Posts tagged ‘talk notes’

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:


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:



  • 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:



  • 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:



  • 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)

L. Manivel: “Quantum Satake for Miniscule Spaces”

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



  • 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:



  • 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:



  • 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:



  • 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.



  • 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