She won for her work on "the dynamics and geometry of Riemann surfaces
and their moduli spaces.
This volume collects lecture notes from nine lecture series delivered at the July 2011 Park City Mathematics Institute on mapping class groups and moduli spaces of Riemann surfaces
Families of Riemann surfaces
and Weil-Petersson geometry.
Wolpert (University of Maryland), and Shing-Tung Yau (Harvard University), "Surveys in Differential Geometry: Volume 14, Geometry of Riemann surfaces
and their moduli spaces" is a collection of lectures delivered at the 2009 JDG conference.
We are interested in studying when Riemann surfaces
equipped with their Poincare metric are Gromov hyperbolic (see e.
For Riemann surfaces
, the same result gotten by Jones and Singerman in 1978.
Chapters cover hyperbolic geometry (illustrated with Escher's models of the hyperbolic plane), complex numbers (so essential in quantum mechanics), Riemann surfaces
, quaternions, n-dimensional manifolds, fibre bundles, Fourier analysis, G6del's theorem, Minkowski space, Lagrangians, Hamiltonians, and other terrifying topics.
By means of the concept of Riemann surfaces
, the six roots may be considered to define a single-valued algebraic function [[xi].
Among the topics are jet schemes of homogeneous hypersurfaces, the universal degenerating family of Riemann surfaces
, invariants of splice quotient singularities, nearby cycles and characteristic classes of singular spaces, two birational invariants in statistical learning theory, and the minimality of hyperplane arrangements and the basis of local system cohomology.
Among the topics are convergence of a formal solution to an ordinary differential equation, isomonodromic deformations on Riemann surfaces
, expansions for solutions of the Schlesinger equation at a singular point, the sixth Painleve transcendent and uniformizable orbifolds, and deriving Painleve equations by anti-quantization.
Ueno begins by describing Riemann spaces and stable curves, including compact Riemann surfaces
and pointed curves, then moves to affine Lie algebras and integrable highest weight representations, with an explanation of the energy-momentum tensor, Uechi then moves to conformal blocks and correlation functions, the sheaf of conformal blocks associated with a family of pointed Reimann surfaces with coordinates, the sheaf's support of projectively flat connections, one of the most important facts of conformal field theory.
The text is intended to serve as a link between basic undergraduate geometry and more theoretical geometry such as Riemann surfaces
, differential manifolds, algebraic topology, and Riemannian geometry.