September 22, 2010, 11:30 am
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) OG(5,10)](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/e51/e51df67019c1dee46072619da7689a0f-ffffff-000000-0.png)
- the quantum Chevalley formula (Fulton–Woodward)
- cohomology of G/P in general
- cohomology of G/P: improvements in the miniscule case
- semisimplicity of
(Chaput–Manivel–Perrin) and connections to work of Kostant (Gorbunov–Petrov)
- quantum Satake for miniscule type-D Grassmannians
September 22, 2010, 11:23 am
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)) QH^\bullet(Gr(k,n+1))](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/481/48184a6c3da6e61be5d1e49c7374600a-ffffff-000000-0.png)
- the geometric Satake correspondence (Drinfeld–Ginzburg)
- miniscule Grassmannians as smooth
-orbits in the affine Grassmannian
- a quantum analog of geometric Satake in this case
September 22, 2010, 11:17 am
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
and ![QH^\bullet(OG(n,2n)) QH^\bullet(OG(n,2n))](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/ffe/ffeebe22a8c8c5066f0d1e08dcccaa14-ffffff-000000-0.png)
- the connection to algebraic combinatorics via Praguez-Rataiski polynomials
- quantum Giambelli and quantum Pieri (Kresch-Tamvarkis)
September 22, 2010, 11:13 am
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:
is rational of dimension
implies that the spectrum of
has no gaps of size greaeter than ![n-2 n-2](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/ba8/ba81144e78f150a37fbe511667f9594b-ffffff-000000-0.png)
- the outlook for 4-dimensional cubics
September 22, 2010, 11:07 am
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