Week 08.12.2024 – 14.12.2024

We consider ‘λ-shaped random matrices’, whose entries are i.i.d. in the boxes of a given Young diagram λ and zero elsewhere. In particular, we study their limiting spectral distribution when the shape λ is dilated by a growing factor N. The moments of such a distribution are a generalisation of Catalan numbers, and enumerate combinatorial objects which we call λ-plane trees: these are trees whose vertices are labelled in a way that is ‘compatible’ with λ. Based on joint works with Elia Bisi and Marilena Ligabò. 

Subordination is the operation that evaluates a Lévy process at a subordinator, giving rise to a path-wise construction of a "time-changed'' process. In probability semigroups, subordination was applied to create the variance gamma process, which is prominently used in financial modelling.

However, subordination may not produce a Lévy process unless the subordinate has independent components or the subordinate has indistinguishable components. A new operation known as weak subordination is introduced that always produces a Lévy process by assigning the distribution of the subordinate conditional on the value of the subordinate, which matches traditional subordination in law in the cases above. Weak subordination is applied to extend the class of variance-generalised gamma convolutions and to construct the weak variance-alpha-gamma process. The latter process exhibits a wider range of dependence than using traditional subordination.

In this presentation I will discuss how I have used mind maps in my teaching. In particular, I will describe an assignment in which students were asked to produce mind maps in a module that heavily featured numerical methods. I will explain my reasons for developing this kind of assignment and what skills I wanted students to develop. I will also reflect on the implementation of the assignment, and on student engagement with it. I will use this as a springboard to briefly discuss challenges and possibilities of helping students develop a richer understanding of mathematics.

On the circle of radius R centred at the origin, consider a ``thin'' sector about the fixed line y = \alpha x with edges given by the lines y = (\alpha \pm \epsilon) x, where \epsilon = \epsilon_R \rightarrow 0 as R \to \infty. We discuss an asymptotic count for S_{\alpha}(\epsilon,R), the number of integer lattice points lying in such a sector, and moreover present results concerning the variance of such lattice points across sectors.

Geometric learning on probability distributions with the Fisher-Rao metric

Information geometry is a differential geometric approach to probability theory and statistics. In this approach, probability distributions are seen as elements of a differentiable manifold, and the Fisher information is used to define a Riemannian metric. The induced geometric tools, such as geodesics, geodesic distances and intrinsic means, have proven useful to interpolate, compare, average or perform segmentation between objects modeled by probability densities.
In this talk, we will give an introduction to geometric learning and information geometry. In particular, we will investigate the Fisher-Rao geometry of beta and Dirichlet distributions, showing that they are negatively curved and geodesically complete. These properties, also shared by other parametric families such as Gaussian distributions, guarantee the existence and uniqueness of geodesics and means. This makes the Fisher-Rao metric a suitable metric, in these parametric families, to use in learning procedures such as K-means clustering.

The assessment and quantification of spatial correlations in complex
systems is of central relevance for a variety of processess, ranging
from chemical reactions to the emergence of social deprivation. The
traditional approaches to the quantification of spatial correlation and
heterogeneity are either based on the comparison with a "well-mixed"
case, which is often pretty artificial and almost irrelevant, or depend
on the choice of a scale parameter, or are affected by the actual size
and peculiarity of the system at hand. In this talk we will explore a
family of methods, inspired by dynamical systems and statistical
physics, to quantify spatial correlations from first principles. These
methods are intrinsically non-parametric, and are able to encompass
information about segregation at all the relevant scales of a system. We
show how these methods allow to uncover spatial patterns in different
real-world systems, and how they correlate with exogenous measures of
segregation and social deprivation.

Concentration-dependent cytotoxicity experiments are frequently used in toxicology. Although it has been reported that an adequate choice of concentrations, i.e., the design, substantially improves the quality of the statistical inference, a recent literature review of three major toxicological journals has shown that the corresponding methods are rarely used in toxicological practice.
In this talk, we address the optimal design problem in cytotoxicity experiments both from an applied and a theoretical point of view. On the one hand, we present strategies and concrete examples of how established statistical methodology can be made more accessible to potential users, especially to biologists. On the other hand, we consider a specific biological challenge in cytotoxicity experiments from the statistician’s point of view: identifying alert concentrations where a pre-specified threshold of the response variable is exceeded. We develop a model-based testing procedure for that purpose and address the corresponding optimal design problem. We construct an optimal design criterion to improve the model-based testing procedure concerning its power. Thus, an optimal design minimizes the maximum variance of the alert concentration estimator. Optimal design theory is developed, and the results are illustrated in several examples where the alert concentration is identified under the assumption of different concentration-response relationships. In particular, it is demonstrated within a simulation study that using the optimal design results in more powerful tests for identifying alerts than using other “non-optimal” designs.

15:00 - Jeremy Mann: "Semiclassical N-body Problem in AdS at Large Spin" //
15:20 - Azadeh Maleknejad: "Stochastic Fermion Creation: Remnant of Gravitational Chiral Anomaly"
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15:40 - Refreshments
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16:10 - Ofer Lahav (UCL): "The Status of Dark Energy Observations"
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17:00 - Pub