Corinna Ulcigrai

Slide Presentations - Surveys/colloquium style:

• Inaugural Lecture: Billiards, Pretzels... and Chaos

  These are the slides of my professorial inaugural lecture: the talk was aimed at presenting to a general audience some of the highlights of research area and achievements (the actual audience consisted of about 150 people, which were a quite equal mixture of colleagues, undergraduate students and general public). You can also listen to a audio recording of the lecture.

• Mathematical billiards and Flows on Surfaces

  This is a survey/colloquium style talk on mathematical billiards and area preserving flows on surfaces, with particular focus on billiards in rational polygons and their unfolding to translation surfaces and locally Hamiltonian flows on surfaces. It is meant as an introduction to Teichmueller dynamics and it presents my works on locally Hamiltonian flows on surfaces (Annals of Math 2011, J. Modern Dynamics 2009 and Erg. Theory and Dyn. Systems 2007) and my joint works with J. Smillie (Proc. Lond. Math. Soc. 2011 and AMS Contemporary Math. 2010)

• Shearing and Mixing in Parabolic flows

  This is a colloquium style talk which surveys some results on mixing for different type of parabolic flows. It seems that shearing is a key geometric mechanism in producing mixing in several parabolic flows. We survey and compare mixing results and mechanisms (and spectrum) for time-changes of horocycle flows (joint work with Forni), mixing for locally Hamiltonian flows and mixing for time-changes of Heisenberg nilflows (joint work with Avila and Forni).

• NEW: On Birkhoff sums and Roth type conditions for interval exchange maps

  This talk was given at the conference in memory of Jean-Christophe Yoccoz at College de France, in June 2017. It presents some work in progress with Jean-Christophe Yoccoz and Stefano Marmi, but it starts with an accessible introduction to the study of Birkhoff sums over interval exchange transformations, in particular deviations, the Kontsevich-Zorich conjecture and limit shapes of Birkhoff sums.

• Cutting sequences, Regular Polygons and Veech groups

  This is a accessible introductory talk on Sturmian sequences and cutting sequences of linear trajectories in regular polygons. The first focus on the characterization of Sturmian sequences (that is, sequences that code how an irrational line cross a square grid) in the spirit of C. Series (Math. Intelligencer) and their relation with continued fractions. The second part focus on results and ideas of proofs based on my joint works (Proc. Lond. Math. Soc. 2011 and AMS Contemporary Math. 2010) with J. Smillie (Cornell University).

Slide Presentations - conferences/seminar research talks:

• A Central Limit Theorem for cocycles over rotations

  NEW: This talk starts with an introduction on limit theorems in dynamical systems, then focus on my joint work with Michael Bromberg (currently my postdoc in Bristol) on a generalization of a (temporal) Central Limit Theorem for cocycles over rotations, generalizing earlier work by J.Beck and D.Dolgopyat and O. Sarig (see preprint). A sketch of the symbolic coding and main ideas in the proof is given.

• Mixing Time-Changes of Heisenberg nilflows

  This is a more specialized talk which focus on my joint paper with Giovanni Forni (Maryland) and Artur Avila (IMPA and CNRS) on mixing time-changes of Heisenberg Nilflows (see Journal of Differential Geometry, 2011).

• Diagonal changes algorithms

  This talk, which should also be accessible, focus on my joint work with Vincent Delecroix on Diagonal Changes algorithms for translation surfaces in hyperelliptic components (see Geometriae Dedicata, 2015 or the preprint ).

---

Videos of Talks:

• Billiards, Surfaces and Chaos

  This is a Colloquium I gave at the University of Padova (April 2017) in which I give an overview of the study of (rational) polygonal billiards and some of the highlights of my research on slowly chaotic systems (such as infinite polygonal billiards and smooth area-preserving flows on surfaces).

• Billiards in polygons and flows on periodic surfaces

  This is a survey talk on polygonal mathematical billiards. It starts from billiards in bounded rational polygons and then moves to periodic planar billiards such as the Ehrenfest model. The unfolding procedure is explained both for bounded and periodic billiards, showing the connection respectively with compact translation surfaces and Abelian covers of translation surfaces. Both classical results and recent progress on planar billiards such as the Ehrenfest model are mentioned (in particular my joint work with K. Fraczek).

• Ergodic Properties of Surface Flows

  This talk focus on locally Hamiltonian flows on surfaces and Teichmueller dynamics. It presents my early works on ergodic properties of locally Hamiltonian flows and limit theorems for functions with logarithmic singularities over IETs (J. Modern Dynamics 2009 and Erg. Theory and Dyn. Systems 2007)

• Eaton lenses, pseudo-integrable billiards and genericity along curvesEaton lenses, pseudo-integrable billiards and genericity along curves

  This ICTP talk is based on joint work with Frzysztof Fraczek and Ronggang Shi . We consider three problems: a question about ergodic properties of billiards in a pseudointegrable elliptical billiard, a question about the dynamics of light rays in arrays of Eaton lenses and a question about gaps in the sequence of fractional parts of square-roots of n. Answers to all these three questions are based on results about Birkhoff (and Oseledets) genericity along curves in a space of branched covers of lattices.

• Multiple mixing and Ratner property in area-preserving flows

  This CIRM talk is based on joint work with Adam Kanigowski and Joanna Kulaga-Przymus (to appear in the Journal of the European Mathematical Society ). We consider smooth area-preserving flows on surfaces. In the open set in which typical minimal components are mixing, we prove that the flow is actually mixing of all oders. The proof is based on showing that these surface flows with singularities enjoy a variation of the quantitative shearing property first described and exploited by Marina Ratner for the horocyle flow (the switchable Ratner property defined by Fayad-Kanigowski).