There is a wide class of integrable Hamiltonian systems on finite-dimensional coadjoint orbits of the loop algebra g̃l(r) which are represented by r × r Lax equations with a rational spectral parameter. A reduced complex phase space is foliated with open subsets of Jacobians of regularized spectral curves. Under some generic restrictions on the structure of the Lax matrix, we propose an algebrai...

In this paper, we make specific conjectures about the distribution of Jacobians of random graphs with their canonical duality pairings. Our conjectures are based on a Cohen-Lenstra type heuristic saying that a finite abelian group with duality pairing appears with frequency inversely proportional to the size of the group times the size of the group of automorphisms that preserve the pairing. We...

We approach the Torelli problem of reconstructing a curve from its Jacobian computational point view. Following Dubrovin, we design machinery to solve this effectively, which builds on methods in numerical algebraic geometry. verify method via experiments with curves up genus 7.