# Dr Peter Hochs

I got M.Sc. degrees in mathematics and physics from Utrecht University in the Netherlands in 2003. After that I did a Ph.D. project at the Radboud University in NIjmegen, the Netherlands, supervised by Klaas Landsman and Gert Heckman. That project was about the relations between index theory and representation theory of Lie groups, as well as the problem of quantising a classical theory in physics. After working in industry for a few years I came back to pure mathematics, working at the Leibniz University Hanover in Germany, and at the University of Adelaide.

I am most interested in research into relations between different areas in mathematics. My main area is index theory, where geometry, topology and analysis interact in fundamental ways. I am also interested in links to representation theory of Lie groups, and to noncommutative geometry, K-theory and KK-theory.

*✓ Eligible to supervise Masters and PhD (as Co-Supervisor) — email supervisor to discuss availability.*

I work in index theory, which is the study of relations between geometry, analysis and topology through differential equations on geometric spaces. I am particularly interested in equivariant index theory for noncompact groups and manifolds, and its relations with geometric quantisation, noncommutative geometry, differential geometry and geometric analysis, Lie theory, links between K-theory, K-homology and representation theory.

## Appointments

Date | Position | Institution name |
---|---|---|

2013 | Lecturer | The University of Adelaide |

2013 - 2016 | Marie Curie Fellow | Radboud University |

2012 - 2013 | Alexander von Humboldt Fellow | Leibniz University Hanover |

2009 - 2012 | Research Geophysicist | Shell |

2007 - 2009 | Researcher | Netherlands Organisation for Applied Scientific Research TNO |

## Education

Date | Institution name | Country | Title |
---|---|---|---|

2008 | Radboud University | The Netherlands | PhD |

## Research Interests

## Journals

Year | Citation |
---|---|

2018 | Hochs, P., & Wang, H. (2018). A fixed point formula and Harish-Chandra's character formula. Proceedings of the London Mathematical Society, 116(1), 1-32.DOI |

2018 | Hochs, P., & Song, Y. (2018). An equivariant index for proper actions II: Properties and applications. JOURNAL OF NONCOMMUTATIVE GEOMETRY, 12(1), 157-193.DOI |

2017 | Hochs, P., & Song, Y. (2017). On the Vergne conjecture. Archiv der Mathematik, 108(1), 99-112.DOI WoS1 |

2017 | Hochs, P., & Song, Y. (2017). An equivariant index for proper actions I. Journal of Functional Analysis, 272(2), 661-704.DOI Scopus1 WoS3 |

2017 | Hochs, P., & Mathai, V. (2017). Quantising proper actions on Spin$^c$-manifolds. Asian Journal of Mathematics, 21(4), 631-686.DOI Scopus3 WoS3 |

2017 | Hochs, P., & Song, Y. (2017). Equivariant indices of Spin(c)-Dirac operators for proper moment maps. Duke Mathematical Journal, 166(6), 1125-1178.DOI Scopus2 WoS3 |

2016 | Hochs, P., & Varghese, M. (2016). Formal geometric quantisation for proper actions. Journal of Homotopy and Related Structures, 11(3), 409-424.DOI Scopus1 WoS1 |

2016 | Hochs, P., & Mathai, V. (2016). Spin-structures and proper group actions. Advances in Mathematics, 292, 1-10.DOI Scopus2 WoS2 |

2016 | Hochs, P., & Song, Y. (2016). An equivariant index for proper actions III: the invariant and discrete series indices. Differential Geometry and its Applications, 49, 1-22.DOI Scopus4 WoS4 |

2015 | Hochs, P. (2015). Quantisation of presymplectic manifolds, K-theory and group representations. Proceedings of the American Mathematical Society, 143(6), 2675-2692.DOI Scopus7 WoS7 |

2015 | Hochs, P., & Mathai, V. (2015). Geometric quantization and families of inner products. Advances in Mathematics, 282, 362-426.DOI Scopus10 WoS10 |

2009 | Hochs, P. (2009). Quantisation commutes with reduction at discrete series representations of semisimple groups. Advances in Mathematics, 222(3), 862-919.DOI Scopus12 WoS12 |

2008 | Hochs, P., & Landsman, N. (2008). The Guillemin-Sternberg conjecture for noncompact groups and spaces. Journal of K-Theory, 1(3), 473-533.DOI Scopus10 WoS10 |

— | Hochs, P., & Wang, H. (n.d.). A fixed point theorem on noncompact manifolds. Th., 3, 235-286.DOI |

— | Hochs, P., Song, Y., & Yu, S. (n.d.). A geometric realisation of tempered representations restricted to maximal compact subgroups. |

— | Hochs, P., & Wang, H. (n.d.). Shelstad's character identity from the point of view of index theory. |

— | Hochs, P., & Wang, H. (n.d.). Orbital integrals and $K$-theory classes. |

— | Hochs, P., Song, Y., & Yu, S. (n.d.). A geometric formula for multiplicities of $K$-types of tempered representations. |

— | Hochs, P., Kaad, J., & Schemaitat, A. (n.d.). Algebraic $K$-theory and a semi-finite Fuglede-Kadison determinant. Th., 3, 193-206.DOI |

## Conference Papers

Year | Citation |
---|---|

2011 | Hochs, P. (2011). Quantisation commutes with reduction at nontrivial representations. In Oberwolfach Reports. Mathematisches Forschungsinstitut Oberwolfach. |

Marie Curie International Outgoing Fellowship, European Union, 2012 (personal fellowship).

Alexander von Humboldt Postdoctoral Fellowship, 2011 (personal fellowship).

Support for Workshop on Geometric Quantisation, AMSI/AustMS, 2015.

Support for workshop Representation theory and Operator algebras, AMSI, 2013

## Current Higher Degree by Research Supervision (University of Adelaide)

Date | Role | Research Topic | Program | Degree Type | Student Load | Student Name |
---|---|---|---|---|---|---|

2018 | Co-Supervisor | I will investigate convergence and continuity of functionals defined by orbital integrals, on subalgebras of group C*-algebras, generalising results by Samurkas from discrete to non-discrete groups. I will also investigate applications to K-theory. | Master of Philosophy | Master | Full Time | Mr Nicholas Lawson McLean |

2018 | Co-Supervisor | Computations in Higher Twisted K-theory | Master of Philosophy | Master | Full Time | Mr David Leonard Brook |

2017 | Co-Supervisor | Non-Commutative Index Theory on Singular Spaces | Master of Philosophy | Master | Full Time | Mr Samuel Mills |

## Other Supervision Activities

Date | Role | Research Topic | Location | Program | Supervision Type | Student Load | Student Name |
---|---|---|---|---|---|---|---|

2015 - 2016 | Co-Supervisor | Determinants by K-theory | Radboud University | — | Master | Full Time | André Schemaitat |

2015 - 2016 | Co-Supervisor | Towards faster internet connections: connecting cabinets to the fiber glass network | Radboud University | — | Master | Full Time | Niels Nuemasnn |