Upcoming Seminars: Mathematics and Artificial Intelligence

I will discuss stochastic systems of interacting particles with non-overlapping constraints, which give rise to so-called excluded-volume interactions. The aim is to derive effective macroscopic equations governing the evolution of particle densities from the underlying microscopic dynamics. When particles possess nontrivial size or shape, geometric constraints become essential: they complicate the coarse-graining process and strongly influence the emergent behaviour of the system. I will present two representative examples, hard spheres and infinitely thin needles, highlighting how geometry enters the macroscopic description.

Hill functions are widely used to model biological input-output responses. While long treated as empirical fits, Martinez-Corral, Nam, DePace, and Gunawardena proposed Hill functions as the universal Hopfield barrier for sharpness of systems at thermodynamic equilibrium. Their case rested on numerical evidence for n

Since the invention of the compound microscope in the early seventeenth century, scientists have marvelled over red blood cells and their surprising shape. An influential model of Canham predicts the shapes of blood cells and similar biomembranes come from a variational problem minimizing the “bending energy” of these surfaces. Because observed cells have the same shape in humans, it is natural to ask whether the model admits a unique solution. Yu and Chen reduced solution uniqueness for the genus one Canham problem (for a range of isoperimetric ratios) to proving positivity of a P-recursive sequence defined by an explicit linear recurrence relation with polynomial coefficients. In this talk we discuss a method of proving this positivity property, joint with Marc Mezzarobba, and its generalization, with Mezzarobba and Ruiwen Dong, to a wide range of P-recursive sequences. We combine rigorous numeric analytic continuation of D-finite functions with classic bounds from singularity analysis to derive an effective index where the asymptotic behaviour of the sequence, which is positive, dominates the sequence behaviour. Positivity of the finite number of remaining terms can then be checked computationally. Our work has been incorporated into the SageMath ore_algebra package, and can be used by researchers to automatically prove positivity for “generic” positive P-recursive sequences.

TBA

The flows of tissues of epithelial cells often involve neighbour exchanges called T1 transitions. Mechanically, they are irreversible rearrangements crossing an energy barrier. In this talk, I will deploy geometric constructions of classical Euclidean geometry to calculate this energy barrier for general, isolated T1 transitions dominated by line tensions. I will show how regularity of cell packings, tension fluctuations, and nonlinear tensions increase this energy barrier, providing the basis for coarse-grained understanding of cell neighbour exchanges in continuum models of epithelia.