serre's finiteness for the n-th stable stem of the "negative" motivic sphere over a field. Also, they show that the "negative" part of the rationalization of SH(F) can be described as Witt motives.
The Borel character, Frédéric Déglise and Jean Fasel https://arxiv.org/abs/1903.11679
Develops a theory of characteristic classes and a total Borel character for symplectically-oriented motivic cohomology theories. Also discusses formal ternary laws associated to symplectic orientations.
Quadratic Riemann-Roch formulas, Frédéric Déglise and Jean Fasel https://arxiv.org/abs/2403.09266
Establishes a very general Riemann-Roch type formula before specializing to symplectic orientations. Connects with formal ternary laws and Hermitian K-theory.
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Stable operations in motivic homotopy theory, Tom Bachmann and Mike Hopkins http://tom-bachmann.com/ops.pdf
Brief note comparing different power operations on motivic cohomology.
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Motivic Brown--Peterson invariants of the rationals, Kyle Ormsby and Paul Arne Ostvaer https://arxiv.org/abs/1208.5007
Uses the motivic Adams spectral sequence and a local-to-global principle to compute the homotopy of BPGL and its truncations over the rationals and the 2-adics. Reintroduces some theorems on algebraic K-theory in a new light.
Stable operations and cooperations in derived Witt theory with rational coefficients, Alexey Ananyevskiy https://arxiv.org/abs/1504.04848
Computes the rational operations and cooperations algebras for KW, which give a similar description to Adams work on KO operations and cooperations. Also computes the integral structure as a module over KW of a point.
Witt sheaves and the η-inverted sphere spectrum, Alexey Ananyevskiy, Marc Levine, and Ivan Panin https://arxiv.org/abs/1504.04860
Establishes a version of Serre's finiteness for the n-th stable stem of the "negative" motivic sphere over a field. Also, they show that the "negative" part of the rationalization of SH(F) can be described as Witt motives.
Textbook accounts
Online writings/Minicourse notes
The motivic slice spectral sequence, Oliver Rondigs https://bpb-us-e1.wpmucdn.com/s.wayne.edu/dist/0/60/files/2019/11/echt-lecture.pdf
eCHT minicourse notes. Great introduction and overview on these topics.
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Motivic stable homotopy groups, Dan Isaksen and Paul Arne Ostvaer https://arxiv.org/pdf/1811.05729
Excellent survey article on computations of the motivic stable homotopy groups of spheres.
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The integral motivic dual Steenrod algebra, Bjorn Dundas and Paul Arne Ostvaer https://arxiv.org/abs/2311.13304
Investigates the homotopy of HZ \otimes HZ over a variety of base schemes
The motivic Steenrod algebra in positive characteristic, Marc Hoyois, Shane Kelly, and Paul Arne Ostvaer https://arxiv.org/abs/1305.5690
Computes the structure of the motivic Steenrod algebra and its dual over positive characteristic schemes, and shows that they are indeed all of the sbitable cohomology operations.
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Motivic stable stems and Galois approximations of cellular motivic categories, Tom Bachmann, Robert Burklund, and Zhouli Xu https://arxiv.org/abs/2503.12060
Reconstructs cellular stable motivic homotopy theory via synthetic methods and the absolute Galois group of the base field.
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Motivic twisted K-theory, Markus Spitzweck and Paul Arne Ostvaer https://arxiv.org/abs/1008.4915
Initiates the study of a twisted KGL and a twisted rational Chern character.
Twisted K-theory in motivic homotopy theory, Elden Elmanto, Denis Nardin, and Maria Yakerson https://arxiv.org/abs/2110.09203
More geared towards the Brauer group and algebraic geometry than Spitzweck--Ostvaer
Textbook accounts/Course notes
Unstable motivic homotopy theory, Thomas Brazelton https://github.com/tbrazel/math266-motivic/blob/main/main.pdf
This has probably everything you would ever want.
Online writings
A primer for unstable motivic homotopy theory, Ben Antieau and Elden Elmanto https://arxiv.org/pdf/1605.00929
Robust. Lots of details on model categories. Builds up to algebraic vector bundles.
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Unstable motivic homotopy theory, Kirsten Wickelgren and Ben Williams https://arxiv.org/abs/1902.08857
Brief introduction to unstable methods, then a nice discussion of some fun topics in the field.
Textbook accounts
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On the relation of special linear algebraic cobordism to Witt groups, Alexey Ananyevskiy https://arxiv.org/abs/1212.5780
Proves a Conner--Floyd style isomorphism for expressing Witt theory in terms of MSL, after inverting eta.
Stable operations and cooperations in derived Witt theory with rational coefficients, Alexey Ananyevskiy https://arxiv.org/abs/1504.04848
Computes the rational operations and cooperations algebras for KW, which give a similar description to Adams work on KO operations and cooperations. Also computes the integral structure as a module over KW of a point.
Witt sheaves and the η-inverted sphere spectrum, Alexey Ananyevskiy, Marc Levine, and Ivan Panin https://arxiv.org/abs/1504.04860
Establishes a version of Serre's finiteness for the n-th stable stem of the "negative" motivic sphere over a field. Also, they show that the "negative" part of the rationalization of SH(F) can be described as Witt motives.