arXiv research

We translate a classification scheme for periodic CMC surfaces developed by J. Dorfmeister and the author to discrete CMC surfaces in the sense of A. Bobenko and U. Pinkall. The scheme uses the dressing action on discrete CMC surfaces to arrive at a classification for periodic discrete CMC surfaces.

Researchers develop Orlov-Schulman symmetries for self-dual conformal structures.

We introduce the technique combining the features of integration schemes for SDYM equations and multidimensional dispersionless integrable equations to get SDYM equations on the conformally self-dual background. Generating differential form is defined, the dressing scheme is developed. Some special cases and reductions…

The paper explores reductions of self-dual conformal structure equations.

We demonstrate that SDYM equations for the Lie algebra of one-dimensional vector fields represent a natural reduction in the framework of general linearly degenerate dispersionless hierarchy. We define the reduction in terms of wave functions, introduce generating relation, Lax-Sato equations and the dressing scheme fo…

In this note we consider the relationship between the dressing action and the holonomy representation in the context of constant mean curvature surfaces. We characterize dressing elements that preserve the topology of a surface and discuss dressing by simple factors as a means of adding bubbles to a class of non finite…

The aim of this paper is to give a new link between integrable systems and minimal surface theory. The dressing operation uses the associated family of flat connections of a harmonic map to construct new harmonic maps. Since a minimal surface in 3-space is a Willmore surface, its conformal Gauss map is harmonic and a d…

All polynomial invariants of links for two dimensional solutions of Yang-Baxter equation is constructed by employing Turaev's method. As a consequence, it is proved that the best invariant so constructed is the Jones polynomial and there exist three solutions connecting to the Alexander polynomial. Invariants for highe…

We present a loop group description for curves in $\mathbb{R}^3$, and apply it to classify the circletons: Circles dressed by simple factors.

We investigate the dressing action on surfaces of constant mean curvature (CMC surfaces) in Euclidean space. In particluar, we show that for CMC surfaces with umbilics the isotropy group under dressing is always trivial. This result is applied to the investigation of CMC surfaces whose metric is invariant under a group…

We characterize Bianchi-Bäcklund transformations of surfaces of positive constant Gauss curvature in terms of dressing actions of the simplest type on the space of harmonic maps.

We give a description of recently introduced Doubrov-Ferapontov general heavenly equation in terms of closed differential Plücker two-form, rationally depending on the spectral parameter. We demonstrate that general heavenly equation is an important generating equation in the context of Takasaki hyper-Kähler hierarchy,…

We deal with the problem of description of nonsingular pairs of compatible flat metrics for the general $N$-component case. We describe the scheme of the integrating the nonlinear equations describing nonsingular pairs of compatible flat metrics (or, in other words, nonsingular flat pencils of metrics). It is based on …

We use the dressing method to construct transformations of constrained Willmore surfaces in arbitrary codimension. An adaptation of the Terng--Uhlenbeck theory of dressing by simple factors to this context leads us to define Bäcklund transforms of these surfaces for which we prove Bianchi permutability. Specialising to…

We define and study the Burnside quotient Green ring of a Mackey functor. Some refinements of Dress induction theory are presented, together with applications to computation results for $K$-theory and $L$-theory of finite and infinite groups.

The paper studies conformal and projective structures using 2-frame bundles.

We classify the simplest rational elements in a twisted loop group, and prove that dressing actions of them on proper indefinite affine spheres give the classical Tzitzéica transformation and its dual. We also give the group point of view of the Permutability Theorem, construct complex Tzitzéica transformations, and di…

We will first clarify the loop group formulations for both hyperbolic and elliptic definite affine spheres in R^3. Then we classify the rational elements with 3 poles or 6 poles in a real twisted loop group, and compute dressing actions of them on such surfaces. Some new examples with pictures will be produced at last.

We define a transformation on harmonic maps from a Riemann surface into the 2-sphere which depends on a complex parameter, the so-called mu-Darboux transformation. In the case when the harmonic map N is the Gauss map of a constant mean curvature surface f and the parameter is real, the mu-Darboux transformation of -N i…

Unified representation for minimal and constant mean curvature surfaces.

We propose a Poisson-Lie analog of the symplectic induction procedure, using an appropriate Poisson generalization of the reduction of symplectic manifolds with symmetry. Having as basic tools the equivariant momentum maps of Poisson actions, the double group of a Poisson-Lie group and the reduction of Poisson manifold…

We discuss constant mean curvature bubbletons in Euclidean 3-space via dressing with simple factors, and prove that single bubbletons are not embedded.

Robot-assisted dressing offers an opportunity to benefit the lives of many people with disabilities, such as some older adults. However, robots currently lack common sense about the physical implications of their actions on people. The physical implications of dressing are complicated by non-rigid garments, which can r…

Gauge symmetries are a cornerstone of modern physics but they come with technical difficulties when it comes to quantization, to accurately describe particles phenomenology or to extract observables in general. These shortcomings must be met by essentially finding a way to effectively reduce gauge symmetries. We propos…

Unified view of integrable systems linking CMC, isothermic, and Willmore surfaces.

The quantum cohomology of CP^1 is generated by some potential (Frobenius manifold) that also has an interpretation as a potential of some harmonic map. Actually, the potential induces harmonic maps into three different symmetric spaces and each of these harmonic maps induces an immersion of an integrable surface. The f…

We study the harmonic map equations for maps of a Riemann surface into a Riemannian symmetric space of compact type from the point of view of soliton theory. There is a well-known dressing action of a loop group on the space of harmonic maps and we discuss the orbits of this action through particularly simple harmonic …

In this paper, we develop the mathematical tools needed to explore isotopy classes of tilings on hyperbolic surfaces of finite genus, possibly nonorientable, with boundary, and punctured. More specifically, we generalize results on Delaney-Dress combinatorial tiling theory using an extension of mapping class groups to …

This paper is an expository account of the development of soliton mathematics, from its inception in famous numerical experiments of Fermi-Pasta-Ulam and Zabusky-Kruskal to the recent synthesis of Terng-Uhlenbeck (dg-ga/9707004) that explains hidden symmetries of soliton equations in terms of loop-groups acting by dres…

A geometric account explains why 'The Dress' is ambiguous, predicting observable signatures in image processing.

We give a survey of the following six closely related topics: (i) a general method for constructing a soliton hierarchy from a splitting of a loop algebra into positive and negative subalgebras, together with a sequence of commuting positive elements, (ii) a method---based on (i)---for constructing soliton hierarchies …

The L-equivalent counterpart of the M-LXIX equation is found. This L-equivalent equation is the Gauss-Codazzi equation which is integrable by the dressing method. This means that the M-LXIX equation is also integrable in this sense.

With the fast growth in the visual surveillance and security sectors, thermal infrared images have become increasingly necessary ina large variety of industrial applications. This is true even though IR sensors are still more expensive than their RGB counterpart having the same resolution. In this paper, we propose a d…

General formulas for the construction of exact solutions of the equation of the minimal surface in $R^3$, which appears in various physical problems, have been derived by the Zakharov-Shabat "dressing" method. Particular examples are considered.

We investigate CMC-surfaces with periodic metric in a dressing orbit of the cylinder. It is shown, that such surfaces are always of finite type. Using the periodicity conditions for the extended frame of a CMC-surface, we develop an alternative approach to the classification of CMC-tori given by Pinkall and Sterling.

The purpose of this paper is to describe certain natural 4-vector fields on quaternionic flag manifolds, which geometrically determine the Bruhat cell decomposition. This structure naturally descends from the symplectic group, where it is related to the dressing action given by the Iwasawa decomposition of the general …

Paper untwists gauge fields, linking two bundles.

After giving the most general formulation to date of the notion of integrability for axially symmetric harmonic maps from R^3 into symmetric spaces, we give a complete and rigorous proof that, subject to some mild restrictions on the target, all such maps are integrable. Furthermore, we prove that a variant of the inve…

We review the recent approach of correlation based networks of financial equities. We investigate portfolio of stocks at different time horizons, financial indices and volatility time series and we show that meaningful economic information can be extracted from noise dressed correlation matrices. We show that the metho…

We find generators for the full rational loop group of GL(n,C) as well as for the subgroup consisting of loops that satisfy the reality condition with respect to the noncompact real form GL(n,R). We calculate the dressing action of some of those generators on the positive loop group, and apply this to the ZS-AKNS flows…

We solve the problem of description for nonsingular pairs of compatible flat metrics in the general N-component case. The integrable nonlinear partial differential equations describing all nonsingular pairs of compatible flat metrics (or, in other words, nonsingular flat pencils of metrics) are found and integrated. Th…

Symplectic manifolds which are homogeneous spaces of Poisson-Lie groups are studied in this paper. We show that these spaces are, under certain assumptions, covering spaces of dressing orbits of the Poisson-Lie groups which act on them. The effect of the Poisson induction procedure on such spaces is also examined, thus…

Accurate forecasts of electricity spot prices are essential to the daily operational and planning decisions made by power producers and distributors. Typically, point forecasts of these quantities suffice, particularly in the Nord Pool market where the large quantity of hydro power leads to price stability. However, wh…

This work draws inspiration from three important sources of research on dissimilarity-based clustering and intertwines those three threads into a consistent principled functorial theory of clustering. Those three are the overlapping clustering of Jardine and Sibson, the functorial approach of Carlsson and Mémoli to par…

We describe a method to classify crystallographic tilings of the Euclidean and hyperbolic planes by tiles whose stabiliser group contains translation isometries or whose topology is not that of a closed disk. We tackle this problem from two different viewpoints, one with constructive techniques to enumerate such tiling…

We give an account of the classical and integrable geometry of isothermic surfaces in arbitrary co-dimension. We show that the classical transformation theory of Darboux, Bianchi and Calapso goes through unchanged in arbitrary co-dimension as does the connection with the "curved flats" of Ferus and Pedit. Moreover, we …

We propose an extension of the differential system for constant mean curvature (CMC) surfaces in a three dimensional space form to an associated hierarchy of evolution equations by the higher-order commuting symmetries. The infinite sequence of higher-order conservation laws of CMC surfaces admit the corresponding exte…

E. Cartan proved that conformally flat hypersurfaces in S^{n+1} for n>3 have at most two distinct principal curvatures and locally envelop a one-parameter family of (n-1)-spheres. We prove that the Gauss-Codazzi equation for conformally flat hypersurfaces in S^4 is a soliton equation, and use a dressing action from sol…