A GROMOV-HAUSDORFF HYPERTOPOLOGY OVER THE CLASS OF PROPER QUANTUM METRIC SPACES

Speaker(s)

FRÉDÉRIC LATRÉMOLIÈRE

Affiliation

University of Denver, USA

Language of the talk

English

Date

Dec. 17, 2025, 5:15 p.m.

Information about the event

IMPAN - Room 405

Title in Polish

A GROMOV-HAUSDORFF HYPERTOPOLOGY OVER THE CLASS OF PROPER QUANTUM METRIC SPACES

Seminar

North Atlantic Noncommutative Geometry Seminar

---

The field of noncommutative metric geometry has grown to provide a framework for the convergence, à la Gromov, of compact quantum metric spaces and various associated structures. However, the question of extending this theory to noncompact quantum metric spaces remained difficult and unaddressed. Even defining the right class of quantum metric spaces in the realm of a Gromov-Hausdorff-like topology proved elusive. In this presentation, we will propose our first answer to this problem by defining a hypertopology on the class of proper quantum metric spaces. Borrowing from our earlier work on locally compact quantum metric spaces, a proper quantum metric space generalizes metric spaces that are boundedly compact and locally compact to the noncommutative world. The new hypertopology is induced by a "metametric" that simultaneously generalizes the propinquity between compact quantum metric spaces and the Gromov-Hausdorff distance between proper metric spaces. Better still, it also enables the presentation of new examples of convergence of noncommutative noncompact quantum metric spaces. We hope that this progress will start the exploration of locally compact aspects of noncommutative metric geometry.

 

---