# UB Geometry and Topology Seminar

### 2017-2018

Unless noted, all seminars are on Friday at 4pm, in Mathematics 122. 2016-17 seminar listing can be found here.

Sep 8 Piotr Hajac (Polish Academy of Sciences): There and back again: from the Borsuk-Ulam theorem to quantum spaces Abstract: Assuming that both temperature and pressure are continuous functions, we can conclude that there are always two antipodal points on Earth with exactly the same pressure and temperature. This is the two-dimensional version of the celebrated Borsuk-Ulam Theorem which states that for any continuous map from the n-dimensional sphere to n-dimensional real Euclidean space there is always a pair of antipodal points on the sphere that are identified by the map. Our quest to unravel topological mysteries in the Middle Earth of quantum spaces will begin with gentle preparations in the Shire of elementary topology. Then, after riding swiftly through the Rohan of $C^\ast$-algebras and Gelfand-Naimark Theorems, and carefully avoiding the Mordor of incomprehensible technicalities, we shall arrive in the Gondor of compact quantum groups acting freely on unital $C^\ast$-algebras. It is therein that the generalized Borsuk-Ulam-type statements dwell waiting to be proven or disproven. To end with, we will pay tribute to the ancient quantum group $SUq(2)$, and show how the proven non-trivializability of Pflaum's $SUq(2)$-principal instanton bundle is a special case of two different noncommutative Borsuk-Ulam-type conjectures. Time permitting, we shall also explain a general method to extend the non-trivializability result from the Pflaum quantum instanton bundle to an arbitrary finitely iterated equivariant join of $SUq(2)$ with itself. The latter is a quantum sphere with a free $SUq(2)$-action whose space of orbits defines a quantum quaternionic projective space. (Based on joint work with Paul F. Baum, Ludwik Dabrowski, Tomasz Maszczyk and Sergey Neshveyev.) Devin Murray (Brandeis): Groups, rigidity, and the Morse boundary Abstract: In geometric group theory a finitely presented group is often identified with its Cayley graph. Unfortunately, the Cayley graph depends on the generating set of the group. When the generating set is changed the homeomorphism type of the graph can change quite a lot. However, there is a metric equivalence relation called a quasi-isometry (QI) and the Cayley graph *is* well defined up to quasi-isometries. Unfortunately quasi-isometries are a very weak geometric equivalence (in general there may be no continuous bijections between quasi- isometric spaces!). In this context a very natural question to ask is, if two groups are quasi-isometric how similar are they as groups? For some classes of groups the answer is quite surprising! For example, if a group $G$ is QI to $\pi_1(S)$ where $S$ is a surface group, then $G$ is (virtually) isomorphic to a surface group! A class of groups which exhibits this property is called quasi-isometrically rigid. Many classes of groups are quasi-isometrically rigid, free abelian groups, finite volume discrete subgroups of a non-compact Lie group, surface groups, etc. Often, some form of negative curvature plays an important role in proving rigidity style theorems. The Morse property is a natural generalization of the behavior of geodesics in negatively curved manifolds. The Morse boundary is a quasi-isometry invariant for finitely generated groups that serves a similar role that the boundary of hyperbolic n-space serves for hyperbolic manifold groups. I will introduce all of the objects in question and talk about some results for the morse boundary that make it a promising tool to study the quasi-isometric rigidity phenomena of some classes of groups, in particular some CAT(0) groups, Mapping Class groups, some artin/coxeter groups etc. This is joint work with Ruth Charney and Matt Cordes.

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