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College of Arts & Sciences

Marcel Bischoff

Marcel Bischoff in an outdoor setting

Assistant Professor

Mathematics
Morton 521
bischoff@ohio.edu
740-593-1261

Education

Ph.D. in Mathematics at the University of Rome “Tor Vergata,” 2012. Ph.D. thesis “Construction of Models in low-dimensional Quantum Field Theory using Operator Algebraic Methods” supervised by Prof. Roberto Longo.

Diploma in Physics at the Georg-August-Universität Göttingen, 2009. Diploma thesis: “On the Pole Structure of Higher Correlation Functions in Globally Conformal Invariant Quantum Field Theory” (in german) supervised by Prof. Karl-Henning Rehren.

Research Interests

  • Operator Algebras, in particular Subfactors
  • Algebraic Quantum Field Theory, in particular Conformal Field Theory
  • Tensor Categories

Personal website

NSF Grant

Bischoff is the PI on an NSF grant titled Quantum Symmetries and Conformal Nets.

Abstract: Symmetries, which can be mathematically described by groups, play an important role in science. In quantum physics, the fundamental theory of physics at small scales, operator algebras provide a mathematical framework to study quantum systems and their symmetries. To describe quantum particles and matter in low dimensions (for example, on a thin layer) one needs a generalization of symmetry that goes beyond groups, often referred to as quantum symmetries. A main focus of this project is to find models realizing such symmetries in quantum field theory, which combines the principles of quantum physics and the theory of special relativity. The goal is to use the rich interplay between operator algebras and quantum field theory via conformal nets and to better understand possible quantum symmetries in mathematics and low-dimensional physics. One potential application is topological quantum computing, where the goal is to use quantum operations coming from non-trivial quantum symmetries to perform computations.

Conformal nets give rise to interesting quantum symmetries in terms of subfactors and unitary modular tensor categories. The project will focus on three main directions. First, the principal investigator will extend the understanding of boundaries or defects between conformal nets, generalize the abstract description of defects in terms of braided subfactors, and explore the relation to Jones' planar algebras. The second project focus is to provide new structural results and examples of rational conformal nets realizing the quantum doubles subfactors. Lastly, the principal investigator will develop methods for using operator-algebraic techniques to construct conformal nets from their quantum symmetries and give relations between the operator algebraic approach of conformal nets and the purely algebraic approach of vertex operator algebras.

Positions

2014/08 – now Assistant Professor (non-tenure track, Postdoc) of Mathematics, Vanderbilt University, Nashville, TN. Supervisor: Vaughan F. R. Jones.

2012/10 – 2014/08 Postdoctoral researcher in the DFG (German Research Foundation) Research Training Group 1493 “Mathematical Structures in Modern Quantum Physics”, University of Göttingen, Germany.

Scientific/Academic Honors and Grants

2015-2016 Summer research support through NSF Grant DMS-1362138, principal investigator Vaughan F.R. Jones.

2014-2017 Vanderbilt University, College of Arts and Science annual research fund and Postdoc appointment with reduced (1-1) teaching load.

2009-2012 PhD scholarship in Mathematics at the University of Rome “Tor Vergata”.

Publications

Articles and Books

M. Bischoff, Generalized Orbifold Construction for Conformal Nets, to appear in Reviews in Mathematical Physics Vol. 29, No. 1 (2017) 1750002. DOI: 10.1142/S0129055X17500027, (arXiv: 1608.00253).

M. Bischoff, A Remark on CFT Realization of Quantum Doubles of Subfactors: Case Index < 4, Letters in

Mathematical Physics 106 (3), 2016, p. 341–363. DOI: 10.1007/s11005-016-0816-z, (arXiv: 1506.02606).

M. Bischoff, Y. Kawahigashi, R. Longo, K.-H. Rehren: Phase boundaries in algebraic conformal QFT, Communications in Mathematical Physics 342 (1), 2016, p. 1–45, DOI: 10.1007/s00220-015-2560-0, (arXiv: 1405.7863).

M. Bischoff, Y. Kawahigashi, R. Longo: Characterization of 2D rational local conformal nets and its boundary conditions: the maximal case, Documenta Mathematica 20 (2015) 1137–1184.(open access).

M. Bischoff, Y. Tanimoto: Integrable QFT and Longo-Witten endomorphisms, Annales Henri Poincaré 16 (2), 2015, p. 569–608 DOI: 10.1007/s00023-014-0337-1 (open access).

M. Bischoff, Y. Tanimoto: Construction of wedge-local nets of observables through Longo-Witten endomorphisms. II, Communications in Mathematical Physics 317 (3), 2013 p. 667–695 DOI: 10.1007/s00220-012-1593-x (open access).

M. Bischoff: Models in Boundary Quantum Field Theory Associated with Lattices and Loop Group Models, Communications in Mathematical Physics 315 (3), 2012 p. 827–858 DOI: 10.1007/s00220-012-1511-2, (arXiv: 1108.4889).

M. Bischoff, D. Meise, K.-H. Rehren, I. Wagner: Conformal quantum field theory in various dimensions, Bulg. J. Phys. 36 (2009) 170–185. http://www.bjp-bg.com/paper.php?id=516 (open access).

Preprints

M. Bischoff, K.-H. Rehren: The Hypergroupoid Of Boundary Conditions for Local Quantum Observables, (arXiv: 1612.02972).

M. Bischoff, Conformal Net Realizability of Tambara–Yamagami Categories (preliminary title), In Progress. Proceedings

M. Bischoff, The Relation between Subfactors arising from Conformal Nets and the Realization of Quantum Doubles, To appear in Volume 46 of the Proceedings of the Centre for Mathematics and its Applications, in honor of Vaughan F. R. Jones’ 60th birthday. (arXiv: 1511.08931).

M. Bischoff (joint with Yasuyuki Kawahigashi, Roberto Longo, Karl-Henning Rehren): An Algebraic Conformal Quantum Field Theory Approach to Defects, p. 906–908. In: Oberwolfach Reports, Subfactors and Conformal Field Theory, Volume 12, Issue 2, 2015, p. 849–926. DOI: 10.4171/OWR/2015/16.


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