Unless noted, all seminars are on Friday at 4pm, in Mathematics 122. 201617 seminar listing can be found here.
Sep 8  Piotr Hajac (Polish Academy of Sciences): There and back again: from the BorsukUlam 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 twodimensional version of the celebrated BorsukUlam Theorem which states that for any continuous map from the ndimensional sphere to ndimensional 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 GelfandNaimark 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 BorsukUlamtype 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 nontrivializability of Pflaum's $SUq(2)$principal instanton bundle is a special case of two different noncommutative BorsukUlamtype conjectures. Time permitting, we shall also explain a general method to extend the nontrivializability 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.)


Sep 22  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 quasiisometry
(QI) and the Cayley graph *is* well defined up to quasiisometries.
Unfortunately quasiisometries 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 quasiisometric 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 quasiisometrically rigid.
Many classes of groups are quasiisometrically rigid, free abelian
groups, finite volume discrete subgroups of a noncompact 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
quasiisometry invariant for finitely generated groups that serves a
similar role that the boundary of hyperbolic nspace 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 quasiisometric 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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