arXiv research

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

We study the existence of lattices in almost abelian Lie groups that admit left invariant locally conformal Kähler or locally conformal symplectic structures in order to obtain compact solvmanifolds equipped with these geometric structures. In the former case, we show that such lattices exist only in dimension $4$, whi…

Based on \cite{DH94}, we introduce a bijective correspondence between first order differential calculi and the graph structure of the symmetric lattice that allows one to encode completely the interconnection structure of the graph in the exterior derivative. As a result, we obtain the Grassmannian character of the lat…

New lattice path method for statistical inference of persistent diagrams.

Unified framework for complex financial networks using lattice theory.

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…

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…

Generalizes Kauffman's clock theorem to surfaces.

This paper solves PDEs for embedding discrete lattices into smooth manifolds.

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

This paper introduces lattice representations for efficient discrete learning.

Motivated by Kohno's result on the holonomy Lie algebra of a hyperplane arrangement, we define the holonomy Lie algebra of a finite geometric lattice in a combinatorial way. For a solvable pair of lattices, we show that the holonomy Lie algebra is an almost-direct product of the holonomy Lie algebra of the sublattice a…

We investigate variations of Brieskorn lattices over non-compact parameter spaces, and discuss the corresponding limit objects on the boundary divisor. We study the associated variation of twistors and the corresponding limit mixed twistor structures. We construct a compact classifying space for regular singular Briesk…

New minimal surfaces found using Toda lattice and integrable systems.

L-CNNs learn gauge invariant quantities on lattices.

Lattice formulation captures Atiyah-Patodi-Singer index.

Cube category simplifies set modeling.

New method uses neural maps to efficiently sample lattice QCD distributions.

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…

This paper investigates the geometry of compact contact manifolds that are uniformized by contact Lie groups, i.e., compact manifolds that are the quotient of some Lie group G with a left invariant contact structure and a uniform lattice subgroup. We re-examine Alexander's criteria for existence of lattices on solvable…

Flows on (or variations of) discrete curves in $\R^2$ give rise to flows on a subalgebra of functions on that curve. For a special choice of flows and a certain subalgebra this is described by the Toda lattice hierachy. In the paper it is shown that the canonical symplectic structure on $\R^{2N}$, which can be interpre…

We give an overview on the tt*-geometry defined for isolated hypersurface singularities and tame functions via Brieskorn lattices. We discuss nilpotent orbits in this context, as well as classifying spaces of Brieskorn lattices and (limits of) period maps.

Study connects K3 surfaces to holomorphic metrics, solving complex structure variation.

We study tt*-geometry on the classifying space for regular singular TERP-structures, e.g., Fourier-Laplace transformations of Brieskorn lattices of isolated hypersurface singularities. We show that (a part of) this classifying space can be canonically equipped with a hermitian structure. We derive an estimate for the h…

Study LCP structures on solvmanifolds, complete list up to 5 dimensions.

New manifold types defined on quotient spaces.

The Phi- relationship also known as Phi-factor appears in a number of lattice structures, mostly considering the lines within several separate circles or polygons. The paper considers a regular hexagonal tessellation as a lattice with the highest specific mechanical stiffness.

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…

Abstract framework for no-arbitrage concepts in topological vector lattices.

New combinatorial structure for hierarchically hyperbolic spaces.

Knots and links have been considered to be useful models for structural analysis of molecular chains such as DNA and proteins. One quantity that we are interested on molecular links is the minimum number of monomers necessary to realize them. In this paper we consider every link in the cubic lattice. Lattice stick numb…

Study compact symplectic solvmanifolds' hard Lefschetz property.

Deep convolutional neural networks (CNNs) have shown outstanding performance in the task of semantically segmenting images. However, applying the same methods on 3D data still poses challenges due to the heavy memory requirements and the lack of structured data. Here, we propose LatticeNet, a novel approach for 3D sema…

We announce results on the structure of CAT(0) groups, CAT(0) lattices and of the underlying spaces. Our statements rely notably on a general study of the full isometry groups of proper CAT(0) spaces. Classical statements about Hadamard manifolds are established for singular spaces; new arithmeticity and rigidity state…

New invariant unifies two theories of 3-manifolds, recovering quantum invariants.

We establish two exact sequences for the lattice cohomology associated with non-degenerate plumbing graphs. The first is the analogue of the surgery exact triangle proved by Ozsvath and Szabo for the Heegaard-Floer invariant HF^+; for the lattice cohomology over Z_2-coefficients it was proved by J. Greene. Here we prov…

Using nonlinear pde techniques, we construct a new family of globally smooth tt* structures. This includes tt* structures associated to the (orbifold) quantum cohomology of a finite number of complex projective spaces and weighted projective spaces. The existence of such "magical solutions" of the tt* equations, namely…

New method learns dependencies in high-dimensional data without graph assumptions.

The first Betti number for a lattice in a classifying space for variations of Hodge structures vanishes.

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

We give an algorithmic computation for the height of Kauffman's clock lattice obtained from a knot diagram with two adjacent regions starred and without crossing information specified. We show that this lattice is more familiarly the graph of perfect matchings of a bipartite graph obtained from the knot diagram by over…

In this paper we prove that every H-type Lie algebra possesses a basis with respect to which the structure constants are integers. Existence of such an integral basis implies via the Mal'cev criterion that all simply connected H-type Lie groups contain cocompact lattices. Since the Campbell-Hausdorff formula is very si…

For a compact Lie group $G$ we consider a lattice gauge model given by the $G$-Hamiltonian system which consists of the cotangent bundle of a power of $G$ with its canonical symplectic structure and standard moment map. We explicitly construct a Fedosov quantization of the underlying symplectic manifold using the Levi-…

The lattice of integer flows of a graph is known to determine the graph up to 2-isomorphism (work of Su--Wagner and Caporaso--Viviani). In this paper we give an algorithmic construction of the graphic matroid $\calM(G)$ of a graph $G$, given its lattice of integer flows $\calF(G)$. The algorithm can then be applied to …

In this continuation of \cite{BM}, we prove the following: Let $Γ\subset \text{SL}(2,{\mathbb C})$ be a cocompact lattice, and let $ρ: Γ\rightarrow \text{GL}(r,{\mathbb C})$ be an irreducible representation. Then the holomorphic vector bundle $E_ρ\longrightarrow \text{SL}(2,{\mathbb C})/Γ$ associated to $ρ$ is polystab…

Quantum codes on hyperbolic lattices outperform Euclidean ones with higher rates and lower overhead.

Tropical Geometry and Mathematical Morphology share the same max-plus and min-plus semiring arithmetic and matrix algebra. In this chapter we summarize some of their main ideas and common (geometric and algebraic) structure, generalize and extend both of them using weighted lattices and a max-$\star$ algebra with an ar…

We continue the study of the distribution of closed geodesics on nilmanifolds constructed from a simply connected 2-step nilpotent Lie group with a left invariant metric and a lattice. We consider a Lie group with an associated 2-step nilpotent Lie algebra constructed from an irreducible representation of a compact sem…