Differential Geometry, Billiards, and Geometric Optics
Géométrie différentielle, billards et optique géométrique
4 – 8 October 2021
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Scientific Committee
Comité scientifique Vered Rom-Kedar (Weizmann Institute) |
Organizing Committee
Comité d’organisation Alexey Glutsyuk (CNRS – ENS-Lyon) |
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Mathematical theory of billiards has close relations to mechanics of particles, Riemannian geometry, geometric analysis, Hamiltonian mechanics, and symplectic geometry. Theoretical results on billiards have a potential for real-life applications in geometric optics, such as free form optical design and manufacturing of ideal lenses.
There are several longstanding open problems in the field, such as Birkhoff’s conjecture, Ivrii’s conjecture, their analogs for geodesic flows, existence of periodic billiard trajectories in polygonal billiards, characterization of symplectomorphisms on the space of lines which can be realized as composition of billiard reflections,etc. Recently, a relation was found between billiards, Mahler’s conjecture in convex geometry, and Viterbo’s conjecture in symplectic geometry. There has been a lot of research on billiards in the last years, and some sub-stantial progress was made towards the solution of the above mentioned problems. Also, various generalizations of the classical billiard reflection law were introduced. During the conference, the recent developments in the theory of billiards will be presented. We expect many fruitful interactions on the “cross-roads” between different areas of geometry, mechanics, and optics. |
La théorie mathématique des billards est fortement liée à la mécanique des particules, à la géométrie Riemannienne, à l’analyse géométrique, à la mécanique Hamiltonienne et à la géométrie symplectique. Les résultats théoriques sur les billards auraient potentiellement des applications dans l’optique géométrique : notamment, dans le design optique de forme libre et dans la production des lentilles idéales.
Il y a quelques vieux problèmes ouverts dans le domaine des billards, comme par exemple la Conjecture de Birkhoff, la Conjecture d’Ivrii, leurs analogues pour les flots géodésiques, la question d’existence des orbites périodiques dans les billards polygonaux, la caractérisation des symplectomorphismes de l’espace de droites réalisables comme compositions des réflections de billard, etc. Une relation a été récemment trouvée entre les billards, la Conjecture de Mahler dans la géométrie convexe et la Conjecture de Viterbo dans la géométrie symplectique. Au cours des dernières années il y a eu beaucoup de recherches sur les billards et un certain progrès a été atteint vers les solutions des problèmes mentionnés ci-dessus. En plus, des généralisations diverses de la loi classique de réflection de billard ont été introduites. Pendant la conférence, on présentera les développements récents de la théorie des billards. Nous espérons qu’il y aura de nombreuses intéractions fructueuses au croisement des domaines différents de la géométrie, de la mécanique et de l’optique. |
Arseniy Akopyan (IITP, RAS, Moscow) The beauty of random polytopes inscribed in the 2-sphere
Peter Albers (University of Heidelberg) A symplectic dynamics proof of the degree-genus formula
Michael Berry (University of Bristol) Geometric optics of four illusions
Michael Bialy (Tel Aviv University) Birkhoff-Poritsky conjecture for centrally-symmetric billiards
Sergei Bolotin (Steklov Institute of Mathematics, Moscow) Degenerate billiards
Gil Bor (Centro de Investigación en Matemáticas, Guanajuato) Geometry and new symmetries of the Kepler problem
Leonid Bunimovich (Georgia Institute of Technology) Elliptic flowers
Andrew Clarke (Polytechnic University of Catalonia) Arnold diffusion in multi-dimensional convex billiards
Diana Davis (Swarthmore College) Billiards on regular polygons
Vladimir Dragovic (University of Texas at Dallas) Integrable billiards, Chebyshev dynamics, and isoharmonic and isomonodromic deformations
Vadim Kaloshin (University of Maryland) Deformational spectral rigidity of analytic Bunimovich stadia and Bunimovich squashes
Mark Levi (Pennsylvania State University) Spectral asymptotics and Lamé spectrum in periodic potentials
Vladimir Matveev (Friedrich Schiller University Jena) Superintegrable metrics on surfaces
Marco Mazzucchelli (CNRS, ENS-Lyon) C^2 structurally stable Riemannian geodesic flows of closed surfaces are Anosov
Richard Montgomery (University of California Santa Cruz) Billiards arising in N-body problems
Ron Perline (Drexel University) Free-form mirror design and the prescribed mean curvature problem
Alexander Plakhov (University of Aveiro) Newton’s problem of minimal resistance for convex bodies
Olga Paris-Romaskevich (CNRS, Aix-Marseille Université) Novikov’s problem and tiling billiards
Alfonso Sorrentino (University of Rome Tor Vergata) On the persistence of completely periodic tori for symplectic twist maps
Dmitry Treschev (Moscow State University) On isochronous dynamics
Alexander Veselov (Loughborough University) Geodesic scattering on hyperboloids
Vadim Zharnitsky (University of Illinois at Urbana-Champaign) Whispering gallery orbits in Sinai oscillator trap
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