Talk

Venue: Endenicher Allee 60, Room 1.007, Math. Center, University of Bonn

Lectures 3 & 4 by Carlo 

Thursday 26 June, 10:30-12:00 & 13:00-14:30 

These lectures focus on the use of elliptic curves in proving H10/R for any finitely generated ring R; using many existing reductions, one arrives at a problem of controlling Selmer groups of an analogue of the Manin elliptic curve, which is done by new tools using additive combinatorics.

By a log-symplectic form I mean a meromorphic closed non-degenerate 2-form omega on a complex manifold X that has only simple poles along a divisor D. I will discuss how the de Rham class of omega in H^2(X\D) determines the deformations of the triple (X,D,omega). Some open problems about topology of the divisor deforming D will be discussed.

Seminar webpage:  https://guests.mpim-bonn.mpg.de/bianchi/wiqi.html

We present two important actions of free groups on hyperbolic 3-space and Hermitian symmetric space via Kleinian group theory

and surface group representations into Hermitian Lie groups. The first one is related to the Thurston's conjecture of generalization of the double limit 

theorem to 3-manifolds with compressible boundary, and the second one is related to the Atiyah-Patodi-Singer index theory on flat bundles over surfaces with boundary.

In the talk I report on joint work with Beaudry, Hermele, Moreno, Qi and Spiegel, where a homotopy theoretic framework for studying state spaces of quantum lattice spin systems has been introduced using the language of C*-algebraic quantum mechanics. First some old and new results about the state space of the quasi-local algebra of a quantum lattice spin system when endowed with either the natural metric topology or the weak* topology will be presented. Switching to the algebraic topological side, the homotopy groups of the unitary group of a UHF algebra will then be determined and it will be indicated that the pure state space of any UHF algebra in the weak* topology is weakly contractible. In addition, I will show at the example of non-commutative tori that also in the case of a not commutative C*-algebra, the homotopy type of the state space endowed with the weak* topology can be non-trivial and is neither deformation nor Morita invariant. Finally, I indicate how such tools together with methods from higher homotopy theory such as E_infinity spaces may lead to a framework for constructing Kitaev’s loop-spectrum of bosonic invertible gapped phases of matter.

Orbifold data are categorical symmetries that can be gauged in oriented defect topological quantum field theories. We review the general construction and apply it to 2-group symmetries of 3-dimensional TQFTs; upon further specialisation this leads to equivariantisation of G-crossed braided fusion categories. We also describe a proposal, via higher dagger categories, to gauging categorical symmetries in the context of other tangential structures. This is based on separate projects with Benjamin Haake and Tim Lüders.