Seminários e cursos curtos

Axiomatic definition of blowups; Cartier divisors; base change property; persistence properties.

The famous Burau representation of the braid group is known to be unfaithful for braids with at least five strands. In the early 2000s, two constructions were provided to fix faithfulness: the first being the Lawrence–Krammer–Bigelow linear representation, hence proving linearity of braid groups, and the second being the Khovanov–Seidel categorical representation. In this talk, based on joint work in progress with Licata, Queffelec, and Wagner, I will investigate the interplay between these two representations.

We study the fluctuations of the size (that is, the number of vertices) of the giant component in the Erdős–Rényi random graph process. The functional CLT in the supercritical case was recently obtained by Enriquez, Faraud and Lemaire. Our approach is based on an exploration algorithm called the simultaneous breadth-first walk, introduced by Limic in 2019, which encodes the dynamics of the evolution of the sizes of the connected components of random graph processes. We will also discuss how our method can be adapted to establish a similar functional CLT in the barely supercritical regime.

This is joint work with Vlada Limic and Sophie Lemaire.

First examples of resolutions; curves and surfaces; effect of blowups on polynomials; singularity invariants; lexicographic orderings.

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

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

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).

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 varitey of possible asymptotic states which in some cases are described by partially hyperbolic lines of equilibria, bands of periodic orbits or generalised Liénard systems.