• 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.
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.
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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 3): 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.

References


Department of Mathematics