arXiv research

Newly discovered 5 triangle-free intrinsically knotted graphs with 22 edges.

New research finds six bipartite intrinsically knotted graphs with 23 edges.

A graph is intrinsically knotted if every embedding contains a knotted cycle. It is known that intrinsically knotted graphs have at least 21 edges and that the KS graphs, $K_7$ and the 13 graphs obtained from $K_7$ by $\nabla Y$ moves, are the only minor minimal intrinsically knotted graphs with 21 edges. This set incl…

Study Hodge Cousin groups for geometric properties.

Efficiently accelerates Feldman-Cousins method using Gaussian processes.

Study Hodge decomposition for special geometric manifolds.

New ${ m Spin}^h$-manifolds studied for their properties.

A graph is called intrinsically knotted if every embedding of the graph contains a knotted cycle. Johnson, Kidwell and Michael showed that intrinsically knotted graphs have at least 21 edges. Recently Lee, Kim, Lee and Oh, and, independently, Barsotti and Mattman, showed that $K_7$ and the 13 graphs obtained from $K_7$…

We investigate the close relationship between minimal surfaces in Euclidean 3-space and constant mean curvature 1 surfaces in hyperbolic 3-space. Just as in the case of minimal surfaces in Euclidean 3-space, the only complete connected embedded constant mean curvature 1 surfaces with two ends in hyperbolic space are we…

In this paper we shall establish that properly embedded constant mean curvature one surfaces in H^3 of finite topology are of finite total curvature and each end is regular. In particular, this implies the horosphere is the only simply connected such example, and the catenoid cousins the only annular examples of this n…

The paper proves macroscopic versions of conjectures about scalar curvature and volume bounds.

In this paper we study the maximal stable domains on minimal catenoids in Euclidean and hyperbolic spaces and in $H^2 \times R$. We in particular investigate whether half-vertical catenoids are maximal stable domains (\emph{Lindelöf's property}). We also consider stable domains on catenoid-cousins in hyperbolic space. …

Until now, the only known maximal surfaces in Minkowski 3-space of finite topology with compact singular set and without branch points were either genus zero or genus one, or came from a correspondence with minimal surfaces in Euclidean 3-space given by the third and fourth authors in a previous paper. In this paper, w…

We present a global representation for surfaces in 3-dimensional hyperbolic space with constant mean curvature 1 (CMC-1 surfaces) in terms of holomorphic spinors. This is a modification of Bryant's representation. It is used to derive explicit formulas in hypergeometric functions for CMC-1 surfaces of genus 0 with thre…

This preliminary report studies immersed surfaces of constant mean curvature in $H^3$ through their {\it adjusted Gauss maps} (as harmonic maps in $S^2$) and their {\it adjusted frames} in SU(2). Lawson's correspondence between Euclidean CMC surfaces and their hyperbolic cousins is interpreted here under a different pe…

Various transformations of isothermic surfaces are discussed and their interrelations are analyzed. Applications to cmc-1 surfaces in hyperbolic space and their minimal cousins in Euclidean space are presented: the Umehara-Yamada perturbation, the classical and Bryant's Weierstrass type representations, and the duality…

Witten's approach to Khovanov homology of knots is based on the five-dimensional system of partial differential equations, which we call Haydys-Witten equations. We argue for a one-to-one correspondence between its solutions and solutions of the seven-dimensional system of equations. The latter can be formulated on any…

Study mean curvature forms on Kähler foliations with restrictions on basic cohomology.

Suppose $G$ is a connected complex Lie group and $Γ$ is a discrete subgroup such that $X := G/Γ$ is Kähler and the codimension of the top non--vanishing homology group of $X$ with coefficients in $\mathbb Z_2$ is less than or equal to two. We show that $G$ is solvable and a finite covering of $X$ is biholomorphic to a …

We show that the emerging field of discrete differential geometry can be usefully brought to bear on crystallization problems. In particular, we give a simplified proof of the Heitmann-Radin crystallization theorem (R. C. Heitmann, C. Radin, J. Stat. Phys. 22, 281-287, 1980), which concerns a system of $N$ identical at…

The Steklov problem is an eigenvalue problem with the spectral parameter in the boundary conditions, which has various applications. Its spectrum coincides with that of the Dirichlet-to-Neumann operator. Over the past years, there has been a growing interest in the Steklov problem from the viewpoint of spectral geometr…

We compute the flux of Killing fields through ends of constant mean curvature 1 in hyperbolic space, and we prove a result conjectured by Rossman, Umehara and Yamada : the flux matrix they have defined is equivalent to the flux of Killing fields. We next give a geometric description of embedded ends of finite total cur…

Following Riemann's idea, we prove the existence of a minimal disk in Euclidean space bounded by three lines in generic position and with three helicoidal ends of angles less than $π$. In the case of general angles, we prove that there exist at most four such minimal disks, we give a sufficient condition of existence i…

CV inference can be invalid for relatively unstable model comparisons.

We prove that many complete, noncompact, constant mean curvature (CMC) surfaces $f:Σ\to \R^3$ are nondegenerate; that is, the Jacobi operator $Δ_f + |A_f|^2$ has no $L^2$ kernel. In fact, if $Σ$ has genus zero and $f(Σ)$ is contained in a half-space, then we find an explicit upper bound for the dimension of the $L^2$ j…

The paper uses belief propagation to analyze rankings and partial orders from partial information.

Stability of Morse index for Yang-Mills connections in 4D.

New metrics for SPD matrices explore affine invariance and symmetry principles.

Study on deformations of pre-symplectic structures using an L-infinity algebra.

Paper constructs solutions for a class of overdetermined systems.

Transformers predict scattering amplitudes in theoretical physics.

Exploring distance functions on spacetime models.

Nonlinear analysis has played a prominent role in the recent developments in geometry and topology. The study of the Yang-Mills equation and its cousins gave rise to the Donaldson invariants and more recently, the Seiberg-Witten invariants. Those invariants have enabled us to prove a number of striking results for low …

We study how finite Bayesian neural networks adapt their hidden representations.

Scaling relations, such as the IPAT equation and the Kaya identity, are useful for quickly gauging the scale of economic, technological, and demographic changes required to reduce environmental impacts and pressures; in the case of the Kaya identity, the environmental pressure is greenhouse gas emissions. However, when…

Quantum complexity lowerbound proved using differential geometry.

Bayesian rLASSO improves model selection and prediction.

We show that the index of a constant mean curvature 1 surface in hyperbolic 3-space is completely determined by the compact Riemann surface and secondary Gauss map that represent it in Bryant's Weierstrass representation. We give three applications of this observation. Firstly, it allows us to explicitly compute the in…

The paper explores theoretical insights into WGANs for better understanding and stability.

Let $V$ be a complex algebraic hypersurface defined by a polynomial $f$ with Newton polytope $Δ$. It is well known that the spine of its amoeba has a structure of a tropical hypersurface. We prove in this paper that there exists a complex tropical hypersurface $V_{\infty, f}$ such that its coamoeba is homeomorphic to t…

Machine learning and statistical modeling complement each other in healthcare analytics.

A new method simplifies contact Hamiltonian mechanics.

We propose the Bayesian bridge estimator for regularized regression and classification. Two key mixture representations for the Bayesian bridge model are developed: (1) a scale mixture of normals with respect to an alpha-stable random variable; and (2) a mixture of Bartlett--Fejer kernels (or triangle densities) with r…

(CMC) 1-immersions in hyperbolic 3-manifolds often develop singularities.

Generative AI can solve in-context learning problems using a martingale perspective.

The Weyl problem is extended to hyperbolic and anti-de Sitter spaces, connecting geometry, analysis, and group theory.

Line graph transformation aids graph isomorphism tests by excluding challenging graph properties.

Proposes MGMN for end-to-end graph similarity learning.