Searcharxiv⌕ Search

arXiv subjects

Gonçalo Oliveira

Publications and source records attributed to Gonçalo Oliveira.

18 recordsLinked to original sources
arXiv2026

Deformed Hermitian-Yang-Mills equation on the manifold of full flags

We construct the first example of a higher rank, irreducible deformed Hermitian-Yang-Mills (dHYM) connection in the small radius regime. We also construct these in the large radius regime on infinitely many different bundles and make some contributions to the rank one equation as well. In particular, we investigate solutions away from the supercritical regime, showing the existence of solutions with any possible angle, and rule out some possible stability conditions.

math.DG↗
arXiv2025

A ZeNN architecture to avoid the Gaussian trap

We propose a new simple architecture, Zeta Neural Networks (ZeNNs), in order to overcome several shortcomings of standard multi-layer perceptrons (MLPs). Namely, in the large width limit, MLPs are non-parametric, they do not have a well-defined pointwise limit, they lose non-Gaussian attributes and become unable to perform feature learning; moreover, finite width MLPs perform poorly in learning high frequencies. The new ZeNN architecture is inspired by three simple principles from harmonic analysis: i) Enumerate the perceptons and introduce a non-learnable weight to enforce convergence; ii) Introduce a scaling (or frequency) factor; iii) Choose activation functions that lead to near orthogonal systems. We will show that these ideas allow us to fix the referred shortcomings of MLPs. In fact, in the infinite width limit, ZeNNs converge pointwise, they exhibit a rich asymptotic structure beyond Gaussianity, and perform feature learning. Moreover, when appropriate activation functions are chosen, (finite width) ZeNNs excel at learning high-frequency features of functions with low dimensional domains.

cs.LG↗
arXiv2025

Counting Equilibria of the Electrostatic Potential

In 1873, James C. Maxwell conjectured that the electric field generated by $n$ point charges in generic position has at most $(n-1)^2$ isolated zeroes. The first (non-optimal) upper bound was only obtained in 2007 by Gabrielov, Novikov and Shapiro, who also posed two additional interesting conjectures. In this article, we give the best upper bound known to date on the number of zeroes of the electric field, and construct a counterexample to a conjecture of Gabrielov, Novikov and Shapiro that the number of equilibria cannot exceed those of the distance function defined by the unit point charges. Finally, we note that it is quite possible that Maxwell's quadratic upper bound is not tight, so it is prudent to find smaller bounds. Hence, we also explore examples and construct configurations of charges achieving the highest ratios of the number of electric field zeroes by point charges found to this day.

cs.CG↗
arXiv2024

Hermitian, Ricci-flat toric metrics on non-compact surfaces à la Biquard-Gauduchon

Biquard-Gauduchon have shown that conformally Kähler, Ricci-flat, ALF toric metrics on the complement of toric divisors are: the Taub-NUT metric with reversed orientation, in the Kerr-Taub-bolt family or in the Chen-Teo family. The same authors have also given a unified construction for the above families relying on an axi-symmetric harmonic function on $\mathbb{R}^3$. In this work, we reverse this construction and use methods from a paper of the second named author, "Uniqueness among scalar-flat Kähler metrics on non-compact toric 4-manifolds", to show that all conformally Kähler, Ricci-flat, toric metrics on the complement of toric divisors, under some mild assumptions on the associated moment polytope, are among the families above. In particular all such metrics are ALF.

math.DG↗
arXiv2024

The Positivity of the Neural Tangent Kernel

The Neural Tangent Kernel (NTK) has emerged as a fundamental concept in the study of wide Neural Networks. In particular, it is known that the positivity of the NTK is directly related to the memorization capacity of sufficiently wide networks, i.e., to the possibility of reaching zero loss in training, via gradient descent. Here we will improve on previous works and obtain a sharp result concerning the positivity of the NTK of feedforward networks of any depth. More precisely, we will show that, for any non-polynomial activation function, the NTK is strictly positive definite. Our results are based on a novel characterization of polynomial functions which is of independent interest.

cs.LG↗
arXiv2024

Einstein metrics from the Calabi ansatz via Derdziński duality

Drawing on results of Derdziński's from the 80's, we classify conformally Kähler, $U(2)$-invariant, Einstein metrics on the total space of $\mathcal{O}(-m)$, for all $m \in \mathbb{N}$. This yields infinitely many $1$-parameter families of metrics exhibiting several different behaviours including asymptotically hyperbolic metrics (more specifically of Poincaré type), ALF metrics, and metrics which compactify to a Hirzebruch surface $\mathbb{H}_m$ with a cone singularity along the "divisor at infinity". This allows us to investigate transitions between behaviours yielding interesting results. For instance, we show that a Ricci--flat ALF metric known as the Taub-bolt metric can be obtained as the limit of a family of cone angle Einstein metrics on $\mathbb{CP}^2 \# \overline{\mathbb{CP}}^2$ when the cone angle converges to zero. We also construct Einstein metrics which are asymptotically hyperbolic and conformal to a scalar-flat Kähler metric. Such metrics cannot be obtained by applying Derdziński's theorem.

math.DG↗
arXiv2023

On the bifurcation theory of the Ginzburg-Landau equations

We construct nonminimal and irreducible solutions to the Ginzburg-Landau equations on closed manifolds of arbitrary dimension with trivial first real cohomology. Our method uses bifurcation theory where the "bifurcation points" are characterized by the eigenvalues of a Laplace-type operator. To our knowledge these are the first such examples on nontrivial line bundles.

math.AP↗
arXiv2023

Wide neural networks: From non-gaussian random fields at initialization to the NTK geometry of training

Recent developments in applications of artificial neural networks with over $n=10^{14}$ parameters make it extremely important to study the large $n$ behaviour of such networks. Most works studying wide neural networks have focused on the infinite width $n \to +\infty$ limit of such networks and have shown that, at initialization, they correspond to Gaussian processes. In this work we will study their behavior for large, but finite $n$. Our main contributions are the following: (1) The computation of the corrections to Gaussianity in terms of an asymptotic series in $n^{-\frac{1}{2}}$. The coefficients in this expansion are determined by the statistics of parameter initialization and by the activation function. (2) Controlling the evolution of the outputs of finite width $n$ networks, during training, by computing deviations from the limiting infinite width case (in which the network evolves through a linear flow). This improves previous estimates and yields sharper decay rates for the (finite width) NTK in terms of $n$, valid during the entire training procedure. As a corollary, we also prove that, with arbitrarily high probability, the training of sufficiently wide neural networks converges to a global minimum of the corresponding quadratic loss function. (3) Estimating how the deviations from Gaussianity evolve with training in terms of $n$. In particular, using a certain metric in the space of measures we find that, along training, the resulting measure is within $n^{-\frac{1}{2}}(\log n)^{1+}$ of the time dependent Gaussian process corresponding to the infinite width network (which is explicitly given by precomposing the initial Gaussian process with the linear flow corresponding to training in the infinite width limit).

cs.LG↗
arXiv2022

Nonminimal solutions to the Ginzburg-Landau equations on surfaces

We prove the existence of novel, nonminimal and irreducible solutions to the (self-dual) Ginzburg-Landau equations on closed surfaces. To our knowledge these are the first such examples on nontrivial line bundles, that is, with nonzero total magnetic flux. Our method works with the 2-dimensional, critically coupled Ginzburg-Landau theory and uses the topology of the moduli space. The method is nonconstructive, but works for all values of the remaining coupling constant. We also prove the instability of these solutions.

math.DG↗
arXiv2022

The asymptotic geometry of $\rm{G}_2$-monopoles

This article investigates the asymptotics of $\rm{G}_2$-monopoles. First, we prove that when the underlying $\rm{G}_2$-manifold is nonparabolic (i.e. admits a positive Green's function), finite intermediate energy monopoles with bounded curvature have finite mass. The second main result restricts to the case when the underlying $\rm{G}_2$-manifold is asymptotically conical. In this situation, we deduce sharp decay estimates and that the connection converges, along the end, to a pseudo-Hermitian--Yang--Mills connection over the asymptotic cone. Finally, our last result exhibits a Fredholm setup describing the moduli space of finite intermediate energy monopoles on an asymptotically conical $\rm{G}_2$-manifold.

math.DG↗
arXiv2022

The Kapustin--Witten equations on ALE and ALF gravitational instantons

We study solutions to the Kapustin--Witten equations on ALE and ALF gravitational instantons. On any such space and for any compact structure group, we prove asymptotic estimates for the Higgs field. We then use it to prove a vanishing theorem in the case when the underlying manifold is $\mathrm{R}^4$ or $\mathrm{R}^3 \times \mathbb{S}^1$ and the structure group is $\mathrm{SU} (2)$.

math.DG↗
arXiv2022

The Haydys monopole equation

We study complexified Bogomolny monopoles using the complex linear extension of the Hodge star operator; these monopoles can be interpreted as solutions to the Bogomolny equation with a complex gauge group. Alternatively, these equations can be obtained from dimensional reduction of the Haydys instanton equations to three dimensions, thus we call them Haydys monopoles. We find that (under mild hypotheses) the smooth locus of the moduli space of finite energy Haydys monopoles on $\mathbb{R}^3$ is a Kähler manifold containing the ordinary Bogomolny moduli space as a minimal Lagrangian submanifold -- an $A$-brane. Moreover, using a gluing construction we construct an open neighborhood of this submanifold modeled on a neighborhood of the zero section in the tangent bundle to the Bogomolny moduli space. This is analogous to the case of Higgs bundles over a Riemann surface, where the (co)tangent bundle of holomorphic bundles canonically embeds into the Hitchin moduli space. These results contrast immensely with the case of finite energy Kapustin--Witten monopoles for which we have shown a vanishing theorem in [12].

math.DG↗
arXiv2020

Minimal Lagrangian tori and action-angle coordinates

We investigate which orbits of an $n$-dimensional torus action on a $2n$-dimensional toric Kähler manifold $M$ are minimal. In other words, we study minimal submanifolds appearing as the fibres of the moment map on a toric Kähler manifold. Amongst other questions we investigate and give partial answers to the following: (1) How many such minimal Lagrangian tori exist? (2) Can their stability, as critical points of the area functional, be characterised just from the ambient geometry? (3) Given a toric symplectic manifold, for which sets of orbits $S$, is there a compatible toric Kähler metric whose set of minimal Lagrangian orbits is $S$?

math.DG↗
arXiv2020

Refined compartmental models, asymptomatic carriers and COVID-19

The goal of this article is to analyze some compartmental models specially designed to model the spread of a disease whose transmission has the same features as COVID-19. The major contributions of this article are: (1) Rigorously find sufficient conditions for the outbreak to only have one peak, i.e. for no second wave of infection to form; (2) Investigate the formation of other waves of infection when such conditions are not met; (3) Use numerical simulations to analyze the different roles asymptomatic carriers can have. We also argue that dividing the population into non-interacting groups leads to an effective reduction of the transmission rates. As in any compartmental model, the goal of this article is to provide qualitative understanding rather than exact quantitative predictions.

q-bio.PE↗
arXiv2019

${\rm Spin}(7)$-Instantons from evolution equations

In this paper we study ${\rm Spin}(7)$-instantons on asymptotically conical ${\rm Spin}(7)$-orbifolds (and manifolds) obtained by filling in certain squashed $3$-Sasakian $7$-manifolds. We construct a $1$-parameter family of explicit ${\rm Spin}(7)$-instantons. Taking the parameter to infinity, the family (a) bubbles off an ASD connection in directions transverse to a certain Cayley submanifold $Z$, (b) away from $Z$ smoothly converges to a limit ${\rm Spin}(7)$-instanton that extends across $Z$ onto a topologically distinct bundle, (c) satisfies an energy conservation law for the instantons and the bubbles concentrated on $Z$, and (d) determines a Fueter section, in the sense of Donaldson and Segal, Haydys and Walpuski.

math.DG↗
arXiv2018

From vortices to instantons on the Euclidean Schwarzschild manifold

The first irreducible solution of the $\mathrm{SU} (2)$ self-duality equations on the Euclidean Schwarzschild (ES) manifold was found by Charap and Duff in 1977, only 2 years later than the famous BPST instantons on $\mathbb{R}^4$ were discovered. While soon after, in 1978, the ADHM construction gave a complete description of the moduli spaces of instantons on $\mathbb{R}^4$, the case of the Euclidean Schwarzschild manifold has resisted many efforts for the past 40 years. By exploring a correspondence between the planar Abelian vortices and spherically symmetric instantons on ES, we obtain: a complete description of a connected component of the moduli space of unit energy $\mathrm{SU} (2)$ instantons; new examples of instantons with non-integer energy (and non-trivial holonomy at infinity); a complete classification of finite energy, spherically symmetric, $\mathrm{SU} (2)$ instantons. As opposed to the previously known solutions, the generic instanton coming from our construction is not invariant under the full isometry group, in particular not static. Hence disproving a conjecture of Tekin.

math.DG↗
arXiv2016

Automatic Graphic Logo Detection via Fast Region-based Convolutional Networks

Brand recognition is a very challenging topic with many useful applications in localization recognition, advertisement and marketing. In this paper we present an automatic graphic logo detection system that robustly handles unconstrained imaging conditions. Our approach is based on Fast Region-based Convolutional Networks (FRCN) proposed by Ross Girshick, which have shown state-of-the-art performance in several generic object recognition tasks (PASCAL Visual Object Classes challenges). In particular, we use two CNN models pre-trained with the ILSVRC ImageNet dataset and we look at the selective search of windows `proposals' in the pre-processing stage and data augmentation to enhance the logo recognition rate. The novelty lies in the use of transfer learning to leverage powerful Convolutional Neural Network models trained with large-scale datasets and repurpose them in the context of graphic logo detection. Another benefit of this framework is that it allows for multiple detections of graphic logos using regions that are likely to have an object. Experimental results with the FlickrLogos-32 dataset show not only the promising performance of our developed models with respect to noise and other transformations a graphic logo can be subject to, but also its superiority over state-of-the-art systems with hand-crafted models and features.

cs.CV↗