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.

Cubic fourfolds: some rational examples

by Zakaria Brahimi

May 22, 2024 – 16:00 – Room M3

Abstract. Projective spaces are the most basic algebraic varieties that we know and rational varieties are those that are the closest possible to being projective spaces. The question of rationality of algebraic varieties has led to the development of a rich and beautiful theory and many questions are still open. Historically, the rationality problem for quadric and cubic surfaces was settled classically over one hundred years ago. In this seminar, we will talk about some rational cubic fourfolds that are known. We look for them inside Hassett Divisors; These divisors are codimension one varieties in the Moduli space of cubic fourfolds.

Realisability: from constructive proofs to program specification

by Adrien Ragot (Université Paris Nord)

April 10, 2024 – 16:00 – Room M3

Abstract. In a first part - after recalling some basics of proof theory, namely what is a formal proof - we will introduce the key concept of Realisability. The Brouwer-Heyting-Kolmogorov (BHK) interpretation of proofs - originally formulated for intuistionistic logic - describes fundamental properties that should enjoy proofs. In particular it implies that the space in which proofs lives should be equipped with a product and a notion of interaction. Realisability is the implementation of the BHK-interpretation inside an untyped model of computation - first inside recursive function, variants of the lambda calculus and more recently in the context of linear logic in sets of permutations, operators on an Hilbert space, or weighted graphs. We will point out how the first versions of Realisability fall short if one wants to adopt an interactive approach and how this limit relates to consistency. We exhibit the existing modern solutions that constitutes Interactive Realisability, briefly we expose how interactive realisability is relevant for program specification.
In a second part, we will present parts of our current work on Realisability for Linear Logic. We show how a simple algebraic structure, a polarized self-operand, is suited to realize multiplicative linear logic. We relate the problem of completeness in realisability to the correctness criterions of proof structures for Linear Logic. In order to study correctness, we present an algebraic framework of "path algebra" to study the connectedness and acyclicity of a graph. If times allow we will discuss in particular how one can handle second order quantifiers.

3D Human Pose Estimation: Real-time Performance and Applications

by Muhammad Usman

March 6, 2024 – 16:00 – Room 108 (Physics Dept.)

Abstract. Human pose estimation is a task that involves identifying the location of specific points in an image, usually referred to as keypoints. The keypoints can represent various parts of the object such as joints, landmarks, or other distinctive features. YOLO (You Only Look Once), a current state-of-the-art deep learning object detection model, simultaneously identifies and predicts the keypoints that define the human body's pose. In our talk, we will present a real-time 3D human pose estimation system by employing two cameras and triangulation methods to calculate the 3D pose from the 2D pose predicted by YOLO. Our results showcase the potential of the proposed method as a robust tool for real-time 3D human pose estimation. Beyond accuracy and efficiency, this approach opens up numerous possibilities across various domains, including human-computer interaction, sports analytics, and surveillance.

An overview on Coble surfaces, their moduli, and the Coble conjecture

by Federico Pieroni

January 31, 2024 – 16:00 – Room B (Physics Dept.)

Abstract. Since the late 19^th century, a classical problem in Algebraic Geometry was to find a nice description of the group of birational transformations of the projective plane \(\mathbb{P}^2\) to itself. Coble surfaces were introduced with this aim. After a brief introduction on their definition and main properties, we will reconstruct the moduli space of Coble surfaces \(M_{\mathrm{Coble}}\), compute its dimension, and show that every Coble surface \(X\) admits a distinguished Coble curve \(C \subset X\). We will conclude the discussion talking about the Coble conjecture: for a fixed Coble surface \(X\), can we find transformations \(\varphi : X \rightarrow X\) acting identically on the Coble curve? How many surfaces inside \(M_{\mathrm{Coble}}\) have this property?

Unlikely Intersections in diophantine geometry

by Luca Ferrigno

December 12, 2023 – 10:00 – Room B (Physics Dept.)

Abstract. We talk about "unlikely intersections" whenever we have a non-empty intersection between algebraic varieties that, for dimensional reasons, we do not expect to intersect: for example, if \(X, Y\) are varieties of dimensions \(r\) and \(s\) in a space of dimension \(n > r + s\), we usually expect \(X \cap Y\) to be empty.
This expectation is the basis of many problems in diophantine geometry, which usually study the finiteness of the number of points satisfying some arithmetical conditions on algebraic varieties.
After an overview of some problems of unlikely intersections in algebraic tori, we will focus on unlikely intersections in families of elliptic curves.

The rationality problem for the cubic fourfold

by Elena Sammarco

November 14, 2023 – 10:30 – Room B (Physics Dept.)

Abstract. One of the main open problems of Birational Geometry is to determine the rationality of the cubic hypersurfaces in \(\mathbb{P}^5\), the so-called cubic fourfolds: to date there are few explicit results and several conjectures, but no universal criteria. After an overview of the state of the art of the problem on the complex field \(\mathbb{C}\), we briefly present a new geometric construction of a divisor in the moduli space of cubic fourfolds which could represent a significant contribution in this general discussion.

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