AnalysisAffine Moser-Trudinger and Morrey inequalities
Resum: In this talk we establish affine versions of fractional Moser-Trudinger and Morrey inequalities, that are stronger than the Euclidean. These results are limiting cases of affine fractional Sobolev inequalities obtained recently by Haddad and Ludwig. This is a joint work with S. Tikhonov (CRM), Y. Li, D. Yang, W. Yuan (Beijing).
Convex sets can have interior hot spots
Resum: A homogeneous, insulated object with a non-uniform initial temperature will eventually reach thermal equilibrium. The Hot Spots conjecture addresses which point in the object takes the longest to reach this equilibrium: Where is the maximum temperature attained as time progresses?
StatisticsDescobrint el Perquè: presentació del llibre "Causal Inference for Data Science"
Resum: Els tests A/B i els assaigs controlats són sovint costosos i inviables en entorns empresarials. La Inferència Causal per a la Ciència de Dades mostra tècniques per identificar causes a partir de dades, fins i tot sense experiments. Aquesta metodologia permet entendre la causalitat per fer prediccions basades en dades i intervenir per influir en resultats. Veurem alguns resultats pràctics a escenaris empresarials reals.
GeometryAbout an instanton-type PDE for Poisson geometry
Resum: In this talk, I will present an instanton-type PDE associated with a Poisson manifold M. After reviewing its role in an underlying field theory, we present the main theorem showing existence and classification of its solutions. Finally, we discuss its geometric significance leading to a generating function for a symplectic groupoid, Lie-theoretic, integration of M.
Systolic inequalities for S1-invariant contact forms in dimension three, and applications
Resum: In contact geometry, a systolic inequality aims to give a uniform upper bound on the shortest period of a periodic Reeb orbit for contact forms with fixed volume on a given manifold. This generalizes a well-studied notion in Riemannian geometry. It is known that there is no systolic inequality valid for all contact forms on any given contact manifold. In this talk, I will state a systolic inequality for contact forms that are invariant under a circle action in the three-dimensional case, and will discuss applications to a class of Finsler geodesic flows and to a conjecture of Viterbo.
Curvatura escalar i radi d'injectivitat
Resum: Als anys 1960, L. Green va demostrar que el radi d'injectivitat d'una varietat amb curvatura escalar superior a n(n-1) està acotat per π, amb igualtat només per a l'esfera estàndard. Una pregunta natural és llavors si una varietat amb curvatura escalar superior a n(n-1) i un radi d'injectivitat gairebé igual a π s'assembla a l'esfera. Mostraré que a la dimensió 3, si una varietat amb curvatura escalar superior a n(n-1) té un radi d'injectivitat superior a 2π/3, llavors és un quocient de S^3 per un grup cíclic de cardinal senar. La prova utilitza superfícies mínimes i mu-bombolles. En dimensions superiors, aquests mètodes s'apliquen per donar millors límits al radi d'injectivitat de mètriques amb curvatura escalar positiva a S^2xT^kxR^l amb l≤2 i 2+k+l≤7.
Sthochastic and Deterministic ModelInterval-scale classification: a theoretical and practical approach
Resum: In supervised machine learning, the use of appropriate performance metrics enables the evaluation of classification models and facilitates their comparison and selection. In this talk, we focus on the domain of ordinal classification, revisiting performance metrics based on the confusion matrix and proposing a theoretical framework for adapting these metrics to the interval-scale scenario, where measurements are recorded at intervals and both their order and length assume significance. To enhance the applicability of the proposed metrics, we investigate the case of unbounded rightmost intervals, combining both theoretical approaches and simulations. Additionally, we provide an overview of the results obtained from experiments conducted on real-world datasets.
Stochastic analysis meets Euclidean quantum field theory
Resum: An Euclidean (quantum) field describes the spatial statistics associated to the imaginary-time fluctuations of a quantum field theory in Minkowski space. While this seems very remote from probability, it turns out that Euclidean field are beautiful and natural generalisation of Markov processes. Unfortunately they have turned out to be quite complicated to study, so much that even in very simple situations we do not have a good theory for them, just some glimpses and partial results. I would like to discuss insights from stochastic analysis into this problem of modern mathematics.
Dynamical SystemsHistory of special relativity theory: 120 years on
Resum: The aim of this talk is to show that as early as may 1905 (before Einstein had published his famous article) Henri Poincaré has already laid the foundations of the special relativity theory.
Energy function of 2D \& 3D dynamical systems
Resum: The aim of this talk is to prove that the jerk equation of three-dimensional autonomous dynamical systems must be multiplied by the second time derivative of the state variable and not by the first like in dimension two. We then provide an interpretation of the new term appearing in the energy function and called jerk energy. Two and three-dimensional Van der Pol models are then used to exemplify these main results. Applications to Lorenz and Chua's models confirms their validity. Then, we will focus on the energy function of two and three-dimensional coarse systems and prove that it can be directly deduced from the so-called flow curvature manifold.
Limit cycles of canard type of non linear regularizations of piecewise smooth vector fields
Resum: Let $Z$ be a planar piecewise smooth vector field and denote its switching boundary by $\Sigma$. Suppose that $p\in\Sigma$ is a fold-fold singularity. We study limit cycles of nonlinear regularizations of $Z$ around $p$ in the case in which $\Sigma\backslash\{p\}$ consists of sewing points. We give a simple criterion for lower bounds and the existence of limit cycles of canard type, expressed in terms of zeros of slow divergence integral. Using the criterion, we prove that it there exist a quadratic regularization of non-smooth center such that, for any integer $k>0$, it has at least $k$ hyperbolic limit cycles, for a suitably chosen monotonic transition function $\varphi_k:\mathbb{R}\rightarrow\mathbb{R}$ and $\varepsilon > 0$ sufficiently small. We prove a similar result concerning codimension 1 non-smooth focus II$_2$, providing more limit cycles than the known results concerning the Sotomayor--Teixeira regularization. This is a work in progress with Peter de Maesschalck (UHasselt) and Renato Huzak (UHasselt).
Ring TheoryA monoid-theoretical approach to infinite direct-sum decompositions of modules
Resum: It has long been understood that many questions about finite direct-sum decompositions in a class of modules $C$ (closed under isomorphisms and direct sums) translate into factorization questions in the associated monoid $V(C)$. Recently there has been a lot of progress in studying countably generated projective modules, in which case it is also sensible to consider infinite direct sums. We introduce the notion of $\kappa$-monoids, with $\kappa$ a cardinal, to capture such infinite direct sums via a monoid-theoretical approach. A first question is then which $\kappa$-monoids actually arise in such a fashion. This leads to a universal extension property for $\kappa$-monoids and an equivalent braiding property for families of modules. In particular, we see that for a hereditary algebra, the monoid of finitely generated projective modules $V(R)$ completely determines the $\kappa$-monoid $V^{\kappa}(R)$ of $\kappa$-generated projective modules. Together with Bergman's classical realization result for $V(R)$, it follows that the possible $V^{\kappa}(R)$ for hereditary algebras are precisely the universal $\kappa$-extensions of reduced commutative monoids with order-unit.
TopologyTBA
TBA
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