Abstract: In a convex domain of Euclidean space,
the Dirichlet heat flow transmits log-concavity from the initial time to any time.
I first introduce a notion of generalized concavity and specify a concavity preserved by the Dirichlet heat flow.
Then I show that in a totally convex domain of a Riemannian manifold,
if some concavity is preserved by the Dirichlet heat flow,
then the sectional curvature must vanish on the domain.
The first part is based on joint work with Kazuhiro Ishige and Paolo Salani,
and the second part is based on joint work with Kazuhiro Ishige and Haruto Tokunaga.
4月15日 -- 現地開催 (056号室) & オンライン中継, 17:00 -- 18:30
坂井 健人 (東京大学大学院数理科学研究科)
Harmonic maps and uniform degeneration of hyperbolic surfaces with boundary
Abstract: If holomorphic quadratic differentials on a punctured Riemann surface have poles of order >1 at the punctures,
they correspond to hyperbolic surfaces with geodesic boundary via harmonic maps.
This correspondence is known as the harmonic map parametrization of hyperbolic surfaces.
In this talk, we use this parametrization to describe the degeneration of hyperbolic surfaces via Gromov-Hausdorff convergence.
As an application, we study the limit of a one-parameter family of hyperbolic surfaces in the Thurston boundary of Teichmüller space.
Coarse coding theory and discontinuous groups on homogeneous spaces
Abstract: Let $M$ and $＼mathcal{I}$ be sets, and consider a surjective map
＼[
R : M ＼times M ＼to ＼mathcal{I}.
＼]
For each subset $＼mathcal{A} ＼subseteq ＼mathcal{I}$, we define $＼mathcal{A}$-free codes on $M$ as subsets $C ＼subseteq M$ satisfying
＼[
R(C ＼times C) ＼cap ＼mathcal{A} = ＼emptyset.
＼]
This definition encompasses various types of codes, including error-correcting codes, spherical codes, and those defined on association schemes or homogeneous spaces.
In this talk, we introduce a "pre-bornological coarse structure" on $＼mathcal{I}$ and define the notion of coarsely $＼mathcal{A}$-free codes on $M$.
This extends the concept of $＼mathcal{A}$-free codes introduced above. As a main result, we establish relationships between coarse coding theory on Riemannian homogeneous spaces $M = G/K$ and discontinuous group theory on non-Riemannian homogeneous spaces $X = G/H$.
