Integrable systems, Frobenius manifolds and cohomological field theories

This dissertation studies the underlying geometry of tau-symmetric bi-Hamiltonian hierarchies of evolutionary PDEs.
First, we give a new proof of the Witten conjecture, which constructs the string tau-function of the KdV 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.
Third, we propose a conjectural formula for the simplest non-trivial product of double ramification cycles DRg(1,−1)λg in terms of cohomology classes represented by standard strata, refining the one point case of the Buryak–Guéré–Rossi conjectural tautological relations.
Finally, 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. This greatly complicates the analysis of the Stokes phenomenon, which we perform by splitting the space of solutions into infinitely many two-dimensional subspaces.