Frobenius manifold structures on the spaces of abelian integrals

Cyclic stratum of Frobenius manifolds, Borel-Laplace $(\boldsymbolα,\boldsymbolβ)$-multitransforms, and integral representations of solutions of Quantum Differential Equations

In the first part of this paper, we introduce the notion of "cyclic stratum" of a Frobenius manifold $M$. This is the set of points of the extended manifold $\mathbb C^*\times M$ at which the unit vector field is a cyclic vector for the isomonodromic system defined by the flatness condition of the extended deformed connection. The study of the geometry of the complement of the cyclic stratum is addressed. We show that at points of the cyclic stratum, the isomonodromic system attached to $M$ can be reduced to a scalar differential equation, called the "master differential equation" of $M$. In t…

关于2-可解Frobenius李代数的分类

On the classification of 2-solvable Frobenius Lie algebras

本文研究2-可解Frobenius李代数的分类，证明其可分解为向量空间与极大Abel子代数的半直和，并给出非减次自同态及类2极大Abel幂零子代数情形的完整分类，同时修正了sl(4,R)的MASA分类表。

On some geometric properties of currents and Frobenius theorem

In this note we announce some results, due to appear in [2], [3], on the structure of integral and normal currents, and their relation to Frobenius theorem. In particular we show that an integral current cannot be tangent to a distribution of planes which is nowhere involutive (Theorem 3.6), and that a normal current which is tangent to an involutive distribution of planes can be locally foliated in terms of integral currents (Theorem 4.3). This statement gives a partial answer to a question raised by Frank Morgan in [1].

A non-abelian Seiberg-Witten invariant for integral homology 3-spheres

A new diffeomorphism invariant of integral homology 3-spheres is defined using a non-abelian 'quaternionic' version of the Seiberg-Witten equations.

Linear estimate for the number of zeros of Abelian integrals

We prove a linear in $\degω$ upper bound on the number of real zeros of the Abelian integral $I(t)=\int_{δ(t)}ω$, where $δ(t)\subset\R^2$ is the real oval $x^2y(1-x-y)=t$ and $ω$ is a one-form with polynomial coefficients.