**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. - [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). - [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 24-31 – Elden/Tom]
**The St. Pete school’s computation of framed motives of varieties:**go through the details of the St. Pete papers. - [Nov 7 – Jeremy (?)]
**The -category of framed correspondences**: define normally framed and tangentially framed correspondences; prove descent properties, additivity and comparison between these correspondence; sketch the construction of the -category of framed correspondences and define its composition law. Reference: deloop1. - [Nov 14 – TBD]
**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. - [Nov 21 – TBD]
**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. - [Dec 5 – TBD] ?