Ralph Martin Kaufmann (born August 4, 1969) is a German-American mathematician working in the United States.

Career Kaufmann studied mathematics, physics and philosophy at the University of Bonn. He obtained a master's degree in Physics in 1994 under the supervision of Werner Nahm and a master's degree in philosophy under the supervision of Rainer Stuhlmann-Laeisz in 1996. His doctoral studies were carried out at the Max Planck Institute for Mathematics under the supervision of Yuri Manin, and he graduated summa cum laude from University of Bonn in 1997 with a thesis entitled "The geometry of the moduli space of pointed curves, the tensor product in the theory of Frobenius manifolds and the explicit Künneth formula in quantum cohomology".Bonner Mathematische Schriften 312, 95 p., Bonn 1997.

He remained at the Max Planck Institute for one year after his graduation as a researcher before moving to the Institut des Hautes Études Scientifiques for a year with a Marie Curie Fellowship from the European Union.

In 1999 Kaufmann moved to the United States, where he has held several positions. He arrived at his current institution, Purdue University, in 2007 as an associate professor, being promoted to full professor in 2012.

Kaufmann has been a member of the Institute for Advanced Study in Princeton, and has held visiting positions at the Max Planck Institute for Mathematics in Bonn, the Institut des Hautes Études Scientifiques, the Mittag-Leffler Institute in Stockholm, the Isaac Newton Institute in Cambridge, the Mathematical Sciences Research Institute in Berkeley, California and the Hamburg Institute for Advanced Study.hias-hamburg.de

Research

Kaufmann's research has covered different areas of mathematics and theoretical physics. After briefly working on the Virasoro algebra,Kaufmann, Ralph M. "Path Space Decompositions for the Virasoro Algebra and its Verma Modules". Internat. J. of Modern Phys. A 10 (1995), 943-961. he started to work on the quantum cohomology of a productMaxim Kontsevich and Yuri Manin with appendix by R. Kaufmann. "Quantum cohomology of a product". Invent. Math. 124 (1996), 313-339. and the Künneth formula, obtaining results on explicit formulasKaufmann, Ralph M. "The intersection form in H*(M0,n) and the explicit Künneth formula in quantum cohomology". Internat. Math. Res. Notices 19 (1996), 929-954. and global results.Kaufmann, Ralph M. "The tensor product in the theory of Frobenius manifolds". Internat. J. of Math. 10 (1999), 159-206 In collaboration with Yuri Manin and Don Zagier he started the study of higher Weil–Peterson volumes,Kaufmann, R.; Manin, Yu.; and Zagier, D. "Higher Weil-Petersson Volumes of Moduli Spaces of Stable n-pointed Curves". Comm. Math. Phys. 181 (1996), 763-787. later continued by Losef–Manin and Maryam Mirzakhani. Kaufmann then started to study stringy and mirror phenomena for orbifoldsKaufmann, Ralph M. "Orbifolding Frobenius Algebras". Internat. J. of Math. 14 (2003), 573-619 and singularities.Kaufmann, Ralph M. "Singularities with Symmetries, Orbifold Frobenius algebras and Mirror Symmetry" Contemp. Math., 403 (2006), 67-116 This led to the invention of stringy K-theory.Jarvis, T.; Kaufmann, R. and Kimura, T. "Stringy K-theory and the Chern character". Inventiones Math. 168, 1 (2007), 23-81Kaufmann, Ralph M. "Global Stringy Orbifold Cohomology, K-Theory and de Rham Theory". Letters in Mathematical Physics, 94, 2 (2010) 165-195.

Kaufmann has also worked on string topology, invented by Moira Chas and Dennis Sullivan, and operad theory.Kaufmann, Ralph M.; Livernet, Muriel and Penner, Robert C. "Arc Operads and Arc Algebras". Geometry and Topology 7 (2003), 511-568.Kaufmann, Ralph M. "On Spineless Cacti, Deligne's Conjecture and Connes-Kreimer's Hopf Algebra." Topology 46, 1 (2007), 39-88.Kaufmann, Ralph M. "Operads, Moduli of Surfaces and Quantum Algebras", in: N. Tongring and R. C. Penner "Woods Hole Mathematics. Perspectives in Mathematics and Physics", Series on Knots and Everything - Vol. 34, World Scientific 2004. Here he first proved a cyclic version of Pierre Deligne's conjecture in deformation theoryKaufmann, Ralph M. "A proof of a cyclic version of Deligne's conjecture via Cacti" Math. Res. Letters 15, 5 (2008), 901--921. and provided an extension of string topology to moduli spaces.Kaufmann, Ralph M. "Moduli space actions on the Hochschild cochain complex I: cell models". Journal of Noncommutative Geometry 1, 3 (2007) 333-384., Kaufmann, Ralph M. "Moduli space actions on the Hochschild cochain complex II: correlators". Journal of Noncommutative Geometry 2, 3 (2008), 283-332.Kaufmann, Ralph M. "Open/Closed String Topology and Moduli Space Actions via Open/Closed Hochschild Actions". SIGMA 6 (2010) 036, 33 pages. Kaufmann, Ralph M."Graphs, strings and actions" in: Algebra, Arithmetic and Geometry Volume II: In Honor of Yu. I. Manin. Progress in Mathematics 270, 127--178. Birhauser, Boston (2010).Kaufmann, Ralph M. "Arc Geometry and Algebra: Foliations, Moduli Spaces, String Topology and Field Theory". Chapter in Handbook of Teichmüller Theory IV. Ed: Athanase Papadopoulos. European Mathematical Society 2014.

More recently Kaufmann has introduced the notion of Feynman categoriesKaufmann, Ralph M. and Ward, Benjamin C. "Feynman categories". Astérisque 387 (2017), x+161 pages. to give a common framework for various aspects of algebra, geometry, topology, and category theory.Kaufmann, Ralph M., Ward, Benjamin C. and Zuniga, J. Javier "The odd origin of Gerstenhaber brackets, Batalin-Vilkovisky operators and the master equations". Journal of Math. Phys. 56 (2015), 103504.Kaufmann, Ralph M., Lucas, Jason. "Decorated Feynman categories". arXiv:1602.00823 J. of Noncommutative Geometry to appear.Berger, Clemens and Kaufmann, Ralph M. "Comprehensive Factorization Systems". arXiv:1710.09438. Special Issue of Tbilisi Mathematical Journal in honor of Peter J. Freyd and F.W. Lawvere on the occasion of their 80th birthdays to appear

In mathematical physics, he has also studied the geometry of wire networksKaufmann, Ralph M., Khlebnikov, Sergei and Wehefritz-Kaufmann, Birgit. "The geometry of the Double Gyroid wire network: Quantum and Classical". Journal of Noncommutative Geometry, 6 (2012), 623-664. as well as periodic systemsKaufmann, Ralph M., Khlebnikov, Sergei and Wehefritz-Kaufmann, Birgit. "Singularities, swallowtails and Dirac points. An analysis for families of Hamiltonians and applications to wire networks, especially the Gyroid". Annals of Physics, 327 (2012), 2865-2884.Kaufmann, Ralph M., Khlebnikov, Sergei and Wehefritz-Kaufmann, Birgit. "Re-gauging groupoid, symmetries and degeneracies for Graph Hamiltonians and applications to the Gyroid wire network". Ann. H. Poincaré 7, 6 (2016), 1383-1414. and topological insulators.Kaufmann, Ralph M., Li, Dan, and Wehefritz-Kaufmann, Birgit. "Notes on topological insulators". Reviews in Math. Physics, Vol. 28, No. 10 (2016) 1630003

In philosophy, he has recently worked on Hegel's theory of mathematicsKaufmann, Ralph M. and Yeomans, Christopher L. "Math by Pure Thinking: R First and the Divergence Of Measures in Hegel's Philosophy of Mathematics". European Journal of Philosophy, ahead of print.Kaufmann, Ralph M. and Yeomans, Christopher L. "Hegel on Calculus". History of Philosophy Quarterly, to appear. and on Friedrich Hölderlin.Kaufmann, Ralph M. Der Dichter spricht: Eine Rezeption Hölderlins 'Verfahrensweise des Poetischen Geistes'", Zeitschrift für Kulturphilosophie, 2017/1 "Sprache und Gestalt", Meiner Verlag, Hamburg.

Honors and awards Heinrich Hörlein Gedächtnis Preis for his dissertation. Simons Fellow

Publications (books authored)

References

External links Homepage at Purdue Papers on the eprints ArXiv