Mon 19th-Wed 21st Oct 2009 at Warwick
Extremal Laurent polynomials -- new approaches to mirror symmetry
and classification of Fanos
A follow-up meeting to
2007-08 Warwick EPSRC symposium on Algebraic Geometry
Organisers: Corti, Golyshev, Reid, van Straten
The aims include but are not necessarily restricted to Golyshev and van
Straten's original proposal (appended below).
Preliminary draft of program (prepared by Corti and Golyshev)
All lectures in D1.07
Social event, after hours discussion in Common Room and D1.07
Mon 19th Oct
11:00--11:50 V Golyshev "Intro I"
12:00--12:50 A Corti "Intro II"
13:00 Sandwich lunch
14:00 Hofmann "Monodromy calculations"
15:00 After Hours
18:00 Wine and Buffet supper
Tue 20th Oct
10:00--10:50 V Batyrev "Toric Degenerations"
11:00--11:50 V Przyjalkowski "LG models for Fano 3-folds"
12:00--12:50 L Katzarkov "HMS and algebraic cycles"
13:00 find your own lunch on Warwick campus
14:00--14:50 M Kreuzer "PALP and the classification of reflexive polytopes"
15:00 After Hours
19:00 Curry and pint in Rootes Hall bar
Wed 21st Oct
10:00--10:50 TBA
11:00--11:50 V Golyshev "Monodromy"
12:00--12:50 A Craw "Quiver flag varieties and derived categories"
13:00 Sandwich lunch
14:00 After Hours
Participants (as of Tue 13th Oct)
Batyrev, Michael Bognor, Brown, Cheltsov, Corti, Craw, Galkin,
Golyshev, Heinrich Hartmann, Joerg Hofmann, Zheng Hua, Kaloghiros,
Katzarkov, Dmitra Kosta, Kreuzer, LIN Haijian Kevin, Logvinenko, Pavel
Metelitsyn, Dmitri Panov, Markus Perling, Przyjalkowski, Max Pumperla,
Reid, Konstanz Rietsch, Michael Semmel, Shramov, Maxim Smirnov,
Wendland, Sarah Davis, Sohail Iqbal, ZHOU Shengtian, Umar Hayat,
Eduardo Dias
Please register at
https://www.warwick.ac.uk/mrc/events.php
Original circular
=================
In a recent development, Sergei Galkin and Viktor Przyjalkowski
confirmed that the study of extremal Laurent polynomials is immediately
related to the approach to classifying Fano varieties suggested by V.
Batyrev.
An extremal Laurent polynomial should be considered, with a bit of
stretching, as a higher dimensional generalization of the classical
dessin d'enfant. We are interested in the functions on tori whose
critical values come together `as strongly as possible'. More precisely,
we now know that with each polytope P one should associate the linear
space X_P of Laurent polynomials L supported on P, and stratify X_P
according to the Euler characteristic of the (star extended) local
system of the cohomology of the pencil of the level hypersurfaces L=t.
The extremal Laurent polynomials are the ones that sit in the highest
codimension stratum of this stratification. Of special importance are
the Euler characteristic zero cases.
The privileged role of these extremal Laurent polynomials is in that
they correspond to Frobenius manifolds which may come from Fano
varieties and stacks. In order to quantify Batyrev's theory of toric
degenerations of Fanos we set up a systematic search of the extremal
Laurent polynomials. As the procedure is algorithmic, the search itself
is largely a computational question. At a further, more conceptual stage
one interprets these polynomials by the methods similar to the ones
proposed by van Enckevort and van Straten. One studies the respective
monodromies and makes predictions concerning the topology of the
geometric or quasigeometric objects whose Gromov-Witten calculus would
generate the same Frobenius manifold as the Frobenius manifold of the
isomonodromic deformation of the local system associated to an extremal
Laurent polynomial.
Again, this program should be thought of as shifting emphasis in the
discipline of Fano classification from initial geometric methods to
Batyrev's toric degeneration insight.