Courses: upcoming

This course is available for registration under the Western Dean's Agreement but registrations must be approved by the course instructor. Please contact the instructor (using the email link to the left) including details of how you meet the course prerequisites. Next, you must complete the Western Deans' Agreement form , with the following course details:

Completed forms should be returned to your graduate advisor who will sign it and take the required steps. For students at PIMS sites, please see this list to find your graduate advisor, for other sites, please see the Western Deans' Agreement website .

The Western Deans' Agreement provides an automatic tuition fee waiver for visiting students. Graduate students paying normal required tuition fees at their home institution will not pay tuition fees to the host institution. However, students will typically be be required to pay other ancillary fees to the host institution, or explicitly request exemptions (e.g. Insurance or travel fees). Details vary by university, so please contact the graduate student advisor at your institution for help completing the form. Links to fee information and contact information for PIMS member universities is available below in the WDA section.

Students at universities not covered by the WDA but which are part of the Canadian Association for Graduate Studies (CAGS) may still be eligible to register for some courses under the terms of the Canadian University Graduate Transfer Agreement (CUGTA). Details of this program vary by university and registration is also typically subject to ancillary fees. Both your local and the hosting university must participate in the agreement (e.g. UBC does not participate in CAGS). Please contact the relevant graduate student advisors for more information.

Translation surfaces and their moduli spaces have been the objects of extensive recent study and interest, with connections to widely varied fields including (but not limited to) geometry and topology; Teichmüller theory; low-dimensional dynamical systems; homogeneous dynamics and Diophantine approximation; and algebraic and complex geometry. This course will serve as an introduction to some of the big ideas in the field, centered on the ergodic properties of translation flows and counting problems for saddle connections, and associated renormalization techniques, without attempting to reach the full state of the art (an aim that is in any case impossible given the speed at which the field is evolving).

We will start by introducing the important motivating example of the flat torus, exploring its geometry, and its associated dynamical and counting problems. The linear flow on the torus and its associated first return map, a rotation of a circle, are amongst the first dynamical systems ever studied. The counting of closed orbits is intricately tied to number theory. We discuss, as motivation, the moduli space of translation surfaces on a torus, a bundle over the well-known modular curve and the action of $GL^+(2,\mathbb R)$ on this space of translation surfaces. Translation surfaces are higher-genus generalizations of flat tori. We will define translation surfaces from three perspectives (Euclidean geometry, complex analysis, and geometric structures), and show how some translation surfaces arise from unfolding billiards in rational polygons. We will give a short introduction to Teichmüller theory and its relation to the study of translation surfaces, and discuss the natural dynamical systems associated to translation surfaces, namely, linear flows and their first return maps, interval exchange transformations. We will explore their ergodicity and mixing properties, and will study an important example of a translation surface flow for which every orbit is dense but not every orbit is equidistributed with respect to Lebesgue measure, a phenomenon that does not occur in the case of linear flows on the torus. We will show how information about the recurrence properties of an orbit of a translation surface under the positive diagonal subgroup of $SL(2, \mathbb R)$ (the Teichmüller geodesic flow) can be used to get information about the ergodic properties of the associated linear flow on an individual translation surface. As another example of the strength of renormalization ideas, we will show how the ergodic properties of the $SL(2, \mathbb R)$-action can be used to obtain counting results for saddle connections and, subsequently. Finally we will discuss examples, characterizations, and properties of surfaces with large affine symmetry groups, known as lattice or Veech surfaces.

Remote Access

The instructor will use a tablet and Zoom. The tablet will be displayed locally in the classroom and via zoom. Lecture notes will be distributed in PDF format.

Class Schedule

This class will meet every Monday, Wednesday and Friday from 1:30-2:20 (Pacific time), starting on March 31st.