# Robert Penner

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**Robert Clark Penner** is an American [mathematician](/source/Mathematician) whose work in [geometry](/source/Geometry) and [combinatorics](/source/Combinatorics) has found applications in [high-energy physics](/source/High-energy_physics) and more recently in [theoretical biology](/source/Theoretical_biology). He is the son of [Sol Penner](/source/Stanford_S._Penner), an aerospace engineer.

## Biography

Robert Clark Penner received his [B.S.](/source/Bachelor_of_Science) degree from [Cornell University](/source/Cornell_University) in 1977 and his [Ph.D.](/source/Ph.D.) from the [Massachusetts Institute of Technology](/source/Massachusetts_Institute_of_Technology) in 1981, the latter under the direction of [James Munkres](/source/James_Munkres) and [David Gabai](/source/David_Gabai). In his doctoral studies, he solved a 50 year old problem posed by [Max Dehn](/source/Max_Dehn) on the action of the [mapping class group](/source/Mapping_class_group) on curves and arcs in surfaces, developed combinatorial aspects of Thurston's theory of [train tracks](/source/Train_track_(mathematics)) and generalized Thurston's construction of [pseudo-Anosov maps](/source/Pseudo-Anosov_map).[1]

After postdoctoral positions at [Princeton University](/source/Princeton_University) and at the [Mittag-Leffler Institute](/source/Mittag-Leffler_Institute), Penner spent most of the period of 1985–2003 at the [University of Southern California](/source/University_of_Southern_California). From 2004 until 2012, he worked at [Aarhus University](/source/Aarhus_University), where he co-founded with [Jørgen Ellegaard Andersen](/source/J%C3%B8rgen_Ellegaard_Andersen) the [Center for the Quantum Geometry of Moduli Spaces](/source/Center_for_the_Quantum_Geometry_of_Moduli_Spaces).[2] Since 2013 Penner has held the position of the [René Thom](/source/Ren%C3%A9_Thom) Chair in Mathematical Biology at the [Institut des Hautes Etudes Scientifiques](/source/Institut_des_Hautes_Etudes_Scientifiques).[3]

Throughout his career Penner held various visiting positions around the world including [Harvard University](/source/Harvard_University), [Stanford University](/source/Stanford_University), [Max-Planck-Institut für Mathematik](/source/Max-Planck-Institut_f%C3%BCr_Mathematik) at [Bonn](/source/Bonn), [University of Tokyo](/source/University_of_Tokyo), [Mittag-Leffler Institute](/source/Mittag-Leffler_Institute), [Caltech](/source/Caltech), [UCLA](/source/UCLA), [Fields Institute](/source/Fields_Institute), [University of Chicago](/source/University_of_Chicago), [ETH Zurich](/source/ETH_Zurich), [University of Bern](/source/University_of_Bern), [University of Helsinki](/source/University_of_Helsinki), [University of Strasbourg](/source/University_of_Strasbourg), [University of Grenoble](/source/University_of_Grenoble), [Nonlinear Institute of Nice-Sophia Antipolis](/source/Nonlinear_Institute_of_Nice-Sophia_Antipolis).

## Contributions to mathematics, physics, and biology

Penner's research began in the theory of [train tracks](/source/Train_track_(mathematics)) including a generalization of [Thurston](/source/William_Thurston)'s original construction of [pseudo-Anosov maps](/source/Pseudo-Anosov_map) to the so-called Penner-Thurston construction, which he used to give estimates on least dilatations. He then co-discovered the so-called Epstein-Penner decomposition of non-compact complete hyperbolic manifolds with [David Epstein](/source/David_B._A._Epstein), in dimension 3 a central tool in knot theory. Over several years he developed the decorated [Teichmüller theory](/source/Teichm%C3%BCller_theory) of punctured surfaces including the so-called Penner matrix model, the basic partition function for Riemann's moduli space. Extending the foregoing to orientation-preserving homeomorphisms of the circle, Penner developed his model of universal [Teichmüller theory](/source/Teichm%C3%BCller_theory) together with its Lie algebra. He discovered combinatorial cocycles with [Shigeyuki Morita](/source/Shigeyuki_Morita) for the first and with [Nariya Kawazumi](/source/Nariya_Kawazumi) for the higher Johnson homomorphisms. Penner has also contributed to theoretical biology in joint work with [Jørgen E. Andersen](/source/J%C3%B8rgen_E._Andersen) et al. discovering a priori geometric constraints on protein geometry, and with [Michael S. Waterman](/source/Michael_S._Waterman), [Piotr Sulkowski](/source/Piotr_Sulkowski), [Christian Reidys](/source/Christian_Reidys) et al. introducing and solving the matrix model for RNA topology.

## Main journal publications

- Penner, R. C. (1987). "The decorated Teichmüller space of punctured surfaces". *[Communications in Mathematical Physics](/source/Communications_in_Mathematical_Physics)*. **113** (2): 299–339. [doi:10.1007/BF01223515](https://doi.org/10.1007/BF01223515). [S2CID 120198031](https://api.semanticscholar.org/CorpusID:120198031)
- Epstein, D.B.A. & Penner, R. C. (1988). "Euclidean decompositions of noncompact hyperbolic manifolds". *[Journal of Differential Geometry](/source/Journal_of_Differential_Geometry)*. **27** (1): 67–80. [doi:10.4310/jdg/1214441650](https://doi.org/10.4310/jdg/1214441650)
- Penner, R. C. (1988). "Perturbative series and the moduli space of Riemann surfaces". *[Journal of Differential Geometry](/source/Journal_of_Differential_Geometry)*. **27** (1): 35–53. [doi:10.4310/jdg/1214441648](https://doi.org/10.4310/jdg/1214441648)
- Penner, Robert C. (1988). "A construction of pseudo-Anosov homeomorphisms". *[Transactions of the American Mathematical Society](/source/Transactions_of_the_American_Mathematical_Society)*. **310** (1): 179–197. [doi:10.1090/S0002-9947-1988-0930079-9](https://doi.org/10.1090/S0002-9947-1988-0930079-9)
- Penner, R. C. (1991). "Bounds on least dilatations". *[Proceedings of the American Mathematical Society](/source/Proceedings_of_the_American_Mathematical_Society)*. **113** (2): 443–450. [doi:10.1090/S0002-9939-1991-1068128-8](https://doi.org/10.1090/S0002-9939-1991-1068128-8)
- Penner, R. C. (1992). "Weil-Petersson volumes". *[Journal of Differential Geometry](/source/Journal_of_Differential_Geometry)*. **35** (3): 559–608. [doi:10.4310/jdg/1214448257](https://doi.org/10.4310/jdg/1214448257)
- Penner, R. C. (1993). "Universal constructions in Teichmüller theory". *[Advances in Mathematics](/source/Advances_in_Mathematics)*. **98** (2): 143–215. [doi:10.1006/aima.1993.1015](https://doi.org/10.1006/aima.1993.1015)
- Penner, R. C. (1996). "The geometry of the Gauss product". *Journal of Mathematical Sciences*. **81** (3): 2700–2718. [doi:10.1007/BF02362336](https://doi.org/10.1007/BF02362336)
- Penner, R. C. & Waterman, Michael S. (1993). "Spaces of RNA secondary structures". *[Advances in Mathematics](/source/Advances_in_Mathematics)*. **101** (1): 31–49. [doi:10.1006/aima.1993.1039](https://doi.org/10.1006/aima.1993.1039)
- Papadopoulos, Athanase & Penner, R. C. (1991). "La forme symplectique de Weil-Petersson et le bord de Thurston de l'espace de Teichmüller". *[Comptes rendus de l'Académie des Sciences](/source/Comptes_rendus_de_l'Acad%C3%A9mie_des_Sciences)*. **312**: 871–874. Série I.
- Kaufmann, Ralph & Penner, R. C. (2006). "Closed/open string diagrammatics". *[Nuclear Physics B](/source/Nuclear_Physics_B)*. **748** (3): 335–379. [arXiv:math/0603485](https://arxiv.org/abs/math/0603485). [doi:10.1016/j.nuclphysb.2006.03.036](https://doi.org/10.1016/j.nuclphysb.2006.03.036). [S2CID 14428342](https://api.semanticscholar.org/CorpusID:14428342)
- Morita, S. & Penner, R. C. (2008). "Torelli groups, extended Johnson homomorphisms, and new cycles on the moduli space of curves". *[Mathematical Proceedings of the Cambridge Philosophical Society](/source/Mathematical_Proceedings_of_the_Cambridge_Philosophical_Society)*. **144** (3): 651–671. [arXiv:math/0602461](https://arxiv.org/abs/math/0602461). [doi:10.1017/S0305004107000990](https://doi.org/10.1017/S0305004107000990). [S2CID 13633256](https://api.semanticscholar.org/CorpusID:13633256)
- Bene, Alex James; Kawazumi, Nariya; Penner, R. C. (2009). "Canonical extensions of the Johnson homomorphisms to the Torelli groupoid". *[Advances in Mathematics](/source/Advances_in_Mathematics)*. **221** (2): 627–659. [doi:10.1016/j.aim.2009.01.004](https://doi.org/10.1016/j.aim.2009.01.004)
- Penner, Robert C.; Andersen, Ebbe S.; Jensen, Jens L.; Kantcheva, Adriana K.; Bublitz, Maike; Nissen, Poul; Rasmussen, Anton M. H.; Svane, Katrine L.; Hammer, Bjørk; Rezazadegan, Reza; Nielsen, Niels Chr.; Nielsen, Jakob T.; Andersen, Jørgen E. (2014). "Hydrogen bond rotations as a uniform structural tool for analyzing protein architecture". *[Nature Communications](/source/Nature_Communications)*. **5**: 5803. [doi:10.1038/ncomms6803](https://doi.org/10.1038/ncomms6803). [PMID 25517704](https://pubmed.ncbi.nlm.nih.gov/25517704)
- Reidys, Christian M.; Huang, Fenix W. D.; Andersen, Jørgen E.; Penner, Robert C.; Stadler, Peter F.; Nebel, Markus E. (2011). "Topology and prediction of RNA pseudoknots". *[Bioinformatics](/source/Bioinformatics_(journal))*. **27** (8): 1076–1085. [doi:10.1093/bioinformatics/btr090](https://doi.org/10.1093/bioinformatics/btr090). [PMID 21335320](https://pubmed.ncbi.nlm.nih.gov/21335320)
- Andersen, Jørgen E.; Chekhov, Leonid O.; Penner, R. C.; Reidys, Christian M.; Sułkowski, Piotr (2012). "Topological recursion for chord diagrams, RNA complexes, and cells in moduli spaces". *[Nuclear Physics B](/source/Nuclear_Physics_B)*. **866** (3): 414–443. [arXiv:1205.0658](https://arxiv.org/abs/1205.0658). [doi:10.1016/j.nuclphysb.2012.09.012](https://doi.org/10.1016/j.nuclphysb.2012.09.012). [S2CID 10826116](https://api.semanticscholar.org/CorpusID:10826116)
- Penner, R. C. (2016). "Moduli spaces and macromolecules". *[Bulletin of the American Mathematical Society](/source/Bulletin_of_the_American_Mathematical_Society)*. **53** (2): 217–268. [doi:10.1090/bull/1524](https://doi.org/10.1090/bull/1524)
- Penner, R. C. & Zeitlin, Anton M. (2019). "Decorated super-Teichmüller space". *[Journal of Differential Geometry](/source/Journal_of_Differential_Geometry)*. **111** (3): 527–566. [arXiv:1509.06302](https://arxiv.org/abs/1509.06302). [doi:10.4310/jdg/1552442609](https://doi.org/10.4310/jdg/1552442609). [S2CID 119664779](https://api.semanticscholar.org/CorpusID:119664779)

## Books

- with the assistance of [J. L. Harer](/source/J._L._Harer): *Combinatorics of Train Tracks*, *[Annals of Mathematics Studies](/source/Annals_of_Mathematics_Studies)* 125, *[Princeton University Press](/source/Princeton_University_Press)* (1992); second printing (2001).
- *Perspectives in Mathematical Physics*, *[International Press](/source/International_Press)*, edited by R. C. Penner and [Shing-Tung Yau](/source/S.T._Yau) (1994).
- *Discrete Mathematics--proof techniques and mathematical structures*, *[World Scientific Publishing Company](/source/World_Scientific_Publishing_Company)* (1999); second printing (2001).
- *Woods Hole Mathematics: perspectives in math and physics*, edited by [N. Tongring](/source/N._Tongring) and R. C. Penner, foreword by [Raul Bott](/source/R._Bott), *[World Scientific Publishing Company](/source/World_Scientific_Publishing_Company)* (2004).
- *Groups of Diffeomorphisms-in honor of Shigeyuki Morita on the occasion of his 60th birthday*, *[Advanced Studies in Pure Mathematics](/source/Advanced_Studies_in_Pure_Mathematics)* 52 (2008), *[Mathematical Society of Japan](/source/Mathematical_Society_of_Japan)*, edited by R. C. Penner, [D. Kotschick](/source/D._Kotschick), [T. Tsuboi](/source/T._Tsuboi), [N. Kawazumi](/source/N._Kawazumi), [T. Kitano](/source/T._Kitano), [Y. Mitsumatsu](/source/Y._Mitsumatsu).
- *Decorated Teichmüller theory,* (with a foreword by [Yuri I. Manin](/source/Yuri_Manin)), *[QGM Master Class Series](/source/QGM_Master_Class_Series)*, *[European Mathematical Society](/source/European_Mathematical_Society)*, [Zürich](/source/Z%C3%BCrich), 2012, xviii+360 pp. ISBN 978-3-03719-075-3.
- *Topology and K-theory: Lectures by Daniel Quillen,* Notes by Robert Penner, Springer-Verlag [Lecture Notes in Mathematics](/source/Lecture_Notes_in_Mathematics) (2020)

## Patents

*Methods of Digital Filtering and Multi-Dimensional Data Compression Using the Farey Quadrature and Arithmetic, Fan, and Modular Wavelets*, US Patent 7,158,569 (granted 2Jan07)[4]

## Philanthropy

In 2018 Penner endowed the Alexzandria Figueroa and Robert Penner Chair at the IHES in memoriam of Alexzandria Figueroa.[5]

## References

1. Penner, Robert Clark (March 5, 1982). [*A computation of the action of the mapping class group on isotopy classes of curves and arcs in surfaces*](https://dspace.mit.edu/handle/1721.1/15618). Massachusetts Institute of Technology. [hdl:1721.1/15618](https://hdl.handle.net/1721.1/15618). – via dspace.mit.edu.

1. ["qgm.au.dk"](https://qgm.au.dk/). *qgm.au.dk*

1. ["Robert C. Penner"](https://www.ihes.fr/en/professeur/robert-c-penner-2/)

1. ["Methods of digital filtering and multi-dimensional data compression using the farey quadrature and arithmetic, fan, and modular wavelets"](https://patents.google.com/patent/WO2000044104A1/en)

1. ["The Alexzandria Figueroa and Robert Penner Chair established at the Institut des Hautes Etudes Scientifiques"](https://www.ihes.fr/en/the-alexzandria-figueroa-and-robert-penner-associated-professor-chair-established/). February 1, 2019.

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