Abstract:
Surface parameterization is fundamental to computational geometry and plays an important role in various applications ranging from texturing in computer graphics to the analysis and comparison of anatomical shapes in evolutionary biology. In many cases, it is desired, or even essential, that such a parameterization is invertible. The focus of this talk is invertibility, which while desirable, often presents a challenge in the computation of parameterizations.

Remarkably, in the special case of harmonic parameterization onto a convex subset, invertibility is guaranteed by Tutte’s graph embedding and its continuous analog, the Rado-Kneser-Choquet Theorem. I will discuss our work on generalizing these results to the non-convex case and present simple geometric conditions for invertibility.

Joint work with Noam Aigerman, Misha Kazhdan, Ingrid Daubechies, Jianfeng Lu and Stefan Steinerberger

Let $X$ be a smooth variety (e.g., affine space) over a finite field (e.g., the integers modulo a prime). In the course of proving the last of Weil’s conjectures on zeta functions of varieties over finite field, Deligne studied a certain category of representations of the fundamental group of $X$ which carry information about these zeta functions. He also made a far-reaching conjecture to the effect that such objects always look as if they “come from geometry”. We will state the conjecture, describe some of its more concrete consequences, and discuss some results of various authors (L. Lafforgue, V. Lafforgue, Deligne, Drinfeld, T. Abe, Abe-Esnault, and the speaker) which very recently have led to a resolution of this 40-year-old open problem.