arXiv research

The paper characterizes arithmetic metrics in coarsely geometric settings.

In this article, we investigate when the set of primitive geodesic lengths on a Riemannian manifold have arbitrarily long arithmetic progressions. We prove that in the space of negatively curved metrics, a metric having such arithmetic progressions is quite rare. We introduce almost arithmetic progressions, a coarsific…

We extend the Faltings modular heights of abelian varieties to general arithmetic varieties and show direct relations with the Kahler-Einstein geometry, the Minimal Model Program, heights of Bost and Zhang, and give some applications. Along the way, we propose arithmetic Yau-Tian-Donaldson conjecture, an equivalence of…

New method constructs non-arithmetic hyperbolic orbifolds from complex arithmetic ball quotients.

A "hidden symmetry" of a Riemannian manifold M is an isometry of a d-sheeted, 1<d<\infty, Riemannian cover of M which is not the lift of any isometry. In this paper we characterize the locally symmetric metric(s) on a closed, arithmetic manifold as the unique metric with infinitely many hidden symmetries.

For singular metrics, there is no Quillen metric formalism on cohomology determinant. In this paper, we develop an admissible theory, with which the arithmetic Deligne-Riemann-Roch isometry can be established for singular metrics. As an application, we first study Weil-Petersson metrics and Takhtajan-Zograf metrics on …

R. Zimmer proved that, on a compact manifold, a foliation with a dense leaf, a suitable leafwise Riemannian symmetric metric and a transverse Lie structure has arithmetic holonomy group. In this work we improve such result for totally geodesic foliations by showing that the manifold itself is arithmetic. This also give…

Block and Weinberger show that an arithmetic manifold can be endowed with a positive scalar curvature metric if and only if its $\rationals$-rank exceeds 2. We show in this article that these metrics are never in the same coarse class as the natural metric inherited from the base Lie group. Furthering the coarse $C^\as…

In this article we verify an orbifold version of a conjecture of Nimershiem from 1998. Namely, for every flat $n$-manifold $M$, we show that the set of similarity classes of flat metrics on $M$ which occur as a cusp cross-section of a hyperbolic $(n+1)$-orbifold is dense in the space of similarity classes of flat metri…

We show that the finiteness length of an $S$-arithmetic subgroup $Γ$ in a noncommutative isotropic absolutely almost simple group $G$ over a global function field is one less than the sum of the local ranks of $G$ taken over the places in $S$. This determines the finiteness properties for arithmetic subgroups in isotro…

We prove an arithmetic Hilbert-Samuel type theorem for semi-positive singular hermitian line bundles of finite height. In particular, the theorem applies to the log-singular metrics of Burgos-Kramer-Kühn. Our theorem is thus suitable for application to some non-compact Shimura varieties with their bundles of cusp forms…

The purpose this article is to try to understand the mysterious coincidence between the asymptotic behavior of the volumes of the Moduli Space of closed hyperbolic surfaces of genus $g$ with respect to the Weil-Petersson metric and the asymptotic behavior of the number of arithmetic closed hyperbolic surfaces of genus …

In this paper we will discuss local coordinates canonically corresponding to a Kahler metric. We will also discuss and prove the $C^\infty$ convergence of Bergman metrics following Tian's result on $C^2$ convergence of Bergman metrics. At the end, we present an interesting characterization of ample line bundle that cou…

A new metric framework for weighted projective spaces improves clustering and analysis.

We study lattices in non-positively curved metric spaces. Borel density is established in that setting as well as a form of Mostow rigidity. A converse to the flat torus theorem is provided. Geometric arithmeticity results are obtained after a detour through superrigidity and arithmeticity of abstract lattices. Residua…

In this paper, we extend Deligne's functorial Riemann-Roch isomorphism for hermitian holomorphic line bundles on Riemann surfaces to the case of flat, not necessarily unitary connections. The Quillen metric and star-product of Gillet-Soule are replaced with complex valued logarithms. On the determinant of cohomology si…

Faces of quasi-arithmetic Coxeter polytopes are also quasi-arithmetic.

The study quantifies distances between certain hyperbolic surfaces and bounds their number.

New metrics found on complex solvmanifolds.

Develops arithmetic PDE geometry concepts like curvature and cohomology.

Sharp bounds on Fano varieties' heights proven for specific cases.

New geometric invariant limits the number of semi-arithmetic groups.

Course on arithmetic lattices at EPFL.

Paper shows non-arithmetic surface with unique geometric property.

New classification of hyperbolic Coxeter prisms.

Arithmetic Dijkgraaf-Witten theory constructs analogues in Chern-Simons TQFT.

In this article, we prove that every arithmetic locally symmetric orbifold of classical type without Euclidean or compact factors has arbitrarily long arithmetic progressions in its primitive length spectrum. Moreover, we show the stronger property that every primitive length occurs in arbitrarily long arithmetic progr…

We show that a locally symmetric space of noncompact type and with finite volume is quasi-isometric to the euclidean cone over a finite simplicial complex. A detailed analysis of metric properties yields a proof of a conjecture of Siegel.

We show that the non-arithmetic lattices in PO(n,1) of Belolipetsky and Thomson (2011), obtained as fundamental groups of closed hyperbolic manifolds with short systole, are quasi-arithmetic in the sense of Vinberg, and, by contrast, the well-known non-arithmetic lattices of Gromov and Piatetski-Shapiro are not quasi-a…

The paper explores subspaces in hyperbolic lattices and their arithmetic properties.

Define an arithmetic variety to be the quotient of a bounded symmetric domain by an arithmetic group. An arithmetic variety is algebraic, and the theorem in question states that when one applies an automorphism of the field of complex numbers to the coefficients of an arithmetic variety the resulting variety is again a…

Geodesics on modular surface yield arithmetic 3-manifolds.

New property identifies arithmetic lattices from nonuniform lattices.

New proof shows maximal arithmetic groups are finite.

We study the arithmeticity of the Couwenberg-Heckman-Looijenga lattices in PU(n,1), and show that they contain a non-arithmetic lattice in PU(3,1) which is not commensurable to the non-arithmetic Deligne-Mostow lattice in PU(3,1).

Develops arithmetic PDE geometry using Fermat quotients.

Study general hyperbolic gluings, proving quasi-arithmeticity of building blocks.

The study of systoles in arithmetic hyperbolic manifolds.

We explore hybrid subgroups of certain non-arithmetic lattices in $\mathrm{PU}(2,1)$. We show that all of Mostow's lattices are virtually hybrids; moreover, we show that some of these non-arithmetic lattices are hybrids of two non-commensurable arithmetic lattices in $\mathrm{PU}(1,1)$.

New research shows certain arithmetic lattices can't be LERF.

Arithmetic spaces' thin parts are negligible, impacting Betti numbers.

We apply G. Prasad's volume formula for the arithmetic quotients of semi-simple groups and Bruhat-Tits theory to study the covolumes of arithmetic subgroups of SO(1,n). As a result we prove that for any even dimension n there exists a unique compact arithmetic hyperbolic n-orbifold of the smallest volume. We give a for…

Develops Kleinian Sphere Packings and Bugs, proving their arithmetic origins.

Conditions for flat manifolds as cusp cross-sections in arithmetic hyperbolic manifolds.

Study on Alexander polynomials in braids, linking number theory and topology.

Study finds bounds for systole length on arithmetic punctured spheres.

The study finds infinitely many semi-arithmetic Riemann surfaces with dense systoles and distinct invariant trace fields.

Neural Power Unit (NPU) learns arbitrary power functions on real numbers.