Title: Combinatorial models for CAT(0) spaces
Speaker: Abdul Zalloum（University of Toronto）
Time: Monday, April 30, 2024, 10:00-11:30
Location: Mingde Building, B201-1
Abstract: A metric space X is said to be CAT(0) if triangles in X are no fatter than triangles in the Euclidean plane. These were introduced by Alexandrov in the 1950s and were given prominence by Gromov who showed that a great deal of the theory of manifolds of non-positive curvature could be developed without using much more than the CAT(0) condition.
Typically, in order to understand an object in math, one would like to break it into its most-simple building blocks; understand such blocks individually, and how they fit together to produce the underlying object (for instance, this is what one does with integers and primes!). In fact, the primary reason behind our very good understanding of CAT(0) cube complexes (a subclass of CAT(0) spaces) and mapping class groups of finite type surfaces is precisely that combinatorial building blocks for the aforementioned objects exist: hyperplanes for the former and curve graphs for the latter.
I will talk about recent work with Petyt and Spriano where we attempt to bring the class of CAT(0) spaces in the picture by introducing two combinatorial objects for studying them: curtains which are analogues of hyperplanes in CAT(0) cube complexes, and the curtain model which is a counter part of the curve graph of a finite-type surface. The talk will be completely self-contained!
