Week 16.10.2023 – 22.10.2023

Lecture 3 in the minicourse by Prof. Leonid Pastur

see

https://dsadvancedlectures.weebly.com/

The replica method, together with Parisi symmetry breaking mechanism, is a powerful tool which allows to compute the limiting free energy of virtually any mean field disordered system. Unfortunately, the tool is dramatically flawed from a mathematical point of view. I will discuss a truly elementary procedure which allows to rigorously implement two (out of three) steps of the procedure, and which allows to represent the free energy of virtually any model from statistical mechanics as a Gaussian mixture model. I will then conclude with some remarks on the ensuing Babylonian formulas and their relation with :
1) work by Dellacherie-Martinez-San Martin on M-matrices, potential theory and ultrametricity, the latter being the key yet unjustified assumption of the whole Parisi theory\DSEMIC
2) work of Mezard-Virasoro suggesting that the onset of scales and the universal hierarchical self-organisation of mean field random systems is intimately linked to hidden geometrical properties of large random matrices which satisfy rules reminiscent of the popular SUDOKU game.

The geometry and topology of negatively curved manifolds are subtly reflected in a geometric bound for the Laplace eigenvalues, a connection that has been explored since the 1980s. Building upon these foundational studies in the case of the Laplacian, we investigate the Steklov eigenvalues of pinched negatively curved manifolds with totally geodesic boundary. These eigenvalues are associated with a first-order elliptic pseudodifferential operator known as the Dirichlet-to-Neumann operator. We discuss how the results for Laplace eigenvalues can be extended to Steklov eigenvalues. In particular, we show a spectral gap for the Steklov eigenvalue problem in negatively curved manifolds with dimensions of at least three. This talk is based on joint work with Ara Basjmaian, Jade Brisson, and Antoine Métras.

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.

An elliptic curve over a characteristic zero field is said to have complex multiplication when its endomorphism ring is larger than Z ("E has extra endomorphisms"). Generic elliptic curves don't have complex multiplication. Often one tries to understand elliptic curves via their Tate modules. When the Tate module of E has extra endomorphisms we say E has formal complex multiplication. Over a number field, E has formal complex multiplication if and only if it has complex multiplication. Over a local field this need not be the case. How often does this happen?