Abstract

It is 20 years since the publication of "Geometry of 2D topological field theories" in volume 1620 of Lecture Notes in Mathematics by Boris Dubrovin.  In this tour de force of ideas and examples, the theory of Frobenius manifolds was firmly established. 

Frobenius manifolds are differential geometric objects, motivated by deformations of quantum field theories, unfoldings of singularities, and quantum cohomology.  Dubrovin's definition introduced a common framework for these rather different subjects.  Then he applied tools from the theory of integrable systems to obtain deep and surprising relations between them.  

In these lectures we shall try to explain what Dubrovin did, focusing on the case of 2-dimensional Frobenius manifolds where everything can be calculated explicitly by elementary methods. For Lectures 1 and 2, knowledge of 2nd order linear o.d.e. and linear algebra will be sufficient. Ideally, the listener will not notice the gradual submersion into a difficult subject (mirror symmetry).  In particular, prior knowledge of physics or algebraic geometry will not be assumed. In Lecture 3 we shall mention more advanced topics and reflect on what the theory of Frobenius manifolds has achieved.