arXiv research

New minimal surfaces found from vortex crystals.

Researchers create topologically protected knots in a realizable system.

Topologically protected vortex knots and links are proposed and proven.

This article introduces proximal planar vortex 1-cycles, resembling the structure of vortex atoms introduced by William Thomson (Lord Kelvin) in 1867 and recent work on the proximity of sets that overlap either spatially or descriptively. Vortex cycles resemble Thomson's model of a vortex atom, inspired by P.G. Tait's …

Researchers create crystallizations of lens spaces.

Machine learning predicts molecular crystal stability.

This paper explores vortices and harmonic flows on compact surfaces, using Hodge decomposition.

ShotgunCSP predicts crystal structures using machine learning, achieving high accuracy with minimal computation.

We constructed physically stable sp2 negatively curved cubic carbon structures which reticulate a Schwarz P-like surface. The method for constructing such crystal structures is based on the notion of the standard realization of abstract crystal lattices. In this paper, we expound on the mathematical method to construct…

We present a Hamiltonian framework for higher-dimensional vortex filaments (or membranes) and vortex sheets as singular 2-forms with support of codimensions 2 and 1, respectively, i.e. singular elements of the dual to the Lie algebra of divergence-free vector fields. It turns out that the localized induction approximat…

We demonstrate that the five vortex equations recently introduced by Manton ariseas symmetry reductions of the anti-self-dual Yang--Mills equations in four dimensions. In particular the Jackiw--Pi vortex and the Ambjørn--Olesen vortex correspond to the gauge group $SU(1, 1)$, and respectively the Euclidean or the $SU(2…

The Machine Recognition of Crystallization Outcomes (MARCO) initiative has assembled roughly half a million annotated images of macromolecular crystallization experiments from various sources and setups. Here, state-of-the-art machine learning algorithms are trained and tested on different parts of this data set. We fi…

Study vortex loops as coadjoint orbits of diffeomorphisms.

The paper proves estimates for vortex-type equations on compact Riemann surfaces.

Vortex solutions on flat surfaces map to harmonic spinors on Nappi-Witten space.

Minimal crystallizations of simply connected PL 4-manifolds are very natural objects. Many of their topological features are reflected in their combinatorial structure which, in addition, is preserved under the connected sum operation. We present a minimal crystallization of the standard PL K3 surface. In combination w…

The paper studies special crystallizations of 4-manifolds to minimize certain PL-invariants.

This article introduces vortex nerve complexes in CW (Closure finite Weak) topological spaces, which first appeared in works by P. Alexandroff, H. Hopf and J.H.C. Whitehead during the 1930s. A vortex nerve is a CW complex containing one or more intersecting path-connected cycles. Each vortex nerve has its own distincti…

One-dimensional crystals have convex shapes under certain conditions.

We give a complete solution to the extremal topological combinatorial problem of finding the minimum number of tiles needed to construct a polyomino with $h$ holes. We denote this number by $g(h)$ and say that a polyomino is crystallized if it has $h$ holes and $g(h)$ tiles. We analyze structural properties of crystall…

CRYSPNet predicts crystal structures from chemical compositions.

A fundamental problem in applying machine learning techniques for chemical problems is to find suitable representations for molecular and crystal structures. While the structure representations based on atom connectivities are prevalent for molecules, two-dimensional descriptors are not suitable for describing molecula…

The paper studies point vortex dynamics on specific Kähler twistor spaces.

Solved Demailly systems for Vortex bundles on manifolds.

The article studies crystallizations of small covers over simple polytopes and finds unique crystallizations for the $n$-simplex.

We have defined weight of the pair $(\langle S \mid R \rangle, R)$ for a given presentation $\langle S \mid R \rangle$ of a group, where the number of generators is equal to the number of relations. We present an algorithm to construct crystallizations of 3-manifolds whose fundamental group has a presentation with two …

Let L --> X be a complex line bundle over a compact connected Riemann surface. We consider the abelian vortex equations on L when the metric on the surface has finitely many point degeneracies or conical singularities and the line bundle has parabolic structure. These conditions appear naturally in the study of vortex …

The paper solves a thermodynamics problem about crystal shape.

Alternative construction of quasi-Fuchsian flows using vortex equations.

Paper solves vortex equations on complex surfaces, linking to Higgs bundle stability.

Consider a finite connected graph possibly with multiple edges and loops. In discrete geometric analysis, Kotani and Sunada constructed the crystal associated to the graph as a standard realization of the maximal abelian covering of the graph. As an application of what the author showed in an earlier paper with Seshadr…

We obtain a Hitchin-Kobayashi-type correspondence for symplectic vortex equations, with the target a Kahler cone over a compact Sasakian manifold. We show that the correspondence reduces to studying the existence and uniqueness of Kazdan-Warner equations. Using this, we construct a map between the moduli space of solut…

Imaging techniques are essential tools for inquiring a number of properties from different materials. Liquid crystals are often investigated via optical and image processing methods. In spite of that, considerably less attention has been paid to the problem of extracting physical properties of liquid crystals directly …

Theory of point vortices extended to closed surfaces.

Study how large-scale flows align small-scale vortices in 3D Euler equations.

We show how the smooth geometry of Calabi-Yau manifolds emerges from the thermodynamic limit of the statistical mechanical model of crystal melting defined in our previous paper arXiv:0811.2801. In particular, the thermodynamic partition function of molten crystals is shown to be equal to the classical limit of the par…

Minimal crystallizations bound for 3-manifolds with boundary.

For $d\geq 2$, the regular genus of a closed connected PL $d$-manifold $M$ is the least genus (resp., half of the genus) of an orientable (resp., a non-orientable) surface into which a crystallization of $M$ imbeds regularly. The regular genus of every orientable surface equals its genus, and the regular genus of every…

Machine learning predicts perovskite formability and classifies crystal structures.

Simple crystallizations are edge-coloured graphs representing PL 4-manifolds with the property that the 1-skeleton of the associated triangulation equals the 1-skeleton of a 4-simplex. In the present paper, we prove that any (simply-connected) PL $4$-manifold $M$ admitting a simple crystallization admits a special hand…

Study phase transition in liquid crystal droplets using mathematical analysis.

In this note we quantize the usual symplectic (Kähler) form on the vortex moduli space by modifying the Quillen metric of the Quillen determinant line bundle.

The topology and the geometry of a surface play a fundamental role in determining the equilibrium configurations of thin films of liquid crystals. We propose here a theoretical analysis of a recently introduced surface Frank energy, in the case of two-dimensional nematic liquid crystals coating a toroidal particle. Our…

We consider the vortex equations for a U(n) gauge field coupled to a Higgs field with values on the n times n square matrices. It is known that when these equations are defined on a compact Riemann surface, their moduli space of solutions is closely related to a moduli space of tau-stable holomorphic n-pairs on that su…

Machine-learning models are capable of capturing the structure-property relationship from a dataset of computationally demanding ab initio calculations. Over the past two years, the Organic Materials Database (OMDB) has hosted a growing number of calculated electronic properties of previously synthesized organic crysta…

The Seiberg-Witten equations are defined on certain complex line bundles over smooth oriented four manifolds. When the base manifold is a complex Kahler surface, the Seiberg-Witten equations are essentially the Abelian vortex equations. Using known non-abelian generalizations of the vortex equations as a guide, we expl…

Theory of packing diabolic domains in liquid crystals.

We present the census of all non-orientable, closed, connected 3-manifolds admitting a rigid crystallization with at most 30 vertices. In order to obtain the above result, we generate, manipulate and compare, by suitable computer procedures, all rigid non-bipartite crystallizations up to 30 vertices.