Extra topics

Extra topics to be added as appendices

The primary assessment in this course is a writing assignment, to add an appendix to the notes. Here are some suggested topics. They are roughly arranged from what I expect to be easiest to what I expect to be hardest, but difficulty is fully subjective and depends on your background. Most of these are totally open-ended, so I encourage you to write about whatever you find interesting! I'll update this page with more references and ideas as we keep going.

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🟢 $\mathbb{A}^1$-Euler characteristics (Atticus Wang)

Give an introduction to the Euler characteristic of a dualizable motivic space/spectrum, valued in the Grothendieck--Witt ring of symmetric bilinear forms. Provide a cheatsheet of computational tricks and an overview of some known formulas and examples.

• Levine-Raksit: Motivic Gauss-Bonnet
• Levine: Aspects of enumerative geometry with quadratic forms

🟢 $\mathbb{A}^1$-contractibility of varieties (Sofía Marlasca Aparicio)

Discuss $\mathbb{A}^1$-contractibility, giving some examples and computations

• Asok-Østvær: A^1-homotopy theory and contractible varieties: a survey

🟢 Unimodular rows (Hamilton Wan)

Explore the theory of unimodular rows from the perspective of $\mathbb{A}^1$-homotopy theory. Show how $\mathbb{A}^2-0$ represents unimodular rows. Some references include:

• Lerbet: Motivic stable cohomotopy and unimodular rows
• Syed: Symplectic orbits of unimodular rows

🟢 Matsumoto's theorem (Jonathan Buchanan)

Give a writeup of the isomorphism $K_2^M \cong K_2^Q$ between Milnor and Quillen $K$-theory

• Try doing this via Matsumoto's original thesis. A more modern approach is due to [Hutchinson](A new approach to Matsumoto's theorem)

🟠 Jouanolou's device

Write some exposition about constructing Jouanolou devices, and the settings in which they occur. References should include Jouanolou's original work, Thomason's expansion of it, and:

• Asok: The Jouanolou-Thomas homotopy lemma
• BHQW for construction of Jouanolou devices: Making the motivic group structure on the endomorphisms of the projective line explicit

🟠 The James construction motivically (Matthew Niemiro)

Discuss the motivic James construction

• Section 4.2 in Neisendorfer
• Devalapurkar and Haine: On the James and Hilton–Milnor Splittings, & the metastable EHP sequence
• Wickelgren and Williams: The simplicial EHP sequence in A1-algebraic topology

🟠 Real Betti realization (Howard Beck)

Discuss Dugger's theorem comparing real motivic and $C_2$-equivariant homotopy theory.

🟠 Hermitian $K$-theory

Explore the Hermitian $K$-theory spectrum $\textbf{KO}$ and the properties of its homotopy sheaves

• Hornbostel: A1-representability of hermitian $K$-theory and Witt groups

🔴 Algebraic cobordism (Keita Allen)

Discuss the construction of $\text{MGL}$ and its basic properties.

• Levine: overview, book

🔴 Model structures on simplicial presheaves (Peter Chon)

Contrast the various model structures on simplicial presheaves (local, injective/projective, BG, etc.) and indicate how they relate to the $\infty$-categorical perspective we presented in this class

• There's a massive diagram on nLab, and referencee can be found there

🔴 $\eta$-inverted cohomology

Explore motivic cohomology theories where the Hopf element $\eta$ is inverted. Discuss their various properties

• Bachmann and Hopkins: $\eta$-periodic motivic stable homotopy theory over fields