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Abstracts for Summer Tropical Seminar

Alternatively have a look at the program.

The conifold point, conjecture O, and related problems

Posted in
Speaker: 
Sergey Galkin
Affiliation: 
HSE - Moscow
Date: 
Tue, 09/06/2015 - 16:30 - 18:00
Location: 
MPIM Lecture Hall
Parent event: 
Summer Tropical Seminar

Conjecture O describes the geometry in the complex line of the eigenvalues u_i
of the operator of quantum multiplication by the first Chern class acting on the
cohomology of a Fano manifold. In particular, it says that eigenvalues with maximal
absolute value have multiplicity one and one of them is real and positive number T.

Fano manifolds tend to have mirror dual Ginzburg-Landau potentials f, which
tend to have a distinguished non-degenerate critical point which we name the

Tropical Differential Equations

Posted in
Speaker: 
D. Grigoriev
Date: 
Tue, 23/06/2015 - 16:30 - 18:00
Location: 
MPIM Lecture Hall
Parent event: 
Summer Tropical Seminar

Tropical differential equations are introduced. Similar to usual
tropical algebraic equations which provide necessary conditions for
solvability of systems of polynomial equations in Puiseux series,
tropical differential equations express necessary conditions for
solvability of systems of differential equations in power (or Hahn)
series.

For a system of tropical linear differential equations we prove the
existence of the minimal among its solutions. A polynomial
complexity algorithm is designed for solving such systems. For

Real deformations of Cayley octads and their links

Posted in
Speaker: 
Sergey Finashin
Affiliation: 
METU, Ankara
Date: 
Fri, 17/07/2015 - 15:30 - 17:00
Location: 
MPIM Lecture Hall
Parent event: 
Summer Tropical Seminar

Cayley octads are 8-configurations in $P^3$ that are complete intersections of three quadrics (i.e., the base locus of a net of quadrics). The locus formed by singular quadrics of the net form a Hessian (or spectral) quartic endowed with an even theta characteristic. An octad is called regular if this quartic is non-singular. Using this correspondence one can obtain a deformation classification of real regular Cayley octads which will be discussed. Namely, I will present the 8 deformation classes of maximal real regular Cayley octads and discuss their monodromy groups and degenerations.

On the total curvature of minimizing geodesics on convex surfaces

Posted in
Speaker: 
Anton Petrunin
Affiliation: 
Penn State/MPI
Date: 
Tue, 21/07/2015 - 16:30 - 18:00
Location: 
MPIM Lecture Hall
Parent event: 
Summer Tropical Seminar

We give an upper bound for the total curvature of minimizing geodesics on convex surfaces.

Growth rate, congruences, and other qualitative aspects of counting real rational curves on real K3 surfaces

Posted in
Speaker: 
Viatcheslav Kharlamov
Affiliation: 
Strasbourg/MPIM
Date: 
Tue, 28/07/2015 - 16:30 - 18:00
Location: 
MPIM Lecture Hall
Parent event: 
Summer Tropical Seminar

I intend to start from reminding our, with Rares Rasdeaconu, real version of the Yau-Zaslow formula for the number of complex ones, and then will discuss some asymptotic and arithmetical properties of the lower bounds provided by our formula. In particular, I will discuss
under what assumptions on the surface a certain abundance of real rational curves holds.

Projective spaces in Fermat varieties

Posted in
Speaker: 
Alex Degtyarev
Affiliation: 
Bilkent Univ. / MPIM
Date: 
Tue, 04/08/2015 - 16:30 - 18:00
Location: 
MPIM Lecture Hall
Parent event: 
Summer Tropical Seminar 
In 1981, T. Shioda showed that the straight lines contained in a
Fermat surface of degree prime to 6 generate the Picard group of
the surface *over the rationals*, and he conjectured that the same
lines also generate the Picard group *over the integers*.

Enumeration of singular hypersurfaces via tropical geometry

Posted in
Speaker: 
Eugenii Shustin
Affiliation: 
Tel Aviv Univ./MPIM
Date: 
Tue, 11/08/2015 - 16:30 - 18:00
Location: 
MPIM Lecture Hall
Parent event: 
Summer Tropical Seminar 
We discuss a tropical approach to enumeration of
singular complex and real hypersurfaces in toric varieties. In particular,
we establish a correspondence theorem between singular tropical and
algebraic hypersurfaces and demonstrate a multi-dimensional version of
a lattice path algorithm. As application we obtain a lower bound to the
maximal possible number of real singular hypersurfaces in a generic real pencil.

Homotopy quantization of real curve enumeration

Posted in
Speaker: 
Grigory Mikhalkin
Affiliation: 
Geneva/MPIM
Date: 
Tue, 18/08/2015 - 16:30 - 18:00
Location: 
MPIM Lecture Hall
Parent event: 
Summer Tropical Seminar 
The complexification of a real rational curve is composed
of two holomorphic disks glued along the real locus of the
curve. It turns out that each of these disks realizes an element
of a certain (non-commutative) Heisenberg-type group which
appears as the second homotopy group \pi_2(X,L,p) for some X
and L. Using this group we may recover the quantum index of
the real curve which in its turn is responsible for the refined
enumeration of complex rational curves in the plane.
Joint work with Sergey Galkin.

Square-tiled surfaces of fixed combinatorial type: equidistribution, counting, volumes of the ambient strata

Posted in
Speaker: 
Anton Zorich
Affiliation: 
Paris 7/MPIM
Date: 
Tue, 25/08/2015 - 16:30 - 17:30
Location: 
MPIM Lecture Hall
Parent event: 
Summer Tropical Seminar

We  prove  that  square-tiled  surfaces (correspondingly pillowcase
covers)  tiled  with  tiny squares sharing a fixed combinatorics of
cylinder  gluings are asymptotically equidistributed in the ambient
stratum  in the moduli space of Abelian (correspondingly quadratic)
differentials.   We  prove  similar  equidistribution  results  for
rational interval exchange transformation.

 

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