Triangulations of hyperbolic 3-manifolds admitting strict angle structures

It is conjectured that every cusped hyperbolic 3-manifold has a decomposition into positive volume ideal hyperbolic tetrahedra (a "geometric" triangulation of the manifold). Under a mild homology assumption on the manifold we construct topological ideal triangulations which admit a strict angle structure, which is a necessary condition for the triangulation to be geometric. In particular, every knot or link complement in the 3-sphere has such a triangulation. We also give an example of a triangulation without a strict angle structure, where the obstruction is related to the homology hypothesis, and an example illustrating that the triangulations produced using our methods are not generally geometric.Comment: 28 pages, 9 figures. Minor edits and clarification based on referee's comments. Corrected proof of Lemma 7.4. To appear in the Journal of Topolog

Voronoi conjecture for five-dimensional parallelohedra

We prove the Voronoi conjecture for five-dimensional parallelohedra. Namely, we show that if a convex five-dimensional polytope $P$ tiles $\mathbb{R}^5$ with translations, then $P$ is an affine image of the Dirichlet-Voronoi cell for a five-dimensional lattice

Micro- and nanoengineering approaches to control stem cell-biomaterial interactions.

As our population ages, there is a greater need for a suitable supply of engineered tissues to address a range of debilitating ailments. Stem cell based therapies are envisioned to meet this emerging need. Despite significant progress in controlling stem cell differentiation, it is still difficult to engineer human tissue constructs for transplantation. Recent advances in micro- and nanofabrication techniques have enabled the design of more biomimetic biomaterials that may be used to direct the fate of stem cells. These biomaterials could have a significant impact on the next generation of stem cell based therapies. Here, we highlight the recent progress made by micro- and nanoengineering techniques in the biomaterials field in the context of directing stem cell differentiation. Particular attention is given to the effect of surface topography, chemistry, mechanics and micro- and nanopatterns on the differentiation of embryonic, mesenchymal and neural stem cells