Related papers: Rigidity of square-tiled interval exchange transfo…
We show that the water waves system is locally wellposed in weighted Sobolev spaces which allow for interfaces with corners. No symmetry assumptions are required. These singular points are not rigid: if the initial interface exhibits a…
We give conditions for minimality of $\mathbb Z/N\mathbb Z$ extensions of a rotation of angle $\alpha$ with one marked point, solving the problem for any prime $N$: for $N=2$, these correspond to the Veech 1969 examples, for which a…
A mixed quasi-\'etale quotient is the quotient of the product of a curve of genus at least 2 with itself by the action of a group which exchanges the two factors and acts freely out of a finite subset. A mixed quasi-\'etale surface is the…
In this paper the authors find examples of translation surfaces that have infinitely generated Veech groups, satisfy the topological dichotomy property that for every direction either the flow in that direction is completely periodic or…
We study nonperiodic tilings of the line obtained by a projection method with an interval projection structure. We obtain a geometric characterisation of all interval projection tilings that admit substitution rules and describe the set of…
The mean ($\kappa$) and Gaussian ($\bar{\kappa}$) bending rigidities of liquid-liquid interfaces, of importance for shape fluctuations and topology of interfaces, respectively, are not yet established: even their signs are debated. Using…
This paper studies singularities of mean curvature flows with integral mean curvature bounds $H \in L^\infty L^p_{loc}$ for some $p \in ( n, \infty]$. For such flows, any tangent flow is given by the flow of a stationary cone $\mathbf{C}$.…
We obtain structural results on translational tilings of periodic functions in $\mathbb{Z}^d$ by finite tiles. In particular, we show that any level one tiling of a periodic set in $\mathbb{Z}^2$ must be weakly periodic (the disjoint union…
In this article we investigate rigidity properties of integrable area-preserving twist maps of the cylinder. More specifically, we prove that if a deformation of the standard integrable map preserves rotational invariant circles (i.e.,…
We study C^2 weakly order preserving circle maps with a flat interval. In particular we are interested in the geometry of the mapping near to the singularities at the boundary of the flat interval. Without any assumption on the rotation…
For a given permutation $\pi \in S_N$, Fulton proves that the matrix Schubert variety $\overline{X_{\pi}} \cong Y_{\pi} \times \mathbb{C}^q$ can be defined via certain rank conditions encoded in the Rothe diagram of $\pi$. In the case where…
Square-tiled surfaces can be classified by their number of squares and their cylinder diagrams (also called realizable separatrix diagrams). For the case of $n$ squares and two cone points with angle $4 \pi$ each, we set up and parametrize…
We prove that there exists a residual set of (non-rational) polygons such the billiard flow is weakly mixing with respect to the Liouville measure (on the unit tangent bundle to the billiard). This follows, via a Baire category argument,…
We know that tilesets that can tile the plane always admit a quasi-periodic tiling [4, 8], yet they hold many uncomputable properties [3, 11, 21, 25]. The quasi-periodicity function is one way to measure the regularity of a quasi-periodic…
The process of interaction between nonlinear waves on a free surface of a nonconducting fluid in a strong tangential electric field is simulated numerically (effects of the force of gravity and capillarity are neglected). It is shown that…
We characterize sandwiched singularities in terms of their link in two different settings. We first prove that such singularities are precisely the normal surface singularities having self-similar non-archimedean links. We describe this…
Structurally stable (rough) flows on surfaces have only finitely many singularities and finitely many closed orbits, all of which are hyperbolic, and they have no trajectories joining saddle points. The violation of the last property leads…
In this paper we investigate the flow of surfaces by a class of symmetric functions of the principal curvatures with a mixed volume constraint. We consider compact surfaces without boundary that can be written as a graph over a sphere. The…
We prove that on the typical translation surface the flow in almost every pair of directions are not isomorphic to each other and are in fact disjoint. It was not known if there were any translation surfaces other than torus covers with…
For an integrable hierarchy which possesses a bihamiltonian structure with semisimple hydrodynamic limit, we prove that the linear reciprocal transformation with respect to any of its symmetry transforms it to another bihamiltonian…