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# Professor Corinna Ulcigrai

## Professor Corinna Ulcigrai

Ph.D., M.A.(Prin.)

Professor of Pure Mathematics
## Summary

## Biography

## Memberships

### Organisations

### Pure Mathematics

### Probability, Analysis and Dynamics

### Research themes

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## Recent publications

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Ph.D., M.A.(Prin.)

Office Room 4.09

Howard House,

Queen's Ave,
Bristol
BS8 1SD

(See a map)

+44 (0) 117 928 7999

+44 (0) 117 3315249

corinna.ulcigrai@

bristol.ac.uk

Ergodic theory and Teichmuller dynamics

My main research interests are in ergodic theory and Teichmuller dynamics, an area of ̈research in dynamical systems which has developed and bloomed in the last decades and that often involves arithmetical, geometrical and combinatorial tools. I am especially interested in parabolic dynamical systems, which are systems which display a "slow" form of chaotic evolution.

Examples of systems studied in Teichmuller dynamics are: ̈

• Polygonal billiards, in which a point-particle moves in a planar polygon bouncing elastically at sides.

• Geodesics on translation surfaces, which are locally Euclidean surfaces but at some conical singularities.

• Maps of the interval which are piecewise isometries, called interval exchange transformations (IETs).

One is interested in investigating ergodic properties that describe how chaotic these systems are.

The properties of these elementary systems are beautifully and deeply connected with the dynamics on an abstract space of deformations, more precisely with the Teichmuller geodesic flow and the SL(2,R) action on moduli spaces. At the level of interval exchange transformations, this connections can be exploited at a more combinatorial level, using a continued fraction algorithm called Rauzy-Veech induction.

Some areas of research for possible PhD projects:

• Area preserving flows on surfaces: in a natural class of flows locally given by a Hamiltonian, I studied properties like mixing and weak mixing; many interesting questions about spectral properties are open;

• Infinite periodic polygonal billiards and periodic infinite translation surfaces; while compact translation surfaces are well understood, the study of infinite covers has just recently started and there are many open questions on the ergodic properties of respectively the billiard and linear flow in this infinite ergodic theory setup.

• Cutting sequences: in a joint work with Smillie, we gave a characterization of the symbolic sequences which code linear trajectories in regular polygons. Similar questions could be addressed in other translation surfaces with the lattice property.

• Interval exchange transformations: questions about distributions of orbit points, as spacings and discrepancy, in the spirit of limit theorems or for special classes of IETs (like bounded type).

If you are interested in knowing more about this research area and potential projects, feel free to contact me by email. You might also want to browse my webpage, in the Slides section you will find slides and videos of talks that I've given explaining my research.

See personal webpage linked to the right hand side.

- Davis, D, Pasquinelli, I & Ulcigrai, C, 2018, ‘Cutting sequences on Bouw-Moeller surfaces: an S-adic characterization’.
*Annales Scientifiques de l'École Normale Supérieure*. - Hubert, P, Lelievere, S, Marchese, L & Ulcigrai, C, 2018, ‘The Lagrange spectrum of some square-tiled surfaces’.
*Israel Journal of Mathematics*, vol 225., pp. 553-607 - Fraczek, K, Shi, R & Ulcigrai, C, 2018, ‘Genericity on curves and applications: pseudo-integrable billiards, Eaton lenses and gap distributions’.
*Journal of Modern Dynamics*, vol 12., pp. 55-122 - Bromberg, M & Ulcigrai, C, 2018, ‘A temporal Central Limit Theorem for real-valued cocycles over rotations’.
*Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques*, vol 54., pp. 2304-2334 - Kanigowski, A, Kułaga-Przymus, J & Ulcigrai, C, 2017, ‘Multiple mixing and parabolic divergence in smooth area-preserving flows on higher genus surfaces’.
*Journal of the European Mathematical Society*. - Artigiani, M, Marchese, L & Ulcigrai, C, 2016, ‘The Lagrange spectrum of a Veech surface has a Hall ray’.
*Groups, Geometry and Dynamics*, vol 10., pp. 1287-1337 - Hubert, P, Marchese, L & Ulcigrai, C, 2015, ‘Lagrange Spectra in Teichmüller Dynamics via Renormalization’.
*Geometric and Functional Analysis*, vol 25., pp. 180-255 - Delecroix, V & Ulcigrai, C, 2015, ‘Diagonal changes for surfaces in hyperelliptic components - A geometric natural extension of Ferenczi-Zamboni moves’.
*Geometriae Dedicata*, vol 176., pp. 117?174 - Fraczek, K & Ulcigrai, C, 2014, ‘Non-ergodic Z-periodic billiards and infinite translation surfaces’.
*Inventiones Mathematicae*, vol 197., pp. 241 - Fraczek, K & Ulcigrai, C, 2014, ‘Ergodic Directions for Billiards in a Strip with Periodically Located Obstacles’.
*Communications in Mathematical Physics*, vol 327., pp. 643-663

View complete publications list in the University of Bristol publications system

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