arXiv subjects

Nicolau C. Saldanha

Publications and source records attributed to Nicolau C. Saldanha.

At least 19 recordsLinked to original sources

arXiv2026

Stability of compact actions and a result on divided differences

We study smooth locally free actions of ${\mathbb R}^n$ on manifolds $M$ of dimension $n+1$. We are interested in compact orbits and in compact actions: actions with all orbits compact. Given a compact orbit in a neighborhood of compact orbits, we give necessary and sufficient conditions for the existence of a $C^k$ perturbation with noncompact orbits in the given neighborhood. We prove that if such a perturbation exists it can be assumed to differ from the original action only in a smaller neighborhood of the initial orbit. As an application, for each $k$, we give examples of compact actions which admit $C^{k-1}$-perturbations with noncompact orbits but such that all $C^k$-perturbations are compact. The main result generalizes for $k > 1$ a previous result for the case $C^1$. A critical auxiliary result is an estimate on divided differences.

math.DS↗

arXiv2026

A CW complex homotopy equivalent to spaces of locally convex curves

Locally convex curves in the sphere $S^n$ have been studied for several reasons, including the study of linear ordinary differential equations. Taking Frenet frames obtains corresponding curves $Γ$ in the group $Spin_{n+1}$; $Π: Spin_{n+1} \to Flag_{n+1}$ is the universal cover of the space of flags. Determining the homotopy type of spaces of such curves $Γ$ with prescribed initial and final points appears to be a hard problem. We may focus on $L_n$, the space of locally convex curves $Γ: [0,1] \to Spin_{n+1}$ with $Γ(0) = 1$, $Π(Γ(1)) = Π(1)$. Convex curves form a contractible connected component of $L_n$; there are $2^{n+1}$ other components, one for each endpoint. The homotopy type of $L_n$ has so far been determined only for $n=2$. This paper is a step towards solving the problem for larger values of $n$. The itinerary of $Γ$ belongs to $W_n$, the set of finite words in the alphabet $S_{n+1} \setminus \{e\}$. The itinerary of a curve lists the non open Bruhat cells crossed. Itineraries stratify the space $L_n$. We construct a CW complex $D_n$ which is a kind of dual of $L_n$ under this stratification: the construction is similar to Poincaré duality. The CW complex $D_n$ is homotopy equivalent to $L_n$. The cells of $D_n$ are naturally labeled by words in $W_n$; $D_n$ is locally finite. Explicit glueing instructions are described for lower dimensions. We describe an open subset $Y_n \subset L_n$, a union of strata of $L_n$. In each non convex component of $L_n$, the intersection with $Y_n$ is connected and dense. Most connected components of $L_n$ are contained in $Y_n$. For $n > 3$, in the other components the complement of $Y_n$ has codimension at least $2$. The set $Y_n$ is homotopy equivalent to the disjoint union of $2^{n+1}$ copies of $ΩSpin_{n+1}$. For all $n \ge 2$, all connected components of $L_n$ are simply connected.

math.GT↗

arXiv2026

Homotopy type of spaces of locally convex curves in the sphere S^3

Locally convex (or nondegenerate) curves in the sphere $S^n$ have been studied for several reasons, including the study of linear ordinary differential equations of order $n+1$. Taking Frenet frames allows us to obtain corresponding curves $Γ$ in the group $Spin_{n+1}$. Let $L_n(z_0;z_1)$ be the space of such curves $Γ$ with prescribed endpoints $Γ(0) = z_0$, $Γ(1) = z_1$. The aim of this paper is to determine the homotopy type of the spaces $L_3(z_0;z_1)$ for all $z_0, z_1 \in Spin_4$. As a corollary, we obtain the homotopy type of the space of closed locally convex curves in either $S^3$ or $P^3$. There are many previous papers addressing related questions. An early paper solves the corresponding problem for curves in $S^2$. Another previous result (with B. Shapiro) reduces the problem to $z_0 = 1$ and $z_1 \in Quat_4$ where $Quat_4 \subset Spin_4$ is a finite group of order $16$. A more recent paper shows that for $z_1 \in Quat_4 \smallsetminus Z(Quat_4)$ we have a homotopy equivalence $L_3(1;z_1) \approx ΩSpin_4$. In this paper we compute the homotopy type of $L_3(1;z_1)$ for $z_1 \in Z(Quat_4)$: it is equivalent to the wedge of $ΩSpin_4$ with an infinite countable family of spheres (as for the case $n = 2$). The structure of the proof can be compared to that of the case $n = 2$ but some of the steps require the creation of new theories, involving algebra and combinatorics. We construct explicit subsets $Y \subset L_n(z_0;z_1)$ for which the inclusion $Y \subset ΩSpin_{n+1}(z_0;z_1)$ is a homotopy equivalence. For $n = 2$, there is a simple geometric description of $Y$; for $n = 3$, the far less natural construction is based on the theory of itineraries of such curves. The itinerary of a curve in $L_n(1;z_1)$ is a finite word in the alphabet $S_{n+1} \smallsetminus \{e\}$ of nontrivial permutations.

math.GT↗

arXiv2025

Homotopy type of spaces of curves with constrained curvature on flat surfaces

Let $S$ be a complete flat surface, such as the Euclidean plane. We determine the homeomorphism class of the space of all curves on $S$ which start and end at given points in given directions and whose curvatures are constrained to lie in a given open interval, in terms of all parameters involved. Any connected component of such a space is either contractible or homotopy equivalent to an $n$-sphere, and every $n\geq 1$ is realizable. Explicit homotopy equivalences between the components and the corresponding spheres are constructed.

math.GT↗

arXiv2025

Domino tilings beyond 2D

There is a rich history of domino tilings in two dimensions. Through a variety of techniques we can answer questions such as: how many tilings are there of a given region or what does the space of all tilings look like? These questions and their answers become significantly more difficult in dimension three and above. Despite this curse of dimensionality, there have been exciting recent advances in the theory. Here we briefly review foundational results of two-dimensional domino tilings and highlight where challenges arise when generalizing to higher dimensions. We then survey results in higher dimensional domino tilings, focusing on the question of connectivity of the space of tilings, and the open problems that still remain.

math.CO↗

arXiv2025

Relative Helicity and Tiling Twist

We consider domino tilings of 3D cubiculated regions. The tilings have two invariants, flux and twist, often integer-valued, which are given in purely combinatorial terms. These invariants allow one to classify the tilings with respect to certain elementary moves, flips and trits. In this paper we present a construction associating a divergence-free vector field $ξ_t$ to any domino tiling $t$, such that the flux of the tiling $t$ can be interpreted as the (relative) rotation class of the field $ξ_t$, while the twist of $t$ is proved to be the relative helicity of the field $ξ_t$.

math.CO↗

arXiv2025

Slab tilings, flips and the triple twist

A \textit{domino} is a $2\times 1\times 1$ parallelepiped formed by the union of two unit cubes and a \textit{slab} is a $2\times 2\times 1$ parallelepiped formed by the union of four unit cubes. We are interested in tiling regions formed by the finite union of unit cubes. Domino tilings have been studied before; here we investigate \textit{slab tilings}. As for domino tilings, a flip in a slab tiling is a local move: two neighboring parallel slabs are removed and placed back in a different position. Inspired by the twist for domino tilings, we construct a flip invariant for slab tilings: the \textit{triple twist}, assuming values in $\mathbb{Z}^3$. We show that if the region is a large box then the triple twist assumes a large number of possible values, roughly proportional to the fourth power of the volume. We also give examples of smaller regions for which the set of tilings is connected under flips, so that the triple twist assumes only one value.

math.CO↗

arXiv2023

Stratification of spaces of locally convex curves by itineraries

Locally convex (or nondegenerate) curves in the sphere (or projective space) have been studied for several reasons, including the study of linear ordinary differential equations. Taking Frenet frames allows us to translate such curves into corresponding curves in the flag space, the orthogonal group or its cover $Spin_{n+1}$. Determining the homotopy type of the space of such closed curves or, more generally, of spaces of such curves with prescribed initial and final jets appears to be a hard problem, which has been solved for $n=2$ but otherwise remains open. This paper is a step towards solving the problem for larger values of $n$. In the process, we prove a related conjecture of B. Shapiro and M. Shapiro. We define the itinerary of a locally convex curve $Γ:[0,1]\to Spin_{n+1}$ as a word $w$ in the alphabet of non-trivial permutations. This word encodes the succession of non-open Bruhat cells of $Spin_{n+1}$ pierced by $Γ$. We prove that, for each word $w$, the subspace of curves of itinerary $w$ is an embedded contractible (topological) submanifold of finite codimension, thus defining a stratification of the space of curves. We show how to obtain explicit (topologically) transversal sections for each of these submanifolds. We study both a space of curves with minimum regularity hypotheses, where only topological transversality applies, and spaces of sufficiently regular curves. In both cases we study the neighboring relation between strata. This is an important step in the construction of CW cell complexes mapped into the original space of curves by weak homotopy equivalences. Our stratification is not as nice as might be desired, lacking for instance the Whitney property. The differentiability class of the curves affects some properties of the stratification. The necessary ingredients for the construction of a dual CW complex are proved.

math.GT↗

arXiv2023

Linearizing Toda and SVD flows on large phase spaces of matrices with real spectrum

We consider different phase spaces for the Toda flows and the less familiar SVD flows. For the Toda flow, we handle symmetric and non-symmetric matrices with real simple eigenvalues, possibly with a given profile. Profiles encode, for example, band matrices and Hessenberg matrices. For the SVD flow, we assume simplicity of the singular values. In all cases, an open cover is constructed, as are corresponding charts to Euclidean space. The charts linearize the flows, converting it into a linear differential system with constant coefficients and diagonal matrix. A variant construction transform the flows into uniform straight line motion. Since limit points belong to the phase space, asymptotic behavior becomes a local issue. The constructions rely only on basic facts of linear algebra, making no use of symplectic geometry.

math.SP↗

arXiv2022

On the homotopy type of intersections of two real Bruhat cells

Real Bruhat cells give an important and well studied stratification of such spaces as $GL_{n+1}$, $Flag_{n+1} = SL_{n+1}/B$, $SO_{n+1}$ and $Spin_{n+1}$. We study the intersections of a top dimensional cell with another cell (for another basis). Such an intersection is naturally identified with a subset of the lower nilpotent group $Lo_{n+1}^{1}$. We are particularly interested in the homotopy type of such intersections. In this paper we define a stratification of such intersections. As a consequence, we obtain a finite CW complex which is homotopically equivalent to the intersection. We compute the homotopy type for several examples. It turns out that for $n \le 4$ all connected components of such subsets of $Lo_{n+1}^1$ are contractible: we prove this by explicitly constructing the corresponding CW complexes. Conversely, for $n \ge 5$ and the top permutation, there is always a connected component with even Euler characteristic, and therefore not contractible. This follows from formulas for the number of cells per dimension of the corresponding CW complex. For instance, for the top permutation $S_6$, there exists a connected component with Euler characteristic equal to $2$. We also give an example of a permutation in $S_6$ for which there exists a connected component which is homotopically equivalent to the circle $S^1$.

math.AT↗

arXiv2022

Domino tilings of cylinders: the domino group and connected components under flips

We consider domino tilings of three-dimensional cubiculated regions. A flip is a local move: two neighboring parallel dominoes are removed and placed back in a different position. The twist is an integer associated to each tiling, which is invariant under flips. A balanced quadriculated disk $D$ is regular if whenever two tilings $t_0$ and $t_1$ of $D \times [0,N]$ have the same twist then $t_0$ and $t_1$ can be joined by a sequence of flips provided some extra vertical space is allowed. We define the domino group of a quadriculated disk and prove that $D$ is regular if and only if its domino group is isomorphic to $Z \oplus Z/(2)$. We prove that a rectangle $D = [0,L] \times [0,M]$ with $LM$ even is regular if and only if $\min\{L,M\} \ge 3$ and conjecture that in general "large" disks are regular. In the cases where $D$ is not regular we prove partial results concerning the structure of the domino group: the group is not abelian and has exponential growth. We also prove that if $D$ is regular then the extra vertical space necessary to join by flips two tilings of $D \times [0,N]$ with the same twist depends only on $D$, not on the height $N$.

math.CO↗

arXiv2021

Domino tilings and flips in dimensions 4 and higher

In this paper we consider domino tilings of bounded regions in dimension $n \geq 4$. We define the twist of such a tiling, an elements of ${\mathbb{Z}}/(2)$, and prove it is invariant under flips, a simple local move in the space of tilings. We investigate which regions $D$ are regular, i.e. whenever two tilings $t_0$ and $t_1$ of $D \times [0,N]$ have the same twist then $t_0$ and $t_1$ can be joined by a sequence of flips provided some extra vertical space is allowed. We prove that all boxes are regular except $D = [0,2]^3$. Furthermore, given a regular region $D$, we show that there exists a value $M$ (depending only on $D$) such that if $t_0$ and $t_1$ are tilings of equal twist of $D \times [0,N]$ then the corresponding tilings can be joined by a finite sequence of flips in $D \times [0,N+M]$. As a corollary we deduce that, for regular $D$ and large $N$, the set of tilings of $D \times [0,N]$ has two twin giant components under flips, one for each value of the twist.

math.CO↗

arXiv2021

Finiteness of rank for Grassmann convexity

The Grassmann convexity conjecture gives a conjectural formula for the maximal total number of real zeros of the consecutive Wronskians of an arbitrary fundamental solution to a disconjugate linear ordinary differential equation with real time. The conjecture can be reformulated in terms of convex curves in the nilpotent lower triangular group. The formula has already been shown to be a correct lower bound and to give a correct upper bound in several small dimensional cases. In this paper we obtain a general explicit upper bound.

math.CA↗

arXiv2021

On the connectivity of spaces of three-dimensional tilings

We consider domino tilings of three-dimensional cubiculated manifolds with or without boundary, including subsets of Euclidean space and three-dimensional tori. In particular, we are interested in the connected components of the space of tilings of such regions under local moves. Building on the work of the third and fourth authors, we allow two possible local moves, the flip and trit. These moves are considered with respect to two topological invariants, the twist and flux. Our main result proves that, up to refinement: Two tilings are connected by flips and trits if and only if they have the same flux. Two tilings are connected by flips alone if and only if they have the same flux and twist.

math.CO↗

arXiv2021

Domino tilings of cylinders: connected components under flips and normal distribution of the twist

We consider domino tilings of $3$-dimensional cubiculated regions. A three-dimensional domino is a 2x2x1 rectangular cuboid. We are particularly interested in regions of the form $R_N = D \times [0,N]$ where $D$ is a fixed quadriculated disk. In dimension 3, the twist associates to each tiling $t$ an integer $Tw(t)$. We prove that, when $N$ goes to infinity, the twist follows a normal distribution. A flip is a local move: two neighboring parallel dominoes are removed and placed back in a different position. The twist is invariant under flips. A quadriculated disk $D$ is regular if, whenever two tilings $t_0$ and $t_1$ of $R_N$ satisfy $Tw(t_0) = Tw(t_1)$, $t_0$ and $t_1$ can be joined by a sequence of flips provided some extra vertical space is allowed. Many large disks are regular, including rectangles $D = [0,L] \times [0,M]$ with $LM$ even and $L,M \ge 3$. For regular disks, we describe the larger connected components under flips of the set of tilings of the region $R_N = D \times [0,N]$. As a corollary, let $p_N$ be the probability that two random tilings $T_0$ and $T_1$ of $D \times [0,N]$ can be joined by a sequence of flips conditional to their twists being equal. Then $p_N$ tends to 1 if and only if $D$ is regular. Under a suitable equivalence relation, the set of tilings has a group structure, the domino group. These results illustrate the fact that the domino group dictates many properties of the space of tilings of the cylinder $R_N = D \times [0,N]$, particularly for large $N$.

math.CO↗

arXiv2020

Locally convex curves and the Bruhat stratification of the spin group

We study the lifting of the Schubert stratification of the homogeneous space of complete real flags of $R^{n+1}$ to its universal covering group $Spin_{n+1}$. We call the lifted strata the Bruhat cells of $Spin_{n+1}$, in keeping with the homonymous classical decomposition of reductive algebraic groups. We present explicit parameterizations for these Bruhat cells in terms of minimal-length expressions $σ=a_{i_1}... a_{i_k}$ for permutations $σ\in S_{n+1}$ in terms of the $n$ generators $a_i=(i,i+1)$. These parameterizations are compatible with the Bruhat orders in the Coxeter-Weyl group $S_{n+1}$. This stratification is an important tool in the study of locally convex curves; we present a few such applications.

math.GT↗

arXiv2018

Spaces of curves with constrained curvature on hyperbolic surfaces

Let $ S $ be a hyperbolic surface. We investigate the topology of the space of all curves on $ S $ which start and end at given points in given directions, and whose curvatures are constrained to lie in a given interval $ (κ_1,κ_2) $. Such a space falls into one of four qualitatively distinct classes, according to whether $ (κ_1,κ_2) $ contains, overlaps, is disjoint from, or contained in the interval $ [-1,1] $. Its homotopy type is computed in the latter two cases. We also study the behavior of these spaces under covering maps when $ S $ is arbitrary (not necessarily hyperbolic nor orientable) and show that if $ S $ is compact then they are always nonempty.

math.GT↗

arXiv2018

Results on the homotopy type of the spaces of locally convex curves on $S^3$

A curve $γ: [0,1] \rightarrow S^n$ of class $C^k$ ($k \geqslant n$) is locally convex if the vectors $γ(t), γ'(t), γ"(t), \cdots, γ^{(n)}(t)$ are a positive orthonormal basis to $R^{n+1}$ for all $t \in [0,1]$. Given an integer $n \geq 2$ and $Q \in SO_{n+1}$, let $LS^n(Q)$ be the set of all locally convex curves $γ: [0,1] \rightarrow S^n$ with fixed initial and final Frenet frame $F_γ(0)=I$ and $F_γ(1)=Q$. Saldanha and Shapiro proved that there are just finitely many non-homeomorphic spaces among $LS^n(Q)$ when $Q$ varies in $SO_{n+1}$ (in particular, at most $3$ for $n=3$). For any $n \geqslant 2$, the homotopy type of one of these spaces is well-known, but not of the others. For $n=2$, Saldanha determined the homotopy type of the spaces $LS^2(Q)$. The purpose of this work is to study the case $n=3$. We will obtain information on the homotopy type of one of these $2$ other spaces, allowing us to conclude that its connected components are not homeomorphic to the connected components of the known space.

math.GT↗