arXiv subjects

Eduardo Fernández

Publications and source records attributed to Eduardo Fernández.

At least 19 recordsLinked to original sources

arXiv2026

Spaces of tight contact structures in dimension 3

We study spaces of tight contact structures and contactomorphism groups in dimension $3$. We introduce a microfibration method for studying spaces of convex surfaces in contact $3$-manifolds and apply it to describe spaces of convex disks and spheres in tight contact $3$-manifolds. With these ingredients in hand, we prove a parametric gluing theorem for contact handle attachments: the weak homotopy type of the space of tight contact structures is invariant under contact $0$-, $1$-, and $3$-handle attachments. Based on these new techniques, we present a substantial number of independent applications. First, we obtain explicit descriptions of the weak homotopy type of several spaces of tight contact structures. We determine the weak homotopy type of the contactomorphism group of every standard contact handlebody, of the standard $\mathbb{S}^1\times\mathbb{S}^2$, and of every Legendrian circle bundle with non-empty boundary, partially resolving a conjecture of Giroux. Additionally, we compute the weak homotopy type of an infinite family of spaces of Legendrian and transverse unknots, proving Legendrian and transverse analogues of Hatcher's theorem for smooth unknots in the $3$-sphere.

math.SG↗

arXiv2026

Masked Autoencoders Learn Perception-Relevant Representations from Resting State Neural Data

Clinical neuroprosthetics face a data bottleneck: labeled perception trials are scarce while hours of spontaneous neural activity are largely underutilized. Here, we test whether self-supervised learning can use these unlabeled datasets to improve perception decoding. We pretrained a masked autoencoder on 14.6 hours of spontaneous multiunit activity from an intracortical array in a blind participant's V1. The model captured interpretable brain structure without supervision: V1's spatial organization and perceptual state separation both emerged purely from its latent representations. To test these features, we used linear probing (logistic regression on the frozen latents) to measure performance on the data with stimulation. Perception decoding accuracy reached 84.1% on a general psychometric task. On the more difficult threshold-level task, accuracy reached 64.0%. This work shows that spontaneous cortical activity is not noise; it contains rich, task-relevant structure. Unsupervised pretraining on this data is a promising strategy to improve neural decoding.

q-bio.NC↗

arXiv2026

The h-principle fails for prelegendrians in corank 2 fat distributions

We investigate the $h$-principle problem for fat distributions. These are maximally non-integrable distributions with natural symplectisations and contactisations, that generalize contact distributions to higher corank. We focus on the corank-$2$ case, where we study a natural class of submanifolds, which we call prelegendrians. Their key feature is that they admit a canonical Legendrian lift to the contactisation. Our main results state that the $h$-principle fails for these submanifolds in all dimensions. To the best of our knowledge, this is the first example of rigidity in the study of maximally non-integrable distributions, outside of contact topology. First, we find an infinite family of $(2n+1)$-tori in the standard fat $(\mathbb{C}^{2n+1},\mathcal{D}_{\mathrm{std}})$, with the following two properties: (1) They all represent the same formal prelegendrian class, (2) but they are not prelegendrian isotopic because they are distinguished by pseudoholomorphic curve invariants of their Legendrian lift. Secondly, we define the notion of prelegendrian stabilization in $(\mathbb{C}^{2n+1},\mathcal{D}_{\mathrm{std}})$. This allows us to take an arbitrary prelegendrian and produce another one, in the same formal class, whose Legendrian lift is loose. In order to prove these results we also develop the fundamentals of the theory of prelegendrians. This includes: (1) introducing the notion of front projection in $(\mathbb{C}^{2n+1},\mathcal{D}_{\mathrm{std}})$, (2) proving that pseudoholomorphic curve invariants are robust under perturbations of the fat structure, allowing us to transport our results to non-standard fat structures, (3) introducing a zooming argument showing that any fat structure in dimension $6$ admits prelegendrians.

math.SG↗

arXiv2025

Spaces of Legendrian cables and Seifert fibered links

We determine the homotopy type of the spaces of several Legendrian knots and links with the maximal Thurston--Bennequin invariant. In particular, we give a recursive formula of the homotopy type of the space of Legendrian embeddings of sufficiently positive cables, and determine the homotopy type of the space of Legendrian embeddings of Seifert fibered links, which include all torus knots and links, in the standard contact 3-sphere, except when one of the link components is a negative torus knot. In general, we prove that the space of contact structures on the complement of a sufficiently positive Legendrian cable with the maximal Thurston-Bennequin invariant is homotopy equivalent to the space of contact structures on the complement of the underlying Legendrian knot, and prove that the space of contact structures on a Legendrian Seifert fibered space over a compact oriented surface with boundary is contractible. From this result, we find infinitely many new components of the space of Legendrian embeddings in the standard contact 3-sphere that satisfy an injective h-principle. These include the spaces of Legendrian embeddings of an algebraic link with the maximal Thurston--Bennequin invariant. In particular, the inclusion of these Legendrian embedding spaces into the corresponding formal Legendrian embedding spaces is a homotopy injection.

math.GT↗

arXiv2025

Strongly overtwisted contact 3-manifolds

We prove the existence of a subclass of overtwisted contact structures, called strongly overtwisted, on a 3-manifold that satisfy a complete h-principle without prescribing the contact structures over any subset of the 3-manifold. As a consequence, the homotopy type of the space of overtwisted disk embeddings into a strongly overtwisted contact 3-manifold is determined. A complete h-principle for a subclass of loose Legendrians is also derived from the main result. In general, the method allows us to deduce an h-principle for overtwisted disks that are fixed near the boundary in an arbitrary overtwisted contact 3-manifold.

math.SG↗

arXiv2025

Engel CR submanifolds of $\mathbb{C}^3$

We give a sufficient condition for an $\mathbb{S}^1$-bundle over a $3$-manifold to admit an immersion (or embedding) into $\mathbb{C}^3$ so that its complex tangencies define an Engel structure. In particular, every oriented $\mathbb{S}^1$-bundle over a closed, oriented $3$-manifold admits such an immersion. If the bundle is trivial, this immersion can be chosen to be an embedding and, moreover, infinitely many pairwise smoothly non-isotopic embeddings of this type can be constructed. These are the first examples of compact submanifolds of $\mathbb{C}^3$ whose complex tangencies are Engel, answering a question of Y. Eliashberg.

math.DG↗

arXiv2025

Convex disks with Legendrian boundary in overtwisted contact 3-manifolds

We classify convex disks with a fixed characteristic foliation and Legendrian boundary, up to contact isotopy relative to the boundary, in every closed overtwisted contact 3-manifold. This classification covers cases where the neighborhood of such a disk is tight or where the boundary violates the Bennequin-Eliashberg inequality. We show that this classification coincides with the formal one, establishing an h-principle for these disks. As a corollary, we deduce that the space of Legendrian unknots that lie in some Darboux ball in a closed overtwisted contact 3-manifold satisfies the h-principle at the level of fundamental groups. Finally, we determine the contact mapping class group of the complement of each Legendrian unknot with non-positive tb invariant in an overtwisted 3-sphere.

math.GT↗

arXiv2024

Grasper families of spheres in $S^2 \times D^2$ and barbell diffeomorphisms of $S^1\times S^2 \times I$

We show that the fundamental group of framed circles in $S^1 \times D^3$ injects into the fundamental group of framed spheres in $S^2\times D^2$, so that the cokernel is the fundamental group of framed neat disks in $D^4$. In particular, grasper families of circles give rise to countably many nontrivial families of spheres. Ambient extensions of either of these two types of families induce the same barbell diffeomorphisms of $S^1\times S^2\times I$. We give two proofs that these diffeomorphisms are nontrivial and pairwise distinct. This implies infinite generation of the abelian group of isotopy classes of diffeomorphisms of $S^1\times S^2\times I$ that are pseudo-isotopic to the identity, recovering a result of Singh.

math.GT↗

arXiv2024

Exotic Dehn twists on sums of two contact 3-manifolds

We exhibit the first examples of exotic contactomorphisms with infinite order as elements of the contact mapping class group. These are given by certain Dehn twists on the separating sphere in a connected sum of two closed contact 3-manifolds. We detect these by a combination of hard and soft techniques. On the one hand, we make essential use of an invariant for families of contact structures which generalises the Kronheimer--Mrowka contact invariant in monopole Floer homology. We then exploit an h-principle for families of convex spheres in tight contact 3-manifolds, from which we establish a parametric version of Colin's decomposition theorem. As a further application, we also exhibit new exotic 1-parametric phenomena in overtwisted contact 3-manifolds.

math.SG↗

arXiv2024

A bypass in the middle

We provide a topological characterization for a family of bypasses with a fixed attaching arc to be contractible. This characterization is formulated in terms of the existence of a bypass that is disjoint from the given family away from the attaching region. As an application, we provide new proofs of several h-principles in overtwisted contact 3-manifolds.

math.GT↗

arXiv2024

Parametric satellites and connected-sums in the space of Legendrian embeddings

This article introduces two new constructions at the higher homotopy level in the space of Legendrian embeddings in $(\mathbb{R}^3, ξ_{\operatorname{std}})$. We first introduce the parametric Legendrian satellite construction, showing that the satellite operation works for parametric families of Legendrian embeddings. This yields new invariants at the higher-order homotopy level. We then introduce the parametric connected-sum construction. This operation takes as inputs two $n$-spheres based at Legendrian embeddings $K_1$ and $K_2$, respectively, and produces a new $n$-sphere based at $K_1\# K_2$. As a main application we construct new infinite families of loops of Legendrian embeddings with non-trivial LCH monodromy invariant.

math.SG↗

arXiv2022

Generalized-alpha scheme in the PFEM for velocity-pressure and displacement-pressure formulations of the incompressible Navier-Stokes equations

Despite the increasing use of the Particle Finite Element Method (PFEM) in fluid flow simulation and the outstanding success of the Generalized-alpha time integration method, very little discussion has been devoted to their combined performance. This work aims to contribute in this regard by addressing three main aspects. Firstly, it includes a detailed implementation analysis of the Generalized-alpha method in PFEM. The work recognizes and compares different implementation approaches from the literature, which differ mainly in the terms that are alpha-interpolated (state variables or forces of momentum equation) and the type of treatment for the pressure in the time integration scheme. Secondly, the work compares the performance of the Generalized-alpha method against the Backward Euler and Newmark schemes for the solution of the incompressible Navier-Stokes equations. Thirdly, the study is enriched by considering not only the classical velocity-pressure formulation but also the displacement-pressure formulation that is gaining interest in the fluid-structure interaction field. The work is carried out using various 2D and 3D benchmark problems such as the fluid sloshing, the solitary wave propagation, the flow around a cylinder, and the collapse of a cylindrical water column.

physics.flu-dyn↗

arXiv2022

Topology optimization for additive manufacturing with length scale, overhang, and building orientation constraints

This paper presents a density-based topology optimization approach considering additive manufacturing limitations. The presented method considers the minimum size of parts, the minimum size of cavities, the inability of printing overhanging parts without the use of sacrificial supporting structures, and the printing directions. These constraints are geometrically addressed and implemented. The minimum size on solid and void zones is imposed through a well-known filtering technique. The sacrificial support material is reduced using a constraint that limits the maximum overhang angle of parts by comparing the structural gradient with a critical reference slope. Due to the local nature of the gradient, the chosen restriction is prone to introduce parts that meet the structural slope but that may not be self-supporting. The restriction limits the maximum overhang angle for a user-defined printing direction, which could reduce structural performance if the orientation is not properly selected. To ease these challenges, a new approach to reduce the introduction of such non-self-supporting parts and a novel method that includes different printing directions in the maximum overhang angle constraint are presented. The proposed strategy for considering the minimum size of solid and void phases, maximum overhang angle, and printing direction, is illustrated by solving a set of 2D benchmark design problems including stiff structures and compliant mechanisms. We also provide MATLAB codes in the appendix for educational purposes and for replication of the results.

cs.CE↗

arXiv2021

Topology Optimization for Large-Scale Additive Manufacturing: Generating designs tailored to the deposition nozzle size

Additive Manufacturing (AM) processes intended for large scale components deposit large volumes of material to shorten process duration. This reduces the resolution of the AM process, which is typically defined by the size of the deposition nozzle. If the resolution limitation is not considered when designing for Large-Scale Additive Manufacturing (LSAM), difficulties can arise in the manufacturing process, which may require the adaptation of the deposition parameters. This work incorporates the nozzle size constraint into Topology Optimization (TO) in order to generate optimized designs suitable to the process resolution. This article proposes and compares two methods, which are based on existing TO techniques that enable control of minimum and maximum member size, and of minimum cavity size. The first method requires the minimum and maximum member size to be equal to the deposition nozzle size, thus design features of uniform width are obtained in the optimized design. The second method defines the size of the solid members sufficiently small for the resulting structure to resemble a structural skeleton, which can be interpreted as the deposition path. Through filtering and projection techniques, the thin structures are thickened according to the chosen nozzle size. Thus, a topology tailored to the size of the deposition nozzle is obtained along with a deposition proposal. The methods are demonstrated and assessed using 2D and 3D benchmark problems.

cs.CG↗

arXiv2021

Analytical relationships for imposing minimum length scale in the robust Topology Optimization formulation

The robust topology optimization formulation that introduces the eroded and dilated versions of the design has gained increasing popularity in recent years, mainly because of its ability to produce designs satisfying a minimum length scale. Despite its success in various topology optimization fields, the robust formulation presents some drawbacks. This paper addresses one in particular, which concerns the imposition of the minimum length scale. In the density framework, the minimum size of the solid and void phases must be imposed implicitly through the parameters that define the density filter and the smoothed Heaviside projection. Finding these parameters can be time consuming and cumbersome, hindering a general code implementation of the robust formulation. Motivated by this issue, in this article we provide analytical expressions that explicitly relate the minimum length scale and the parameters that define it. The expressions are validated on a density-based framework. To facilitate the reproduction of results, MATLAB codes are provided. As a side finding, this paper shows that to obtain simultaneous control over the minimum size of the solid and void phases, it is necessary to involve the 3 fields (eroded, intermediate and dilated) in the topology optimization problem. Therefore, for the compliance minimization problem subject to a volume restriction, the intermediate and dilated designs can be excluded from the objective function, but the volume restriction has to be applied to the dilated design in order to involve all 3 designs in the formulation.

math.OC↗

arXiv2021

A numerical scheme for filter boundary conditions in topology optimization on regular and irregular meshes

In density-based topology optimization, design variables associated to the boundaries of the design domain require unique treatment to negate boundary effects arising from the filtering technique. An effective approach to deal with filtering boundary conditions is to extend the design domain beyond the borders using void densities. Although the approach is easy to implement, it introduces extra computational cost required to store the additional Finite Elements (FEs). This work proposes a numerical technique for the density filter that emulates an extension of the design domain, thus, it avoids boundary effects without demanding additional computational cost, besides it is very simple to implement on both regular and irregular meshes. The numerical scheme is demonstrated using the compliance minimization problem on two-dimensional design domains. In addition, this article presents a discussion regarding the use of a real extension of the design domain, where the Finite Element Analysis (FEA) and volume restriction are influenced. Through a quantitative study, it is shown that affecting FEA or volume restriction with an extension of the FE mesh promotes the disconnection of material from the boundaries of the design domain, which is interpreted as a numerical instability promoting convergence towards local optimums.

math.NA↗

arXiv2020

Imposing minimum and maximum member size, minimum cavity size, and minimum separation distance between solid members in topology optimization

This paper focuses on density-based topology optimization and proposes a combined method to simultaneously impose Minimum length scale in the Solid phase (MinSolid), Minimum length scale in the Void phase (MinVoid) and Maximum length scale in the Solid phase (MaxSolid). MinSolid and MinVoid mean that the size of solid parts and cavities must be greater than the size of a prescribed circle or sphere. This is ensured through the robust design approach based on eroded, intermediate and dilated designs. MaxSolid seeks to restrict the formation of solid parts larger than a prescribed size, which is imposed through local volume restrictions. In the first part of this article, we show that by proportionally restricting the maximum size of the eroded, intermediate and dilated designs, it is possible to obtain optimized designs satisfying, simultaneously, MinSolid, MinVoid and MaxSolid. However, in spite of obtaining designs with crisp boundaries, some results can be difficult to manufacture due to the presence of multiple rounded cavities, which are introduced by the maximum size restriction with the sole purpose of avoiding thick solid members in the structure. To address this issue, in the second part of this article we propose a new geometric constraint that seeks to control the minimum separation distance between two solid members, also called the Minimum Gap (MinGap). Differently from MinVoid, MinGap introduces large void areas that do not necessarily have to be round. 2D and 3D test cases show that simultaneous control of MinSolid, MinVoid, MaxSolid and MinGap can be useful to improve the manufacturability of maximum size constrained designs.

cs.CE↗

arXiv2020

Fundamental groups of formal Legendrian and horizontal embedding spaces

We compute the fundamental group of each connected component of the space of formal Legendrian embeddings in R3. We use it to show that previous examples in the literature of non trivial loops of Legendrian embeddings are already non trivial at the formal level. Likewise, we compute the fundamental group of the different connected components of the space of formal horizontal embeddings into the standard Engel R4. We check that this computes the fundamental group of the space of horizontal embeddings as well.

math.SG↗