This week's Mathematical Sciences Department colloquium is given by Dr. S.I. Agafonov of Department of Mathematics, Universidade Estadual Paulista, S.J. Rio Preto, Brazil.
The theory of Frobenius manifolds, having its origin in theoretical physics, has deep interrelations with apparently very different areas of mathematics: Witten-Gromov invariants and quantum cohomology, deformation of flat connections, integrable systems, singularity theory etc. We discuss a new aspect of this fruitful and fast developing theory: its relations with the classical chapter of differential geometry, namely the web theory.
Using the structure of a given semi-simple Frobenius 3-fold, we construct a 3-web in the plane. This web enjoys the following properties:
1) it is flat,
2) it admits at least one-dimensional symmetry algebra and
3) its Chern connection remains holomorphic in singular points, where at least 2 web directions coincide.
We present a classification of singularities of 3- webs with such properties and show that any such web is obtained by the presented construction. We give also a geometrical interpretation of the associativity equation, describing the corresponding Frobenius 3-fold.
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