L4

Every population consists of individuals who vary in their traits, and each trait may, or may not, be associated with frailty or fitness. Variation in frailty and fitness traits makes population studies prone to selective depletion bias (SDB). The issue is widespread across scientific domains. When an ageing cohort exhibits declining mortality, is it individuals becoming healthier or selective depletion of the frail? In an epidemic, when growth in cumulative infections decelerates, is it individuals cautiously changing behaviour or selective depletion of the most susceptible? In microbial populations, when mutations increase population vulnerability to stress, it is individuals becoming more vulnerable or mutant populations having higher variance in fitness? In each case, the first explanation invokes individuals changing, while the second recognises that populations change due to selection on pre-existing variation. While the former are intuitive and widely adopted, explanations that rely on selective depletion are more neutral and less commonly considered due to cognitive biases and challenges in estimating all variation that matters. 

Remodelling selection (ReMS) is proposed as a general strategy of study design and analysis to address the SDB problem. I will show how the approach has been employed in specific case studies and how it has been formalised in general.

Phenotypic variation is a ubiquitous feature of biological cell populations, even in genetically identical cells growing in uniform environments. Such variability can have profound consequences for population-level behaviour, particularly under stress, yet it is often neglected in classical modelling frameworks.

In the first part of this talk, I consider mathematical models of bacterial population growth that explicitly incorporate non-heritable variation in individual cell growth rates. I examine how phenotypic heterogeneity and environmental selection shape population growth and the dynamics of phenotypic subpopulations. We derive theoretical results for population growth rates and compare them with predictions from homogeneous models, identifying regimes in which variability qualitatively alters population outcomes.

 In the second part of the talk, I turn to self-generated chemotaxis, a collective process in which cells modify their chemical environment to guide movement. Using a hybrid discrete–continuum model that couples stochastic cell motion with a continuum description of the chemoattractant, I investigate how phenotypic variation in motility, sensing, and chemical degradation affects the robustness of collective migration. The results and tools developed have broader implications for collective behaviour in cell biology, ecology, and evolution.

I will describe a circle-valued Morse theory for simplicial complexes. The central objects of study are partial matchings which admit certain zigzag cycles; these cyclic matchings lift canonically to acyclic matchings on the infinite cyclic cover of the underlying simplicial complex. From the lifted acyclic matchings, we obtain a finitely generated Morse chain complex defined over the Novikov ring, which consists of power series in one variable with finite negative support. We then establish a quasi-isomorphism between this Morse-Novikov complex and the simplicial chain complex of the cyclic cover, duly completed over the Novikov ring. As a pleasant consequence, we can define new computable invariants to detect (obstructions to) the fiberedness of tame knots.

In this talk we summary some recent progress on limit theorems for the Wasserstein distance of empirical measures of Markov processes. For symmetric diffusion processes on Riemannian manifold possibly with reflecting or killing boundary, the sharp convergence rate is derived with renormalization limit formulated by using the spectrum of the generator. Moreover, a general framework is established to estimate the convergence rate in Wasserstein distance of empirical measures for ergodic Markov processes.

Recent advances at the interface of stochastic analysis, rough path theory, stochastic filtering, stochastic control, and mean-field systems have led to a rapidly developing framework for analyzing stochastic dynamics conditioned on common/observation noise. This mini course  will survey how rough stochastic differential equations, introduced in 2021 by A. Hocquet, K. Lê and the speaker, lead to a unifying perspective across several areas of applied probability. (Additional coauthors include F. Bugini, J. Dause, W. Stannat, H. Zhang and P.Zorin-Kranich).

Recent advances at the interface of stochastic analysis, rough path theory, stochastic filtering, stochastic control, and mean-field systems have led to a rapidly developing framework for analyzing stochastic dynamics conditioned on common/observation noise. This mini course  will survey how rough stochastic differential equations, introduced in 2021 by A. Hocquet, K. Lê and the speaker, lead to a unifying perspective across several areas of applied probability. (Additional coauthors include F. Bugini, J. Dause, W. Stannat, H. Zhang and P.Zorin-Kranich).