6533b7d7fe1ef96bd1268c3f

RESEARCH PRODUCT

Integrable systems, Frobenius manifolds and cohomological field theories

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In this dissertation, we study the underlying geometry of integrable systems, in particular tausymmetric bi-Hamiltonian hierarchies of evolutionary PDEs and differential-difference equations.First, we explore the close connection between the realms of integrable systems and algebraic geometry by giving a new proof of the Witten conjecture, which constructs the string taufunction of the Korteweg-de Vries hierarchy via intersection theory of the moduli spaces of stable curves with marked points. This novel proof is based on the geometry of double ramification cycles, tautological classes whose behavior under pullbacks of the forgetful and gluing maps facilitate the computation of intersection numbers of psi classes.Second, we examine the Dubrovin-Zhang hierarchy, an integrable system constructed from a Frobenius manifold by deforming its associated pencil of Poisson structures of hydrodynamic type. This integrable hierarchy was proved to be Hamiltonian and tau-symmetric, and conjectured to be bi-Hamiltonian. We prove a vanishing theorem for the negative degree terms of the second Poisson bracket, thus providing strong evidence to support this conjecture. The proof of this theorem demonstrates the implications the bi-Hamiltonian recursion relation and tautological relations in the cohomology rings of the moduli spaces of stable curves have on the bi-Hamiltonian structure of the Dubrovin-Zhang hierarchies.Third, we propose a conjectural formula for the simplest non-trivial product of doubleramification cycles DR_g(1,1)lambda_g in terms of cohomology classes represented by standard strata. Although there are known formulas relating double ramification cycles to other, more natural tautological classes, they are much more complicated than the one conjectured here. This conjecture refines the one point case of the Buryak-Guéré-Rossi conjectural tautological relations, which are equivalent to the existence of a Miura transformation relating Buryak's double ramification hierarchies and the Dubrovin-Zhang ones.Finally, we analyze the differential geometry of (2 + 1) integrable systems through infinitedimensional Frobenius manifolds. More concretely, we study, both formally and analytically, the Dubrovin equation of the 2D Toda Frobenius manifold at its irregular singularity. The fact that it is infinite-dimensional implies a qualitatively different behavior than its finite-dimensional analogue, the Frobenius manifold underlying the extended Toda hierarchy. The two most remarkable differences are non-uniqueness of formal solutions to the Dubrovin equation and non-completeness of the analytic ones. These features together greatly complicate the analysis of Stokes phenomenon, which we perform by splitting the space of solutions into infinitely many two-dimensional subspaces.