We relate previously defined quantum characteristic classes to Morse theoretic aspects of the Hofer length functional on Ham (M, ω). As an application we prove a theorem which can be interpreted as stating that this functional is “virtually” a perfect Morse-Bott functional. This can be applied to study the topology and Hofer geometry of Ham(M, ω). We also use this to give a prediction for the index of some geodesics for this functional, which was recently partially verified by Yael Karshon and Jennifer Slimowitz[5].

In \cite{rigidity}, Luo introduced a $\psi_{\lambda}$ edge invariant which turns out to be a coordinate of the Teichm\"uller space of a surface with boundary. And he proved that for $\lambda \geq 0$, the image of the Teichm\"uller space under $\psi_{\lambda}$ edge invariant coordinate is an open cell. In this paper we verify his conjecture that for $\lambda<0$, the image of the Teichm\"uller space is a bounded convex polytope.

The Cauchy problem for the homogeneous real/complexMongeAmp`ere equation (HRMA/HCMA) arises from the initial value problem for geodesics in the space of K¨ahler metrics equipped with the Mabuchi metric. This Cauchy problem is believed to be ill-posed and a basic problem is to characterize initial data of (weak) solutions which exist up to time T . In this article, we use a quantization method to construct a subsolution of the HCMA on a general projective variety, and we conjecture that it solves the equation for as long as the unique solution exists. The subsolution, called the “quantum analytic continuation potential,” is obtained by (i) Toeplitz quantizing the Hamiltonian flow determined by the Cauchy data, (ii) analytically continuing the quantization, and (iii) taking a certain logarithmic classical limit. We then prove that in the case of torus invariant metrics (where the HCMA reduces to the HRMA) the quantum analytic continuation potential coincides with the well-known Legendre transform potential, and hence solves the equation as long as it is smooth. In the sequel [29], it is proved that the Legendre transform potential ceases to solve the HCMA once it ceases to be smooth. The results here and in the sequels in particular characterize the initial data of smooth geodesic rays.

Given a hyperbolic 3–manifold with torus boundary, we bound the change in volume under a Dehn filling where all slopes have length at least 2π. This result is applied to give explicit diagrammatic bounds on the volumes of many knots and links, as well as their Dehn fillings and branched covers. Finally, we use this result to bound the volumes of knots in terms of the coefficients of their Jones polynomials.

We show that the complex cohomologies of Bott, Chern, and Aeppli and the symplectic cohomologies of Tseng and Yau arise in the context of type II string theory. Specifically, they can be used to count a subset of scalar moduli fields in Minkowski compactification with RR fluxes in the presence of either O5/D5 or O6/D6 brane sources, respectively. Further, we introduce a new set of cohomologies within the generalized complex geometry framework which interpolate between these known complex and symplectic cohomologies. The generalized complex cohomologies play the analogous role for counting massless fields for a general supersymmetric Minkowski type II compactification with Ramond–Ramond flux.