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Martin Guest (Waseda University) 
2026-04-30  13:00 - 14:30
Room 202, Astronomy and Mathematics Building

In this lecture we shall explain some of the significant progress made on the Dubrovin Conjecture in the series of works by  Galkin-Golyshev-Iritani and Cotti-Dubrovin-Guzzetti in the 2010's.  We shall describe some of the difficulties which occur and some questions which arise.  The first main difficulty is that monodromy data (Stokes and connection matrices) is almost impossible to compute explicitly. It can be computed when solutions of the o.d.e. can be written as contour integrals, and/or in the presence of a high degree of symmetry (as for most classical o.d.e.).  For X = CP^n the quantum differential equations are of this type, so (like the above authors) we shall focus on this case.  The second main difficulty is the lack of a canonical description of the monodromy data - various choices are involved.  This is more subtle, and it leads to an organisational problem: how should this data be presented in order to reduce the ambiguities?  For the Stokes matrices of CP^n this can be done, but the connection matrices still present interesting challenges.  We shall describe some of the results available, and the relevance of real structures and tt* geometry.