Junior Seminars

Nonlinear Spin System

by Mario Morellini

February 26, 2025 – 14:00 – Room A (Physics Dept.)

Abstract. In this seminar, we explore a class of nonlinear dynamics governed by the framework of mass action dynamics, which models the evolution of systems through pairwise interactions. This framework underpins several important nonlinear models across disciplines, including chemical reaction networks, Boltzmann's ideal gas model, and recombination processes in population genetics and genetic algorithms.
Our primary focus is on the nonlinear dynamics of spin systems, with an emphasis on the Ising model. Specifically, we investigate the convergence to equilibrium for spin systems at high temperatures, where non-trivial correlations are present.
Utilizing Kac's Program, we interpret these nonlinear systems as limits of Markov chain particle systems under the propagation of chaos as system size tends to infinity. This perspective enables us to estimate the rate of convergence to equilibrium. A key aspect of our approach is ensuring that the Modified Log-Sobolev inequalities used to study the particle systems remain independent of system size. When this condition is met, the nonlinear system converges to equilibrium at the same rate as its corresponding particle system.
We will present two methods for analyzing this convergence:

1. A direct approach using McKean trees to study the nonlinear system.
2. An indirect approach that derives convergence from particle systems using size-independent Modified Log-Sobolev inequalities.

The Prym-canonical Clifford index

by Martina Miseri

December 11, 2024 – 16:00 – Room C (Physics Dept.)

Abstract. After a brief introduction on algebraic curves and what Brill-Noether Theory is about, I will try to give an idea of the geometric meaning of the Clifford index, a classic invariant for algebraic curves. I will conclude the seminar presenting my research project, that starts with a new definition of Clifford index and focuses on the case of hyperelliptic curves.

Linear Flow on the Infinite Torus

by Lorenzo Baroni

November 20, 2024 – 16:00 – Room C (Physics Dept.)

Abstract. The celebrated Arnold-Liouville theorem provides a beautiful description of integrable Hamiltonian systems stating that the dynamics on compact invariant sets is conjugated to a linear flow on the torus \(\mathbb{T}^n := \mathbb{R}^n/\mathbb{Z}^n\). It is then clear that in trying to generalize this statement to Hamiltonian systems with infinitely many degrees of freedom, one has first to face with the issue of understanding linear flows on the infinite torus. In this talk we discuss the simple characterization occurring in finite dimension and why it fails to apply in an infinite-dimensional setting. Then, we use Pontryagin’s theory of locally compact abelian groups in order to understand the dynamics and we also discuss an analogy with the finite dimensional case, namely unique ergodicity in the absence of resonances.

Junior Seminars

Academic Year 2023/24

Partecipants.

• Zakaria Brahimi – Cubic fourfolds: some rational examples
• Adrien Ragot – Realisability: from constructive proofs to program specification
• Muhammad Usman – 3D Human Pose Estimation: Real-time Performance and Applications
• Federico Pieroni – An overview on Coble surfaces, their moduli, and the Coble conjecture
• Luca Ferrigno – Unlikely Intersections in diophantine geometry
• Elena Sammarco – The rationality problem for the cubic fourfold

Junior Seminars

Academic Year 2022/23

Organised by Armando Capasso and Elia Onofri
with the participation of Martina Miseri and Bruno Renzi.

Partecipants.

• Armando Capasso – What is... a Higgs Bundle?
• Elia Onofri – How machine learning can improve macroscopic models for traffic state estimation and forecast
• Simone Pesatori – Coincident root loci and the moduli space of rational elliptic surfaces
• F. C. Simon Schirren – Virtual Cycles: An Introduction
• Bruno Renzi – The dimer model: phase diagram and universality

G.T.M. – General Topics in Mathematics (informal seminars)

Academic Year 2021/22

Organised by Armando Capasso and Elia Onofri

Partecipants.

• Elia Onofri – On tweakable black-box polinomials: the cube attacks family
• Armando Capasso – Numerically flatness for Higgs bundles over compact Kähler manifolds
• Luca Ferrigno – An Overview of Diophantine Geometry
• Muhammad Usman – Symbol-based preconditioning for Riesz distributed-order space-fractional diffusion equations
• Elia Onofri – Attribute-based colouring and its applications in reducing problem's size
• Armando Capasso – Algebraic hypersurfaces with vanishing Hessian and CW-complexes
• Elia Onofri – The role of the applied mathematician: optimisation of pedestrian flows in crowded museums