arXiv research

The study restricts circle actions on certain manifolds to dimensions 4, 8, and 16.

This is the first part of a series of papers where we compute Euler characteristics, signatures, elliptic genera, and a number of other invariants of smooth manifolds that admit Riemannian metrics with positive sectional curvature and large torus symmetry. In the first part, the focus is on even-dimensional manifolds i…

Let M be a Riemannian manifold and R its curvature tensor. For a unit vector X tangent to M at a point p, the Jacobi operator is defined by R_X = R(X, .) X$. The manifold M is called pointwise Osserman if, for every point p, the spectrum of the Jacobi operator does not depend of the choice of X, and is called globally …

The paper explores connections between quaternionic and Cayley calibrations in dimensions 8 and 16.

This paper is an addition to the book [54] on Compact projective planes. Such planes, if connected and finite-dimensional, have a point space of topological dimension 2, 4, 8, or 16, the classical example in the last case being the projective closure of the affine plane over the octonion algebra. The final result in th…

We implement a Tensor Train layer in the TensorFlow Neural Machine Translation (NMT) model using the t3f library. We perform training runs on the IWSLT English-Vietnamese '15 and WMT German-English '16 datasets with learning rates $\in \{0.0004,0.0008,0.0012\}$, maximum ranks $\in \{2,4,8,16\}$ and a range of core dime…

We give a lower and an upper bound for the conformal dimension of the boundaries of certain small cancellation groups. We apply these bounds to the few relator and density models for random groups. This gives generic bounds of the following form, where $l$ is the relator length, going to infinity. (a) $1 + 1/C < \Cdim(…

We show that pseudo-Riemannian almost quaternionic homogeneous spaces with index 4 and an H-irreducible isotropy group are locally isometric to a pseudo-Riemannian quaternionic Kähler symmetric space if the dimension is at least 16. In dimension 12 we give a non-symmetric example.

We determine all complete projective special real surfaces. By the supergravity r-map, they give rise to complete projective special Kähler manifolds of dimension 6, which are distinguished by the image of their scalar curvature function. By the supergravity c-map, the latter manifolds define in turn complete quaternio…

Researchers found new dimensions for exceptional Lie group realizations.

We consider manifolds $M^{2n}$ which admit smooth maps into a connected sum of $S^1\times S^n$ with only finitely many critical points, for $n\in\{2,4,8\}$, and compute the minimal number of critical points.

New techniques improve 16-bit training accuracy without 32-bit units.

Since there is no hyperbolic Dehn filling theorem for higher dimensions, it is challenging to construct explicit hyperbolic manifolds of small volume in dimension at least four. Here, we build up closed hyperbolic 4-manifolds of volume $\frac{34π^2}{3}\cdot 16$ by using the small cover theory. In particular, we classif…

Two geometric realizations of $F(4)$ Lie superalgebra found.

We explore all warped $AdS_4\times_w M^{D-4}$ backgrounds with the most general allowed fluxes that preserve more than 16 supersymmetries in $D=10$- and $11$-dimensional supergravities. After imposing the assumption that either the internal space $M^{D-4}$ is compact without boundary or the isometry algebra of the back…

Maximal rank Coxeter quotients found for 1.7M knots up to 16 crossings.

Listed 19,513 prime knots with arc index 12-16.

We prove that the supergravity r- and c-maps preserve completeness. As a consequence, any component H of a hypersurface {h=1} defined by a homogeneous cubic polynomial such that -d^2 h is a complete Riemannian metric on H defines a complete projective special Kahler manifold and any complete projective special Kahler m…

An algebraic curvature tensor is called Osserman if the eigenvalues of the associated Jacobi operator are constant on the unit sphere. A Riemannian manifold is called conformally Osserman if its Weyl conformal curvature tensor at every point is Osserman. We prove that a conformally Osserman manifold of dimension $n \ne…

Optimal scaling for proximal MALA in high dimensions confirmed.

We study bi-Lagrangian structures (a symplectic form with a pair of complementary Lagrangian foliations, also known as para-Kähler or Künneth structures) on nilmanifolds of dimension less than or equal to 6. In particular, building on previous work of several authors, we determine which 6-dimensional nilpotent Lie alge…

Notation for spin coefficients for metrics of neutral signature in four dimensions is introduced. The utility and interpretation of spin coefficients is explored through themes in null geometry familiar from (complex) general relativity. Four-dimensional Walker geometry is exploited to provide examples and the generali…

The study proves conditions for complete Riemannian manifolds to be Einstein.

We show that the self-duality defined in [Trautman, Int.J.Theor.Phys.,{\bf 16},561 (1977)] is equivalent to strong self-duality defined in [Bilge, Dereli and Kocak, Lett.Math.Phys., {\bf 36}, 301-309, (1996)] and we obtain an upper bound on $\int (p_2)^{n}$, where $p_2$ is the second Pontrjagin class of an $SO(N)$ bund…

Isomorphism classes of Alexander quandles of order 16 are determined, and classes of connected quandles are identified. This paper extends the list of known distinct connected finite Alexander quandles.

The aim of the present paper is the investigation of $Spin(9)$-structures on 16-dimensional manifolds from the point of view of topology as well as holonomy theory. First we construct several examples. Then we study the necessary topological conditions resulting from the existence of a $Spin(9)$-reduction of the frame …

For a Riemannian manifold $M^n$ with the curvature tensor $R$, the Jacobi operator $R_X$ is defined by $R_XY = R(X,Y)X$. The manifold $M^n$ is called {\it pointwise Osserman} if, for every $p \in M^n$, the eigenvalues of the Jacobi operator $R_X$ do not depend of a unit vector $X \in T_pM^n$, and is called {\it globall…

We show that there are no smooth warped AdS$_3$ solutions in 10- and 11-dimensional supergravities which preserve strictly more than 16 supersymmetries and have internal space a compact without boundary manifold.

We classify the simply-connected supersymmetric parallelisable backgrounds of heterotic supergravity. They are all given by parallelised Lie groups admitting a bi-invariant lorentzian metric. We find examples preserving 4, 8, 10, 12, 14 and 16 of the 16 supersymmetries.

The 4-dimensional abstract Kummer variety K^4 with 16 nodes leads to the K3 surface by resolving the 16 singularities. Here we present a simplicial realization of this minimal resolution. Starting with a minimal 16-vertex triangulation of K^4 we resolve its 16 isolated singularities - step by step - by simplicial blowu…

Gray & Hervella gave a classification of almost Hermitian structures (g,I) into 16 classes. We systematically study the interaction between these classes when one has an almost hyper-Hermitian structure (g,I,J,K). In general dimension we find at most 167 different almost hyper-Hermitian structures. In particular, we ob…

Paper proves optimal decomposition for matrix fields, reducing convex integration steps.

Moebius-Kantor graph connects multiple groups and topological properties.

In M-theory vacua with vanishing 4-form F, one can invoke the ordinary Riemannian holonomy H \subset SO(1,10) to account for unbroken supersymmetries n=1, 2, 3, 4, 6, 8, 16, 32. However, the generalized holonomy conjecture, valid for non-zero F, can account for more exotic fractions of supersymmetry, in particular 16<n…

We use Klyachko's methods to prove that the natural map G to G-hat, where G is a torsion-free group and G-hat is obtained by adding a new generator t and a new relator w, is surjective only if w is conjugate to gt or gt^{-1} for some g in G. This solves a special case of the surjectivity problem for group extensions, r…

In this project we further investigate the idea of reducing the dimensionality of datasets using a Borel isomorphism with the purpose of subsequently applying supervised learning algorithms, as originally suggested by my supervisor V. Pestov (in 2011 Dagstuhl preprint). Any consistent learning algorithm, for example kN…

eDCF estimates intrinsic dimension using local connectivity.

Calibration in 16D disproves Federer's product question.

We prove a relation between the scaling $h^β$ of the elastic energies of shrinking non-Euclidean bodies $S_h$ of thickness $h\to 0$, and the curvature along their mid-surface $S$. This extends and generalizes similar results for plates [BLS16, LRR] to any dimension and co-dimension. In particular, it proves that the na…

We prove the LeBrun-Salamon Conjecture in low dimensions. More precisely, we show that a contact Fano manifold X of dimension 2n+1 that has reductive automorphism group of rank at least n-2 is necessarily homogeneous. This implies that any positive quaternion-Kahler manifold of real dimension at most 16 is necessarily …

Recently, the posit numerical format has shown promise for DNN data representation and compute with ultra-low precision ([5..8]-bit). However, majority of studies focus only on DNN inference. In this work, we propose DNN training using posits and compare with the floating point training. We evaluate on both MNIST and F…

We give an interpretation of the maximal number of linearly independent vector fields on spheres in terms of the Spin(9) representation on R^16. This casts an insight on the role of Spin(9) as a subgroup of SO(16) on the existence of vector fields on spheres, parallel to the one played by complex, quaternionic and octo…

The Teichmüller harmonic map flow deforms both a map from an oriented closed surface $M$ into an arbitrary closed Riemannian manifold, and a constant curvature metric on $M$, so as to reduce the energy of the map as quickly as possible [16]. The flow then tries to converge to a branched minimal immersion when it can [1…

We formulate and discuss two conjectures concerning recursive formulae for Branson's $Q$-curvatures. The proposed formulae describe all $Q$-curvatures on manifolds of all even dimensions in terms of respective lower order $Q$-curvatures and lower order GJMS-operators. They are universal in the dimension of the underlyi…

We prove the following: there are infinitely many finite-covolume (resp. cocompact) Coxeter groups acting on hyperbolic space H^n for every n < 20 (resp. n < 7). When n=7 or 8, they may be taken to be nonarithmetic. Furthermore, for 1 < n < 20, with the possible exceptions n=16 and 17, the number of essentially distinc…

Study K-theory of homogeneous spaces of Lie groups.

For a Spin(9)-structure on a Riemannian manifold M^16 we write explicitly the matrix psi of its Kähler 2-forms and the canonical 8-form Phi. We then prove that Phi coincides up to a constant with the fourth coefficient of the characteristic polynomial of psi. This is inspired by lower dimensional situations, related to…

Flat tori found non-isometric pairs with identical Laplace eigenvalues.