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.
  • [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).
  • [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 24-31 – 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 \infty-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 \infty-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] ?