Posts tagged ‘Imperial’

V. Batyrev: “Toric Deformations of Some Spherical Fano Varieties”

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: SL_2-varieties
  • review: compactification and toric varieties
  • spherical compactification in these examples
  • valuations and coloured cones
  • finding toric degenerations

K. Altmann: “Deformations of Gorenstein Canonical Toric Singularities”

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

B. Siebert: “A Tropical View on Landau-Ginzburg Models”

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 \PP^3 and a pencil of elliptic curves in \PP^2
  • 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: \PP^2, flattening the boundary of the polyhedral complex
  • Broken lines and scattering

V. Golyshev: “The Apery Class and the Gamma Class”

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 Gr(k,n); 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 Gr(k,n)