# Dr Ulrich Pennig

(he/him)

Senior Lecturer

- Available for postgraduate supervision

## Overview

### Research Group

Geometry, Algebra, Mathematical Physics and Topology

### Research Interests

My research interests are connections between algebraic topology, homotopy theory and operator algebras, in particular (twisted) K-theory and index theory. Since I studied theoretical physics before my PhD, I am also interested in topological and conformal field theories with a view towards elliptic cohomology.

### Personal Webpage

### Responsibilities

- Assessment and Feedback Lead for MATHS
- Organiser of the GAPT Seminar
- INI / ICMS correspondent
- External Examiner for the University of Leicester

### Former Responsibilities

- Member of SREC (the School's Research Ethics Committee)
- Member of the Research Committee (from 2017-2021)
- Year Abroad Coordinator (from October 2020 - August 2022)

## Publication

### 2024

- Pennig, U. and Sogabe, T. 2024. Spanier–Whitehead K-duality and duality of extensions of C*-algebras. International Mathematics Research Notices (10.1093/imrn/rnae242)
- Dadarlat, M., McClure, J. E. and Pennig, U. 2024. Computing cohomology groups that classify bundles of strongly self-absorbing C*-algebras. Journal of Topology and Analysis (10.1142/S1793525324500110)
- Dădărlat, M. and Pennig, U. 2024. Bundles of strongly self-absorbing C*-algebras with a Clifford grading. Journal of the Mathematical Society of Japan

### 2023

- Torras Casas, A. and Pennig, U. 2023. Interleaving Mayer-Vietoris spectral sequences. Algebraic and Geometric Topology

### 2022

- Evans, D. and Pennig, U. 2022. Equivariant higher Dixmier-Douady theory for circle actions on UHF-algebras. Advances in Mathematics 410(Part B), article number: 108745. (10.1016/j.aim.2022.108745)
- Evans, D. E. and Pennig, U. 2022. Equivariant higher twisted K-theory of SU(n) for exponential functor twists. Journal of Topology 15(2), pp. 896-949. (10.1112/topo.12219)

### 2019

- Lechner, G., Pennig, U. and Wood, S. 2019. Yang-Baxter representations of the infinite symmetric group. Advances in Mathematics 355, article number: 106769. (10.1016/j.aim.2019.106769)
- Pennig, U. 2019. Exponential functors, R-matrices and twists. Algebraic & Geometric Topology 20, article number: 3. (10.2140/agt.2020.20.1279)

### 2017

- Dadarlat, M. and Pennig, U. 2017. Connective C*-algebras. Journal of Functional Analysis 272(12), pp. 4919-4943. (10.1016/j.jfa.2017.02.009)
- Dadarlat, M., Pennig, U. and Schneider, A. 2017. Deformations of wreath products. Bulletin of the London Mathematical Society 49(1), pp. 23-32. (10.1112/blms.12008)
- Dadarlat, M. and Pennig, U. 2017. Deformations of nilpotent groups and homotopy symmetric C*-algebras. Mathematische Annalen 367, pp. 121-134. (10.1007/s00208-016-1379-0)

### 2016

- Evington, S. and Pennig, U. 2016. Locally trivial W*-bundles. International Journal of Mathematics 27, article number: 1650088. (10.1142/S0129167X16500889)
- Dadarlat, M. and Pennig, U. 2016. A Dixmier-Douady theory for strongly self-absorbing C*-algebras II: the Brauer group. Journal of Noncommutative Geometry 9(4), pp. 1137-1154. (10.4171/JNCG/218)
- Dadarlat, M. and Pennig, U. 2016. A Dixmier--Douady theory for strongly self-absorbing C*-algebras. Journal für die reine und angewandte Mathematik 2016(718), pp. 153-181. (10.1515/crelle-2014-0044)
- Meyer, R. and Pennig, U. 2016. Crossed module actions on continuous trace C*-algebras. Communications in Mathematical Physics 346(1), pp. 115-142. (10.1007/s00220-016-2638-3)
- Pennig, U. 2016. A non-commutative model for higher twisted K-theory. Journal of Topology 9(1), pp. 27-50. (10.1112/jtopol/jtv033)

### 2015

- Dadarlat, M. and Pennig, U. 2015. Unit spectra of K-theory from strongly self-absorbing C*-algebras. Algebraic and Geometric Topology 15(1), pp. 137-168. (10.2140/agt.2015.15.137)

### 2014

- Pennig, U. 2014. Twisted K-theory and obstructions against positive scalar curvature metrics. Journal of K-theory: K-theory and its Applications to Algebra, Geometry, and Topology 14(1), pp. 47-71. (10.1017/is014003008jkt259)

### 2011

- Pavlov, A., Pennig, U. and Schick, T. 2011. Quasi-multipliers of Hilbert and Banach C*-bimodules. Mathematica Scandinavica 109(1), pp. 71-92.

### 2007

- Kawahigashi, Y., Longo, R., Pennig, U. and Rehren, K. 2007. The classification of non-local chiral CFT with c < 1. Communications in Mathematical Physics 271(2), pp. 375-385. (10.1007/s00220-007-0199-1)

### Articles

- Pennig, U. and Sogabe, T. 2024. Spanier–Whitehead K-duality and duality of extensions of C*-algebras. International Mathematics Research Notices (10.1093/imrn/rnae242)
- Dadarlat, M., McClure, J. E. and Pennig, U. 2024. Computing cohomology groups that classify bundles of strongly self-absorbing C*-algebras. Journal of Topology and Analysis (10.1142/S1793525324500110)
- Dădărlat, M. and Pennig, U. 2024. Bundles of strongly self-absorbing C*-algebras with a Clifford grading. Journal of the Mathematical Society of Japan
- Torras Casas, A. and Pennig, U. 2023. Interleaving Mayer-Vietoris spectral sequences. Algebraic and Geometric Topology
- Evans, D. and Pennig, U. 2022. Equivariant higher Dixmier-Douady theory for circle actions on UHF-algebras. Advances in Mathematics 410(Part B), article number: 108745. (10.1016/j.aim.2022.108745)
- Evans, D. E. and Pennig, U. 2022. Equivariant higher twisted K-theory of SU(n) for exponential functor twists. Journal of Topology 15(2), pp. 896-949. (10.1112/topo.12219)
- Lechner, G., Pennig, U. and Wood, S. 2019. Yang-Baxter representations of the infinite symmetric group. Advances in Mathematics 355, article number: 106769. (10.1016/j.aim.2019.106769)
- Pennig, U. 2019. Exponential functors, R-matrices and twists. Algebraic & Geometric Topology 20, article number: 3. (10.2140/agt.2020.20.1279)
- Dadarlat, M. and Pennig, U. 2017. Connective C*-algebras. Journal of Functional Analysis 272(12), pp. 4919-4943. (10.1016/j.jfa.2017.02.009)
- Dadarlat, M., Pennig, U. and Schneider, A. 2017. Deformations of wreath products. Bulletin of the London Mathematical Society 49(1), pp. 23-32. (10.1112/blms.12008)
- Dadarlat, M. and Pennig, U. 2017. Deformations of nilpotent groups and homotopy symmetric C*-algebras. Mathematische Annalen 367, pp. 121-134. (10.1007/s00208-016-1379-0)
- Evington, S. and Pennig, U. 2016. Locally trivial W*-bundles. International Journal of Mathematics 27, article number: 1650088. (10.1142/S0129167X16500889)
- Dadarlat, M. and Pennig, U. 2016. A Dixmier-Douady theory for strongly self-absorbing C*-algebras II: the Brauer group. Journal of Noncommutative Geometry 9(4), pp. 1137-1154. (10.4171/JNCG/218)
- Dadarlat, M. and Pennig, U. 2016. A Dixmier--Douady theory for strongly self-absorbing C*-algebras. Journal für die reine und angewandte Mathematik 2016(718), pp. 153-181. (10.1515/crelle-2014-0044)
- Meyer, R. and Pennig, U. 2016. Crossed module actions on continuous trace C*-algebras. Communications in Mathematical Physics 346(1), pp. 115-142. (10.1007/s00220-016-2638-3)
- Pennig, U. 2016. A non-commutative model for higher twisted K-theory. Journal of Topology 9(1), pp. 27-50. (10.1112/jtopol/jtv033)
- Dadarlat, M. and Pennig, U. 2015. Unit spectra of K-theory from strongly self-absorbing C*-algebras. Algebraic and Geometric Topology 15(1), pp. 137-168. (10.2140/agt.2015.15.137)
- Pennig, U. 2014. Twisted K-theory and obstructions against positive scalar curvature metrics. Journal of K-theory: K-theory and its Applications to Algebra, Geometry, and Topology 14(1), pp. 47-71. (10.1017/is014003008jkt259)
- Pavlov, A., Pennig, U. and Schick, T. 2011. Quasi-multipliers of Hilbert and Banach C*-bimodules. Mathematica Scandinavica 109(1), pp. 71-92.
- Kawahigashi, Y., Longo, R., Pennig, U. and Rehren, K. 2007. The classification of non-local chiral CFT with c < 1. Communications in Mathematical Physics 271(2), pp. 375-385. (10.1007/s00220-007-0199-1)

## Research

### My research interests in a nutshell

My main research interest is algebraic topology. A topology on a space allows us to talk about closeness or neighbourhoods of points without actually having to talk about distances between them. This enables us to define continuity of maps as well. Many geometric objects, like surfaces or higher dimensional manifolds, are in particular topological spaces. The goal of algebraic topology is to associate algebraic invariants, often groups, rings or modules, to these spaces in a natural way to distinguish them up to homotopy equivalence, i.e. up to continuous deformations.

An important example of such an invariant is topological K-theory. It is a generalised cohomology theory, which means that it satisfies axioms that allow its computation from a decomposition of the space into smaller parts. It can be defined as an invariant associated to the algebra of continuous functions on the space (vanishing at infinity) and as such it can be generalised to noncommutative algebras, which should be considered as "noncommutative topological spaces". These operator algebras not only play an important role in mathematical physics, they also can be used to describe highly singular topological spaces, like orbifolds or foliations.

K-theory has many different variants, like equivariant K-theory or twisted K-theory. The outcome of one of my research projects, which was joint work with Marius Dadarlat, was a description of twisted K-theory via operator algebras that includes the higher twists, a feature of the theory, which was formerly only accessible via homotopy theory.

I am interested in these and other interactions between algebraic topology, homotopy theory and operator algebras. Since I studied mathematical physics before my PhD, I am also interested in topological and conformal field theories and their connection with another generalised cohomology theory, called elliptic cohomology.

### Recent Conference Organisation

06/09/2024 - 07/09/2024: Operator Algebras in the South of the UK

(with Veronique Fischer, Jan Spakula and Stuart White)

15/03/2024 - 16/03/2024: Operator Algebras in the South of the UK

24/11/2023 - 24/11/2023: Operator Algebras in the South of the UK

19/05/2023 - 19/05/2023: Operator Algebras in the South of the UK

19/05/2023 - 19/05/2023: Operator Algebras in the South of the UK

03/03/2023 - 03/03/2023: Operator Algebras in the South of the UK

27/07/2022 - 02/08/2022: Operator Algebras: Subfactors, K-Theory, Conformal Field Theory

(with Masaki Izumi, Gandalf Lechner, Mathew Pugh)

08/05/2022 - 20/05/2022: Research in Groups at the ICMS

(with Christian Voigt)

### Selected Conference and Seminar Talks

06/2024: GeNoCAS 2024

06/2024: UK Operator Algebras Conference

11/2023: Twinned Conference on C*-Algebras and Tensor Categories

09/2023: British Topology Meeting

03/2023: Lunch time seminar, Southampton

03/2023: Analysis Seminar, Glasgow

06/2022: Non-commutative Geometry along the North Sea (Lorentz Center, Leiden)

Bundles of C*-Algebras and Higher Dixmier-Douady Theory

02/2022: Oberseminar Global Analysis and Geometry (Regensburg)

Equivariant Higher Dixmier-Douady Theory for Circle Actions

01/2022: Oberseminar Topologie Göttingen

Equivariant higher twists over SU(n)

11/2021 University of Aberdeen

Exponential Functor Twists for K-theory over SU(n)

11/2021 Functional Analysis Seminar, Oxford

Equivariant higher twists over SU(n)

04/2021: Operator Algebra Seminar Oslo

Equivariant higher twists over SU(n)

04/2021: Regensburg Seminar in Homotopy Theory and Related Areas

Equivariant higher twists over SU(n)

01/2021: Institute for Pure & Applied Mathematics (IPAM)

Fell bundles, Dixmier-Douady theory and higher twists (video)

11/2020: Sheffield Pure Maths Colloquium

Bundles of Algebras - Dixmier-Douady Theory and Beyond

08/2020: UK Virtual Operator Algebra Seminar

Bundles of C*-Algebras - An introduction to Dixmier-Douady Theory

05/2020: Online Algebraic Topology Seminar (OATS)

Loops, Group and Twists - The role of K-theory in mathematical physics (video)

10/2019: Pure Maths Colloquium, Southampton

Loops, Groups and Twists - the role of K-theory in mathematical physics

10/2019: Topology Seminar, Sheffield

Equivariant higher twisted K-theory of SU(n) via exponential functors

10/2019: Operator Algebra Seminar, Glasgow

Equivariant higher twisted K-theory of SU(n) via exponential functors

09/2019: 34th British Topology Meeting, Warwick

Equivariant higher twisted K-theory of SU(n) via exponential functors

06/2019: Operator Algebras and Quantum Physics, Simons Center New York

Equivariant higher twisted K-theory of SU(n) via exponential functors

11/2018: Barcelona Weekend on Operator Algebras

Connective C*-algebras

10/2018: Algebra Seminar, Glasgow

The Yang-Baxter equation and higher twists over SU(n)

11/2018: Operator Algebra Seminar, KU Leuven

Connective C*-algebras

04/2018: Operator Algebra Seminar, Copenhagen

Connective C*-algebras

01/2018: Kyoto Operator Algebra Seminar

A Dixmier-Douady Theory for strongly self-absorbing C*-algebras

10/2017: Analysis Seminar, Glasgow

Connective C*-algebras

03/2017: Newton Institute, Cambridge

Equivariant Higher Twisted K-theory

11/2016: 101st Meeting of the Transpennine Topology Triangle, Leicester

Unit Spectra of K-theory via strongly self-absorbing C*-algebras

11/2016: Analysis Seminar, Loughborough

Twisted index obstructions to positive scalar curvature

10/2016: Algebra and Topology Seminar, Swansea

Higher twisted K-theory via strongly self-absorbing C*-algebras

08/2016: 31st Summer Conference on Topology and its Applications, Leicester

A non-commutative model for higher twisted K-theory

04/2016: Wabash Analysis Seminar

Deformations of nilpotent groups and homotopy symmetric C*-algebras

03/2016: Analysis Seminar, Glasgow

Deformations of nilpotent groups and homotopy symmetric C*-algebras

03/2016: Topology Seminar, Aberdeen

Higher twisted K-theory via strongly self-absorbing C*-algebras

10/2015: Operator Algebra Seminar, Copenhagen

Deformations of nilpotent groups and homotopy symmetric C*-algebras

06/2015: Oberwolfach Workshop “Noncommutative Geometry”

A Dixmier-Douady Theory for strongly self-absorbing C*-algebras

03/2015: Operator Algebra Seminar, Copenhagen (invited by Dominic Enders)

A noncommutative model for higher twisted K-theory and K-homology

02/2015: String Geometry Network Meeting, Greifswald

Crossed module actions on continuous trace C*-algebras

11/2014: Geometry and Quantum Theory, Zandvoort (invited by Maarten Solleveld)

Twisted index obstructions to positive scalar curvature

10/2014: Methods of Noncommutative Geometry in Analysis and Topology, Hannover

A Dixmier-Douady Theory for strongly self-absorbing C*-algebras

09/2014: Quantum physics and non-commutative geometry

during the trimester: Non-commutative Geometry and its Applications, HIM Bonn

Twisted K-theory via strongly self-absorbing C*-algebras

07/2014: C*-Algebras and Dynamical Systems, Fields Institute, Toronto

Topological invariants of C(X)-algebras

11/2013: Analysis Seminar, Glasgow (invited by Christian Voigt)

Unit spectra of K-theory via strongly self-absorbing C*-algebras

10/2013: Twists, generalised cohomology and applications, Münster

Unit spectra of K-theory via strongly self-absorbing C*-algebras

10/2013: String Geometry Network Meeting, MPI, Bonn

Unit spectra of K-theory via strongly self-absorbing C*-algebras

09/2013: Séminaires de l'équipe de Topologie Algébrique, Paris (invited by Christian Ausoni)

Unit spectra of K-theory via strongly self-absorbing C*-algebras

09/2013: Operator Algebras Seminar at Purdue (invited by Marius Dadarlat)

Unit spectra of K-theory via strongly self-absorbing C*-algebras

09/2013: Berkeley Topology Seminar (invited by Constantin Teleman)

K-theory, BU and C*-algebras (Introductory Talk)

Unit spectra of K-theory via strongly self-absorbing C*-algebras (Main Talk)

04/2013: Operator Algebras Seminar at Purdue (invited by Marius Dadarlat)

Operator algebraic K-theory via spectra

04/2013: ICMS-Workshop: The structure and classification of nuclear C*-algebras, Edinburgh

A Dixmier-Douady Theory for strongly self-absorbing C*-algebras

08/2012: Strings and Automorphic Forms in Topology, Bochum

Higher Twisted K-theory

06/2012: Young Topologists Meeting 2012, Copenhagen

Higher Twisted K-theory

05/2012: 17th NRW Topology, Münster

Twisted 2-vector bundles

### Research visits

04/11/2024 - 06/12/2024: Isaac Newton Instiute, Cambridge

14/10/2024 - 23/10/2024: Universität Regensburg

29/09/2024 - 13/10/2024: Georg-August-Universität Göttingen

17/05/2024 - 23/05/2024: Dublin Institute of Advanced Studies (DIAS)

26/06/2023 - 30/06/2023: Isaac Newton Institute, Cambridge

20/09/2021 - 23/09/2021: Isaac Newton Instiute, Cambridge

09/06/2019 - 14/06/2019: Simons Center for Geometry and Physics, New York

09/04/2018 - 13/04/2018: University of Copenhagen, Denmark

16/01/2018 - 26/01/2018: Kyoto University, Japan

14/06/2017 - 17/06/2017: Isaac Newton Institute, Cambridge

05/04/2017 - 29/04/2017: Isaac Newton Institute, Cambridge

26/03/2017 - 31/03/2017: Isaac Newton Institute, Cambridge

14/04/2016 – 28/04/2016: Purdue University, West Lafayette

18/01/2016 – 07/02/2016: Institut Mittag-Leffler, Djursholm

01/10/2015 – 22/12/2015: Centre for Symmetry and Deformation, Copenhagen

23/02/2015 – 27/02/2015: String Geometry Network Meeting, Greifswald

01/09/2014 – 30/09/2014: HIM, Bonn (trimester on non-commutative geometry)

20/07/2014 – 26/07/2014: Purdue University, West Lafayette

24/02/2014 – 28/02/2014: String Geometry Network Meeting, ESI, Vienna

07/10/2013 – 11/10/2013: String Geometry Network Meeting, MPI, Bonn

30/09/2013 – 04/10/2013: Charles University, Prague

14/09/2013 – 20/09/2013: Purdue University, West Lafayette

08/09/2013 – 13/09/2013: University of California, Berkeley

28/04/2013 – 09/05/2013: Purdue University, West Lafayette

18/02/2013 – 22/02/2013: String Geometry Network Meeting, Regensburg

03/12/2012 – 08/12/2012: Charles University, Prague

22/05/2012 – 26/05/2012: String Geometry Network Meeting, Hamburg

20/04/2012 – 27/04/2012: Purdue University, West Lafayette

## Teaching

### Teaching

MA3008 - Algebraic Topology (Spring 2024)

MA1005 - Foundations of Mathematics I (Autumn 2023)

MA3008 - Algebraic Topology (Spring 2023)

MA1005 - Foundations of Mathematics I (Autumn 2022)

MA3008 - Algebraic Topology (Spring 2022)

MA1005 - Foundations of Mathematics I (Autumn 2021)

MA3008 - Algebraic Topology (Spring 2021)

MA1005 - Foundations of Mathematics I (Autumn 2020)

MA3008 - Algebraic Topology (Spring 2020)

MA2008 - Linear Algebra II (Autumn 2019)

MA3008 - Algebraic Topology (Spring 2019)

MA2008 - Linear Algebra II (Autumn 2018)

MA3008 - Algebraic Topology (Spring 2018)

MA2007 - Linear Algebra (Autumn 2017)

MA3008 - Algebraic Topology (Spring 2017)

### Previous Teaching

K-theory and the Hopf invariant (summer semester 2015, University of Münster)

Topological Quantum Field Theories (winter semester 2011/12, University of Münster)

## Biography

### Qualifications

07/2016 – Habilitation (University of Münster)

08/2009 – Doctor rer. nat. in Mathematics (University of Göttingen)

05/2006 – Diploma in Theoretical Physics (University of Göttingen)

### Honours and awards

05/2017: Nomination for the Philip Leverhulme Prize

10/2010: Dissertation Prize 2009 of the Universitätsbund Göttingen

08/2006 – 06/2009: PhD Scholarship of the German National Academic Foundation

04/2003 – 05/2006: Diploma Scholarship of the German National Academic Foundation

### Academic positions

11/2010 – 04/2016: Akademischer Rat auf Zeit (University of Münster)

08/2010 – 10/2010: Research Assistant (University of Münster)

12/2009 – 07/2010: Akademischer Rat auf Zeit (University of Göttingen)

09/2009 – 11/2009: Research Assistant (Courant Research Center, Göttingen)

### Committees and reviewing

**Reviewer for**

- Inventiones Mathematicae
- Advances in Mathematics
- Journal of Functional Analysis
- Algebraic & Geometric Topology
- Mathematical Physics, Analysis and Geometry
- Transactions of the AMS
- International Mathematics Research Notices
- Reviews in Mathematical Physics

## Supervisions

### Research Areas for Potential PhD projects:

- algebraic topology, operator-algebraic and topological K-theory and its variants (twisted, equivariant, ...)
- operator algebras, applications of homotopy theory to the theory of C*-algebras
- index theory

### Current supervision

## Valerio Bianchi

Research student

## Declan Bays

Research student

### Past projects

- Sam Richardson (viva in 2022) worked on the computation of rational characteristic classes associated to maps induced by exponential functors on vector spaces (thesis)
- Álvaro Torras Casas (viva in 2022) worked on distributed computation of persistent homology using the Mayer-Vietoris spectral sequence (paper in Discrete & Computational Geometry, arxiv link)

## Contact Details

+44 29206 88852

Abacws, Room 3.64, Senghennydd Road, Cathays, Cardiff, CF24 4AG

## Research themes

## Specialisms

- Algebraic Topology
- Operator algebras and functional analysis
- Topological Data Analysis
- Topology
- K-Theory