Study on mapping class groups for low genus surfaces.
problem Arithmetic image of homological representations for low genus mapping class groups.
method Examined centralizers of finite groups acting on surfaces with genus at most 3.
result Complete answer to the arithmetic image question for low genus surfaces.
In this paper we produce many examples of thin subgroups of special linear groups that are isomorphic to the fundamental groups of non-arithmetic hyperbolic manifolds. Specifically, we show that the non-arithmetic lattices in SO(n,1) constructed by Gromov and Piatetski-Shapiro can be embedded into $\mathrm{S…
Better neural arithmetic logic units improve cell counting model generalization.
problem Neural networks struggle with high cell counts outside training data range.
method Introduced Neural Arithmetic Logic Units (NALU) for arithmetic operations in existing architectures.
result Improved cell counting accuracy for higher numeric ranges with better generalization.
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 …
This study examines arithmetic properties of GIB manifolds and their monodromy representations.
problem Arithmetic structure of generalized Inoue--Bombieri manifolds.
method Study of monodromy representations and their arithmetic properties.
result The image of the monodromy representation is a subgroup of a cocompact arithmetic lattice.
By considering appropriate finite covering spaces of closed non-orientable surfaces, we construct linear representations of their mapping class group which have finite index image in certain big arithmetic groups.
Study actions of mapping class groups on surface representations, proving finite image for certain representations.
problem Finite image of representations of mapping class groups on surfaces.
method Hodge-theoretic and arithmetic techniques, including non-abelian Hodge theory and isomonodromic deformations.
result Proves finite image for representations with specific properties.
The paper studies representations of braid groups via curves and finds conditions for their Zariski closure and arithmeticity.
problem Representations of braid groups via specific families of Riemann surfaces.
method Consider families of Riemann surfaces defined by plane curves and study their monodromy representations into symplectic groups.
result Criterions for the Zariski closure of the image of the representation to be maximal and for the image to be an arithmetic lattice.
The paper disproves a conjecture about isomorphic subgroups in finite groups.
problem The existence of non-isomorphic subgroups that are isomorphic in extensions of finite groups.
method Constructing extensions of finite groups to show non-isomorphic pre-images of subgroups.
result Subgroups of finite groups that are isomorphic in extensions are not conjugate.
Achieving machine intelligence requires a smooth integration of perception and reasoning, yet models developed to date tend to specialize in one or the other; sophisticated manipulation of symbols acquired from rich perceptual spaces has so far proved elusive. Consider a visual arithmetic task, where the goal is to car…
A new deep learning method using Boolean logic reduces training and inference energy.
problem High computational and energy costs in deep learning training and inference.
method Introduces Boolean weights and inputs for efficient training using Boolean logic.
result Achieves full-precision accuracy in ImageNet classification and surpasses state-of-the-art results in semantic segmentation.
Faces of quasi-arithmetic Coxeter polytopes are also quasi-arithmetic.
problem Characterizing faces of quasi-arithmetic Coxeter polytopes.
method Proof of quasi-arithmetic property of faces and sufficient condition for arithmetic faces.
result Lower-dimensional faces of quasi-arithmetic Coxeter polytopes are quasi-arithmetic.
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…
Develops arithmetic PDE geometry concepts like curvature and cohomology.
problem Creating a geometry framework for arithmetic PDEs.
method Introducing arithmetic analogues of Levi-Civita and Chern connections, then developing curvature and characteristic classes.
result Arithmetic analogues of curvature and characteristic classes have been developed.
We prove that every homomorphism from the elementary Chevalley group over a finitely generated unital commutative ring associated with reduced irreducible classical root system of rank at least 2, and ME analogues of such groups, into acylindrically hyperbolic groups has an absolutely elliptic image. This result provid…
New geometric invariant limits the number of semi-arithmetic groups.
problem Understanding the structure of semi-arithmetic Fuchsian groups.
method Introducing a new geometric invariant called stretch and using the arithmetic Margulis lemma.
result There exist only finitely many conjugacy classes of semi-arithmetic groups with bounded arithmetic dimension, stretch, and coarea.
Course on arithmetic lattices at EPFL.
problem Understanding arithmetic lattices.
method Introductory course on arithmetic lattices.
result Introduction to arithmetic lattices.
Convolutional neural networks (CNNs) have achieved remarkable performance in many applications, especially in image recognition tasks. As a crucial component of CNNs, sub-sampling plays an important role for efficient training or invariance property, and max-pooling and arithmetic average-pooling are commonly used sub-…
Paper shows non-arithmetic surface with unique geometric property.
problem Non-arithmetic surfaces with unique geometric properties.
method Example of a non-arithmetic surface with marked length variety rigidity.
result Found a non-arithmetic surface with marked length variety rigidity.
New classification of hyperbolic Coxeter prisms.
problem Classifying hyperbolic Coxeter prisms.
method Determine which prisms are quasi-arithmetic or arithmetic.
result New insights into commensurability and systoles of associated orbifolds.
Arithmetic Dijkgraaf-Witten theory constructs analogues in Chern-Simons TQFT.
problem Developing arithmetic analogues in Chern-Simons TQFT.
method Constructing arithmetic analogues of Chern-Simons 1-cocycle, prequantization bundle, and Chern-Simons functional.
result Decomposition and gluing formulas for arithmetic Chern-Simons invariants and arithmetic Dijkgraaf-Witten partition functions.
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…
New method constructs non-arithmetic hyperbolic orbifolds from complex arithmetic ball quotients.
problem Creating non-arithmetic lattices in projective orthogonal groups.
method Using anti-holomorphic involutions on complex arithmetic ball quotients, gluing fixed loci along geodesic subspaces.
result Explicit calculation of the volume of constructed non-arithmetic orbifolds.
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.
problem Arithmeticity criterion for hyperbolic lattices and suborbifolds.
method Analysis of totally geodesic suborbifolds and Vinberg's commensurability invariants.
result Arithmeticity of hyperbolic orbifolds is linked to the existence of infinitely many fc-subspaces.
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.
problem Understanding arithmetic properties of modular surfaces.
method Constructing geodesics and analyzing their lifts.
result Complements of canonical lifts are arithmetic 3-manifolds.
New property identifies arithmetic lattices from nonuniform lattices.
problem Characterizing arithmetic lattices among nonuniform lattices.
method Introduced Bounded Clustering (B-C) property.
result B-C property uniquely identifies arithmetic lattices.
New proof shows maximal arithmetic groups are finite.
problem Finiteness of maximal arithmetic reflection groups.
method Arithmetic Margulis lemma without automorphic forms.
result Finiteness of maximal arithmetic reflection groups proven.
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.
problem Creating an arithmetic PDE analogue of Riemannian geometry.
method Using Fermat quotients and Frobenius elements in the absolute Galois group of a p-adic field. result Existence and uniqueness of geodesics and connections proved.
Study general hyperbolic gluings, proving quasi-arithmeticity of building blocks.
problem Proving quasi-arithmeticity of building blocks in hyperbolic gluings.
method Generalized gluings of hyperbolic orbifolds, proving quasi-arithmeticity.
result Building blocks of quasi-arithmetic gluings must also be quasi-arithmetic.
The study of systoles in arithmetic hyperbolic manifolds.
problem Understanding the systoles of arithmetic hyperbolic manifolds.
method Construction and analysis of arithmetic hyperbolic manifolds.
result Explicit bounds on volumes and systoles of arithmetic hyperbolic manifolds.
We explore hybrid subgroups of certain non-arithmetic lattices in 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 PU(1,1).
New research shows certain arithmetic lattices can't be LERF.
problem Determining if arithmetic lattices are LERF.
method Analyzing trialitarian arithmetic lattices in PSO7,1(R). result Trialitarian arithmetic lattices in PSO7,1(R) are not LERF. Arithmetic spaces' thin parts are negligible, impacting Betti numbers.
problem Understanding the structure of arithmetic locally symmetric spaces.
method Analyzing thin parts and deducing asymptotic results on Betti numbers.
result 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.
problem Understanding sphere packings and their arithmetic origins in various dimensions.
method Introduces Kleinian Sphere Packings and Bugs, extending Arithmeticity Theorem.
result Kleinian packings and Bugs come from Q-arithmetic lattices of simplest type.
Conditions for flat manifolds as cusp cross-sections in arithmetic hyperbolic manifolds.
problem Determining when a flat manifold can be a cusp cross-section in arithmetic hyperbolic manifolds.
method Analyzing rational representations of holonomy groups and quasi-arithmetic manifolds.
result Conditions for a flat manifold to appear as a cusp cross-section in every commensurability class of arithmetic hyperbolic manifolds.
Study on Alexander polynomials in braids, linking number theory and topology.
problem Distribution of Alexander polynomials in specific families of braids.
method Exploration of arithmetic invariants and analogies with number theory.
result New directions in arithmetic topology and statistics.
Study finds bounds for systole length on arithmetic punctured spheres.
problem Finding the shortest essential curve on arithmetic punctured spheres.
method Correspondence between surfaces and planar triangulations to bound systole length.
result Arithmetic surfaces do not achieve maximal systole length for n=7,10,11. The study finds infinitely many semi-arithmetic Riemann surfaces with dense systoles and distinct invariant trace fields.
problem Existence and properties of semi-arithmetic Riemann surfaces.
method Combining number theory and hyperbolic geometry to prove existence and properties of semi-arithmetic Riemann surfaces.
result Existence of 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.
problem Neural Networks struggle with generalizing beyond seen data and arithmetic operations.
method Introduces Neural Power Unit (NPU) that operates on real numbers and learns arbitrary power functions.
result NPU outperforms competitors in accuracy and sparsity on arithmetic datasets and discovers governing equations from data.
We prove that any arithmetic hyperbolic n-manifold of simplest type can either be geodesically embedded into an arithmetic hyperbolic (n+1)-manifold or its universal mod 2 Abelian cover can.
The paper characterizes arithmetic metrics in coarsely geometric settings.
problem Characterizing arithmetic metrics in coarsely geometric settings.
method Using coarse-geometric commensurators and under the Hilbert-Smith conjecture.
result Positive answer in general and unconditional for specific cases.
We prove, under the assumption of the virtual fibration conjecture for arithmetic hyperbolic 3-manifolds, that all arithmetic lattices in O(n,1), n> 4, and different from 7, are non-coherent. We also establish noncoherence of uniform arithmetic lattices of the simplest type in SU(n,1), n> 1, and of uniform lattices in …
Study C-Fuchsian subgroups of non-arithmetic lattices.
problem Understand structure and fundamental domains of C-Fuchsian subgroups. method General procedure to analyze structure and show fundamental domains lie on a complex geodesic.
result Fundamental domains of C-Fuchsian subgroups lie on a complex geodesic homeomorphic to the unit disk. Explains how arithmetic manifolds solve geometric questions about systole and kissing number.
problem Geometric questions about systole and kissing number in hyperbolic manifolds.
method Use of arithmetic manifolds to solve geometric questions.
result Answers geometric questions about systole and kissing number for dimension 2 and higher dimensions.