arXiv research

We outline the theory of sets with distributive operations: multishelves and multispindles, with examples provided by semi-lattices, lattices and skew lattices. For every such a structure we define multi-term distributive homology and show some of its properties. The main result is a complete formula for the homology o…

New rigidity theorem for product of lattices.

Proves effective slope gaps for lattice surfaces.

Study Poisson boundaries of building lattices and generalize rigidity results.

Regular integer lattices are characterized by k unit vectors that build up their generator matrices. These have rank k for D-lattices, and are rank-deficient for A-lattices, for E_6 and E_7. We count lattice points inside hypercubes centered at the origin for all three types, as if classified by maximum infinity norm i…

L-CNNs learn gauge invariant quantities on lattices.

Study answers arithmeticity question for normal subgroup of lattices.

Discrete knot theory models use lattice-filtered graphs to detect merging knot components.

Complete classification of Deligne-Mostow lattice representations into PGL(3,C).

The asymptotic lattices and their transformations are studied within the line geometry approach. It is shown that the discrete asymptotic nets are represented by isotropic congruences in the Plucker quadric. On the basis of the Lelieuvre-type representation of asymptotic lattices and of the discrete analog of the Mouta…

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

New methods classify convex lattice polygons for affine dimers.

Study lattice paths from twist knots and double twist knots.

The (isothermic) compressibility of lattice knots can be examined as a model of the effects of topology and geometry on the compressibility of ring polymers. In this paper, the compressibility of minimal length lattice knots in the simple cubic, face centered cubic and body centered cubic lattices are determined. Our r…

We prove that any action of a higher rank lattice on a Gromov-hyperbolic space is elementary. More precisely, it is either elliptic or parabolic. This is a large generalization of the fact that any action of a higher rank lattice on a tree has a fixed point. A consequence is that any quasi-action of a higher rank latti…

New bounds on diameters and generators for specific lattices and graphs.

Knot lattice homology invariant of smooth knot type in rational homology spheres.

Study BSΔE on lattices for asset price analysis.

Study $\mathbb{C}$-Fuchsian subgroups of non-arithmetic lattices.

Study area-minimizing subgraphs in integer lattices.

Equivariant neural networks improve performance and generalization in lattice field theory tasks.

New minimal surfaces found using Toda lattice and integrable systems.

New MCMC method samples from lattice distributions efficiently.

A hyperbolic lattice is called \textit{$1.2$-reflective} if the subgroup of its automorphism group generated by all $1$- and $2$-reflections is of finite index. The main result of this article is a complete classification of $1.2$-reflective maximal anisotropic lattices of rank $4$.

Lattice formulation captures Atiyah-Patodi-Singer index.

The study connects lattices, Garside structures, and weakly modular graphs.

We show the rank (i.e. minimal size of a generating set) of lattices cannot grow faster than the volume.

New property identifies arithmetic lattices from nonuniform lattices.

Novel trading strategy for generalized lattice markets ensures positive profits.

We study geometric consistency relations between angles on 3-dimensional (3D) circular quadrilateral lattices -- lattices whose faces are planar quadrilaterals inscribable into a circle. We show that these relations generate canonical transformations of a remarkable ``ultra-local'' Poisson bracket algebra defined on di…

Classifies lattices from knot surgeries, defining a concordance invariant.

A Riemannian symmetric space is a Riemannian manifold in which it is possible to reflect all geodesics through a point by an isometry of the space. On such spaces, we introduce the notion of a distributional lattice, generalizing the notion of lattice. Distributional lattices exist in any Riemannian symmetric space: th…

Principal binets generalize curvature line surfaces to square lattices and are a discrete integrable system.

The paper introduces lattice homology for integrally closed submodules and applies it to geometric invariants.

The paper proposes a method to sample quantum field configurations using neural operators and flows.

Research finds bounds for knots in hexagonal lattice and classifies 11-stick knots.

Unified framework for complex financial networks using lattice theory.

Course on arithmetic lattices at EPFL.

The lattice stick number $s_L(L)$ of a link $L$ is defined to be the minimal number of straight line segments required to construct a stick presentation of $L$ in the cubic lattice. Hong, No and Oh found a general upper bound $s_L(K) \leq 3 c(K) +2$. A rational link can be represented by a lattice presentation with exa…

We give a simple example showing that a knot or link diagram that lies in the ${\mathbb{Z}}^2$ lattice is not necessarily the projection of a lattice stick knot or link in the ${\mathbb{Z}}^3$ lattice, and we give a necessary and sufficient condition for when a knot or link diagram that lies in the ${\mathbb{Z}}^2$ lat…

We introduce a criterion that a given bihamiltonian structure allows a local coordinate system where both brackets have constant coefficients. This criterion is applied to the bihamiltonian open Toda lattice in a generic point, which is shown to be locally isomorphic to a Kronecker odd-dimensional pair of brackets with…

We derive results on the distribution of directions of saddle connections on translation surfaces using only the Birkhoff ergodic theorem applied to the geodesic flow on the moduli space of translation surfaces. Our techniques, together with an approximation argument, also give an alternative proof of a weak version of…

The paper defines and studies discrete p-density and compression-radius profiles of lattice knots.

We prove that the rank (that is, the minimal size of a generating set) of lattices in a general connected Lie group is bounded by the co-volume of the projection of the lattice to the semi-simple part of the group. This was proved by Gelander for semi-simple Lie groups and by Mostow for solvable Lie groups. Here we con…

The study examines how perturbations of lattice actions on group boundaries behave.

We define and prove properties of link lattice complexes for plumbed links.

Generalizes Kauffman's clock theorem to surfaces.

In the present article, we provide examples of fake quadrics, that is, minimal complex surfaces of general type with the same numerical invariants as the smooth quadric in $\PP ^3$ which are quotients of the bidisc by an irreducible lattice of automorphisms. Moreover, we list classes of arithmetic lattices over a real …