An introduction to kinematic formulas (and why they can be studied on manifolds)

Giacomo Di Paolo, Goethe University Frankfurt

When: September 28th, 2026 at 5pm

Where: Seminario 1 (Dip. Mat. UNIBO)

Abstract: The kinematic formulas express the expected measure of the intersection of two convex bodies under random rigid motions as a sum of products of their intrinsic volumes. In the Euclidean setting, these formulas follow directly from Hadwiger’s characterization theorem and can be localized via Federer’s curvature measures. Between 2005 and 2007, Alesker laid the foundations of valuation theory on manifolds to extend integral geometry to the non-flat setting.

After a brief introduction to the fundamental elements of classical integral geometry, we use tools coming from Alesker’s theory to study local kinematic formulas on an isotropic manifold (M, G), developing an algebraic operation that encodes this formulas in full generality. 

This work is part of my PhD thesis, supervised by Andreas Bernig at Goethe Universit¨at Frankfurt.

Alberto Amaduzzi, Unibo Physics department 

When: November 9th, 2026 at 5pm

Where: Seminario 1 (Dip. Mat. UNIBO)

Abstract: Standard models of adaptive vehicle dynamics on a single long road admit steady-flow solutions that become unstable over certain density ranges. Small density fluctuations then evolve, at first order, according to a Korteweg–de Vries perturbation, known to admit soliton solutions in density. When studying congestion in a road network, however, intersections introduce bifurcations that prevent a purely PDE-based description of congestion. Several coarser models have therefore been proposed, exploiting the relation between speed and density established by macroscopic traffic theories, and using road capacity (maximum flow) as a proxy quantity of interest. Using a GPS dataset, we construct a Markov process that describes a steady-state solution for the number of travelers across the road network. We develop a method to estimate the associated transition matrix, moving toward a description of the system in terms of a Fokker–Planck equation.