Anastasia Matveeva defended her PhD thesis on Poisson Structures on Moduli Spaces and Group Actions, supervised by Professor Eva Miranda Galceran, on October 3, 2022, within the UPC doctoral program in Applied Mathematics. Currently, she leads a team of analysts at a Californian startup developing tools for mobile performance optimization.
Thesis summary
Symplectic and Poisson geometry fields arise at the intersection of geometry and physics. Motivated by understanding the dynamics of mechanical systems, they consider the phase space of such a system as a manifold with a prescribed geometric structure. Understanding the geometric properties of these manifolds brings insights into mechanical systems’ behavior. Symplectic structures cover a large part of the examples coming from classical mechanics and provide very applied techniques. Poisson manifolds, more general, can be viewed through the prism of the symplectic foliation. One of the good examples where symplectic methods shine at their best is the problem of finding periodic orbits (if they exist). Another splendid application comes from the simple idea that any symmetry of a system reduces the number of its degrees of freedom, simplifying the system itself. In physical language, this would be formulated as conservation laws and first integrals. In geometric language, this concept can be encoded as a reduction theorem. The celebrated Marsden-Weinstein reduction reveals an exciting phenomenon that for a group of dimension k, the reduction can be doubled: the system can be simplified by 2k degrees of freedom [1].
Marsden-Weinstein quotients are naturally connected to certain moduli spaces. In their seminal article, Michael Atiyah and Raoul Bott unveiled the symplectic structure on the space of flat connections. The Riemann-Hilbert problem explores the correspondence between the moduli space of flat connections of Fuchsian systems (i.e., differential systems with simple poles) on a sphere and the monodromy data’s moduli space (i.e., representations of the fundamental group of a punctured sphere). There are few cases where the solution can be constructed explicitly. For Riemann spheres, positive results of a classical Riemann-Hilbert problem are usually existence theorems. In that case, Riemann-Hilbert correspondence turns out to be a Poisson morphism.
In recent years, there has been an increasing interest in b-symplectic (together with more general bm– and E-symplectic) geometry. The corresponding manifolds can be viewed as stepping out of the symplectic category toward Poisson, allowing certain types of singularities in the 2-form, which is no longer symplectic. This approach enables a careful transfer of symplectic techniques to larger classes of Poisson structures while tracking which properties break or change. Many aspects of b, bm and E-symplectic geometry are studied in numerous works of Eva Miranda and her collaborators [2–4].
This thesis studies the analog of Marsden-Weinstein reduction in the context of singular symplectic and singular quasi-Hamiltonian structures taking as a motivating example a singular version of the Atiyah-Bott structure on the moduli space of flat connections.
In the second part of this thesis, we turn to Poisson structures on moduli spaces of flat connections and monodromy data related to the Riemann-Hilbert correspondence. It turned out recently, in the work of Irina Bobrova and Marta Mazzocco, that another interesting example of non-autonomous b-symplectic structures appears in the context of Painlevé transcendents [5]. Sigma-coordinates for Okamoto Hamiltonian of the second Painlevé equation lead to a natural b-symplectic structure [6]. For other Painlevé equations P(III) − P(VI), the Poisson structure takes a more complex form. We consider Poisson structure on moduli spaces of flat connections and monodromy data related by the Riemann-Hilbert correspondence for P(VI) and P(V). For P(VI) and the other Fuchsian equations, we explicitly construct such a structure on the corresponding character variety leading us to a conjecture for P(V), which can be seen as a counterpart of the same structure on flat connections and coincides with obtained in [7].
Highlighted publication: [8]
References
[1] J. Marsden and A. Weinstein. Reduction of symplectic manifolds with symmetry. Rep. Mathematical Phys., 5(1):121 130, 1974.
[2] V. Guillemin, E. Miranda, and A. R. Pires. Symplectic and Poisson geometry on b-manifolds. Adv. Math., 264:864-896, 2014.
[3] V. Guillemin, E. Miranda, A. R. Pires, and G. Scott. Toric actions on b-symplectic manifolds. Int. Math. Res. Not. IMRN, (14):5818-5848, 2015.
[4] E. Miranda and G. Scott. The geometry of E-manifolds. Rev. Mat. Iberoam., 37(3):1207-1224, 2021.
[5] P. Painlevé. Sur les équations différentielles du second ordre et d’ordre supérieur dont l’intégrale générale est uniforme. Acta Math., 25(1):1–85, 1902.
[6] I. Bobrova and M. Mazzocco. The sigma form of the second Painlevé hierarchy. J. Geom. Phys., 166: Paper No. 104271, 8, 2021.
[7] L. Chekhov, M. Mazzocco, and V. Rubtsov. Algebras of quantum monodromy data and character varieties. In Geometry and physics. Vol. I, pages 3968. Oxford Univ. Press, Oxford, 2018.
[8] A. Matveeva and E. Miranda. Reduction theory for singular symplectic manifolds and singular forms on moduli spaces, to appear in Advances in Mathematics. arXiv pdf.