# Abstracts

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• The participants are kindly requested to send the files with abstracts of their talks to panas@fuw.edu.pl (Andriy Panasyuk) before August 15 in one of the following forms:

• simple text files
• .doc format
• the source Latex files

• Sergey Agafonov

On implicit ODEs with hexagonal web of solutions

Solutions of an implicit ODE form a web. Already for cubic ODEs the 3-web of solutions has a nontrivial local invariant, namely the curvature form. Thus any local classification of implicit ODEs necessarily has functional moduli if no restriction on the class of ODEs is imposed. Here the most symmetric case of hexagonal 3-web of solutions is discussed, i.e. the curvature is supposed to vanish identically. A finite list of normal forms is established under some natural regularity assumptions. Geometrical meaning of these assumptions is that the surface, defined by ODE in the space of 1-jets, is smooth as well as the criminant, which is the critical set of this surface projection to the plane. One of the normal forms is given by the characteristic web of a specific homogeneous solution of an associativity equation.

Wojciech Domitrz

Symplectic and volume-preserving singularities of varietes and de Rham cohomology

We explain the relations between symplectic and volume-preserving singularities and various types of de Rham complexes of singular varietes. We present the generalization of Darboux-Givental' theorem for singular varietes in a symplectic manifold. We show various applications of this theorem for local symplectic classification problems. We also present the volume preserving classification of singular varietes and maps and explain its cohomological invariants.

Vasily Golyshev

Towards quantum motives: problems and theorems

No theory of quantum motives has been constructed yet. However, a realization of a quantum motive can be turned into a precise concept, parallel to that for a traditional semisimple motive. We discuss the quantum-motivic angle of a few known theorems, and finish by stating problems that arise from the motivic picture.

Yuliya Grishina

Singularities of controllability of simplest dynamic inequalities

Goo Ishikawa

Local symplectic invariants of parametric Lagrangian varieties

A review on local symplectic invariants for parametrized Lagrangian varieties will be presented. I will introduce invariants such as symplectic codimension, symplectic defect, number of isotropic double points, etc., and show their relations with other known invariants. Also I would like to mention their possible relations with Frobenius structures, if possible. This talk is based on joint works with S. Janeczko.

Bumsig Kim

Logarithmic Stable Maps

We introduce the notion of a logarithmic stable map from a minimal log prestable curve to a log twisted semi-stable variety of form xy=0. As an application, we show a modular desingularization of the main component of Kontsevich's moduli space of elliptic stable maps to a projective space.

Dmitry Korotkin

Local and global properties of Bergman tau-function on Hurwitz spaces

The Bergman tau-fuction on Hurwitz spaces is a universal object appearing in isomonodromy deformation problem associated to Frobenius manifolds, random matrices and determinants of Laplacians on Riemann surfaces.Here we review the definition, computation , local and global properties of this object.

Etienne Mann

Smooth toric DM stacks

In 2003, Borisov, Chen and Smith defined smooth toric DM stacks in terms of "stacky fan" which is, as a first approximation, a fan where one marks a point on each ray. With Barbara Fantechi and Fabio Nironi, we defined smooth toric DM stacks via the action of an open dense torus. In particular, we give a geometric interpretation of the combinatorial data contained in a stacky fan. We also give a bottom up classification in terms of simplicial toric varieties and fiber products of root stacks.

Sergey Natanzon

Singularities and noncommutative Frobenius manifolds

We prove that quaternionic miniversal deformations of an $A_n$ singularity have the structure of a noncommutative Frobenius manifold in the sense of the extended cohomological field theory.

Maxim Pavlov

Solutions of WDVV equations and corresponding hydrodynamic reductions of integrable hydrodynamic chains

We describe an infinite set of hydrodynamic type systems equipped by local Hamiltonian structures, which are nothing but hydrodynamic reductions of the Benney hydrodynamic chain.

Pedro de M Rios

A Frobenius-type structure on symmetric symplectic spaces

There is a canonical way of associating the product of (non-infinitesimal) symplectomorphisms (or canonical relations) on a symmetric symplectic space, to a noncommutative product of lagrangian sections of its tangent bundle (for a canonically defined symplectic structure on the tangent bundle). In this talk, I will present and explain this product and its relations to Frobenius structures and singularity theory.

Paolo Rossi

Quantum cohomology of polynomial P1-orbifolds and its mirror model

We compute, with Symplectic Field Theory techniques, the Gromov-Witten theory of $\mathbb{P}^1_{\alpha_1,\ldots,\alpha_a}$, i.e. the complex projective line with $a$ orbifold points. A natural subclass of these orbifolds, the ones with polynomial quantum cohomology, gives rise to a family of (polynomial) Frobenius manifolds and integrable systems of Hamiltonian PDEs, which extend the (dispersionless) bigraded Toda hierarchy (\cite{C}). We then define a Frobenius strucure on the spaces of polynomials in three complex variables of the form $F(x,y,z)=-xyz+P_1(x)+P_2(y)+P_3(z)$ which contains as special cases the ones constructed on the space of Laurent polynomials (\cite{D},\cite{MT}). We prove a mirror theorem stating that these Frobenius structures are isomorphic to the ones found before for polynomial $\mathbb{P}^1$-orbifolds. Finally we link rational Symplectic Field Theory of Seifert fibrations over $\mathbb{P}^1_{a,b,c}$ with orbifold Gromov-Witten invariants of the base, extending a known result (\cite{B}) valid in the smooth case.

Ikuo Satake

Conformal Frobenius structures

We define a notion of a conformal Frobenius structure. This enables us to define a period mapping for a simply elliptic sigularity without fixing a data of primitive form. Then the period mapping is equivariant with respect to an action of a conformal transformation group which changes a primitive form. By using this action and Laplacian, we obtain an explicit description of a period mapping for a simply elliptic singularity.

Artur Sergyeyev

Infinite hierarchies of nonlocal symmetries for the oriented associativity equations

We generalize the results of Chen, Kontsevich, and Schwarz (Nuclear Physics B 730 [PM] (2005) 352-363) and construct infinite hierarchies of nonlocal higher symmetries for the {\em oriented} associativity equations.

BCOV theory via Givental group action

BCOV theory formalized by Barannikov and Kontsevich suggests a way to associate a Frobenius manifold structure to an algebraic structure that captures the basic properties of polyvector fields on Calabi-Yau manifolds. We explain how to reconstruct the cohomological field theory behind this Frobenius manifold using the Givental theory of group action on cohomological field theories.

Vasilisa Shramchenko

Two dual Riemann-Hilbert problems associated to Hurwitz spaces

There are two (dual to each other) Riemann-Hilbert problems naturally associated to every Frobenius manifold. In the case of Frobenius structures on Hurwitz spaces, the corresponding Riemann-Hilbert problems turn out to be solvable in terms of meromorphic bidifferentials defined on the underlying surface. In this talk, I will present these solutions and discuss their monodromy groups. The monodromy transformations for both solutions are related to the monodromy in the appropriate spaces of contours on the Riemann surface.

Ian Strachan

Jacobi group orbit spaces and elliptic (almost dual) Frobenius manifolds

The simplest class of Frobenius manifolds, coming from the original Saito construction, are the orbit spaces $C^N/W$, where $W$ is a Coxeter group. The corresponding prepotential is polynomial in the flat coordinates.

This talk will concern an elliptic version of this construction, where the orbit space involves a Jacobi group (based on some Coxeter group). In particular, the so-called "almost dual" Frobenius manifold will be studied. In this case the (almost dual) prepotential can be written simply in terms of the elliptic trilogarithm introduced by Beilinson and Levin..

Atsushi Takahashi

Homological Mirror Symmetry for Cusp Singularities

After reviewing some necessary basic definitions such as Fukaya categories and categories of coherent sheaves on weighted projective lines, I'll formulate Homological mirror symmetry conjecture and then give my main theorem. I'll explain that it is naturally expected from this HMS conjecture that three constructions of a Frobenius manifold for this setting (via the quantum cohomology for a weighted projective line, via the primitive form for a cusp singularity and via an invariant theory of an extended Weyl group) will give an isomorphic one.

Olga Trubina

To the problem of approximation by polyhedrons of a convex surface with prescribed integral curvature

Dimitri Zvonkine

Givental's quantization, tautological relations, and Witten's $r$-spin conjecture (joint with C. Faber and S. Shadrin)

Tautological relations are identities in the cohomology ring of moduli spaces of stable curves, more precisely, in the so-called "tautological subring" of this ring. The main construction in Givental's theory of quantization of semi-simple Frobenius manifolds is a group action on Gromov-Witten potentials in all genera. The genus expansion of every semi-simple Frobenius manifold is obtained via this group action from the Gromov-Witten potential of a point. We show that Givental's group action respects all relations induced by tautological relations. As an important application we deduce the claim of Witten's $r$-spin conjecture.