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We study ergodic decompositions of Dirichlet spaces under intertwining via unitary order isomorphisms. We show that the ergodic decomposition of a quasi-regular Dirichlet space is unique up to a unique isomorphism of the indexing space.…
We prove that a standard realization of the direct image complex via the so-called Douady-Barlet morphism associated with a smooth complex analytic surface admits a natural decomposition in the form of an injective quasi-isomorphism of…
We study the discretization of Darboux integrable systems. The discretization is done using $x$-, $y$-integrals of the considered continuous systems. New examples of semi-discrete Darboux integrable systems are obtained.
Using an integrable discrete Dirac operator, we construct a discrete version of the Weierstrass representation of time-like surfaces parametrized along isotropic directions in $R^{2,1}$, $R^{3,1}$ and $R^{2,2}$. The corresponding discrete…
A line field on a manifold is a smooth map which assigns a tangent line to all but a finite number of points of the manifold. As such, it can be seen as a generalization of vector fields. They model a number of geometric and physical…
We study the monodromy diffeomorphism of Milnor fibrations of isolated complex surface singularities, by computing the family Seiberg--Witten invariant of Seifert-fibered Dehn twists using recent advances in monopole Floer homology. More…
We define a new theory of discrete Riemann surfaces and present its basic results. The key idea is to consider not only a cellular decomposition of a surface, but the union with its dual. Discrete holomorphy is defined by a straightforward…
The geometry of closed surfaces equipped with a Euclidean metric with finitely many conical points of arbitrary angle is studied. The main result is that the set of closed geodesics is dense in the space of geodesics.
We present a geometric variational discretization of nonlinear elasticity in 2D and 3D in the Lagrangian description. A main step in our construction is the definition of discrete deformation gradients and discrete Cauchy-Green deformation…
In this paper we introduce the concept of Deligne cohomology of an orbifold. We prove that the third Deligne cohomology group of a smooth \'{e}tale groupoid classify gerbes with connection over the groupoid. We argue that the $B$-field and…
In this work, we analyze the Dirichlet Laplacian $-\Delta_{\Omega}^D$ in an unbounded waveguide $\Omega \subset \mathbb R^3$, where the cross-section is translated in a constant direction and rotated along a spatial line. We focus on the…
For matrix analogues of embedded surfaces we define discrete curvatures and Euler characteristics, and a non-commutative Gauss--Bonnet theorem is shown to follow. We derive simple expressions for the discrete Gauss curvature in terms of…
In this work we prove that, for a general polyhedral domain of $\mathbb{R}^3$, the cohomology spaces of the discrete de Rham complex of [Di Pietro and Droniou, An arbitrary-order discrete de Rham complex on polyhedral meshes: Exactness,…
In this paper higher order mimetic discretizations are introduced which are firmly rooted in the geometry in which the variables are defined. The paper shows how basic constructs in differential geometry have a discrete counterpart in…
Given a trivalent graph in the 3-dimensional Euclidean space, we call it a discrete surface because it has a tangent space at each vertex determined by its neighbor vertices. To abstract a continuum object hidden in the discrete surface, we…
We construct compact arbitrary Euler characteristic orientable and non-orientable minimal surfaces in the Berger spheres. Besides we show an interesting family of surfaces that are minimal in every Berger sphere, characterizing them by this…
We give a Weierstrass type representation for semi-discrete minimal surfaces in Euclidean 3-space. We then give explicit parametrizations of various smooth, semi-discrete and fully-discrete catenoids, determined from either variational or…
We show that the theory of isothermic surfaces in $\E^3$ -- one of the oldest branches of differential geometry -- can be reformulated within the modern theory of completely integrable (soliton) systems. This enables one to study the…
The general framework for integrable discrete systems on R in particular containing lattice soliton systems and their q-deformed analogues is presented. The concept of regular grain structures on R, generated by discrete one-parameter…
For oriented surfaces $\Sigma$ with boundary, we consider the infinite-dimensional deformation space of projective structures on $\Sigma$ with nondegenerate boundary, up to isotopies fixing the boundary. We show that this space carries a…