When transitioning from studying Euclidean space to more Riemannian manifolds, one must first unlearn many special properties of the flat world. The same is true in physics: while one can make sense of classical physics on an arbitrary curved background space, many seemingly foundational concepts (like the center of mass) turn out to have no place in the general theory. Freed from the constraints such properties induce, classical physics on a curved background space has many surprises in store. In this talk I will share some stories related to joint work with Brian Day and Sabetta Matsumoto on understanding and simulating such situations, focusing on hyperbolic space when convenient. To give a taste, here are two such surprises:
(1) there is no Galilean relativity: inside a sealed box in hyperbolic geometry it is possible to perform an experiment which detects your precise velocity. And (2): it's possible to ‘swim’ in the vacuum in hyperbolic space - to move your arms and legs in a specific pattern that causes you to translate along a geodesic with no external forces. The arguments for the former are readily accessible to beginning graduate students in geometry, and the latter illustrates a use of gauge theory in classical mechanics, following work of Wilczek and Montgomery.
Vertex operator algebras (VOAs) and their modules define sheaves of conformal blocks over the moduli space of stable curves, generalizing sheaves of conformal blocks attached to Lie algebras. In this talk I will discuss how these sheaves are constructed and which properties they satisfy. I will in particular describe conditions that guarantee that these sheaves are actually vector bundles of finite rank and related open questions. This is based on a joint work with A. Gibney, N. Tarasca and D. Krashen.
The study of real planar curves dates back to antiquity, where the ancient Greeks studied curves defined on the plane cut out by polynomials of two variables. We’ll provide a friendly overview to beautiful formulas of Plücker which govern the “shape” of planar curves. We will discuss the Shapiro—Shapiro conjecture and connections to the real Schubert calculus, and end by presenting some new conjectures and computational evidence joint with Frank Sottile.
I will explain something of the theory of homogeneous Einstein metrics and why certain generalizations of this equation occur naturally in the study of homogeneous spaces.
It is well-known that on a non-projective complex manifold, a
coherent sheaf may not have a resolution by a complex of holomorphic vector
bundles. Nevertheless, J. Block showed that such resolution always exists if
we allow anti-holomorphic flat superconnections which generalize complexes
of holomorphic vector bundles. Block's result makes it possible to study
coherent sheaves with differential geometric and analytic tools. For
example, in a joint work with J.M Bismut, S, Shen, and I, we give an
analytic proof of the Grothendieck-Riemann-Roch theorem for coherent sheaves
on complex manifolds. In this talk I will present the ideas and applications
of anti-holomorphic flat superconnections. I will also talk about analogous
constructions of superconnections in other areas of geometry.
In a work on the geometry of minimal submanifolds written in 1968, Shiing-Shen Chern invited more efforts and reflections to identify relationships between intrinsic and extrinsic curvature invariants of submanifolds in various ambient spaces. After 1993, when Bang-Yen Chen introduced the first of his curvature invariants, namely scal - inf(sec), a lot of work has been done to explore this avenue, which represents an active research area. We will survey some of these results obtained in the last three decades, and conclude our talk with new relationships between intrinsic and extrinsic curvature invariants.