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.
Here to register talks for this seminar. Editable now!
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)
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.
- 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.
- 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.
See Notes.
See Notes.
References
- [Brazelton] Unstable motivic homotopy theory, notes by Thomas Brazelton at Harvard, Fall 2024.
- [Morel] An Introduction to \( \mathbb{A}^1 \)-homotopy Theory, by Fabien Morel (2004).
- [Hoyois] From algebraic cobordism to motivic cohomology, by Marc Hoyois (2015).
- [PCMI 2024] PCMI 2024 notes.
- [Déglise] Notes on motivic homotopy theory, by Frédéric Déglise, PCMI 2021.
- [Déglise_Milnor_K] Notes on Milnor-Witt K-theory, by Frédéric Déglise.
- [Liang] Methods of Homotopy Theory in Algebraic Geometry, Master thesis, by Tongtong Liang (2022).
- [Isaksen-Østvær] Motivic stable homotopy groups, by Daniel C. Isaksen, Paul Arne Østvær (2018).
- [CKQ21] Algebraic slice spectral sequences, by Dominic Leon Culver, Hana Jia Kong, J.D. Quigley (2021).
- [Bachmann] Algebraic K-theory from the Viewpoint of Motivic Homotopy Theory, mini-course by Tom Bachmann.
