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Speaker: Arkajit Pal Choudhury (Ph.D. Student, Stat-Math Unit, ISI Kolkata)
Title: Maximal Gromov Hyperbolic Spaces
Date: 7th November, 2025 (Friday)
Time: 04:00 PM - 05:00 PM (IST)
Abstract: It is well-known that an isometry between real hyperbolic spaces extends naturally to a Möbius homeomorphism - i.e. cross-ratio preserving homeomorphism - between their Gromov (or visual) boundaries. The same phenomenon holds for any boundary continuous Gromov hyperbolic space. The natural question then arises: Does a Möbius homeomorphism between the Gromov boundaries extends to isometry of the underlying spaces? This is the Möbius rigidity problem, related to other problems such as the marked length spectrum rigidity and geodesic conjugacy problem for negatively curved manifolds. Möbius rigidity is known to hold for real hyperbolic spaces, and plays a role in several rigidity results, such us Mostow rigidity. For “good” (i.e. proper, geodesically complete) Gromov hyperbolic spaces the Gromov boundary is a special type of compact metrisable space called quasi-metric antipodal space. In a paper published in 2024, Biswas gave a positive answer to the Möbius rigidity problem for a distinguished sub-class of good Gromov hyperbolic spaces called ‘maximal Gromov hyperbolic spaces’. These Gromov hyperbolic spaces are ‘maximal’ in the sense, any other good Gromov hyperbolic space with the same Gromov boundary isometrically embeds into them, i.e. it is the maximal hyperbolic filling. Biswas showed that given any quasi-metric antipodal space one can naturally construct the maximal Gromov hyperbolic space, hence established an equivalence of categories between the quasi-metric antipodal spaces and maximal Gromov hyperbolic space. Maximal Gromov hyperbolic spaces exhibit rich geometric structures. In this talk, we demonstrate that maximal Gromov hyperbolic spaces with finite boundary admit a polyhedral complex structure. We will also discuss several Gromov–Hausdorff convergence results for these spaces and their boundaries, along with some applications of these results. As a byproduct, we obtain a convergence theorem for CAT($-1$) spaces. This is joint work with Prof. Kingshook Biswas.
[Notes]
Speaker: Dr. Abhishek Pandey (Research Associate, IISc Bangalore)
This is a Crash Course in Metric Geometry. This course is also a part of PMRF student lecture series by ISSS.
Title: BHK Uniformization and Characterization of Gromov Hyperbolic Spaces
Date: 22nd March, 2025 (Saturday)
Time: 11:00 AM - 12:00 PM (IST)
Abstract: In this talk, we will discuss the Bonk-Heinonen-Koskela (BHK) uniformization in detail and explore geometric conditions for Gromov hyperbolicity: the Gehring-Hayman condition and the ball separation condition. This discussion will lead to a characterization of the Gromov hyperbolicity of the quasihyperbolic metric. We will also compare the Gromov hyperbolicity of the hyperbolic and quasihyperbolic metrics in planar hyperbolic domains. If time permits, we will discuss some results on the Gromov hyperbolicity of the Kobayashi metric on convex domains.
Title: Gromov Hyperbolicity
Date: 18th March, 2025 (Tuesday)
Time: 4:00 PM - 5:30 PM (IST)
Abstract: Gromov hyperbolicity is a property of a general metric space to be negatively curved in the sense of coarse geometry. Introduced by Gromov (1987) in geometric group theory, it has since played an increasing role in analysis on general metric spaces with applications to the Martin boundary, invariant metrics in several complex variables. From a complex analysis perspective, Gromov hyperbolicity is significant because:
Many important metrics in complex analysis are frequently Gromov hyperbolic.
The Gromov boundary is a useful concept, both as an alternative way of treating the topological boundary and as a way of defining boundary extensions of maps.
While finding geodesics in invariant metrics is challenging, quasigeodesics are more accessible, and in Gromov hyperbolic spaces, geodesics stay close to quasigeodesics.
This talk will introduce Gromov hyperbolicity for geodesic metric spaces, with examples and non-examples. If time permits, we will discuss geodesic stability and the Gromov boundary.
Title: Notions of Curvature in Metric Spaces
Date: 10th March, 2025 (Monday)
Time: 4:00 PM - 5:30 PM (IST)
Abstract: Some of the important metrics that are used in complex analysis and potential theory, including the Poincaré, Carathéodory, Kobayashi, Hilbert, and quasihyperbolic metrics are quite often negatively curved. The study of nonpositively curved spaces has a long and rich history, dating back to Hadamard’s pioneering work in 1898. The first notion of curvature in metric spaces was introduced by Karl Menger and his student Abraham Wald, laying the foundation for the modern development of this theory. Over time, various formulations of nonpositive and negative curvature have emerged, including Busemann convexity, the CAT(k) condition, and Gromov hyperbolicity. In this talk, we will explore these key concepts, focusing on Busemann convexity and the CAT(k) condition, along with illustrative examples of such spaces.
Title: Hausdorff Measure and Length
Date: 2nd March, 2025 (Sunday)
Time: 11:00 AM - 12:00 PM (IST)
Abstract: In this talk, we will introduce the concept of Hausdorff measure, starting with intuitive examples that motivate its definition. We will then explore its connection to the length of curves. If time permits, we will examine interesting examples, including the Hausdorff measure of curves and Cantor-type sets.
Title: Length and Geodesic Spaces
Date: 22nd February, 2025 (Saturday)
Time: 11:00 AM - 12:30 PM (IST)
Abstract: In this talk, we will begin by exploring conformal metrics, delving into their properties and applications. Following this, we will examine length spaces and geodesic spaces, providing illustrative examples to demonstrate their structure and significance.
Title: Length of Curves, Rectifiability, Length Formula, and Line Integral
Date: 12th February, 2025 (Wednesday)
Time: 04:00 PM - 05:30 PM (IST)
Abstract: In this talk, we will begin by exploring fundamental concepts in metric geometry, including the length of curves, rectifiability, and arc-length parametrization. We will then examine the conditions under which the length formula holds. Finally, we will conclude with a discussion on the line integral.
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