Related papers: A Solution to the Periodic Square Peg Problem

One can formulate the classical Kepler problem on the Heisenberg group, the simplest sub-Riemannian manifold. We take the sub-Riemannian Hamiltonian as our kinetic energy, and our potential is the fundamental solution to the Heisenberg…

In this paper we generalize the periodic unfolding method and the notion of two-scale convergence on surfaces of periodic microstructures to locally periodic situations. The methods that we introduce allow us to consider a wide range of…

An ill-posed problem of synthesis of the Pierce electrodes for a cylindrical beam with a polygonal cross-section is considered. It is assumed that a beam of charged particles is extracted from a space-charge-limited planar diode and the…

The Laplace-Beltrami problem on closed surfaces embedded in three dimensions arises in many areas of physics, including molecular dynamics (surface diffusion), electromagnetics (harmonic vector fields), and fluid dynamics (vesicle…

The symplectomorphism group of a 2-dimensional surface is homotopy equivalent to the orbit of a filling system of curves. We give a generalization of this statement to dimension 4. The filling system of curves is replaced by a decomposition…

We develop an approach to affine symplectic invariant geometry of Lagrangian surfaces by the method of moving frames. The fundamental invariants of elliptic Lagrangian immersions in affine symplectic four-space are derived together with…

Counting interior-disjoint empty convex polygons in a point set is a typical Erd\H{o}s-Szekeres-type problem. We study this problem for 4-gons. Let $P$ be a set of $n$ points in the plane and in general position. A subset $Q$ of $P$, with…

Using a residuum approach, we provide a complete description of the space of the rational spatial curves of given tangent directions. The rational Pythagorean hodograph curves are obtained as a special case when the norm of the direction…

We give a method to resolve 4-dimensional symplectic orbifolds making use of techniques from complex geometry and gluing of symplectic forms. We provide some examples to which the resolution method applies.

The shape invariant of a symplectic manifold encodes the possible area classes of embedded Lagrangian tori. Potentially this is a powerful invariant, but for most manifolds the shape is unknown. We compute the shape for 4 dimensional…

Let $(M,g)$ be a closed Riemannian manifold and $L:TM\rightarrow \mathbb R$ be a Tonelli Lagrangian. In this thesis we study the existence of orbits of the Euler-Lagrange flow associated with $L$ satisfying suitable boundary conditions. We…

An origami (also known as square-tiled surface) is a Riemann surface covering a torus with at most one branch point. Lifting two generators of the fundamental group of the punctured torus decomposes the surface into finitely many unit…

A classical way to construct a Lagrangian in a symplectic manifold $\Sigma$ is to let $\Sigma$ appear as a smooth fiber in a Lefschetz fibration. If this is possible the singularities of the fibration induce Lagrangian spheres in $\Sigma$…

A tromino tiling problem is a packing puzzle where we are given a region of connected lattice squares and we want to decide whether there exists a tiling of the region using trominoes with the shape of an L. In this work we study a slight…

We classify four-dimensional manifolds endowed with symplectic pairs admitting embedded symplectic spheres with non-negative self-intersection, following the strategy of McDuff's classification of rational and ruled symplectic four…

Given a three-manifold with b_1=1 and a nontorsion spin^c structure, we use finite dimensional approximation to construct from the Seiberg-Witten equations two invariants in the form of a periodic pro-spectra. Various functors applied to…

We study the Floer cohomology of the Dehn twist along a real Lagrangian sphere in a symplectic manifold endowed with an anti-symplectic involution. We prove that there exists a distinguished element in the Floer group that is a fixed point…

We derive new obstructions to periodicity of classical knots by employing the Heegaard Floer correction terms of the finite cyclic branched covers of the knots. Applying our results to two fold covers, we demonstrate through numerous…

We prove the Arnold-Givental conjecture for a class of Lagrangian submanifolds in Marsden-Weinstein quotients which are fixpoint sets of some antisymplectic involution. For these Lagrangians the Floer homology cannot in general be defined…

We study solutions of a homogeneous quadratic equation $q(x_0,\dots, x_n)=0$, defined over a field $K$, where the $x_i$ are themselves homogeneous polynomials of some degree $d$ in $r+1$ variables. Equivalently, we are looking at rational…