Quasi-Metric Antipodal Spaces and Maximal Gromov Hyperbolic Spaces

Explore a comprehensive lecture on quasi-metric antipodal spaces and maximal Gromov hyperbolic spaces presented by Kingshook Biswas from the Indian Statistical Institute (Delhi). Delve into advanced geometric concepts, including Gromov inner product, visual metrics on CAT(-1) space boundaries, and Moebius maps and isometries. Examine the marked length spectrum rigidity problem, infinitesimal rigidity, and the circumcenter extension of Moebius maps. Investigate boundary continuous Gromov hyperbolic spaces, good fillings of quasi-metric antipodal spaces, and the Geometric Mean-Value Theorem. Learn about the Moebius space M(Z), discrepancy function, antipodal flow, and antipodalization map. Discover the maxima compactification M(Z), Busemann cocycle, and derivatives for good spaces. Understand the equivalence of antipodal spaces and Moebius spaces, maximal Gromov hyperbolic spaces, and their application to Moebius rigidity. Conclude with an exploration of the convexity problem in this 57-minute lecture from the Workshop on Geometry of Spaces with Upper and Lower Curvature Bounds at the Fields Institute.

Intro
Gromov inner product
Visual metrics on the boundary of a CAT(-1) space
Moebius maps and isometries
Motivation: marked length spectrum rigidity problem
Infinitesimal rigidity
Circumcenter extension of Moebius maps
Boundary continuous Gromov hyperbolic spaces
Good fillings of quasi-metric antipodal spaces
Geometric Mean-Value Theorem
The Moebius space M(Z)
The discrepancy function
The antipodal flow
The antipodalization map
The maxima compactification M(Z)
The Busemann cocycle and derivatives for good
Equivalence of antipodal spaces and Moebius space
Maximal Gromov hyperbolic spaces
Application to Moebius rigidity
Convexity problem