A classical result in spectral theory is that the space of square integrable functions on the modular surface $X = SL(2,\mathbb Z) \backslash SL(2,\mathbb R)$ can be decomposed as the space of Eisenstein series and its orthogonal complements, the cusp forms. The former space corresponds to the spectral projection on the continuous spectrum of the Laplacian on X, and the cusp forms to the projection on the point spectrum. This result is relevant in the geometry of numbers and in dynamics because the modular surface can parameterise the space of all unimodular lattices (and, thus, also the space of all unit area flat tori).

In this talk, I will explain how to extend these ideas to the study of spaces of flat surfaces of higher genus with singularities. We replace the Eisenstein series with the range of the Siegel—Veech transform and in some specific cases can also identify precisely the cusp forms. I will focus on the case of marked flat tori, this space corresponding to the space of affine lattices. In this situation, we can also identify an operator\DSEMIC which is not the Laplacian but a foliated Laplacian\DSEMIC where the natural decomposition corresponds to its spectrum.

This is joint work with Jayadev S. Athreya (Washington), Martin Möller (Frankfurt) and Martin Raum (Chalmers)

A classical result in spectral theory is that the space of square integrable functions on the modular surface $X = SL(2,\mathbb Z) \backslash SL(2,\mathbb R)$ can be decomposed as the space of Eisenstein series and its orthogonal complements, the cusp forms. The former space corresponds to the spectral projection on the continuous spectrum of the Laplacian on X, and the cusp forms to the projection on the point spectrum. This result is relevant in the geometry of numbers and in dynamics because the modular surface can parameterise the space of all unimodular lattices (and, thus, also the space of all unit area flat tori).

In this talk, I will explain how to extend these ideas to the study of spaces of flat surfaces of higher genus with singularities. We replace the Eisenstein series with the range of the Siegel—Veech transform and in some specific cases can also identify precisely the cusp forms. I will focus on the case of marked flat tori, this space corresponding to the space of affine lattices. In this situation, we can also identify an operator, which is not the Laplacian but a foliated Laplacian, where the natural decomposition corresponds to its spectrum.

This is joint work with Jayadev S. Athreya (Washington), Martin Möller (Frankfurt) and Martin Raum (Chalmers)

The two-stage examination method is a variant on exam taking whereby students are asked to take the same exam twice --- once in the 'usual' way, and the second time in small groups of three to four. It has been used in mathematics, physics and engineering since its inception 20 years ago at UBC in Vancouver, but is normally used in basic modules in the first or second year.

I will talk about a trial I am running on two-stage exams in a Masters level class. Here, the focus will be a bit different: I use the second part, in groups, to ask the students slightly more open-ended questions. In this talk I will talk about the concept, my observations, and the challenges that were faces in the first implementation.

Fraser and Schoen have uncovered a beautiful relationship between free boundary minimal surfaces in the unit ball and the Steklov problem: the coordinate functions of such surfaces are Steklov eigenfunctions with eigenvalue 1, and, on the other hand, the eigenfunctions for extremal metrics for the Steklov problem provide embeddings of free boundary minimal surfaces. The Fraser–Li conjecture states that not only are the coordinate functions Steklov eigenfunctions with eigenvalue 1, this eigenvalue is also the smallest non-zero one.

In this talk, I will discuss the history of the problem, the relation with minimal surfaces in the sphere, and explain an elementary proof of special cases of the Fraser–Li conjecture assuming some additional symmetries.