Posts tagged ‘Bonn’

L. Manivel: “Quantum Satake for Miniscule Spaces”

Here are my notes from Manivel’s talk in Bonn:

Manivel_Bonn_Sep_2010

Summary:

  • geometric Satake after Drinfeld and Ginzburg
  • example: OG(5,10)
  • the quantum Chevalley formula (Fulton–Woodward)
  • cohomology of G/P in general
  • cohomology of G/P: improvements in the miniscule case
  • semisimplicity of QH^\bullet(G/P)|_{q=1} (Chaput–Manivel–Perrin) and connections to work of Kostant (Gorbunov–Petrov)
  • quantum Satake for miniscule type-D Grassmannians

V. Golyshev: “Introduction to the Satake Correspondence”

Here are my notes from Golyshev’s second talk in Bonn:

Golyshev_2_Bonn_Sep_2010

Summary:

  • the “spin representation of K3″ and its quantum analog
  • some evidence: the Bertram–Ciocan-Fontanine–Kim result for QH^\bullet(Gr(k,n+1))
  • the geometric Satake correspondence (Drinfeld–Ginzburg)
  • miniscule Grassmannians as smooth L_+ G-orbits in the affine Grassmannian
  • a quantum analog of geometric Satake in this case

A. Kresch: “Quantum Cohomology of Miniscule Type-D Grassmannians”

Here are my notes from Kresch’s talk in Bonn:

Kresch_Bonn_Sep_2010

Summary:

  • the orthogonal Grassmannian
  • Schubert varieties are indexed by strict partitions
  • special Schubert classes
  • ring presentations for H^\bullet(OG(n,2n)) and QH^\bullet(OG(n,2n))
  • the connection to algebraic combinatorics via Praguez-Rataiski polynomials
  • quantum Giambelli and quantum Pieri (Kresch-Tamvarkis)

L. Katzarkov: “Gaps, Spectra, and Applications”

Here are my notes from Katzarkov’s talk in Bonn:

Katzarkov_Bonn_Sep_2010

Summary:

  • Clemens-Griffiths showed that the 3-dimensional cubic is not rational by showing that its intermediate Jacobian is not the Jacobian of a curve; we suggest analogs of this.
  • Homological Mirror Symmetry and the perverse sheaf of vanishing cycles
  • detecting rationality via monodromy properties on the LG mirror (Gross-Katzarkov, Pryzalkowski, Golyshev)
  • spectra of triangulated categories; examples
  • Theorem: X is rational of dimension n implies that the spectrum of D^b(X) has no gaps of size greaeter than n-2
  • the outlook for 4-dimensional cubics

V. Golyshev: “Quantum Motives: Linearizations, Realizations, Use, Detections”

Here are my notes from Golyshev’s talk in Bonn.

Golyshev_1_Bonn_Sep_2010

Summary:

  • a historical analogy: the study of Fano varieties now versus algebraic number theory prior to the proof of the Weil Conjectures
  • linearization and motives
  • Tannakian categories
  • the “quantum Tannakian category”
  • realizations in algebraic number theory: one can detect algebraic varieties by detecting their motives (in practice by detecting their L-functions via Selberg-Stark)
  • the “quantum Tate realization” and quantum detection
  • the Satake correspondence and its quantum parallel