• Motivic Homotopy Theory

    Modern Techniques in Homotopy Theory

  • Motivic Homotopy Theory

    Modern Techniques in Homotopy Theory

Schedule Talks: Wednesdays 10:00-11:30 am Eastern time . Zoom link: https://brown.zoom.us/j/6587728801 and https://upenn.zoom.us/j/9305155993 (for Aug 20 ONLY!).
Problem sessions and discussions: discord channel https://discord.gg/RDfSHNck.

This is a reading course in Modern Techniques in Homotopy Theory. The syllabus can be found here. The topic for this summer will be Motivic Homotopy Theory.

Tentative schedule: see here.
Here to register talks for this seminar. Editable now!
Eastern Standard Time (EST)
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Notes

  • Week 0 (May 29): Lecturer: Mattie Ji
    • This is a motivational/overview talk without many details. Topics covered: invariants under \( - \times \mathbb{A}^1 \) (e.g. Chow rings, algebraic K-theory), Folklore theorem, corank zero splitting (Murthy-Swan, Kuma-Murthy, Murthy), topological proof of the Folklore theorem in the lower case, Asok-Bachmann-Hopkins, motivations for unstable motivic homotopy theory.
    See Notes.
  • Week 1 (Jun 4): Lecturer: Mattie Ji
    • Historic development of algebraic geometry, varieties, Zariski topology. Presheaves, descent conditions, sheaves, sheafification, (locally) ringed spaces, stalks, affine schemes. \( \mathcal{O}_X \)-modules, quasi-coherent sheaves, ideal sheaves, (closed/open) immersions. Examples of schemes (noetherian, quasi-compact, quasi-separated, irreducible, reduced, connected, integral, factorial, normal, smooth, etc.) Ring of rational functions. Morphism of schemes, examples (finite type, closed, universally closed, proper, integral, quasi-finite, finite, smooth, étale, flat, etc.) Dimension theory. Zariski cotangent spaces. Regularity. Yoneda embedding, functor of point perspective. Sheaf cohomology and some related results. Algebraic cycles, rational equivalence, Chow groups/rings, Bézout's theorem, chern classes. Bloch-Quillen formula.
    See Notes.
  • Week 2 (Jun 11): Lecturer: Albert Jinghui Yang
    • Two main motivations for the higher algebra: nice monoidal structure for \( \mathsf{Sp} \) and the detection of higher associativity/commutativity. Comparison between classical algebra and higher algebra. Ideas of how to extract info from higher algebraic objects. Intuition of motivic homotopy theory. Comparison between classical homotopy theory and motivic homotopy theory. Simplicial objects, simplicial sets, faces and degeneracies. Standard n-simplices, \( \Delta \)-complexes (as cosimplicial sets). Kan extensions, geometric realization. Horns, nerves of categories, inner horn extension property, definition of \( \infty \)-categories and their relations to nerves. Functors between \( \infty \)-categories, sub-\( \infty \)-categories. Kan complexes and \( \infty \)-groupoids. \( \infty \)-category of spaces \( \mathcal{S} \). Kernels (fibers) and cokernels (cofibers) in the \( \infty \)-categories. Stable \( \infty \)-categories, stabilization. \( \infty \)-category of spectra \( \mathsf{Sp} \).
    See Notes.
  • Week 3 (Jun 18): Lecturer: Fangji Liu
    • Functor of points revisited. Zariski topology, sheaf properties, Nisnevich topology, comparison theorem. ́Étale/flat/unramified morphisms and their properties, local description, Nisnevich morphism, local rings, Henselian rings, Henselization. Grothendieck topology, sites (Zariski/́etale/Nisnevich/fppf/fpqc), hierarchy, examples. Preshaves & sheaves of sites, examples, stalks, sheaf cohomology, recovery of Galois cohomology, Čech cohomology and its comparison theorem, first cohomology groups, Picard groups, the advantages of Nisnevich topology. \( \infty \)-sheaves/stacks.
    See Notes.
  • Week 4 (Jun 25): Lecturer: David Zhu
    • Category of smooth schemes over a field \( \mathsf{Sm}_k \), presheaves & sheaves over \( \mathsf{Sm}_k \), sheafification, purity theorem, simplicial presheaves, Čech nerves, \( \infty \)-sheaves, Nisnevich presheaves and sheaves, localization. \( \mathbb{A}^1 \)-invariant, examples in algebraic K-theory, \( \mathbb{A}^1 \)-localization, algebraic n-simplices, singular chains, \( \mathrm{Sing} \) functor. Motivic spaces, examples ( \( \mathbb{G}_m \), algebraic K, etc.), motivic localization functor, motivic equivalence, pointed motivic spaces, coproducts, smash product, simplicial spheres, Tate sphere, suspension. \( \mathbb{A}^1 \)-homotopy sheaves, Nisnevich sheaves, Whitehead theorem.
    See Notes.
  • Week 5 (Jul 2): Lecturer: Mattie Ji
    • Vector bundle torsors, strongly homotopy invariant, \( \mathbb{A}^1 \)-invariant, motivic equivalence, Jouanolou-Thomason theorem, sketch of proofs. Group objects, \( \tau \)-localization, \( EG, BG\) are their properties, bar construction, Classifying Spaces in the motivic setting, Dold-Kan, motivic Eilenberg-Maclane spaces, representability, fiber/cofiber sequences, long exact sequences, Asok-Hoyois-Wendt, Freudenthal suspension theorem and the sketch of proof, Bousfield localization, weakly cellular spaces. Homotopy groups of \( B \mathbb{G}_m \), \( \mathbb{A}^1 \)-rigidity, \( BGL \), Quillen’s +-construction, algebraic K-theory via plus construction.
    See Notes.
  • Week 6 (Jul 9): Lecturer: Matthew Stevens
    • Lifting properties, retractions, model categories, examples of model categories, Quillen model structure, (co)fibrant objects, (co)fibrant replacement. Cylinder objects, (left/right) homotopy, path objects, properties of these objects and proofs, more examples.
    See Notes.
  • Week 7 (Jul 16): Lecturer: Mattie Ji
    • Spectra, motivic spheres ( \( S^1, \mathbb{G}_m \) ), motivic spectra. Localization of a category, homotopy category, (pointed) motivic spaces in a concrete sense, model structure on \( \mathsf{Spc}(k) \), \( \mathbb{A}^1 \)-homotopy category, bi-spectra and their category ( \( \mathsf{Spt}_{s,t}(k) \) ), properties of \( \mathsf{Spt}_{s,t}(k) \) and its \( \mathbb{A}^1 \)-structure. \( \infty \)-commutative monoid, symmetric monoidal \( \infty \)-categories, straightening equivalence, cocartesian fibrations, presentable \( \infty \)-categories and their properties, stable motivic \( \infty \)-categories, suspension spectra, bi-graded homotopy groups, motivic homology and cohomology, examples (Milnor-Witt K-theory \( \operatorname{K}_*^{MW} \), Eilenberg-MacLane spectra \( K(n,G) \), algebraic K-theory \( \operatorname{KGL} \)), motivic Bott periodicity.
    Aside: cellular motivic categories, brief intro to synthetic spectra (by Pstrągowski via \( \nu: \mathsf{Sp} \to \mathsf{Syn}_E \) ).
    See Notes.
  • Week 8 (Jul 23): Lecturer: Emerson Hemley
    • Atiyah-Hirzebruch spectral sequences, Beilinson’s dream. Bloch's cycle complexes, higher Chow groups, Milnor K-theory, properties. Finite correspondences and examples, category of finite correspondences \( \mathsf{Cor}_S \) over \( \mathsf{Sm}_S \), A presheaf with transfers, effective motives (Nisnevich sheaves with transfer which are \( \mathbb{A}^1 \)-local), six functor formalism. Suslin complexes, motivic Tate twists \( \mathbb{Z}(n) \), Voevodsky’s definition of motivic cohomology, motivic-Chow Comparison theorem, relations to Milnor K-theory, Beilinson-Soulé vanishing conjecture. Motivic spectral sequences (Bloch-Lichtenbaum, Friedlander-Suslin). \( \mathbb{P}^1 \)-spectra \( \operatorname{KGL}, H\mathbb{Z} \), slice filtrations.
    See Notes.
  • Week 9 (Jul 30): Lecturer: Pengkun Huang
    • Milnor K-theory, elements and properties of Milnor K-theory, étale cohomology and Galois cohomology, detailed proofs. Kummer sequences, Hilbert 90, graded ring structure of étale cohomology, intersection theory. Definition and construction of Voevodsky's motivic cohomology. Leray spectral sequences. Brief proof of norm residue theorem: transfer argument, reduction to char 0, Hilbert 90 condition. Corollaries.
    See Notes with annotations, and Notes without annotations (the raw notes).
  • Week 10 (Aug 6): Lecturer: David Zhu
    • Review of Thom spaces, Thom classes, Thom isomorphism, Thom spectra. Universal Thom spaces, complex Thom spectra, complex oriented cohomology theory, (non)examples, chern classes, easy calculations, formal group laws, formal group laws from orientations, Lazard ring, Lazard theorem. \( \operatorname{MU} \) and Quillen's theorem. \( \mathbb{P}^1 \)-spectra, motivic Thom spaces (via purity theorem) and properties, construction of motivic Thom spectrum \( \operatorname{MGL} \). Motivic ring spectra, oriented motivic spectra, orientation of \( H \mathbb{Z} \), orientation of \( \operatorname{MGL} \), projective bundle theorem, oriented motivic cohomology of Grassmannian, motivic Thom isomorphism, universality of \( \operatorname{MGL} \), motivic Quillen theorem and sketch of proof. Algebraic cobordisms and their associated formal group laws, Morel-Levine.
    See Notes.
  • Week 11 (Aug 13): Lecturer: Mattie Ji
    • Review of chern classes and projective bundle formula. Milnor exact sequence, Mittag-Leffler condition and projective bundle formula. Oriented motivic ring spectra, axioms/properties of motivic chern classes, (refined) Thom classes, stable Thom spaces, motivic Thom isomorphism, splitting of Thom spaces, Whitney sum formula. 6-functor formalism and properties, smooth purity. Bi-variant theory and examples, Fulton-MacPherson formalism. Smoothable lci morphisms, virtual tangent bundles, (refined) fundamental classes, related theorems and sketch proofs, oriented Thom isomorphism, (refined) oriented fundamental classes, Gysin morphisms. Chern character, Todd class, motivic GRR (Grothendieck-Riemann-Roch). Applications to algebraic cobordisms, Beilinson motivic ring spectra, Milnor-Witt K-theory.
    See Notes.

References


Department of Mathematics