Geometry and Topology Seminar[ Edit ]
Moderator: Michael Khanevsky
An aperiodic tiling of R^d has a hull, which typically is a compact foliated space X. The hull comes equipped with an invariant transverse measure, corresponding to a Z^d-invariant measure on a totally disconnected transversal N. Mathematical physicists attach two types of invariants to this situation. On the one hand, the invariant transverse measure gives rise to a real-valued map on H^d(X; R) (Cech cohomology) which corresponds to the diffraction pattern aspect of tilings. On the other hand. it also gives rise to a real-valued map on K_0(C^*G(X)), the topological K-theory of the C^*-algebra of the holonomy groupoid of the tiling, which corresponds to the spectral (of Hamiltonians defined on such tilings) aspect of tilings. We use the Index Theorem for foliated spaces to prove that under some general assumptions that these points of view are equivalent.
This is joint work with Eric Akkermans (physics, the Technion) and Jonathan Rosenberg (math, U. Maryland).
In this talk I will discuss obstructions to having a Riemannian metric with non-positive sectional curvature on a locally CAT(0) manifold. I will focus on the obstruction in dimension = 4 given by Davis-Januszkiewicz-Lafont and show how their method can be extended to construct new examples of locally CAT(0) 4-manifolds M that do not have a Riemannian smoothing. The universal covers of these manifolds satisfy the isolated flats condition and contain a collection of 2-dimensional flats with the property that their boundaries at infinity form non-trivial links in the boundary 3-sphere.
Let M be an orientable hyperbolic surface without boundary and let c be a closed geodesic in M. We prove that any side of any triangle formed by distinct lifts of c in the hyperbolic plane is shorter than c.
The talk will be presented for advanced undergraduate and beginning graduate students.