Computational Mathematics seminars

Michael Borinsky introduces tropical quantum field theory as a simpler deformation limit that preserves useful information about an interacting theory. Its numerical algorithm computes massive-scalar perturbation theory with runtime polynomial in loop order and external-leg count, avoiding the factorial growth of a Feynman-diagram expansion and making very high loop orders feasible. Geometrically, amplitudes become volumes of graph-moduli spaces, whose recursions resemble those for Weil–Petersson volumes. The talk connects the limiting construction, geometry and algorithm.

A Markov chain can sample effectively from a random or data-informed start even when worst-case initialization mixes slowly. This talk uses weak Poincaré inequalities to analyse that regime, with applications to simulated annealing for spherical spin glasses and to mixtures of log-concave distributions.

Continuum Glauber dynamics samples Gibbs point processes with repulsive interactions, including hard spheres. This talk develops continuous spectral-independence and localization tools to establish spectral gaps and efficient sampling for arbitrary-range repulsive potentials. The approach improves fixed-density packing thresholds and works directly in the continuum rather than discretizing the model.

Correlation decay for graph colourings is central to efficient sampling but remains difficult to prove near conjectured thresholds. This talk establishes strong spatial mixing on trees of maximum degree Delta for at least Delta plus three colours, when Delta is at least three. It then obtains optimal Glauber-dynamics mixing for fixed degree and sufficiently large girth under the same colour bound.

A refinement of the product replacement algorithm aims to draw nearly uniform elements from a finite group. This talk studies how many steps are needed for the output distribution to approach uniformity, using spectral-gap estimates and computer-assisted calculations in joint work with Michał Marcinkowski.

Sampling uniformly from a convex body links randomized volume computation to the Kannan–Lovász–Simonovits conjecture. This talk identifies geometric quantities controlling sampler mixing and introduces stochastic localization as a way to bound them, including a simple argument yielding a polylogarithmic, though nonoptimal, bound.