Richard Schwartz | Academic Influence

Richard Schwartz 's AcademicInfluence.com Rankings

Richard Schwartz

Mathematics

#1948

World Rank

#3088

Historical Rank

#783

USA Rank

Measure Theory

#2991

World Rank

#3551

Historical Rank

#869

USA Rank

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Mathematics

Richard Schwartz 's Degrees

PhD Mathematics University of California, Berkeley
Bachelors Mathematics University of California, Berkeley

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Why Is Richard Schwartz Influential?

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According to Wikipedia, Richard Evan Schwartz is an American mathematician notable for his contributions to geometric group theory and to an area of mathematics known as billiards. Geometric group theory is a relatively new area of mathematics beginning around the late 1980s which explores finitely generated groups, and seeks connections between their algebraic properties and the geometric spaces on which these groups act. He has worked on what mathematicians refer to as billiards, which are dynamical systems based on a convex shape in a plane. He has explored geometric iterations involving polygons, and he has been credited for developing the mathematical concept known as the pentagram map. In addition, he is a bestselling author of a mathematics picture book for young children. His published work usually appears under the name Richard Evan Schwartz. In 2018 he is a professor of mathematics at Brown University.

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Richard Schwartz 's Published Works

Number of citations in a given year to any of this author's works

Total number of citations to an author for the works they published in a given year. This highlights publication of the most important work(s) by the author

Published Works

The quasi-isometry classification of rank one lattices (1995) (145)
The Pentagram Map: A Discrete Integrable System (2008) (123)
The Pentagram Map (1992) (95)
Complex hyperbolic triangle groups (2003) (87)
The large-scale geometry of Hilbert modular groups (1996) (78)
Liouville–Arnold integrability of the pentagram map on closed polygons (2011) (61)
The Five-Electron Case of Thomson’s Problem (2013) (58)
The Poncelet grid (2007) (57)
Spherical CR Geometry and Dehn Surgery (2007) (57)
Discrete monodromy, pentagrams, and the method of condensation (2007) (57)
Quasi-isometric rigidity and diophantine approximation (1996) (54)
Ideal triangle groups, dented tori, and numerical analysis (2001) (53)
Outer Billiards on Kites (2007) (52)
Unbounded Orbits for Outer Billiards (2007) (47)
Billiards in ellipses revisited (2020) (46)
Real hyperbolic on the outside, complex hyperbolic on the inside (2003) (42)
Degenerating the complex hyperbolic ideal triangle groups (2001) (41)
Obtuse Triangular Billiards II: One Hundred Degrees Worth of Periodic Trajectories (2009) (38)
Obtuse Triangular Billiards I: Near the (2, 3, 6) Triangle (2006) (33)
The Pentagram Map is Recurrent (2001) (33)
Symmetric patterns of geodesics and automorphisms of surface groups (1997) (30)
Elementary Surprises in Projective Geometry (2009) (29)
A better proof of the Goldman-Parker conjecture (2005) (26)
Pappus’s theorem and the modular group (1993) (22)
The 5 Electron Case of Thomson's Problem (2010) (22)
The Octagonal PETs (2014) (21)
Billiards in nearly isosceles triangles (2008) (20)
A trichotomy for rectangles inscribed in Jordan loops (2018) (20)
Spherical CR Geometry and Dehn Surgery (AM-165) (2007) (19)
Outer Billiards on Kites (AM-171) (2009) (18)
The correlation of length spectra of two hyperbolic surfaces (1993) (17)
The Pentagram Integrals on Inscribed Polygons (2010) (13)
The Pentagram Integrals for Poncelet Families (2015) (13)
Quasiperiodic Motion for the Pentagram Map (2009) (13)
The Projective Heat Map (2017) (13)
Minimax CFAR detection in additive Gaussian noise of unknown covariance (Corresp.) (1969) (13)
Notes on Shapes of Polyhedra (2015) (12)
Outer Billiards, the Arithmetic Graph, and the Octagon (2010) (12)
Outer Billiards, Arithmetic Graphs, and the Octagon (2010) (10)
Four Lines and a Rectangle (2020) (10)
Outer billiards on the Penrose kite: Compactification and renormalization (2011) (9)
Pentagram Spirals (2013) (9)
A projectively natural flow for circle diffeomorphisms (1992) (7)
Really Big Numbers (2014) (7)
Desargues Theorem, Dynamics, and Hyperplane Arrangements (2001) (7)
Outer Billiards and the Pinwheel Map (2010) (7)
Five Point Energy Minimization: A Synopsis (2020) (6)
The Phase Transition in 5 Point Energy Minimization (2016) (5)
Using the Model (2019) (5)
Lengthening a tetrahedron (2015) (5)
The Triangular Bi-Pyramid Minimizes a Range of Power Law Potentials (2015) (5)
Rectangle Coincidences and Sweepouts (2018) (5)
The Density of Shapes in Three-Dimensional Barycentric Subdivision (2003) (4)
Outer Billiards , Quarter Turn Compositions , and Polytope Exchange Transformations (2016) (4)
The spheres of Sol (2019) (4)
Research Announcement: Unbounded Orbits for Outer Billiards (2007) (3)
Polygonal symplectic billiards (2019) (3)
Symmetry Groups of Platonic Solids (2007) (3)
A Conformal Averaging Process on the Circle (2006) (3)
On Area Growth in Sol (2020) (3)
Pan Galactic Division (2015) (3)
Descartes Circle Theorem, Steiner Porism, and Spherical Designs (2018) (3)
Lucy and Lily: A Game of Geometry and Number Theory (2002) (2)
On Spaces of Inscribed Triangles (2019) (2)
The limit sets of some infinitely generated Schottky groups (1991) (2)
Circle quotients and string art (2002) (2)
The Larger Self: Discovering the core within our multiplicity (2004) (2)
Inscribed rectangle coincidences (2021) (2)
Technical Note - An Integer Maximization Problem (1971) (2)
Pentagram Rigidity for Centrally Symmetric Octagons (2021) (2)
Quarter turn compositions (2014) (2)
Affine Subdivision, Steerable Semigroups, and Sphere Coverings (2007) (2)
A Textbook Case of Pentagram Rigidity (2021) (2)
An improved bound on the optimal paper Moebius band (2020) (2)
The Smart Mine Simulator User's Guide and Algorithm Description (1993) (1)
Countable and Uncountable Sets (2007) (1)
Square turning maps and their compactifications (2013) (1)
The Farthest Point Map on the Regular Octahedron (2020) (1)
Unbounded orbits for the Plaid Model (2015) (1)
Étale cohomology for non-Archimedean analytic spaces . Rigidité différentiable des groupes fuchsiens . Pappus's theorem and the modular group . The explosion of singular cycles (1993) (1)
Rectangles, curves, and Klein bottles (2021) (1)
A Hyperbolic View of the Seven Circles Theorem (2019) (1)
Notes on Tensor Products (2014) (1)
A Very Short Proof that Every Right Triangle has a Periodic Billiard Path (2004) (1)
Mostow Rigidity (2013) (1)
Five Point Energy Minimization : A Summary (2018) (1)
Definition of a surface (2011) (1)
You Can Count on Monsters: The First 100 Numbers and Their Characters (2015) (1)
The Orbital Period of the ROSAT Cataclysmic Variable S10932 Comae Berenices (1995) (1)
Chapter 4. Period Copying (2009) (0)
Dehn’s dissection theorem (2011) (0)
Conway’s Nightmare: Brahmagupta and Butterflies (2022) (0)
Properties of the Model (2019) (0)
The Cantor set case (2014) (0)
The plaid model and outer billiards on kites (2015) (0)
SIMNET-Based Tests of Antihelicopter Mines (1994) (0)
The Nature of the Compactification (2019) (0)
Introduction (2019) (0)
NSTITUTE FOR DEFENSE ANALYSES IDA Document D-2102 Symmetric Analysis (1998) (0)
Notes on Expanders and Property T (2020) (0)
Chapter 19. The Decomposition Theorem (2009) (0)
Covering spaces and the deck group (2011) (0)
On Projective Evolutes of Polygons (2022) (0)
Chapter 18. The Diophantine Lemma (2009) (0)
How do we find the primes less than 100 (2010) (0)
Three-Dimensional Interpretation (2019) (0)
Gallery of the Infinite (2016) (0)
Geometry of the Graph Grid (2019) (0)
An octahedral tiling on the ideal boundary of the complex hyperbolic plane (2006) (0)
Erratum: Billiards in nearly isosceles triangles (2014) (0)
The raw data (2014) (0)
The Dehn-Sydler Theorem Explained (2010) (0)
Divide and Conquer: A Distributed Approach to Five Point Energy Minimization (2023) (0)
Graph Master Picture Theorem (2019) (0)
The Octagonal PET II: The Topology of the Limit Sets (2012) (0)
Flat cone surfaces (2011) (0)
Chapter 14. The Barrier Theorem (2009) (0)
Chapter 8. The Torus Lemma (2009) (0)
Continued Fractions and the 4-Color Theorem (2022) (0)
Preface (2006) (0)
Chapter 3. The Hexagrid Theorem (2009) (0)
The Horizontal Lemma (2019) (0)
Outer billiards on semiregular octagons (2014) (0)
Chapter 31. Proof of the Low Vertex Theorem (2009) (0)
DISCRETE ISOPERIMETRY AND PROTEIN FOLDING (2019) (0)
Pushing a Rectangle down a Path (2018) (0)
Chapter 6. The Master Picture Theorem (2009) (0)
Particles and Spacetime Diagrams (2019) (0)
Kastelyn ’ s Formula for Perfect Matchings Rich Schwartz April 8 , 2013 1 Pfaffian Orientations on Planar Graphs (2013) (0)
Chapter 15. Proof of Hexagrid Theorem II (2009) (0)
Chapter 30. Hovering Components (2009) (0)
Riemann surfaces and uniformization (2011) (0)
Chapter 13. Proof of Hexagrid Theorem I (2009) (0)
The Orbit Equivalence Theorem (2019) (0)
The Core Case (2019) (0)
The Segment Lemma (2019) (0)
The Affine Shape of a Figure-Eight under the Curve Shortening Flow (2021) (0)
The Illustrating Mathematics Program at ICERM (2020) (0)
Chapter 25. Geometric Consequences (2009) (0)
Chapter 23. The Comet Theorem (2009) (0)
The Plaid Model (2019) (0)
Orbit stability and combinatorics (2014) (0)
The Vertical Lemma (2019) (0)
Topology of Teichmüller space (2011) (0)
Mathematical Thinking and Doing (2016) (0)
Outer Billiards , Arithmetic Graphs and Multigrid Flows (2012) (0)
Proof of the main theorem (2014) (0)
Further symmetries of the tiling (2014) (0)
Quarter Turn Compositions and PETs (2019) (0)
Examples of fundamental groups (2011) (0)
Part 1. Basic Material (2007) (0)
The change of variables formula (2015) (0)
A Market Survey for the ST-4 CCD Star Tracker and Imaging Camera (1989) (0)
Controlling the limit set (2014) (0)
Chapter 16. Proof of the Intersection Lemma (2009) (0)
The alternating grid system (2014) (0)
Bulletin ( New Series ) of the American Mathematical Society (2001) (0)
The Intertwining Lemma (2019) (0)
Rectangle Sweepouts and Coincidences (2018) (0)
N ov 2 01 9 The Spheres of Sol (2019) (0)
Chapter 7. The Pinwheel Lemma (2009) (0)
Dynamics in the arc case (2014) (0)
The Optimal Paper Moebius Band and Finite Tensegrity Calculations. (2020) (0)
Chapter 20. Existence of Strong Sequences (2009) (0)
Chapter 5. Proof of the Erratic Orbits Theorem (2009) (0)
The Illustrating Mathematics Program at ICERM (2020) (0)
RECTANGLES, CURVES, AND (2021) (0)
The renormalization map (2014) (0)
Disk and plane rigidity (2011) (0)
Chapter 12. Extension and Symmetry (2009) (0)
The Schwarz-Christoffel transformation (2011) (0)
Links inside StraightEdge Embeddings of Complete Bipartite Graphs (2011) (0)
Further geometric results (2014) (0)
A Surface Group Can Act Properly on a Product of two Regular Trees (2019) (0)
Infinite Orbits Revisited (2019) (0)
Survey Lecture on Billiards (2021) (0)
The Quasi-Isomorphism Theorem (2019) (0)
Trisecting the 9-vertex complex projective plane (2022) (0)
Part 3. The Applications (2007) (0)
The fundamental group (2011) (0)
Why do the primes go on forever (2010) (0)
Pixellation and Curve Turning (2019) (0)
Definition of the Plaid Model (2019) (0)
The gluing construction (2011) (0)
The Plaid Master Picture Theorem (2019) (0)
Lengthening a tetrahedron (2014) (0)
Chapter 17. Diophantine Approximation (2009) (0)
Existence of Many Large Orbits (2019) (0)
Links inside Straight-Edge Embeddings of Complete Bipartite Graphs (2011) (0)
Chapter 29. Proof of the Period Theorem (2009) (0)
Five Point Energy Minimization: A Synopsis (2020) (0)
Chapter 21. Structure of the Inferior and Superior Sequences (2009) (0)
Chapter 11. Proof of the Embedding Theorem (2009) (0)
Riemannian metrics on surfaces (2011) (0)
Translation surfaces and the Veech group (2011) (0)
A Hyperbolic View of the Seven Circles Theorem (2019) (0)
Existence of Infinite Orbits (2019) (0)
The Banach-Tarski Theorem (2010) (0)
Part 4. Structure of Ideal Triangle Groups (2007) (0)
Part 2. Proof of the HST (2007) (0)
Chapter 9. The Strip Functions (2009) (0)
Coincident optical monitoring of GRB error boxes. (1992) (0)
Modular circle quotients and PL limit sets (2004) (0)
The Farthest Point Map on the Regular Dodecahedron (2021) (0)
Some Elementary Number Theory (2019) (0)
Notes on Shapes of Polyhedra Rich (2015) (0)
The Cauchy rigidity theorem (2011) (0)
The arc case (2014) (0)
The Weak and Strong Case (2019) (0)
Addendum to “A Hyperbolic View of the Seven Circles Theorem” (2023) (0)
Square turning maps and their compactifications (2017) (0)
The covering lemma (2014) (0)
The Volume Entropy of Sol (2020) (0)
Connection to the Truchet Tile System (2019) (0)
Teichmüller space and moduli space (2011) (0)
Properties of the limit set (2014) (0)
Pushing a Rectangle down a Path (2018) (0)
Proof of the Main Result (2019) (0)
Introducing the Plaid Model (2015) (0)
The filling lemma (2014) (0)
Chapter 2. The Arithmetic Graph (2009) (0)
Collaborators Pt. 2 (2014) (0)
A trichotomy for rectangles inscribed in Jordan loops (2020) (0)
The Hodge Star Operator Rich Schwartz April 22 , 2015 1 Basic Definitions We ’ ll start out by defining the (2015) (0)
Collaborators Pt. 1 (2014) (0)
Chapter 26. Proof of the Copy Theorem (2009) (0)
Existence of universal covers (2011) (0)
The forest case (2014) (0)
Chapter 10. Proof of the Master Picture Theorem (2009) (0)
Properties of the tiling (2014) (0)
Notes on NP Completeness Rich Schwartz November 10 , 2013 1 Overview (2013) (0)
Notes on Antoine’s Necklace (2022) (0)
Chapter 22. The Fundamental Orbit (2009) (0)
A primer on complex analysis (2011) (0)
Chapter 27. Pivot Arcs in the Even Case (2009) (0)
2 Single Variable Case 2 . 1 Positive Dominance (2014) (0)
Unit Lengthenings of Tetrahedra (2014) (0)
The Octagonal PET I: Renormalization and Hyperbolic Symmetry (2012) (0)
Chapter 28. Proof of the Pivot Theorem (2009) (0)
Paper Moebius bands with T Patterns (2020) (0)
Trisecting the Nine-Vertex Complex Projective Plane (2022) (0)
PATTERNS , AND THE COMBINATORIAL GALE TRANSFORM (2014) (0)
Chapter 24. Dynamical Consequences (2009) (0)
Hausdorff dimension bounds (2014) (0)
Pinwheels and Quarter Turns (2019) (0)

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