# Thursday Seminar Fall 2019

References:

Origins

Papers from the St. Petersburg (St. Pete) School

Deloop Series (E. E, M. Hoyois, A. Khan, V. Sosnilo, M. Yakerson)

Other references/applications

Schedule

• [Sept 12 – Mike] Organizational meeting.
• [Sept 19 – Peter] Cast of characters from algebraic geometry: define and discuss examples of local complete intersections in terms of the cotangent complex and in terms of cutting by equations; discuss the localization sequence for cotangent complexes and base change for tor-independent squares; discuss Avaramov’s characterization of lci maps; prove that the lci locus in the Hilbert scheme of points on a smooth variety is an (ind-)smooth scheme; introduce the framed Hilbert scheme of points. References: stacks project, Avramov, deloop1. Peter’s Notes.
• [Sept 26] No seminar
• [Oct 3 – Lucy] Discuss classical infinite loop space theory in modern terms: mention the transfer conjecture of Quillen, talk about the recognition principle in terms of $\Gamma$-spaces, give examples of spectra. References: Adams, Segal, Higher Algebra, Gepner-Groth-Nikolaus.
• [Oct 10 – Dexter] Motivic Homotopy Theory: introduce stable, unstable motivic homotopy theory; discuss basic theorems like the purity theorem, localization, sample computations in the stable/unstable world; introduce motivic spectra; introduce inversion of symmetric objects after Robalo, introduce effective and very effective motivic spectra; discuss examples of motivic spectra (sphere, K-theory, cobordism, motivic cohomology). References: Morel-Voevodsky, primer (for sampler of computations), Robalo, Norms (for the optimal constructions). Dexter’s Notes.
• [Oct 17 – Elden] Equationally framed correspondences: motivate and introduce Veovodsky’s original construction of equationally framed correspondences – a kind of motivic Pontrjagin-Thom construction; state the theorem in GP that the framed motive of a variety computes its infinite loop space; sketch Neshitov’s proof that $\pi_{n,n}$ of the sphere is Milnor-Witt $K$-theory using frames; identify some explicit equationally framed correspondences with generators of the Grothendieck-Witt groups. References: deloop0, GP1, pi0.
• [Oct 24 – Jeremy] The recognition principle: tie up the last four talks and prove the recognition principle; prove that the 0-th space of the motivic sphere spectrum is the framed Hilbert scheme of points; miscellaneous topics (time permitting): the motivic bar construction, comparison with equivariant homotopy theory. Reference: deloop1.
• [Oct 31, Nov 7, Nov 14 – Elden/Tom] The St. Pete school’s computation of framed motives of varieties: go through the details of the St. Pete papers.
• [Nov 24 – Dylan] Construction of the $\infty$-category of (twisted) framed correspondences. Constructs, via a Segal space, the $\infty$-category following deloop3.
• [Dec 5 – Peter] Algebraic cobordism and Hilbert schemes: define motivic Thom spectra after Bachmann-Hoyois; introduce twisted variants of framed correspondence; prove that the zero-th space of the algebraic cobordism spectrum over a field is the Hilbert scheme of systemic points. Reference: deloop3.