arXiv research

Deep neural kernels and Laplace kernel have equivalent RKHS on spheres.

Generative models on spheres improve discrete sequence sampling.

The study of stable and index compact minimal submanifolds in Berger spheres.

We consider local CR-immersions of a strictly pseudoconvex real hypersurface $M\subset\bC^{n+1}$, near a point $p\in M$, into the unit sphere $\mathbb S\subset\bC^{n+d+1}$ with $d>0$. Our main result is that if there is such an immersion $f\colon (M,p)\to \mathbb S$ and $d < n/2$, then $f$ is {\em rigid} in the sense t…

Existence proved for Ricci curvature on sphere product.

The Heegaard Floer correction term ($d$-invariant) is an invariant of rational homology 3-spheres equipped with a Spin$^c$ structure. In particular, the correction term of 1-surgeries along knots in $S^3$ is a ($2\mathbb{Z}$-valued) knot concordance invariant $d_1$. In this paper, we estimate $d_1$ for the $(2,q)$-cabl…

The paper verifies deep neural networks' ability to approximate functions on spheres.

In the late 1980's, it was shown that the Casson invariant appears in the difference between the two filtrations of the Torelli group: the lower central series and the Johnson filtration, and that its core part was identified with the secondary characteristic class $d_1$ associated with the fact that the first $\mathrm…

The study shows how to construct $d$-spheres from $(d-1)$-spheres and $d$-balls without additional vertices.

Let $f:\mathbb{S}^{d-1}\times \mathbb{S}^{d-1}\to\mathbb{S}$ be a function of the form $f(\mathbf{x},\mathbf{x}') = g(\langle\mathbf{x},\mathbf{x}'\rangle)$ for $g:[-1,1]\to \mathbb{R}$. We give a simple proof that shows that poly-size depth two neural networks with (exponentially) bounded weights cannot approximate $f…

Study on the topology of tensorial bodies, showing they are homeomorphic to a product space.

The paper calculates involutive Heegaard Floer homology for specific 3-manifolds.

Minimal simplicial maps constructed for spheres and manifolds.

For $d\geq 2$, Walkup's class $\Kd$ consists of the $d$-dimensional simplicial complexes whose vertex-links are stacked $(d-1)$-spheres. Recently Lutz, Sulanke and Swartz have shown that all $\mathbb{F}$-orientable triangulated $d$-manifolds satisfy the inequality $\binom{f_0-d-1}{2} \geq \binom{d+2}{2}β_1$ for $d\geq …

Solves Minkowski problem for affine invariant convex domains.

We prove a discrete Jordan-Brouwer-Schoenflies separation theorem telling that a (d-1)-sphere H embedded in a d-sphere G defines two different connected graphs A,B in G such a way that the intersection of A and B is H and the union is G and such that the complementary graphs A,B are both d-balls. The graph theoretic de…

Consider a simplicial complex that allows for an embedding into $\mathbb{R}^d$. How many faces of dimension $\frac{d}{2}$ or higher can it have? How dense can they be? This basic question goes back to Descartes' "Lost Theorem" and Euler's work on polyhedra. Using it and other fundamental combinatorial problems, we intr…

Random geometric graphs are a popular choice for a latent points generative model for networks. Their definition is based on a sample of $n$ points $X_1,X_2,\cdots,X_n$ on the Euclidean sphere~$\mathbb{S}^{d-1}$ which represents the latent positions of nodes of the network. The connection probabilities between the node…

We introduce the class $Σ_k(d)$ of $k$-stellated (combinatorial) spheres of dimension $d$ ($0 \leq k \leq d + 1$) and compare and contrast it with the class ${\cal S}_k(d)$ ($0 \leq k \leq d$) of $k$-stacked homology $d$-spheres. We have $Σ_1(d) = {\cal S}_1(d)$, and $Σ_k(d) \subseteq {\cal S}_k(d)$ for $d \geq 2k - 1$…

The standard actions of finite groups on spheres S^d are linear actions, i.e. by finite subgroups of the orthogonal group O(d+1). We prove that, in each dimension d>5, there is a finite group G which admits a faithful, topological action on a sphere S^d but is not isomorphic to a subgroup of O(d+1). The situation remai…

This paper characterizes stable polynomial mappings in a specific set.

Unified solution to Goodman-Pollack transversal problem using matroids and topology.

Hexagonal diagrams link complex curves in $\mathbb{CP}^2$ to minimal genus surfaces.

We construct a family of PL triangulations of the $d$-dimensional real projective space $\mathbb{R}P^d$ on $Θ((\frac{1+\sqrt{5}}{2})^{d+1})$ vertices for every $d\geq 1$. This improves a construction due to Kühnel on $2^{d+1}-1$ vertices.

Study of symmetries of sphere divisions induced by functions with isolated critical points.

Affine deformations of convex cones yield special spacetime structures.

We consider the energy supercritical wave maps from $\mathbb{R}^d$ into the $d$-sphere $\mathbb{S}^d$ with $d \geq 7$. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one dimensional semilinear wave equation $$\partial_t^2 u = \partial^2_r u + \frac{(d-1)}{r}\partial_r u - \frac{(d…

We study the Gibbs sampling algorithm for continuous determinantal point processes. We show that, given a warm start, the Gibbs sampler generates a random sample from a continuous $k$-DPP defined on a $d$-dimensional domain by only taking $\text{poly}(k)$ number of steps. As an application, we design an algorithm to ge…

New space of tensorial bodies defined, properties and representatives studied.

The classical Weyl Law says that if $N_M(λ)$ denotes the number of eigenvalues of the Laplace operator on a $d$-dimensional compact manifold $M$ without a boundary that are less than or equal to $λ$, then $N_M(λ)=cλ^d+O(λ^{d-1}).$ In this paper, we show Duistermaat and Guillemin's result allows us to replace the $O(…

Innovates rotation index for matrix pairs, solving group action problems.

Let $SV^{\pmb n}_{\pmb d}$ be the Segre-Veronese given as the image of the embedding induced by the line bundle $\mathcal{O}_{\mathbb{P}^{n_1}\times\dots\times\mathbb{P}^{n_r}}(d_1,\dots, d_r)$. We prove that asymptotically $SV^{\pmb n}_{\pmb d}$ is not $h$-defective for $h\leq n_1^{\lfloor \log_2(d-1)\rfloor}$.

The study classifies discrete pseudomanifolds with up to 2d+7 vertices.

We compute the bi-Hamiltonian cohomology of an arbitrary dispersionless Poisson pencil in a single dependent variable using a spectral sequence method. As in the KdV case, we obtain that $BH^p_d(\hat{F}, d_1,d_2)$ is isomorphic to $\mathbb{R}$ for $(p,d)=(0,0)$, to $C^\infty (\mathbb{R})$ for $(p,d)=(1,1)$, $(2,1)$, $(…

For integers $d \geq 2$ and $ε= 0$ or 1, let $S^{1, d - 1}(ε)$ denote the sphere product $S^{1} \times S^{d - 1}$ if $ε= 0$ and the twisted $S^{d - 1}$ bundle over $S^{1}$ if $ε= 1$. The main results of this paper are: (a) if $d \equiv ε$ (mod 2) then $S^{1, d - 1}(ε)$ has a unique minimal triangulation using $2d + 3$ …

We find the homogenous Kähler isomorphism $FC$ which expresses the Kähler two-form on the Siegel-Jacobi domain $\mathcal{D}^J_1=\mathbb{C}\times\mathcal{D}_1$ as the sum of the Kähler two-form on $\mathbb{C}$ and the one on the Siegel ball $\mathcal{D}_1$. The classical motion and quantum evolution on $\mathcal{D}^J_1$…

The paper constructs a lamination related to minimal hypersurfaces calibrated by a cohomology class.

We construct, for any ``good'' Cantor set $F$ of $S^{n-1}$, an immersion of the sphere $S^n$ with set of points of zero Gauss-Kronecker curvature equal to $F\times D^{1}$, where $D^{1}$ is the 1-dimensional disk. In particular these examples show that the theorem of Matheus-Oliveira strictly extends two results by do C…

The Fock-Bargmann-Hartogs domain $D_{n,m}$ in $\mathbb{C}^{n+m}$ is defined by the inequality $\|w\|^2<e^{-\|z\|^2},$ where $(z,w)\in \mathbb{C}^n\times \mathbb{C}^m$, which is an unbounded non-hyperbolic domain in $\mathbb{C}^{n+m}$. This paper mainly consists of three parts. Firstly, we give the explicit expression o…

The paper determines the structure of Kakimizu complexes for genus one hyperbolic knots.

Currents represent generalized surfaces studied in geometric measure theory. They range from relatively tame integral currents representing oriented compact manifolds with boundary and integer multiplicities, to arbitrary elements of the dual space of differential forms. The flat norm provides a natural distance in the…

The paper studies neural networks with wide layers and finds a deformed semicircle law.

In this paper, we prove the K-theoretical and L-theoretical Farrell-Jones Conjecture with coefficients in an additive category for nearly crystallographic groups of the form $\mathbb{Q}^n \rtimes \mathbb{Z}$, where $\mathbb{Z}$ acts on $\mathbb{Q}^n$ as an irreducible integer matrix with determinant $d$, $|d |>1$.

The paper proves lifting theorems for complex representations of finite groups.

We consider the energy supercritical harmonic heat flow from $\mathbb{R}^d$ into the $d$-sphere $\mathbb{S}^d$ with $d \geq 7$. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one dimensional semilinear heat equation $$\partial_t u = \partial^2_r u + \frac{(d-1)}{r}\partial_r u - \…

We underline some differences between the geometric aspect of Berezin's approach to quantization on homogeneous Kähler manifolds and Bergman's construction for bounded domains in $\mathbb{C}^n$. We construct explicitly the Bergman representative coordinates for the Siegel-Jacobi disk $\mathcal{D}^J_1$, which is a parti…

We construct 2^{Ω(n^{5/4})} combinatorial types of triangulated 3-spheres on n vertices. Since by a result of Goodman and Pollack (1986) there are no more than 2^{O(n log n)} combinatorial types of simplicial 4-polytopes, this proves that asymptotically, there are far more combinatorial types of triangulated 3-spheres …

Study on kernel regression risk in high dimensions using Pinsker bound.