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| Date | Speaker (Affiliation) | Title/Abstract |
| September 8 | NO SEMINAR | --- |
| September 15 | William Goldman (UMCP) | Geodesics on Margulis Spacetimes Abstract: A Margulis spacetime is a 3-manifold which is a quotient of 3-space by a free group of affine transformations. Associated to every such 3-manifold M is a hyperbolic surface S. Generalizing the correspondence between closed geodesics on M and closed geodesics on S, we establish an orbit equivalence between recurrent spacelike geodesics on $M^3$ and recurrent geodesics on S. In contrast, no timelike geodesic recurs in either forward or backwards time. This is joint work with Francois Labourie. |
| September 22 | Joseph Auslander (UMCP) | Regional Proximality, McMahon's Theorem, and the Veech Relation |
| September 29 | Abed Bounemoura (IAS) | Persistence of invariant submanifolds
Abstract: In this talk, we will give a simple and geometrical proof of the classical result on persistence and uniqueness for normally hyperbolic submanifolds. Moreover, the proof gives a new result on persistence for a wider class of submanifolds, which we call topologically normally hyperbolic. This is a joint work with Pierre Berger. |
| October 6 | Pierre Pageault (ENS Lyon) | Functions whose set of critical points is an arc
Abstract: We prove that on a compact connected manifold M with dim(M)>1, the set of C^1 functions whose set of critical points is an arc is dense for the C^0 topology. We then present applications in dynamic, and link them with uniqueness problems of Weak KAM solution associated to Mañé Lagrangians. |
| October 13 | Mark Demers (Fairfield) | A spectral gap for the transfer operator of the Lorentz gas
Abstract: Much attention has been given in recent years to developing a framework to study directly the transfer operator associated with hyperbolic maps on an appropriate Banach space. For the billiard map associated with a Lorentz gas of both finite and infinite horizon, we construct generalized function spaces on which the transfer operator is quasi-compact and has a spectral gap. This framework gives a unified approach to proving the statistical properties and various limit laws associated with billiards, such as exponential decay of correlations, central limit theorem and large deviation estimates. It also has potential applications to many classes of perturbations. This is joint work Hong-Kun Zhang. |
| October 20 | NO SEMINAR | Semi-annual Workshop in Dynamical Systems and Related Topics, Penn St. |
| October 27 | Nikita Selinger (SUNY Stony Brook) | Title: The proof of Pilgrim's conjecture. Abstract: Let $f$ be a postcritically finite branched self-cover of a 2-dimensional topological sphere. Such a map induces an analytic self-map $\sigma_f$ of a finite-dimensional Teichm\"uller space. We prove that this map extends continuously to the augmented Teichm\"uller space and give an explicit construction for this extension. This allows us to characterize the dynamics of Thurston's pullback map near invariant strata of the boundary of the augmented Teichm\"uller space. The resulting classification of invariant boundary strata is used to prove a conjecture by Pilgrim. |
| November 3 | Alex Eskin (Chicago) Note: This speaker will also give the department colloquium on November 2. | Sums of Lyapunov exponents of the Teichmueller geodesic flow, and the
Siegel-Veech constants This is joint work with Maxim Kontsevich and Anton Zorich |
| November 10 | Francesco Cellarosi (IAS) | Title: Random square-free numbers. Abstract: A square-free number is an integer that is not divisible by p^2 for any prime p. I shall discuss two ways of generating 'random' square-free numbers. One construction is inspired by Statistical Mechanics and enjoys some unexpected properties, such as a non-standard limit theorem. The second construction is more classical and can be understood using a 'natural' dynamical system, whose ergodic properties have been recently examined by P. Sarnak in connection with his conjecture on the randomness for the Möbius function. Joint work with Ya.G. Sinai. |
| November 17 | David Aulicino (UMCP) | Title: "Classifying Teichm\”uller Disks with Completely Degenerate Kontsevich-Zorich Spectrum" Abstract: The moduli space of genus g Riemann surfaces is the space of all complex structures on a closed orientable surface of genus g up to orientation preserving diffeomorphisms. The Teichm\"uller geodesic flow is the flow on the cotangent bundle of the Teichm\"uller space of surfaces defined by the direction of minimal dilatation and it descends to the cotangent bundle of the moduli space under the action of the mapping class group. It is well-known that the Lyapunov spectrum of this flow is determined by g numbers 1=\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_g \geq 0. The Kontsevich-Zorich conjecture, proven by Forni and Avila-Viana, showed that generically all the inequalities are strict with respect to the canonical absolutely continuous measures. However, Forni found an example of a measure on the genus three moduli space, and Forni-Matheus found a measure in genus four, with completely degenerate spectrum, i.e. 1=\lambda_1 > \lambda_2 = \cdots = \lambda_g=0. We prove that these are the only such measures in genus three and four. Furthermore, there are no such measures for g=2 and g \geq 13. Finally, if there are no square-tiled surfaces in genus five that determine a measure with completely degenerate spectrum, then there are no examples for g \geq 5. |
| November 24 | NO SEMINAR | THANKSGIVING BREAK |
| December 1 | Barney Bramham (IAS) Please note that this seminar is joint with the Geometry/Topology seminar. | Title: "Approximating Hamiltonian systems by integrable systems using pseudo-holomorphic curves". Abstract: I will talk about an approach, using pseudo-holomorphic curve techniques from symplectic geometry, to the following question in dynamical systems of Anatole Katok: "In low dimensions is every conservative dynamical system with zero topological entropy a limit of integrable systems?" |
| December 8 | Robbie Robinson (George Washington) | Title: Kakaya's theorem and maps of the interval Abstract: In 1924 Soichi Kakeya described a common generalization of decimal and continued fraction expansions. The idea was reinvented by Bissinger and Everett in the 1940s, and in 1957 Renyi recast the idea in terms of maps of the interval, where it is now called f-expansions. In this talk we will discuss Kakeya's original theorem on the subject in the context of interval maps, as well discussing some contributions of Renyi, Parry and others. |
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| 2010-2011 | --- | --- | --- |
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