arXiv research

We show that one can define a spectral curve for the Cauchy-Riemann operator on a punctured elliptic curve if one imposes appropriate boundary conditions. Algebraic curves of the type thus obtained appear as irreducible components of spectral curves of minimal tori with planar ends in R^3. It appears that these curves …

To each non-isotropic almost-complex immersion of a 2-torus into $S ^ 6$ we associate an algebraic curve, called the spectral curve, and a linear flow in the intersection of two Prym varieties on this spectral curve. We show that generically the spectral curve is smooth and compute the dimension of the moduli space o…

New method extrapolates spectral densities from smaller models to larger ones.

We study the hyperkaehler geometry of a regular semisimple adjoint orbit of SL(k,C) via the algebraic geometry of the corresponding reducible spectral curve.

We study the geometric properties of Darboux transforms of constant mean curvature (CMC) surfaces and use these transforms to obtain an algebro-geometric representation of constant mean curvature tori. We find that the space of all Darboux transforms of a CMC torus has a natural subset which is an algebraic curve (call…

Computes the Euler characteristic and rational homology of moduli spaces of multi-monopoles.

We study the space of conformal immersions of a 2-torus into the 4-sphere. The moduli space of generalized Darboux transforms of such an immersed torus has the structure of a Riemann surface, the spectral curve. This Riemann surface arises as the zero locus of the determinant of a holomorphic family of Dirac type opera…

The quaternionic KP hierarchy is the integrable hierarchy of p.d.e obtained by replacing the complex numbers with the quaternions, mutatis mutandis, in the standard construction of the KP hierarchy equations and solutions; it is equivalent to what is often called the Davey-Stewartson II hierarchy. This article studies …

Constant mean curvature (CMC) tori in Euclidean 3-space are described by an algebraic curve, called the spectral curve, together with a line bundle on this curve and a point on $S ^ 1$, called the Sym point. For a given spectral curve the possible choices of line bundle and Sym point are easily described. The space o…

Constrained Willmore surfaces are critical points of the Willmore functional under conformal variations. As shown in [5] one can associate to any conformally immersed constrained Willmore torus f a compact Riemann surface Σ, such that f can be reconstructed in terms of algebraic data on Σ. Particularly interesting exam…

Study shows rapid decay of Hitchin metric from semi-flat metric on Higgs bundles.

In $\mathbb{S}^2 \times \mathbb{R}$ there is a two-parameter family of properly embedded minimal annuli foliated by circles. In this paper we show that this family contains all properly embedded minimal annuli. We use the description of minimal annuli in $\mathbb{S}^2 \times \mathbb{R}$ by periodic harmonic maps $G : \…

Study connects spectral properties to frame flows on curved manifolds.

Study shows non-spectrality of certain curves and line segments.

In this paper we study maps (curved flats) into symmetric spaces which are tangent at each point to a flat of the symmetric space. Important examples of such maps arise from isometric immersions of space forms into space forms via their Gauss maps. Further examples are found in conformal geometry, e.g. the curved flats…

Constructs coordinate systems from spectral curve sheaves.

We study topological recursion on the irregular spectral curve $xy^2-xy+1=0$, which produces a weighted count of dessins d'enfant. This analysis is then applied to topological recursion on the spectral curve $xy^2=1$, which takes the place of the Airy curve $x=y^2$ to describe asymptotic behaviour of enumerative proble…

This article shows that every non-isotropic harmonic 2-torus in complex projective space factors through a generalised Jacobi variety related to the spectral curve. Each map is composed of a homomorphism into the variety and a rational map off it. The same ideas allow one to construct (pluri)-harmonic maps of finite ty…

Study connects spectral and algebraic torsion in geometric contexts.

We study a class of scalar, linear, non-local Riemann-Hilbert problems (RHP) involving finite subgroups of PSL(2,C). We associate to such problems a (maybe infinite) root system and describe the relevance of the orbits of the Weyl group in the construction of its solutions. As an application, we study in detail the lar…

The multiplier spectral curve of a conformal torus in the 4-sphere is essentially, see arXiv:0712.2311, given by all Darboux transforms of the conformal torus. In the particular case when the conformal immersion is a Hamiltonian stationary torus in Euclidean 4-space, the left normal of the immersion is harmonic, hence …

The spectral curve correspondence for finite-type solutions of the sinh-Gordon equation describes how they arise from and give rise to hyperelliptic curves with a real structure. Constant mean curvature (CMC) 2-tori in $S ^ 3$ result when these spectral curves satisfy periodicity conditions. We prove that the spectra…

Construct spectral triples on C*-algebras with group actions.

On curved spaces, viscous fluids reach equilibrium quickly.

New 2-representations link spectral enhancements in link homology.

Study on JNR monopoles, focusing on their spectral curves and energy density.

We describe two-dimensional potential Schrodinger and Dirac operators which are finite-gap at one energy level and have singular spectral curves. It appears that the singularities can be rather complicated. Such Dirac operators appear as the spectral curves of tori immersed into the three-space.

In this note, we present a new way to associate a spectral triple to the noncommutative $C^*$-algebra $C^*(Λ)$ of a strongly connected finite higher-rank graph $Λ$. We generalize a spectral triple of Consani and Marcolli from Cuntz-Krieger algebras to higher-rank graph $C^*$-algebras $C^*(Λ)$, and we prove that these s…

We determine the spectral curve of charge 3 BPS su(2) monopoles with C_3 cyclic symmetry. The symmetry means that the genus 4 spectral curve covers a (Toda) spectral curve of genus 2. A well adapted homology basis is presented enabling the theta functions and monopole data of the genus 4 curve to be given in terms of g…

Study on spectral stability of an embedded annulus under curve shortening and Ricci flows.

We completely describe Wahlquist-Estabrook prolongation structures (coverings) dependent on u, u_x, u_{xx}, u_{xxx} for the Krichever-Novikov equation u_t=u_{xxx}-3u_{xx}^2/(2u_x)+p(u)/u_x+au_x in the case when the polynomial p(u)=4u^3-g_2u-g_3 has distinct roots. We prove that there is a universal prolongation algebra…

A conformal immersion of a 2-torus into the 4-sphere is characterized by an auxiliary Riemann surface, its spectral curve. This complex curve encodes the monodromies of a certain Dirac type operator on a quaternionic line bundle associated to the immersion. The paper provides a detailed description of the geometry and …

Spectral sequence connects knot homologies via algebraic geometry.

Study Higgs bundles on curves with punctures, extending spectral correspondence.

Classifies charge-3 monopoles with symmetry, identifying new spectral curves.

New method deforms function algebras on manifolds using spectral decomposition.

The study finds sparse sets that uniquely determine metrics on negatively curved manifolds.

The study characterizes wobbly rank-2 bundles on Riemann surfaces using spectral curves.

We consider a continuous curve of linear elliptic formally self-adjoint differential operators of first order with smooth coefficients over a compact Riemannian manifold with boundary together with a continuous curve of global elliptic boundary value problems. We express the spectral flow of the resulting continuous fa…

Develops a spectral sequence for Lie group actions on manifolds.

Lie algebroids and curved Lie algebras are equivalent categories.

Constructs universal local deformations for curves and differential forms.

Neural networks are capable of learning rich, nonlinear feature representations shown to be beneficial in many predictive tasks. In this work, we use such models to explore different geographical feature representations in the context of predicting colorectal cancer survival curves for patients in the state of Iowa, sp…

The topological recursion of Eynard and Orantin governs a variety of problems in enumerative geometry and mathematical physics. The recursion uses the data of a spectral curve to define an infinite family of multidifferentials. It has been conjectured that, under certain conditions, the spectral curve possesses a non-c…

The paper describes a parametrisation of harmonic maps from a 2-torus to the 3-sphere.

Let $G$ be a finite group. Noncommutative geometry of unital $G$-algebras is studied. A geometric structure is determined by a spectral triple on the crossed product algebra associated with the group action. This structure is to be viewed as a representative of a noncommutative orbifold. Based on a study of classical o…

Study reveals learning curves and benign overfitting in spectral algorithms for large dimensions.

Given an $SL(3)$ spectral curve over a simply connected Riemann surface, we describe in detail the reduction steps necessary to construct the core of a pre-building with versal harmonic map whose differential is given by the spectral curve.