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Documents  Zorich, Anton | enregistrements trouvés : 14

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Research talks;Dynamical Systems and Ordinary Differential Equations

We consider "higher dimensional Teichmüller discs", by which we mean complex submanifolds of Teichmüller space that contain the Teichmüller disc joining any two of its points. We prove results in the higher dimensional setting that are opposite to the one dimensional behavior: every "higher dimensional Teichmüller disc" covers a "higher dimensional Teichmüller curve" and there are only finitely many "higher dimensional Teichmüller curves" in each moduli space. The proofs use recent results in Teichmüller dynamics, especially joint work with Eskin and Filip on the Kontsevich-Zorich cocycle. Joint work with McMullen and Mukamel as well as Eskin, McMullen and Mukamel shows that exotic examples of "higher dimensional Teichmüller discs" do exist. We consider "higher dimensional Teichmüller discs", by which we mean complex submanifolds of Teichmüller space that contain the Teichmüller disc joining any two of its points. We prove results in the higher dimensional setting that are opposite to the one dimensional behavior: every "higher dimensional Teichmüller disc" covers a "higher dimensional Teichmüller curve" and there are only finitely many "higher dimensional Teichmüller curves" in ...

30F60 ; 32G15

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Outreach;Mathematics Education and Popularization of Mathematics

Jean-Christophe Yoccoz, né le 29 mai 1957 à Paris, est un mathématicien français, lauréat de la médaille Fields en 1994, professeur au Collège de France depuis 1996. Il est notamment connu pour ses travaux sur les systèmes dynamiques.

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Outreach;Mathematics Education and Popularization of Mathematics

Curtis Tracy McMullen (born 21 May 1958) is Professor of Mathematics at Harvard University. He was awarded the Fields Medal in 1998 for his work in complex dynamics, hyperbolic geometry and Teichmüller theory. McMullen graduated as valedictorian in 1980 from Williams College and obtained his Ph.D. in 1985 from Harvard University, supervised by Dennis Sullivan. He held post-doctoral positions at the Massachusetts Institute of Technology, the Mathematical Sciences Research Institute, and the Institute for Advanced Study, after which he was on the faculty at Princeton University (1987-1990) and the University of California, Berkeley (1990-1997), before joining Harvard in 1997. He received the Salem Prize in 1991 and was elected to the National Academy of Sciences in 2007. In 2012 he became a fellow of the American Mathematical Society. Curtis Tracy McMullen (born 21 May 1958) is Professor of Mathematics at Harvard University. He was awarded the Fields Medal in 1998 for his work in complex dynamics, hyperbolic geometry and Teichmüller theory. McMullen graduated as valedictorian in 1980 from Williams College and obtained his Ph.D. in 1985 from Harvard University, supervised by Dennis Sullivan. He held post-doctoral positions at the Massachusetts Institute of Technology, the ...

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Research talks;Dynamical Systems and Ordinary Differential Equations;Algebraic and Complex Geometry;Number Theory

We study cascades of bifurcations in a simple family of maps on the circle, and connect this behavior to the geometry of an absolute period leaf in genus $2$. The presentation includes pictures of an exotic foliation of the upper half plane, computed with the aid of the Möller-Zagier formula.

30F10 ; 30F30

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Research talks;Dynamical Systems and Ordinary Differential Equations

The behaviour of infinite translation surfaces is, in many regards, very different from the finite case. For example, the geodesic flow is often not recurrent or is not even defined for infinite time in a generic direction.
However, we show that if one focuses on a class of infinite translation surfaces that exclude the obvious counter-examples, one can adapted the proof of Kerckhoff, Masur, and Smillie and show that the geodesic flow is uniquely ergodic in almost every direction. We call this class of surface essentially finite.
(joint work with Anja Randecker).
The behaviour of infinite translation surfaces is, in many regards, very different from the finite case. For example, the geodesic flow is often not recurrent or is not even defined for infinite time in a generic direction.
However, we show that if one focuses on a class of infinite translation surfaces that exclude the obvious counter-examples, one can adapted the proof of Kerckhoff, Masur, and Smillie and show that the geodesic flow is ...

37D40 ; 51A40 ; 37A25

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Resarch talks;Dynamical Systems and Ordinary Differential Equations

We give an algebraic proof of the simplicity of the Lyapunov spectrum for the Teichmüller flow on strata of abelian differentials. This proof extends to the Kontsevich Zorich cocycle over strata of quadratic differentials and can also be used to study the algebraic degree of pseudo-Anosov stretch factors.

37F30 ; 37D35

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Research talks;Dynamical Systems and Ordinary Differential Equations;Algebraic and Complex Geometry

We will present a geometric criterion for the ergodicity of the billiard flow in a polygon with non-rational angles and discuss its application to the Diophantine case.

37D40 ; 37D50 ; 30F10 ; 30F60 ; 32G15

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Research talks;Dynamical Systems and Ordinary Differential Equations;Algebraic and Complex Geometry;Topology

We give a necessary and sufficient condition for the existence of infinitely many non-arithmetic Teichmuller curves in a stratum of abelian differentials. This is joint work with Simion Filip and Alex Wright.

30F30 ; 32G15 ; 32G20 ; 14D07 ; 37D25

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Research talks;Dynamical Systems and Ordinary Differential Equations

37A25 ; 37E35

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Research talks;Dynamical Systems and Ordinary Differential Equations;Algebraic and Complex Geometry

We discuss the current status of the problem of understanding the closures of the strata of curves together with a differential with a prescribed configuration of zeroes, in the Deligne-Mumford moduli space of stable curves.

30F10 ; 30F30 ; 14H10 ; 14H70 ; 32G15

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Research talks;Dynamical Systems and Ordinary Differential Equations

We prove a couple of general conditional convergence results on ergodic averages for horocycle and geodesic subgroups of any continuous $SL(2,\mathbb{R})$- action on a locally compact space. These results are motivated by theorems of Eskin, Mirzakhani and Mohammadi on the $SL(2,\mathbb{R})$-action on the moduli space of Abelian differentials. By our argument we can derive from these theorems an improved version of the “weak convergence” of push-forwards of horocycle measures under the geodesic flow and a new short proof of a theorem of Chaika and Eskin on Birkhoff genericity in almost all directions for the Teichmüller geodesic flow. We prove a couple of general conditional convergence results on ergodic averages for horocycle and geodesic subgroups of any continuous $SL(2,\mathbb{R})$- action on a locally compact space. These results are motivated by theorems of Eskin, Mirzakhani and Mohammadi on the $SL(2,\mathbb{R})$-action on the moduli space of Abelian differentials. By our argument we can derive from these theorems an improved version of the “weak convergence” of ...

37D40 ; 37C40 ; 37A17

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Research talks;Dynamical Systems and Ordinary Differential Equations

Tiling billiards is a dynamical system where beams of light refract through planar tilings. It turns out that, for a regular tiling of the plane by congruent triangles, the light trajectories can be described by interval exchange transformations. I will explain this surprising correspondence, give related results, and show computer simulations of the system.

37D50 ; 37B50

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Research talks;Combinatorics;Dynamical Systems and Ordinary Differential Equations;Topology

We show how recent results of the authors on equidistribution of square-tiled surfaces of given combinatorial type allow to compute approximate values of Masur-Veech volumes of the strata in the moduli spaces of Abelian and quadratic differentials by Monte Carlo method.
We also show how similar approach allows to count asymptotical number of meanders of fixed combinatorial type in various settings in all genera. Our formulae are particularly efficient for classical meanders in genus zero.
We construct a bridge between flat and hyperbolic worlds giving a formula for the Masur-Veech volume of the moduli space of quadratic differentials in terms of intersection numbers of $\mathcal{M}_{g,n}$ (in the spirit of Mirzakhani's formula for Weil-Peterson volume of the moduli space of pointed curves).
Joint work with V. Delecroix, E. Goujard, P. Zograf.
We show how recent results of the authors on equidistribution of square-tiled surfaces of given combinatorial type allow to compute approximate values of Masur-Veech volumes of the strata in the moduli spaces of Abelian and quadratic differentials by Monte Carlo method.
We also show how similar approach allows to count asymptotical number of meanders of fixed combinatorial type in various settings in all genera. Our formulae ...

32G15 ; 05C30 ; 05Axx

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- 177 p.
ISBN 978-0-8218-2094-0

American mathematical society translations series 2 , 0197

Localisation : Collection 1er étage

forme fermée de dimension -1 # fraction continue multidimensionnelle # opérateur limite dans le complexe de Novikov # polyhèdre de Klein # problème variationnel dans les espaces de dimension infinie # système dynamique lisse # théorie ergodique # topologie # topologie différentielle # topologie pseudo-périodique

37Axx ; 37Cxx ; 57Rxx ; 58Exx

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