References:

Origins

• Deloop 0: Framed Correspondences (by V. Voevodsky) deloop zero.
• [MVW] Lecture Notes on Motivic Cohomology (by C. Mazza, C. Weibel and V. Voevodsky) The development of framed motives follows the format of Voevodsky’s motives; this is a textbook account. 

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

• [GP] Framed Motives of Algebraic Varieties (after V. Voevodsky) (by G. Garkusha and I. Panin) Computes the infinite -loop space of a variety in terms of equationally framed correspondences.
• [Cone] Framed Motives of Relative Motivic Sphere (by G. Garkusha, A. Neshitov and I. Panin) Identifies the framed “homotopy type” of certain quotients.
• [Cancellation] Cancellation Theorem for Framed Motives of Algebraic Varieties (by A. Ananyevskiy, G. Garkusha, and I. Panin) Proves that the framed -suspension functor is fully faithful.
• [HITR] Homotopy Invariant Sheaves with Framed Transfers (by G. Garkusha and I. Panin) Proves that the Nisnevich localization of certain homotopy invariant framed presheaves are homotopy invariant – implies commutation of loops against localizations.

• [Surj] Surjectivity of the etale excision map for homotopy invariant framed presheaves (A. Druzhinin and I. Panin) Removes assumptions on the base fields in the above papers.

• [pi0] Framed Correspondences and Milnor-Witt K-Theory (by A. Neshitov) Reproves (cases of) Morel’s computation of the 0-th line of the motivic sphere spectrum.

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

• Deloop 1: Motivic Infinite Loop Spaces. An account of the recognition principle.
• Deloop 2: Framed Transfers and Motivic Fundamental Classes. Compares various transfers in motivic homotopy theory.
• Deloop 3: Modules Over Algebraic Cobordism. Proves that modules over the algebraic cobordism spectra are certain presheaves with finite syntomic transfers.

Other references/applications

• [Norms] Norms in Motivic Homotopy Theory (by T. Bachmann and M. Hoyois)
• [HZ] The localization theorem for framed motivic spaces (by M. Hoyois)
• [Perf] Perfection in Motivic Homotopy Theory (by E. E. and A. Khan)
• [Gm-stab] Towards Conservativity of stabilization (by T. Bachmann and M. Yakerson)
• [Framed-rig] Rigdity for linear framed presheaves and generalized motivic cohomology theories (by A. Ananyevskiy and A. Druzhinin)

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 -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 of the sphere is Milnor-Witt -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 -category of (twisted) framed correspondences. Constructs, via a Segal space, the -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.