Seminar Event Detail


Integrable Systems and Random Matrix Theory

Date:  Monday, November 13, 2017
Location:  1866 East Hall (4:00 PM to 5:00 PM)

Title:  Symplectic aspects of a monodromy map

Abstract:   The monodromy map assigns to an ordinary differential equation on a Riemann surface of the form \psi''-u(z)\psi=0 it's monodromy group, a representation of pi_1 into PSL(2,C) by noting how a solution transforms when one travels around a handle of the Riemann surface. This correspondence between the differential equation, where u(z) denotes a holomorphic projective connection, and it's monodromy group is a specific example of the Riemann-Hilbert correspondence, and the work of Gallo-Kapovich-Marden states that every non-elementary representation of pi_1 into PSL(2,C) which is liftable to SL(2,C) is the monodromy group of some differential equation of the form stated above.

In joint work with Bertola and Korotkin, we study the symplectic properties of the map from the space of such equation, i.e. the moduli space of projective connections, to the character variety of representations of pi_1 into PSL(2,C). Is specific we show that the monodromy map, with various choices of base projective connection, is a symplectomorphism from the moduli space of quadratic differentials with the homological bracket to the character variety with the Goldman bracket.

In our case there are no isomonodromy deformations, the number of parameters in the equation matches the number in the representations. Results showing the symplectic properties of a monodromy map provide a perspective to interpret isomonodromy deformations as the direction transversal to the symplectic leaves, arising from the presence of casimirs, giving an intrinsic explanation for the hamiltonian structure arising in isomonodromy deformations.

Files:


Speaker:  Chaya Norton
Institution:  Concordia University and Centre de Recherches Mathematiques (CRM), Montreal

Event Organizer:   Thomas Bothner and Guilherme Silva    bothner@umich.edu, silvag@umich.edu

 

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