This week

A spatial graph is a specific type of graph with spatial attributes associated with the nodes and the edges. It is a smart modelling choice for capturing the skeleton of a shape, a blood vessel network, a porous tissue, and many other data objects with intrinsically complex geometry. In this talk, we describe how spatial graphs can be analysed using a specific metric (the Fused Gromov–Wasserstein metric). We extend a testing procedure between distributions of spatial graphs, a depth measure to describe the distribution of spatial graphs, and a dimensionality reduction procedure based on preserving key topological features. We present this variety of methods on a dataset of cardiac fibrosis tissue and on a dataset of fungus mycelium networks.

When returns are partially predictable and trading is costly, utility maximizing investors track a target portfolio at a constant trading speed. The target portfolio is optimal for a frictionless market, where asset returns are scaled back to account for trading costs and volatilities are adjusted to proxy the “execution risk” of holding assets that are costly to trade and exposed to volatile states. The trading speed solves an optimal execution problem, which describes how the legacy portfolio inherited from the past is traded towards the target portfolio in an optimal manner. Unlike for period-by-period mean-variance preferences as in Garleanu and Pedersen (2013), the target portfolio hedges changes in investment opportunities, and both it and the trading speed are linked and depend on execution risk. We set the problem out first in an “absolute” framework – price shocks independent of the price level and investors have CARA preferences – and then in a “relative” framework, with price shocks scaled by price levels and CRRA preferences.

There are two complementary ways to view probability; namely, we can define and apply it in terms of counting or in terms of betting. Both ways are useful in applications of probability including statistics. I will give a historical account of these two ways, argue that they are in some sense dual to each other, and conclude with a brief review of recent work on game-theoretic statistics.

Our bodies are composed of many distinct cell types, each thought to correspond to an attractor state of an underlying high-dimensional regulatory network. Yet, we lack an explicit bottom-up mathematical theory linking molecular mechanisms to observed cell identity dynamics. I will introduce a mechanistic theoretical model that explains how regulatory interactions generate and control an effective high-dimensional landscape for cell identity. Inspired by dense associative memory models, the framework describes how transcription factors couple through shared chromatin modulation, leading to multistability, hierarchical organisation of cell identities, and controlled transitions between states. The model quantitatively predicts cell fate reprogramming outcomes and reconstructs the differentiation structure of haematopoiesis, including progenitor states and bifurcations, without fitting unobserved parameters. More broadly, the framework explains how molecular perturbations reshape the landscape in both normal and cancer cells, connecting molecular regulation to systems-level control of cell identity.

The past few years have seen major advances in understanding the properties of axions in string theory. This progress is thanks to new computational tools that allow for fast and automated calculations with Calabi-Yau manifolds. I will describe the predictions string theory makes for axion masses, decay constants, and axion-photon couplings, and how these depend precisely on the topology of the Calabi-Yau. I will describe explicit constructions of millions of axiverse models on Calabi-Yaus with Hodge numbers up to 491, across the whole Kreuzer-Skarke database (and some results beyond this). Phenomenology computed includes: black hole superradiance, dark matter relic density, fuzzy dark matter, decaying heavy relics and the intergalactic medium, and the QCD axion mass. I will describe the correlation between QCD axion mass and topology, and how this makes it possible for axion "haloscope" experiments to experimentally infer Hodge numbers, divisor topologies, and moduli space loci. I demonstrate the statistical state of the art by computing a full forward model incorporating likelihoods from the cosmic microwave background and Lyman-alpha forest and find the maximum Bayesian posterior probability region on the moduli space of a given CY favoured by a resolution of the tension in these data by an ultralight axion composing 1% of the dark matter.