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Robert Clark Penner is an American mathematician whose work in geometry and combinatorics has found applications in high-energy physics and more recently in theoretical biology.

Biography
Robert C. Penner received his B.S. degree from from Cornell University in 1977 and his Ph.D. from the Massachusetts Institute of Technology in 1981, the latter under the direction of James Munkres and David Gabai. In his doctoral studies, he solved a 50 year old problem posed by Max Dehn on the action of the mapping class group on curves and arcs in surfaces, developed  combinatorial aspects of Thurston's theory of train tracks and generalized Thurston's construction of pseudo-Anosov maps.

After postdoctoral positions at Princeton University and at the Mittag-Leffler Institute, Penner spent most of the period of 1985-2003 at the University of Southern California. From 2004 until 2012, he worked at Aarhus University, where he co-founded with Jørgen Ellegaard Andersen the Center for the Quantum Geometry of Moduli Spaces. Since 2013 Penner has held the position of the Rene Thom Chair in Mathematical Biology at the Institut des Hautes Etudes Scientifiques.

Throughout his career Penner held various visiting positions around the world including Harvard University, Stanford University, Max-Planck-Institut für Mathematik at Bonn, University of Tokyo, Mittag-Leffler Institute, Caltech, UCLA, Fields Institute, University of Chicago, ETH Zurich, University of Bern, University of Helsinki, University of Strasbourg, University of Grenoble, Nonlinear Institute of Nice-Sophia Antipolis.

Contributions to Mathematics, Physics and Biology
Started research in the theory of train tracks, generalized Thurston's construction of Pseudo-Anosov maps to so-called Penner-Thurston construction.

Co-discovered the so-called Epstein-Penner decomposition of non-compact complete hyperbolic manifolds, in dimension 3 a central tool in knot theory.

Developed the theory of decorated Teichmüller space of punctured surfaces.

Developed the so-called Penner matrix model, the basic partition function for Riemann's moduli space.

Developed the Universal Teichmüller theory giving a model for orientation-preserving homeomorphism of the circle and its Lie algebra.

Developed with Shigeyuki Morita a combinatorial cocycle for the Johnson homeomorphism and with Nariya Kawazumi the higher Johnson homeomorphisms.

Discovered with Jørgen E. Andersen et al. a priori geometric constraints on protein geometry.

Discovered and solved with Michael S. Waterman, Piotr Sulkowski, Christian Reidys et al. the matrix model for RNA topology.

Main Journal Publications

 * The decorated Teichmüller space of punctured surfaces, Comm. Math. Phys. 113 (1987), no. 2, 299-339.


 * with D.B.A. Epstein: Euclidean decompositions of noncompact hyperbolic manifolds, J. Differential Geom. 27 (1988), no. 1, 67-80.


 * Perturbative series and the moduli space of Riemann surfaces, J. Differential Geom. 27 (1988), no. 1, 35-53.


 * A construction of pseudo-Anosov homeomorphisms, Trans. Amer. Math. Soc. 310 (1988), no. 1, 179-197.


 * Bounds on least dilatations, Proc. Amer. Math. Soc. 113 (1991), no. 2, 443-450.


 * Weil-Petersson volumes, J. Differential Geom. 35 (1992), no. 3, 559-608.


 * Universal constructions in Teichmüller theory, Adv. Math. 98 (1993), no. 2, 143-215.


 * The geometry of the Gauss product, Algebraic Geometry 4, (Festschrift for Yuri Manin) J. Math. Sci. 81 (1996), 2700-2718.


 * with M.S. Waterman: Spaces of RNA secondary structures, Adv. Math. 101 (1993), no. 1, 31-49.


 * with A. Papadopoulos: La forme symplectique de Weil-Petersson et le bord de Thurston de l'espace de Teichmüller, Comptes Rendus Acad. Sci. Paris 312 Série I (1991), 871-874.


 * with R. Kaufmann: Closed/open string diagrammatics, Nucl. Phys. B 748 (2006) 335-379.


 * with S. Morita: Torelli groups, extended Johnson homomorphisms, and new cycles on the moduli space of curves,  Math. Proc. Cambridge Philos. Soc. 144 (2008), no. 3, 651-671.


 * with A. Bene, N. Kawazumi: Canonical extensions of the Johnson homomorphisms to the Torelli groupoid, Adv. Math. 221, No.2, (2009) 627-659.


 * with E.S. Andersen, J.L. Jensen, A.K. Kantcheva, M. Bublitz, P. Nissen, A.M.H. Rasmussen, K.L. Svane, B. Hammer, R. Rezazadegan, N.Chr. Nielsen, J.T. Nielsen, J.E. Andersen: Hydrogen bond rotations as a uniform structural tool for analyzing protein architecture, Nature Communications 5, Article number: 5803 (2014).


 * with C. M. Reidys, F. Huang, J. E. Andersen, P. F. Stadler, M. E. Nebel: Topology and prediction of RNA pseudoknots, Bioinformatics 27 (2011) 1076-1085.


 * with J.E. Andersen, L.O. Chekhov, C.M. Reidys, P. Sulkowski: Topological recursion for chord diagrams, RNA complexes, and cells in moduli spaces,  Nucl.Phys. B 866 No. 3 (2012) 414-443.


 * Moduli spaces and macromolecules, Bull. Amer. Math. Soc. 53 (2016) 217-268.


 * with A.M. Zeitlin: Decorated super-Teichmüller space, J. Differential Geom. 111, No. 3, (2019) 527-566.

Books

 * with the assistance of J. L. Harer: Combinatorics of Train Tracks, Annals of Mathematical Studies 125,  Princeton University Press (1992); second printing (2001).


 * Perspectives in Mathematical Physics, International Press, edited by R.C. Penner and Shing-Tung Yau (1994).


 * Discrete Mathematics--proof techniques and mathematical structures, World Scientific Publishing Company (1999); second printing (2001).


 * Woods Hole Mathematics: perspectives in math and physics, edited by N. Tongring and R.C. Penner, foreword by Raul Bott, 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 52 (2008), Mathematical Society of Japan, edited by R.C. Penner, D. Kotschick, T. Tsuboi, N. Kawazumi, T. Kitano, Y. Mitsumatsu.


 * Decorated Teichmüller theory, (with a foreword by Yuri I. Manin), QGM Master Class Series, European Mathematical Society, Zürich, 2012, xviii+360 pp. ISBN: 978-3-03719-075-3.

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).

Philanthropy
In 2018 Robert C. Penner endowed the Alexzandria Figueroa and Robert Penner Chair at the IHES in memoriam of Alexzandria Figueroa.