arXiv research

Study bottom of spectra on orbifolds via coverings.

We discuss the behaviour of the bottom of the spectrum of scalar Schrödinger operators under Riemannian coverings.

For a Riemannian covering $M_1\to M_0$ of complete Riemannian manifolds with boundary (possibly empty) and respective fundamental groups $Γ_1\subseteqΓ_0$, we show that the bottoms of the spectra of $M_0$ and $M_1$ coincide if the right action of $Γ_0$ on $Γ_1\backslashΓ_0$ is amenable.

For a Riemannian covering $π\colon M_1\to M_0$, the bottoms of the spectra of $M_0$ and $M_1$ coincide if the covering is amenable. The converse implication does not always hold. Assuming completeness and a lower bound on the Ricci curvature, we obtain a converse under a natural condition on the spectrum of $M_0$.

Study calculates spectra of minimal hypersurfaces in hyperbolic space.

The study examines arithmetic orbifolds and their length spectra, proving uniform discreteness and linear dependence of geodesic lengths.

Survey on bottom of spectrum of Hodge Laplacian on complete noncompact Kähler manifolds

A bottom tangle is a tangle in a cube consisting of arc components whose boundary points are on a line in the bottom square of the cube. A ribbon bottom tangle is a bottom tangle whose closure is a ribbon link. For every n-component ribbon bottom tangle T, we prove that the universal invariant J_T of T associated to th…

A bottom tangle is a tangle in a cube consisting only of arc components, each of which has the two endpoints on the bottom line of the cube, placed next to each other. We introduce a subcategory B of the category of framed, oriented tangles, which acts on the set of bottom tangles. We give a finite set of generators of…

Bottom-up algorithms outperform top-down in hierarchical community detection at intermediate levels.

Sharp spectral estimates for negatively curved foliations.

We study analytically and numerically Minsky instability as a combination of top-down, bottom-up and peer-to-peer positive feedback loops. The peer-to-peer interactions are represented by the links of a network formed by the connections between firms, contagion leading to avalanches and percolation phase transitions pr…

We study the bottom of the spectrum in Hilbert geometries, we show that it is zero if and only if the geometry is amenable, in other words if and only if it admits a Fölner sequence. We also show that the bottom of the spectrum admits an upper bound, which depends only on the dimension and which is the bottom of the sp…

The study finds the maximum spectrum of 3D manifolds with lower scalar curvature.

Attention mechanism combines bottom-up and top-down signals in neural networks.

We decompose returns for portfolios of bottom-ranked, lower-priced assets relative to the market into rank crossovers and changes in the relative price of those bottom-ranked assets. This decomposition is general and consistent with virtually any asset pricing model. Crossovers measure changes in rank and are smoothly …

Investigates point spectra of vector fields and their properties.

Khovanov spectra are shown to be functorial under certain conditions.

We prove that if a Riemannian covering preserves the bottom of the spectrum of a Schrödinger operator, which belongs to the discrete spectrum of the operator on the base manifold, then the covering is amenable.

The spectrum of certain manifolds matches that of hyperbolic space if the bottom spectrum is maximal.

A link L is called Brunnian if every proper sublink of L is trivial. Similarly, a bottom tangle T is called Brunnian if every proper subtangle of T is trivial. In this paper, we give a small subalgebra of the n-fold completed tensor power of U_h(sl_2) in which the universal sl_2 invariant of n-component Brunnian bottom…

The study of spectral-tightness in Riemannian manifolds and its topological implications.

BUSTLE synthesizes programs by learning from intermediate values.

We give a simple sufficient condition for Quinn's "bordism-type spectra" to be weakly equivalent to strictly associative ring spectra. We also show that Poincare bordism and symmetric L-theory are naturally weakly equivalent to monoidal functors. Part of the proof of these statements involves showing that Quinn's funct…

The universal sl_2 invariant of bottom tangles has a universality property for the colored Jones polynomial of links. Habiro conjectured that the universal sl_2 invariant of boundary bottom tangles takes values in certain subalgebras of the completed tensor powers of the quantized enveloping algebra U_h(sl_2) of the Li…

Method calculates spectra of Rarita-Schwinger operator on symmetric spaces.

Paper sharpens inequality linking curvature and spectrum on manifolds.

Proves spectra equivalence for Riemannian manifolds.

We compute the bridge spectra of cables of 2-bridge knots. We also give some results about bridge spectra and distance of Montesinos knots.

Paper resolves decades-old problem about $L$-spectra.

Closed hyperbolic manifolds and manifolds with nonpositive sectional curvature are geometrically rigid under certain curvature conditions.

A new hierarchical forecasting method improves overall accuracy.

We consider a class of complete Kahler manifolds with a strictly pseudoconvex boundary at infinity. After studying its asymptotic geometry, we formulate a conjecture in the Kahler-Einstein case relating the bottom of spectrum to the CR geometry on the boundary. We prove some partial results.

New metrics compare rational spectra using optimal transport.

In this paper, we numerically investigate the length spectra and the low-lying eigenvalue spectra of the Laplace-Beltrami operator for a large number of small compact(closed) hyperbolic (CH) 3-manifolds. The first non-zero eigenvalues have been successfully computed using the periodic orbit sum method, which are compar…

Tomova, along with results of Bachman and Schleimer, showed that any high distance knot has a stair-step bridge spectrum. In this paper, we compute the bridge spectra and distance of generalized Montesinos knots. In particular, we produce the first example of a class of knots which attain the stair-step bridge spectra …

New ICA method for sources with mixed spectra.

Jones-Wenzl projectors lifted to Khovanov spectra, proving knot conjectures.

Efficient unsupervised training and inference in deep generative models remains a challenging problem. One basic approach, called Helmholtz machine, involves training a top-down directed generative model together with a bottom-up auxiliary model used for approximate inference. Recent results indicate that better genera…

We introduce a framework, twisted parametrized stable homotopy theory, for describing semi-infinite homotopy types. A twisted parametrized spectrum is a section of a bundle whose fibre is the category of spectra. We define these bundles in terms of modules over a stack of parametrized spectra and in terms of diagrams o…

Functor decomposes Khovanov spectra for non-alternating diagrams.

In this paper, we develop Leray-Serre-type spectral sequences to compute the intersection homology of the regular neighborhood and deleted regular neighborhood of the bottom stratum of a stratified PL-pseudomanifold. The E^2 terms of the spectral sequences are given by the homology of the bottom stratum with a local co…

Sketchy reduces memory and compute requirements for adaptive regularization in deep learning.

Paper bounds the lowest spectrum of manifolds with curvature constraints.

Novel construction of Bauer--Furuta invariant using sheaves of spectra.

Study on sine-cones' spectra and stability under Ricci-de Turck flow.

We prove explicit upper and lower bounds for the $L^1$-moment spectra for the Brownian motion exit time from extrinsic metric balls of submanifolds $P^m$ in ambient Riemannian spaces $N^{n}$. We assume that $P$ and $N$ both have controlled radial curvatures (mean curvature and sectional curvature, respectively) as view…

The paper describes correlations of spectra for higher rank Anosov representations.