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\title{Rational degenerations of $m$-curves and totally positive Grassmannians}
\author{\underline{Simonetta Abenda}, Petr Grinevich \\University of Bologna (Italy)\\Landau Institute of Theoretical Physics (Moscow)}
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\maketitle
In this talk I shall connect two areas of mathematics: the theory of totally positive Grassmannians and the rational degenerations of m-curves using the theory of the KP-2 equation.
Thanks to recent papers by Kodama and Williams \cite{KW2,KW3} the relation between the class of line soliton solutions of KP-2 equation and the totally non-negative part of real finite dimensional Grassmannians is well established.
On the other hand such soliton solutions may also obtained from the limit of regular real finite gap solutions of KP-2. Dubrovin and Natanzon \cite{DN} proved in 1988 that the algebro-geometric data of the regular real quasi-periodic solutions are associated to m-curves.
We show how to associate to any point in the totally positive part of $Gr(N,M)$ the algebro-geometric data a la Krichever \cite{Kr1} for the corresponding line soliton solution, i.e. the rational degeneration of a regular $m$-curve of genus
$g=N(M-N)$ and the divisor of poles of the associated KP wave-function.
The results presented are in collaboration with P.G. Grinevich \cite{AG}.
\begin{thebibliography}{1}
\bibitem[1]{AG} Abenda, S., Grinevich, P.G. ``Rational degenerations of $m$-curves and totally positive Grassmannians'', in press (2015).
\bibitem[2]{DN} Dubrovin, B. A., Natanzon S. M., ``Real theta-function solutions of the Kadomtsev-Petviashvili equation'', Izv. Akad. Nauk SSSR Ser. Mat., 52:2 (1988), 267-286
\bibitem[3]{KW2} Kodama, Yuji; Williams, Lauren ``The Deodhar decomposition of the Grassmannian and the regularity of KP solitons'', Adv. Math. 244 (2013), 979–1032
\bibitem[4]{KW3} Kodama, Yuji; Williams, Lauren ``KP solitons and total positivity for the Grassmannian'', Invent. Math. 198 (2014), no. 3, 637-699.
\bibitem[5]{Kr1} Krichever, I. M., ``An algebraic-geometric construction of the Zakharov-Shabat equations and their periodic solutions''. (Russian) Dokl. Akad. Nauk SSSR 227 (1976), no. 2, 291-294.
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