arXiv research

The paper classifies a specific type of hyperbolic lattices using geometric properties.

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 $4$.

An integral hyperbolic lattice is called reflective if its automorphism group is generated by reflections, up to finite index. Since 1981, it is known that their number is essentially finite. We show that K3 surfaces over C with reflective Picard lattices can be characterized in terms of compositions of their self-corr…

The paper improves Vinberg's algorithm for arithmetic hyperbolic lattices.

Picard modular groups are shown to be generated by complex reflections.

Study thin hyperbolic reflection groups and their properties.

Study stabilizers of complex hyperbolic triangle groups, finding generators and signatures.

Constructs hyperbolic reflection groups with 3D limit sets.

The paper defines parabolic subgroups for complex braid groups and proves they form a lattice.

We introduce a new discrete system that arises from ellipsoidal billiards and is closely related to the double reflection nets. The system is defined on the lattice of a uniform honeycomb consisting of rectified hypercubes and cross polytopes. In the $2$-dimensional case, the lattice is regular and it incorporates dyna…

L-CNNs maintain gauge symmetry on non-Abelian lattice theories.

Geometric constraints help classify hyperbolic polytopes.

Ehrenborg and Jung recently related the order complex for the lattice of d-divisible partitions with the simplicial complex of pointed ordered set partitions via a homotopy equivalence. The latter has top homology naturally identified as a Specht module. Their work unifies that of Calderbank, Hanlon, Robinson, and Wach…

A hyperbolic 3-simplex reflection group is a Coxeter group arising as a lattice in the isometry group of hyperbolic 3-space, with fundamental domain a geodesic simplex (possibly with some ideal vertices). The classification of these groups is known, and there are exactly 9 cocompact examples, and 23 non-cocompact examp…

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…

For a finite volume geodesic polyhedron P in hyperbolic 3-space, with the property that all interior angles between incident faces are integral submultiples of Pi, there is a naturally associated Coxeter group generated by reflections in the faces. Furthermore, this Coxeter group is a lattice inside the isometry group …

Researchers prove a conjecture about a specific type of 3D space.

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

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

Let P and Q be convex polyhedra in E3 with face lattices F(P) and F(Q) and symmetry groups G(P) and G(Q), respectively. Then, P and Q are called face equivalent if there is a lattice isomorphism between F(P) and F(Q); P and Q are called symmetry equivalent if the action of G(P) on F(P) is equivalent to the action of G(…

Translationally equivariant neural networks improve performance and generalization in physics problems.

We introduce a new unsupervised representation learning and visualization using deep convolutional networks and self organizing maps called Deep Neural Maps (DNM). DNM jointly learns an embedding of the input data and a mapping from the embedding space to a two-dimensional lattice. We compare visualizations of DNM with…

The paper explores rigidity and proximality in dynamical systems, proving new results about $C^*$-algebras.

We construct a new type of quantum walks on simplicial complexes as a natural extension of the well-known Szegedy walk on graphs. One can numerically observe that our proposing quantum walks possess linear spreading and localization as in the case of the Grover walk on lattices. Moreover, our numerical simulation sugge…

Hybrid subgroups found in non-arithmetic PU(2,1) lattices.

New property identifies arithmetic lattices from nonuniform lattices.

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

This work compares lattice-free and lattice-based training criteria for LVCSR.

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…

Course on arithmetic lattices at EPFL.

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…

Efficiently estimates material parameter space with multifidelity Gaussian process modeling.

New rigidity theorem for product of lattices.

This paper studies the covolumes of nonuniform arithmetic lattices in PU(n, 1). We determine the smallest covolume nonuniform arithmetic lattices for each n, the number of minimal covolume lattices for each n, and study the growth of the minimal covolume as n varies. In particular, there is a unique lattice (up to conj…

The paper finds incommensurable lattices in complex models of Baumslag-Solitar groups.

Proves a lattice version of the Atiyah-Singer index theorem.

Vertex distortion measures how far lattice knots deviate from straight lines.

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…

In this paper we use techniques from convex projective geometry to produce many new examples of thin subgroups of lattices in special linear groups that are isomorphic to the fundamental groups of finite volume hyperbolic manifolds. More specifically, we show that for a large class of arithmetic lattices in SO(n,1) it …

We show that the set of even positive definite lattices that arise from smooth, simply-connected 4-manifolds bounded by a fixed homology 3-sphere can depend on more than the ranks of the lattices. We provide two homology 3-spheres with distinct sets of such lattices, each containing a distinct nonempty subset of the ra…

The paper proves residual finiteness of certain lattices and constructs surfaces with specific fundamental groups.

A celebrated theorem of Hadwiger states that the Euler-Poincaré characteristic is the the unique invariant and continuous valuation on the distributive lattice of compact polyhedra in R^n that assigns value one to each convex non-empty such polyhedron. This paper provides an analogue of Hadwiger's result for finitely p…

The cubic lattice stick index of a knot type is the least number of sticks necessary to construct the knot type in the 3-dimensional cubic lattice. We present the cubic lattice stick index of various knots and links, including all (p,p+1)-torus knots, and show how composing and taking satellites can be used to obtain t…

New method finds lattice polygons that can be dissected into triangles with integer areas.

L-CNNs approximate gauge actions, revealing fixed points with no lattice artifacts.

The paper refines transformations of lattice diagrams and introduces dotted diagrams.

Classifies knots by lattice size, finding unknot ratios and crossing numbers.

Let $G$ be a simply connected, solvable Lie group and $Γ$ a lattice in $G$. The deformation space $\mathcal{D}(Γ,G)$ is the orbit space associated to the action of $\Aut(G)$ on the space $\mathcal{X}(Γ,G)$ of all lattice embeddings of $Γ$ into $G$. Our main result generalises the classical rigidity theorems of Mal'tsev…