Seminars from until

Choice of centers of blowup; descent in dimension; lexicographic decrease of invariant; transversality; obstructions in positive characteristic; resolution of planar vector fields.

How many different problems can a neural network solve? What makes two machine learning problems different? In this talk, we'll show how Topological Data Analysis (TDA) can be used to partition classification problems into equivalence classes, and how the complexity of decision boundaries can be quantified using persistent homology. Then we will look at a network's learning process from a manifold disentanglement perspective. We'll demonstrate why analyzing decision boundaries from a topological standpoint provides clearer insights than previous approaches. We use the topology of the decision boundaries realized by a neural network as a measure of a neural network's expressive power. We show how such a measure of expressive power depends on the properties of the neural networks' architectures, like depth, width and other related quantities.

References

• Ballester et al, Topological Data Analysis for Neural Network Analysis: A Comprehensive Survey
• Papamrakou et al, Position: Topological Deep Learning is the New Frontier for Relational Learning
• Petri and Leitão, On The Topological Expressive Power of Neural Networks

Zoom: https://tecnico-pt.zoom.us/j/93935874388?pwd=QHxbpTCtH00rY4OUsRaay48CgaglgB.1

How many different problems can a neural network solve? What makes two machine learning problems different? In this talk, we'll show how Topological Data Analysis (TDA) can be used to partition classification problems into equivalence classes, and how the complexity of decision boundaries can be quantified using persistent homology. Then we will look at a network's learning process from a manifold disentanglement perspective. We'll demonstrate why analyzing decision boundaries from a topological standpoint provides clearer insights than previous approaches. We use the topology of the decision boundaries realized by a neural network as a measure of a neural network's expressive power. We show how such a measure of expressive power depends on the properties of the neural networks' architectures, like depth, width and other related quantities.

References

• Ballester et al, Topological Data Analysis for Neural Network Analysis: A Comprehensive Survey
• Papamrakou et al, Position: Topological Deep Learning is the New Frontier for Relational Learning
• Petri and Leitão, On The Topological Expressive Power of Neural Networks

Zoom: https://tecnico-pt.zoom.us/j/93935874388?pwd=QHxbpTCtH00rY4OUsRaay48CgaglgB.1

The classical Stein-Tomas theorem extends from the theory of linear Fourier restriction estimates for smooth manifolds to the one of fractal measures exhibiting Fourier decay. In the multilinear “smooth” setting, transversality allows for estimates beyond those implied by the linear theory. The goal of this talk is to investigate the question “how does transversality manifest itself in the fractal world?” We will show, for instance, that it could be through integrability properties of the multiple convolution of the measures involved, but that is just the beginning of the story. In the special case of Cantor-type fractals, we will construct multilinear Knapp examples through certain co-Sidon sets which, in some cases, will give more restrictive necessary conditions for a multilinear theorem to hold than those currently available in the literature. This is work in progress with Ana de Orellana (University of St. Andrews, Scotland).

Dimension 4 is the next horizon for applications of Ricci flow to topology, where the main goal is to understand the topological operations that Ricci flow generically performs at singular times. Shrinking Ricci solitons model these topological operations, and only the stable ones should arise generically.

I will present recent joint works with Olivier Biquard and Keaton Naff that determine the stability of all of the currently known shrinking Ricci solitons in dimension 4. The arguments use structural features unique to dimension four, in particular self-duality.

Decomposable plane curves of degree up to 5 were shown to be quantum homogeneous spaces by Brown and Tabiri. It was conjectured that all decomposable plane curves of any degree are quantum homogeneous spaces. In this talk, we will discuss recent results which show that decomposable surfaces and plane curves of any degree are quantum homogeneous spaces. Other algebras such as the reduced algebra will be constructed and its properties discussed.

I will construct an example of a bounded planar domain with one single hole for which the nodal line of a second Dirichlet eigenfunction is closed and does not touch the boundary. This shows that Payne's nodal line conjecture can at most hold for simply-connected domains in the plane.

In this talk, I would like to present some recent results on the existence and multiplicity of positive solutions in $H^1(\mathbb{R}^N)$, $N\ge3$, with prescribed $L^2$-norm, for the stationary nonlinear Schrödinger equation with Sobolev critical power nonlinearity. It is well known that, in the free case, the associated energy functional has a mountain pass geometry on the $L^2$-sphere. This boils down, in higher dimensions, to the existence of a mountain pass solution which is (a suitable scaling of) the Aubin-Talenti function. In this talk, we consider the problem in bounded domains, in the presence of weakly attractive potentials, or under trapping potentials, and investigate the following questions:

1. whether a local minimum solution appears, thus providing an orbitally stable family of solitons, and
2. if the existence of a mountain-pass solution persists.

We provide positive answers under suitable assumptions. This is based on joint work with Dario Pierotti and Gianmaria Verzini (Politecnico di Milano).

In this talk I will discuss some results obtained in collaboration with Filipe C. Mena and former PhD student Vítor Bessa on the global dynamics of a minimally coupled scalar field interacting with a perfect-fluid through a friction-like term in spatially flat homogeneous and isotropic spacetimes. In particular, it is shown that the late time dynamics contain a rich variety of possible asymptotic states which in some cases are described by partially hyperbolic lines of equilibria, bands of periodic orbits or generalised Liénard systems.

The Hopfield Neural Network has played, ever since its introduction in 1982 by John Hopfield, a fundamental role in the inter-disciplinary study of storage and retrieval capabilities of neural networks, further highlighted by the recent 2024 Physics Nobel Prize.

From its strong link with biological pattern retrieval mechanisms to its high-capacity Dense Associative Memory variants and connections to generative models, the Hopfield Neural Network has found relevance both in Neuroscience, as well as the most modern of AI systems.

Much of our theoretical knowledge of these systems however, comes from a surprising and powerful link with Statistical Mechanics, first established and explored in seminal works of Amit, Gutfreund and Sompolinsky in the second half of the 1980s: the interpretation of associative memories as spin-glass systems.

In this talk, we will present this duality, as well as the mathematical techniques from spin-glass systems that allow us to accurately and rigorously predict the behavior of different types of associative memories, capable of undertaking various different tasks.