L6

The classification of measure preserving actions up to orbit equivalence has attracted a lot of interest in the last 25 years. The goal of this talk is to survey the major discoveries in the field, including Popa's cocycle and orbit equivalence superrigidity theorem and discuss some recent superrigidity results for dense subgroups of Lie groups acting by translation.

A central question in infinite group theory is to determine how much global information about a group is encoded in its set of finite quotients. In this talk, we will discuss this problem in the case of algebraically fibered groups, which naturally generalise fundamental groups of compact manifolds that fiber over the circle. The study of such groups exploits the relationships between the geometry of the classifying space, the dynamics of the monodromy map, and the algebra of the group, and as such draws from all of these areas.

It was shown by Balasubramanya that any acylindrically hyperbolic group (a natural generalisation of a hyperbolic group) must act acylindrically and non-elementarily on some quasi-tree. It is therefore sensible to ask to what extent this is true for trees, i.e. given an acylindrically hyperbolic group, does it admit a non-elementary acylindrical action on some simplicial tree? In this talk I will introduce the concepts of acylindrically hyperbolic and acylindrically arboreal groups and discuss some particularly interesting examples of acylindrically hyperbolic groups which do and do not act acylindrically on trees.

Oka theory is about the validity of the h-principle in complex analysis and geometry. In this expository lecture, I will trace its main developments, from the classical results of Kiyoshi Oka (1939) and Hans Grauert (1958), through the seminal work of Mikhail Gromov (1989), to the introduction of Oka manifolds (2009) and the present state of knowledge. The lecture does not assume any prior exposure to this theory.

We will survey work of Birman-Craggs, Johnson, and Sato on the abelianization of the level 2 congruence group of the mapping class group of a surface, and of the corresponding Torelli group. We will then describe recent work of Lewis providing a common framework for both abelianizations, with applications including a partial answer to a question of Johnson.

I will talk about a family of measures on partitions (specifically, a case of Okounkov's Schur measures) which are in one-to-one correspondence with models of random unitary matrices and lattice fermions. Under these measures, as the expected size of a partition goes to infinity, the first part of a random partition generically exhibits the same universal asymptotic fluctuations as the largest eigenvalue of a GUE random Hermitian matrix. First, I'll describe how we can tune these measures to exhibit new edge fluctuations at a smaller scale, which naturally generalise the GUE edge behaviour. These new fluctuations are universal, having previously been found for trapped fermions, and when a measure is tuned to have them, the corresponding unitary matrix model is "multicritical". Then, I'll describe how our measures can escape these more general universality classes, when tuned to have several cuts in a certain "Fermi sea". In this case, the breakdown in universality arises from an oscillation phenomenon previously observed in multi-cut Hermitian matrix models. Moreover, we have a one-to-one correspondence with multi-cut unitary matrix models. This is partly based on joint work with Dan Betea and Jérémie Bouttier. 

The infinitely wide neural network has been proven a useful and manageable mathematical model that enables the understanding of many phenomena appearing in deep learning. One example is the convergence of random deep networks to Gaussian processes that enables a rigorous analysis of the way the choice of activation function and network weights impacts the training dynamics. In this paper, we extend the seminal proof of Matthews (2018) to a larger class of initial weight distributions (which we call "pseudo i.i.d."), including the established cases of i.i.d. and orthogonal weights, as well as the emerging low-rank and structured sparse settings celebrated for their computational speed-up benefits. We show that fully-connected and convolutional networks initialized with pseudo i.i.d. distributions are all effectively equivalent up to their variance. Using our results, one can identify the Edge-of-Chaos for a broader class of neural networks and tune them at criticality in order to enhance their training.

Abstract: Shifted moments of the Riemann zeta function, introduced by Chandee, are natural generalizations of the moments of zeta. While the moments of zeta capture large values of zeta, the shifted moments also capture how the values of zeta are correlated along the half line. I will describe recent work giving sharp bounds for shifted moments assuming the Riemann hypothesis, improving previous work of Chandee and Ng, Shen, and Wong. I will also discuss some unconditional results about shifted moments with small exponents.