## Geometry Seminar - Abstracts

### Talk

Wednesday 3 May 2017, 16:00-17:00 in HG03.085

**Piotr M. Hajac** (Polish Academy of Sciences)

*Noncommutative Borsuk-Ulam-type conjectures revisited*

### 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*-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*-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.)

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