Metrics with four conic singularities and spherical quadrilaterals

A spherical quadrilateral is a bordered surface homeomorphic to a closed disk, with four distinguished boundary points called corners, equipped with a Riemannian metric of constant curvature 1, except at the corners, and such that the boundary arcs between the corners are geodesic. We discuss the problem of classification of these quadrilaterals and perform the classification up to isometry in the case that two angles at the corners are multiples of pi. The problem is equivalent to classification of Heun's equations with real parameters and unitary monodromy.Comment: 68 pges, 25 figure

Principal Boundary on Riemannian Manifolds

We consider the classification problem and focus on nonlinear methods for classification on manifolds. For multivariate datasets lying on an embedded nonlinear Riemannian manifold within the higher-dimensional ambient space, we aim to acquire a classification boundary for the classes with labels, using the intrinsic metric on the manifolds. Motivated by finding an optimal boundary between the two classes, we invent a novel approach -- the principal boundary. From the perspective of classification, the principal boundary is defined as an optimal curve that moves in between the principal flows traced out from two classes of data, and at any point on the boundary, it maximizes the margin between the two classes. We estimate the boundary in quality with its direction, supervised by the two principal flows. We show that the principal boundary yields the usual decision boundary found by the support vector machine in the sense that locally, the two boundaries coincide. Some optimality and convergence properties of the random principal boundary and its population counterpart are also shown. We illustrate how to find, use and interpret the principal boundary with an application in real data.Comment: 31 pages,10 figure

On the measure and the structure of the free boundary of the lower dimensional obstacle problem

We provide a thorough description of the free boundary for the lower dimensional obstacle problem in $\mathbb{R}^{n+1}$ up to sets of null $\mathcal{H}^{n-1}$ measure. In particular, we prove (i) local finiteness of the $(n-1)$-dimensional Hausdorff measure of the free boundary, (ii) $\mathcal{H}^{n-1}$-rectifiability of the free boundary, (iii) classification of the frequencies up to a set of dimension at most (n-2) and classification of the blow-ups at $\mathcal{H}^{n-1}$ almost every free boundary point

Boundary regularity for pp-harmonic functions and solutions of obstacle problems on unbounded sets in metric spaces

The theory of boundary regularity for $p$-harmonic functions is extended to unbounded open sets in complete metric spaces with a doubling measure supporting a $p$-Poincar\'e inequality, $1<p<\infty$. The barrier classification of regular boundary points is established, and it is shown that regularity is a local property of the boundary. We also obtain boundary regularity results for solutions of the obstacle problem on open sets, and characterize regularity further in several other ways.Comment: 21 page

Quasi-isometries Between Groups with Two-Ended Splittings

We construct `structure invariants' of a one-ended, finitely presented group that describe the way in which the factors of its JSJ decomposition over two-ended subgroups fit together. For groups satisfying two technical conditions, these invariants reduce the problem of quasi-isometry classification of such groups to the problem of relative quasi-isometry classification of the factors of their JSJ decompositions. The first condition is that their JSJ decompositions have two-ended cylinder stabilizers. The second is that every factor in their JSJ decompositions is either `relatively rigid' or `hanging'. Hyperbolic groups always satisfy the first condition, and it is an open question whether they always satisfy the second. The same methods also produce invariants that reduce the problem of classification of one-ended hyperbolic groups up to homeomorphism of their Gromov boundaries to the problem of classification of the factors of their JSJ decompositions up to relative boundary homeomorphism type.Comment: 61pages, 6 figure