September 22, 2010, 12:09 pm

Here are my (incomplete) notes from Batyrev’s talk in the Extremal Laurent Polynomials workshop at Imperial:

Batyrev_London_Sep_2010

If you have complete notes, please post them here.

Partial summary:

- spherical varieties
- a model family of examples: -varieties
- review: compactification and toric varieties
- spherical compactification in these examples
- valuations and coloured cones
- finding toric degenerations

September 22, 2010, 12:04 pm

Here are my notes from Altmann’s talk at the Extremal Laurent Polynomials workshop at Imperial:

Altmann_London_Sep_2010

Summary:

- an example due to Pinkham
- deformations and Minkoski sums; a lattice condition
- constructing the deformation corresponding to a Minkwoski decomposition
- the versal deformation space and the moduli space of generalized Minkowski summands of Q
- Example: the cone over a hexagon
- the equations defining the versal deformation space in the isolated Gorenstein case
- a new point of view: double divisors

September 22, 2010, 11:58 am

Here are my notes from Siebert’s talk at the Extremal Laurent Polynomials workshop at Imperial:

Siebert_London_Sep_2010

Summary:

- An overview of toric degenerations and the Gross–Siebert picture
- Examples: a pencil of quartics in and a pencil of elliptic curves in
- the Gross–Siebert Reconstruction Theorem
- Mirror Symmetry and the discrete Legendre Transform
- Landau–Ginzburg models; the Hori–Vafa mirror
- how to extend the superpotential from the central fiber to the whole of the mirror family in such a way that the resulting superpotential is proper
- Example: , flattening the boundary of the polyhedral complex
- Broken lines and scattering

September 22, 2010, 11:50 am

Here are my notes from Golshev’s talk at the Extremal Laurent Polynomials workshop at Imperial:

Golyshev_London_Sep_2010

Summary:

- a historical analogy: the study of Fano varieties now versus the study of algebraic varieties in the early 1970s
- the key idea: classifying Fanos by detecting and classifying Fano quantum motives and their realizations
- possible approaches
- the Tannakian picture
- the quantum Satake correspondence (Golyshev–Manivel)
- two steps in this direction: Ueda’s proof of the Dubrovin Conjecture for ; Galkin–Golyshev–Iritani’s proof that Apery=Gamma
- irregular monodromy data for Dubrovin’s quantum connection
- the Extended Dubrovin Conjecture and the Apery=Gamma hypothesis
- the Extended Dubrovin Conjecture and the Apery=Gamma hypothesis hold for

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:
- 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
- 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
- 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
- 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