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We consider a variation of the classical proximal-gradient algorithm for the iterative minimization of a cost function consisting of a sum of two terms, one smooth and the other prox-simple, and whose relative weight is determined by a penalty parameter. This so-called fixed-point continuation method allows one to approximate the problem's trade-off curve, i.e. to compute the minimizers of the cost function for a whole range of values of the penalty parameter at once. The algorithm is shown to converge, and a rate of convergence of the cost function is also derived. Furthermore, it is shown that this method is related to iterative algorithms constructed on the basis of the $\epsilon$-subdifferential of the prox-simple term. Some numerical examples are provided.

A well-known result by Larson and Sweedler shows that integrals on a Hopf algebra can be obtained by applying the Structure Theorem for Hopf modules to the rational part of its linear dual. This fact can be rephrased by saying that taking the space of integrals comes from a right adjoint functor from a category of modules to the category of vector spaces. This observation inspired the categorical approach that we advocate in this work, which yields to a new notion of integrals for bialgebras in the linear setting. Despite the novelty of the construction, it returns the classical definition in the presence of an antipode. We test this new concept on bialgebras that satisfy at least one of the following properties: being coseparable as regular module coalgebras, having a one-sided antipode, being commutative, being cocommutative, or being finite-dimensional. One of the main results we obtain in this process is a dual Maschke-type theorem relating coseparability and total integrals. Remarkably, there are cases in which the space of integrals turns out to be isomorphic to that of the associated Hopf envelope. In particular, this space results to be one-dimensional for finite-dimensional bialgebras, providing an existence and uniqueness theorem for integrals in the finite-dimensional case. Furthermore, explicit computations are given for concrete examples including the polynomial bialgebra with one group-like variable, the quantum plane and the coordinate bialgebra of $n$-by-$n$ matrices.

This paper exploit the equivalence between the Schr\"odinger Bridge problem and the entropy penalized optimal transport in order to find a different approach to the duality, in the spirit of optimal transport. This approach results in a priori estimates which are consistent in the limit when the regularization parameter goes to zero. In particular, we find a new proof of the existence of maximizing entropic-potentials and therefore, the existence of a solution of the Schr\"odinger system. Our method extends also when we have more than two marginals: we can provide an alternative proof of the convergence of the Sinkhorn algorithm with two marginals and we show that the Sinkhorn algorithm converges in the multi-marginal case.

We extend the scope of the dynamical theory of extreme values to cover phenomena that do not happen instantaneously, but evolve over a finite, albeit unknown at the onset, time interval. We consider complex dynamical systems, composed of many individual subsystems linked by a network of interactions. As a specific example of the general theory, a model of neural network, introduced to describe the electrical activity of the cerebral cortex, is analyzed in detail: on the basis of this analysis we propose a novel definition of neuronal cascade, a physiological phenomenon of primary importance. We derive extreme value laws for the statistics of these cascades, both from the point of view of exceedances (that satisfy critical scaling theory) and of block maxima.

Let $(F_n)_{n \geq 1}$ be the sequence of Fibonacci numbers. Guy and Matiyasevich proved that \begin{equation*} \log \operatorname{lcm} (F_1, F_2, \dots, F_n) \sim \frac{3 \log \alpha}{\pi^2} \cdot n^2 \quad \text{as } n \to +\infty, \end{equation*} where $\operatorname{lcm}$ is the least common multiple and $\alpha := \big(1 + \sqrt{5}) / 2$ is the golden ratio. We prove that for every periodic sequence $\mathbf{s} = (s_n)_{n \geq 1}$ in $\{-1,+1\}$ there exists an effectively computable rational number $C_{\mathbf{s}} > 0$ such that \begin{equation*} \log \operatorname{lcm} (F_3 + s_3, F_4 + s_4, \dots, F_n + s_n) \sim \frac{3 \log \alpha}{\pi^2} \cdot C_\mathbf{s} \cdot n^2 , \quad \text{as } n \to +\infty . \end{equation*} Moreover, we show that if $(s_n)_{n \geq 1}$ is a sequence of independent uniformly distributed random variables in $\{-1,+1\}$ then \begin{equation*} \mathbb{E}\big[\log \operatorname{lcm} (F_3 + s_3, F_4 + s_4, \dots, F_n + s_n)\big] \sim \frac{3 \log \alpha}{\pi^2} \cdot \frac{15 \operatorname{Li}_2(1 / 16)}{2} \cdot n^2 , \quad \text{as } n \to +\infty , \end{equation*} where $\operatorname{Li}_2$ is the dilogarithm function.

Given a holomorphic or anti-holomorphic involution on a complex variety, the Smith inequality says that the total $\mathbb{F}_2$-Betti number of the fixed locus is no greater than the total $\mathbb{F}_2$-Betti number of the ambient variety. The involution is called maximal when the equality is achieved. In this paper, we investigate the existence problem for maximal involutions on higher-dimensional compact hyper-Kähler manifolds and on Hilbert schemes of points on surfaces. We show that for $n\geq 2$, a hyper-Kähler manifold of K3$^{[n]}$-deformation type admits neither maximal anti-holomoprhic involutions (i.e. real structures), nor maximal holomorphic (symplectic or anti-symplectic) involutions. In other words, such hyper-Kähler manifolds do not contain maximal (AAB), (ABA), (BAA) or (BBB)-branes. For Hilbert schemes of points on surfaces, we show that for a holomorphic (resp. anti-holomorphic) involution $\sigma$ on a smooth projective surface $S$ with $H^1(S, \mathbb{F}_2)=0$, the naturally induced involution on the $n$th Hilbert scheme of points is maximal if and only if $\sigma$ is a maximal involution of $S$ and it acts on $H^2(S, \mathbb{Z})$ trivially (resp. as $-\operatorname{id}$). This generalizes previous results of Fu and Kharlamov-Răsdeaconu.

We analyse the Maxwell's spectrum on thin tubular neighborhoods of embedded surfaces of $\mathbb R^3$. We show that the Maxwell eigenvalues converge to the Laplacian eigenvalues of the surface as the thin parameter tends to zero. To achieve this, we reformulate the problem in terms of the spectrum of the Hodge Laplacian with relative conditions acting on co-closed differential $1$-forms. The result leads to new examples of domains where the Faber-Krahn inequality for Maxwell's eigenvalues fails, examples of domains with any number of arbitrarily small eigenvalues, and underlines the failure of spectral stability under singular perturbations changing the topology of the domain. Additionally, we explicitly produce the Maxwell's eigenfunctions on product domains with the product metric, extending previous constructions valid in the Euclidean case.

We consider the (viscosity) solution $u(x,t)$ of the nonlinear evolution equation $u_t-\Delta^G_p u=0$ in a (not necessarily bounded) domain $\Omega$, such that $u=0$ in $\Omega$ at time $t=0$ and $u=1$ on the boundary of $\Omega$ at all times. Here, $\Delta_p^G$ is the game-theoretic $p$-laplacian, a $1$-homogeneous version of the standard $p$-laplacian. Also, we consider the (viscosity) solution $u^\varepsilon$ of the nonlinear elliptic equation $\varepsilon^2\Delta_p^G u^\varepsilon= u^\varepsilon$ in $\Omega$, satisfying $u^\varepsilon=1$ on its boundary. In this thesis, we establish asymptotic formulas for small positive values of $t$ and $\varepsilon$ involving both the values of $u(x,t)$ and $u^\varepsilon(x)$ and their $q$-means on balls touching the boundary. In the spirit of S.~R.~S.~Varadhan's work, we associate appropriate rescalings of the values of $u(x,t)$ and $u^\varepsilon(x)$ to the distance of $x$ to the boundary of $\Omega$. We also provide accurate uniform estimates of the rate of approximation in these formulas, highlighting the dependence on both the parameter $p$ and the regularity of the domain. The uniform estimates are new results also in the linear case. Also, we connect the asymptotic behavior of $q$-means on balls touching the boundary to a suitable function of principal curvatures. These results generalize and extend formulas for the heat content, obtained by R. Magnanini and S. Sakaguchi for $p=q=2$. Finally, we give a few applications of the asymptotic formulas to geometric and symmetry results. In particular, we characterize time-invariant level surfaces of $u(x,t)$ (or $\varepsilon$-invariant level surfaces of $u^\varepsilon(x)$) as spheres and hyperplanes.

Plug and Play (PnP) methods achieve remarkable results in the framework of image restoration problems for Gaussian data. Nonetheless, the theory available for the Gaussian case cannot be extended to the Poisson case, due to the non-Lipschitz gradient of the fidelity function, the Kullback-Leibler functional, or the absence of closed-form solution for the proximal operator of such term, leading to employ iterative solvers for the inner subproblem. In this work we extend the idea of PIDSplit+ algorithm, exploiting the Alternating Direction Method of Multipliers, to PnP scheme: this allows to provide a closed form solution for the deblurring step, with no need for iterative solvers. The convergence of the method is assured by employing a firmly non expansive denoiser. The proposed method, namely PnPSplit+, is tested on different Poisson image restoration problems, showing remarkable performance even in presence of high noise level and severe blurring conditions.

This set of notes is intended for a short course aiming to provide an (almost) self-contained and (almost) elementary introduction to the topic of Information Geometry (IG) of the probability simplex. Such a course can be considered an introduction to the original monograph by Amari and Nagaoka (2000), and to the recent monographs by Amari (2016} and by Ay et al. (2017). The focus is on a non-parametric approach, that is, I consider the geometry of the full probability simplex and compare the IG formalism with what is classically done in Statistical Physics.

In the realm of complex systems, dynamics is often modeled in terms of a non-linear, stochastic, ordinary differential equation (SDE) with either an additive or a multiplicative Gaussian white noise. In addition to a well-established collection of results proving existence and uniqueness of the solutions, it is of particular relevance the explicit computation of expectation values and correlation functions, since they encode the key physical information of the system under investigation. A pragmatically efficient way to dig out these quantities consists of the Martin-Siggia-Rose (MSR) formalism which establishes a correspondence between a large class of SDEs and suitably constructed field theories formulated by means of a path integral approach. Despite the effectiveness of this duality, there is no corresponding, mathematically rigorous proof of such correspondence. We address this issue using techniques proper of the algebraic approach to quantum field theories which is known to provide a valuable framework to discuss rigorously the path integral formulation of field theories as well as the solution theory both of ordinary and of partial, stochastic differential equations. In particular, working in this framework, we establish rigorously, albeit at the level of perturbation theory, a correspondence between correlation functions and expectation values computed either in the SDE or in the MSR formalism.

Pointwise estimates for the gradient of solutions to the $p$-Laplace system with right-hand side in divergence form are established. They enable us to develop a nonlinear counterpart of the classical Calder\'on-Zygmund theory in terms of Calder\'on-Zygmund singular integrals, for the Laplacian. As a consequence, a flexible, comprehensive approach to gradient bounds for the $p$-Laplace system for a broad class of norms is derived. In particular, new gradient estimates are exhibited, and well-known results in customary function spaces are easily recovered.

We introduce a novel capacity measure 2sED for statistical models based on the effective dimension. The new quantity provably bounds the generalization error under mild assumptions on the model. Furthermore, simulations on standard data sets and popular model architectures show that 2sED correlates well with the training error. For Markovian models, we show how to efficiently approximate 2sED from below through a layerwise iterative approach, which allows us to tackle deep learning models with a large number of parameters. Simulation results suggest that the approximation is good for different prominent models and data sets.

A well-known result by Larson and Sweedler shows that integrals on a Hopf algebra can be obtained by applying the Structure Theorem for Hopf modules to the rational part of its linear dual. This fact can be rephrased by saying that taking the space of integrals comes from a right adjoint functor from a category of modules to the category of vector spaces. This observation inspired the categorical approach that we advocate in this work, which yields to a new notion of integrals for bialgebras in the linear setting. Despite the novelty of the construction, it returns the classical definition in the presence of an antipode. We test this new concept on bialgebras that satisfy at least one of the following properties: being coseparable as regular module coalgebras, having a one-sided antipode, being commutative, being cocommutative, or being finite-dimensional. One of the main results we obtain in this process is a dual Maschke-type theorem relating coseparability and total integrals. Remarkably, there are cases in which the space of integrals turns out to be isomorphic to that of the associated Hopf envelope. In particular, this space results to be one-dimensional for finite-dimensional bialgebras, providing an existence and uniqueness theorem for integrals in the finite-dimensional case. Furthermore, explicit computations are given for concrete examples including the polynomial bialgebra with one group-like variable, the quantum plane and the coordinate bialgebra of $n$-by-$n$ matrices.

Deep Neural Networks are widely used for solving complex problems in several scientific areas, such as speech recognition, machine translation, image analysis. The strategies employed to investigate their theoretical properties mainly rely on Euclidean geometry, but in the last years new approaches based on Riemannian geometry have been developed. Motivated by some open problems, we study a particular sequence of maps between manifolds, with the last manifold of the sequence equipped with a Riemannian metric. We investigate the structures induced trough pullbacks on the other manifolds of the sequence and on some related quotients. In particular, we show that the pullbacks of the final Riemannian metric to any manifolds of the sequence is a degenerate Riemannian metric inducing a structure of pseudometric space, we show that the Kolmogorov quotient of this pseudometric space yields a smooth manifold, which is the base space of a particular vertical bundle. We investigate the theoretical properties of the maps of such sequence, eventually we focus on the case of maps between manifolds implementing neural networks of practical interest and we present some applications of the geometric framework we introduced in the first part of the paper.

We study stability of optimizers and convergence of Sinkhorn's algorithm in the framework of entropic optimal transport. We show entropic stability for optimal plans in terms of the Wasserstein distance between their marginals under a semiconcavity assumption on the sum of the cost and one of the two entropic potentials. When employed in the analysis of Sinkhorn's algorithm, this result gives a natural sufficient condition for its exponential convergence, which does not require the ground cost to be bounded. By controlling from above the Hessians of Sinkhorn potentials in examples of interest, we obtain new exponential convergence results. For instance, for the first time we obtain exponential convergence for log-concave marginals and quadratic costs for all values of the regularization parameter. Moreover, the convergence rate has a linear dependence on the regularization: this behavior is sharp and had only been previously obtained for compact distributions arXiv:2407.01202. Other interesting new applications include subspace elastic costs [Cuturi et al. PMLR 202(2023)], weakly log-concave marginals, marginals with light tails, where, under reinforced assumptions, we manage to improve the rates obtained in arXiv:2311.04041, the case of unbounded Lipschitz costs, and compact Riemannian manifolds.

We show a sufficient criterion to determine if a planar set $\Omega$ is a minimizer of the prescribed curvature functional among all of its subsets. As a special case, we derive a sufficient criterion to determine if $\Omega$ is a self-Cheeger set, i.e. if it minimizes the ratio $P(E)/|E|$ among all of its subsets. Specifically, if a Jordan domain $\Omega$ possesses the interior disk property of radius $|\Omega|/P(\Omega)$, then it is a self-Cheeger set; if it possesses the strict interior disk property then it is a minimal Cheeger set, i.e. the unique minimizer. As a side effect we provide a way to build self-Cheeger sets.

We give a lattice-theoretic classification of non-symplectic automorphisms of prime order of irreducible holomorphic symplectic manifolds of OG10 type. We determine which automorphisms are induced by a non-symplectic automorphism of prime order of a cubic fourfold on the associated LSV manifolds, giving a geometric and lattice-theoretic description of the algebraic and transcendental lattices of the cubic fourfold. As an application we discuss the rationality conjecture for a general cubic fourfold with a non-symplectic automorphism of prime order.