The Celestial Holography program encompasses recent efforts to understand the flat space hologram in terms of a CFT living on the celestial sphere. Here we have fun relating various extrapolate dictionaries in CCFT and examining tools we can apply when perturbing around a 4D CFT in the bulk.

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.

As part of the internal seminar schedule for Stochastic Analysis for this coming term, DPhil students have been invited to present on their works to date. Student talks are 20 minutes, which includes question and answer time. 

Students presenting are:

Sara-Jean Meyer, supervisor Massimiliano Gubinelli

Satoshi Hayakawa, supervisor Harald Oberhauser 

We consider the homogenisation problem for the ϕ4/2 equation on the torus T² , i.e. the behaviour as ϵ → 0 of the solutions to the equations suggestively written

∂_tu_ϵ − ∇ · A(x/ϵ, t/ϵ² )∇uϵ = −u³_ϵ + ξ

where ξ denotes space-time white noise and A : T ² × R is uniformly elliptic, periodic and H¨older continuous. Based on joint work with M. Hairer

The mapping class group of a surface has a hierarchical structure in which the geometry of the group can be seen by examining its action on the curve graph of every subsurface. This behavior was one of the motivating examples for a generalization of hyperbolicity called hierarchical hyperbolicity. Hierarchical hyperbolicity has many desirable consequences, but the definition is long, and proving that a group satisfies it is generally difficult. This difficulty motivated the introduction of a new condition called combinatorial hierarchical hyperbolicity by Behrstock, Hagen, Martin, and Sisto in 2020 which implies the original and is more straightforward to check. In recent work, Hagen, Mangioni, and Sisto developed a method for building a combinatorial hierarchically hyperbolic structure from a (sufficiently nice) hierarchically hyperbolic one. The goal of this talk is to describe their construction in the case of the mapping class group and illustrate some of the parallels between the combinatorial structure and the original. 

You may be surprised to see the generalised Riemann hypothesis appear in algorithmic topology. For example, knottedness was originally shown to be in NP under the assumption of GRH.
Where does this condition come from? We will discuss this in the context of 3-sphere recognition, and examine why the approach fails for higher dimensions.

It is a truth universally acknowledged, that an infinite group in possession of a good algebraic structure, must be in want of a hyperbolic space to act on. For the mapping class group of a surface, one of the most popular choices is the curve graph. This is a combinatorial object, built from curves on the surface and intersection patterns between them.
Hyperbolicity of the curve graph was proved by Masur and Minsky in a celebrated paper in 1999. In the same article, they showed how the geometry of the action on this graph reflects dynamical/topological properties of the mapping class group; in particular, loxodromic elements are precisely the pseudo-Anosov mapping classes.
In light of this, one would like to better understand distances in the curve graph. The graph is locally infinite, and finding a shortest path between two vertices is highly non-trivial. In this talk, we will see how to use the machinery of train tracks to overcome this issue and compute (approximate) distances in the curve graph. If time permits -- which, somehow, it never does -- we will also analyse this construction from an algorithmic perspective.