Scientific

Mathieu Dutour (University of Alberta, Canada)

In the finite-dimensional situation, Lie's third theorem provides a correspondence between Lie groups and Lie algebras. Going from the latter to the former is the more complicated construction, requiring a suitable representation, and taking exponentials of the endomorphisms induced by elements of the group.

As shown by Garland, this construction can be adapted for some Kac-Moody algebras, obtained as (central extensions of) loop algebras. The resulting group is called a loop group. One also obtains a relevant infinite-rank Chevalley lattice, endowed with a metric. Recent work by Bost and Charles provide a natural setting, that of pro-Hermitian vector bundles and theta invariants, in which to study these objects related to loop groups. More precisely, we will see in this talk how to define theta-finite pro-Hermitian vector bundles from elements in a loop group. Similar constructions are expected, in the future, to be useful to study loop Eisenstein series for number fields.

This is joint work with Manish M. Patnaik.

In the theory of continued fractions, the denominator of the truncated fraction (often denoted q) contains a great deal of information important in applications. However, q is a surprisingly complicated object from the point of view of ergodic theory. We will look at a few problems related to q and see how different techniques have overcome these difficulties, including modular properties (Moeckel, Fisher-Schmidt), renewal-type theorems (Sinai-Ulcigrai, Ustinov), and "nonstandard" arrangements of points (Avdeeva-Bykovskii).

The behavior of quadratic twists of modular L-functions at the critical point is related both to coefficients of half integer weight modular forms and data on elliptic curves. Here we describe a proof of an asymptotic for the second moment of this family of L-functions, previously available conditionally on the Generalized Riemann Hypothesis by the work of Soundararajan and Young. Our proof depends on deriving
an optimal large sieve type bound.

This event is part of the PIMS CRG Group on L-Functions in Analytic Number Theory. More details can be found on the webpage here: https://sites.google.com/view/crgl-functions/crg-weekly-seminar

In the past century, the studies of moments of L-functions have been important in number theory and are well-motivated by a variety of arithmetic applications. This talk will begin with two problems in elementary number theory, followed by a survey of techniques in the past and the present. We will slowly move towards the perspectives of period integrals which will be used to illustrate the interesting structures behind moments. In particular, we shall focus on the “Motohashi phenomena”.

Detecting differences and building classifiers between distributions, given only finite samples, are important tasks in a number of scientific fields. Optimal transport (OT) has evolved as the most natural concept to measure the distance between distributions and has gained significant importance in machine learning in recent years. There are some drawbacks to OT: computing OT can be slow, and it often fails to exploit reduced complexity in case the family of distributions is generated by simple group actions.

If we make no assumptions on the family of distributions, these drawbacks are difficult to overcome. However, in the case that the measures are generated by push-forwards by elementary transformations, forming a low-dimensional submanifold of the Wasserstein manifold, we can deal with both of these issues on a theoretical and on a computational level. In this talk, we’ll show how to embed the space of distributions into a Hilbert space via linearized optimal transport (LOT), and how linear techniques can be used to classify different families of distributions generated by elementary transformations and perturbations. The proposed framework significantly reduces both the computational effort and the required training data in supervised learning settings. We demonstrate the algorithms in pattern recognition tasks in imaging and provide some medical applications.

This is joint work with Alex Cloninger, Keaton Hamm, Harish Kannan, Varun Khurana, and Jinjie Zhang.