Richard Evan Schwartz

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Richard Evan Schwartz (born August 11, 1966 in Los Angeles ) is an American mathematician who studies geometric group theory, geometry and dynamic systems of the billiards type.

Life

Schwartz studied at the University of California, Los Angeles (bachelor's degree 1987) and received his doctorate from Princeton University under William Thurston in 1991 ( The limit sets of some infinitely generated Schottky groups ). He has taught at the University of Maryland and is a professor at Brown University .

In 1992 he introduced the pentagram map, a mapping of closed convex polygons (on the polygon that is formed by the intersections of the shortest diagonals) in the real projective plane, which can be viewed as a discrete dynamic system. It can even be precisely integrated

In 1989 he proved a conjecture by Goldman and John Parker that provides a complete description of the modular space of the complex hyperbolic ideal triangle group .

In 2007 he proved the existence of unlimited orbits from outer billards (a dynamic system introduced by Bernhard Neumann in the 1950s as a toy model for celestial mechanics).

He wrote a children's book on math, originally based on comics he drew for his daughter.

In 2002 he was invited speaker at the International Congress of Mathematicians in Beijing ( Complex hyperbolic triangle groups ). He is a fellow of the American Mathematical Society . From 1996 he was a research fellow of the Alfred P. Sloan Foundation ( Sloan Research Fellow ).

Fonts

  • Spherical CR Geometry and Dehn Surgery, Annals of Mathematics Studies 165, Princeton University Press 2007
  • Outer Billiards on Kites, Annals of Mathematics Studies, 171, Princeton University Press 2009
  • You Can Count on Monsters AK Peters Ltd., 2010 (mathematical children's book)
  • Really Big Numbers, American Math Society, 2014 (children's mathematical book)
  • Mostly Surfaces, American Math Society, 2011 unformatted pdf
  • Elementary surprises in projective geometry, Mathematical Intelligencer 2010
  • Pappus' theorem and the modular group. Inst. Hautes Études Sci. Publ. Math. No. 78: 187-206 (1994) (1993).
  • The quasi-isometry classification of rank one lattices. Inst. Hautes Études Sci. Publ. Math. No. 82: 133-168 (1996) (1995).
  • Quasi-isometric rigidity and Diophantine approximation. Acta Math. 177 (1996) no. 1, 75-112.
  • with Benson Farb : The large-scale geometry of Hilbert modular groups. J. Differential Geom. 44 (1996) no. 3, 435-478.
  • Symmetric patterns of geodesics and automorphisms of surface groups. Invent. Math. 128 (1997) no. 1, 177-199.
  • Degenerating the complex hyperbolic ideal triangle groups. Acta Math. 186 (2001), no. 1, 105-154.
  • Ideal triangle groups, dented tori, and numerical analysis. Ann. of Math. (2) 153 (2001) no. 3, 533-598.
  • Complex hyperbolic triangle groups. Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 339-349, Higher Ed. Press, Beijing, 2002.
  • Unbounded orbits for outer billiards. IJ Mod. Dyn. 1 (2007), no. 3, 371-424.
  • with Valentin Ovsienko , Serge Tabachnikov : The pentagram map: A discrete integrable system. Comm. Math. Phys. 299 (2010), no. 2, 409-446.
  • with Valentin Ovsienko, Serge Tabachnikov: Liouville-Arnold integrability of the pentagram map on closed polygons. Duke Math. J. 162 (2013), no. 12, 2149-2196.

literature

  • Marcel Berger : Dynamiser la géométrie élémentaire: introduction à des travaux de Richard Schwartz. Rend. Mat. Appl. (7) 25 (2005), no. 2, 127-153. pdf

Web links

Individual evidence

  1. ^ Mathematics Genealogy Project
  2. ↑ Schwartz The Pentagram Map , Journal of Experimental Mathematics, Volume 1, 1992, pp. 71-81
  3. ↑ Schwartz The pentagram map is recurrent , J. of Exp. Math., Volume 10, 2001, pp. 519-528
  4. ↑ Schwartz Discrete monodromy, pentagrams and the method of condensation , J. Fixed Point Theory and Applications, Volume 3, 2008, pp. 379-409
  5. ↑ Ovsienko, Serge Tabachnikov , Schwartz The pentagram map: a discrete integrable system , Communications in Mathematical Physics, Volume 299, 2010, pp. 409-446, pdf
  6. ↑ Generated by reflections on the sides of an ideal triangle in the hyperbolic plane
  7. ↑ Schwartz Ideal triangle groups, dented tori and numerical analysis , Annals of Mathematics, Volume 153, 2001, pp. 554-598
  8. ↑ Outer billiards map a point P outside a convex restricted area S to a point Q, so that the tangent from P to S divides the segment PQ. For his proof of the existence of unlimited orbits, Schwartz took a Penrose kite for S, among other things.
  9. ↑ Schwartz Unbounded orbits for outer billards , Journal of Modern Dynamics, Volume 3, 2007
personal data
SURNAME Schwartz, Richard Evan
BRIEF DESCRIPTION American mathematician
DATE OF BIRTH August 11, 1966
PLACE OF BIRTH los Angeles