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We develop a "metrically selfdual" variational calculus for $c$-monotone vector fields between general manifolds $X$ and $Y$, where $c$ is a coupling on $X\times Y$. Remarkably, many of the key properties of classical monotone operators…

We completely determine, up to homeomorphism, which simply connected compact oriented 4-manifolds admit scalar-flat, anti-self-dual Riemannian metrics. The key new ingredient is a proof that the connected sum of five reverse-oriented…

We show that the monodromy of a spherical conical metric is reducible if and only if it has a real-valued eigenfunction with eigenvalue 2 in the holomorphic extension of the associated Laplace--Beltrami operator. Such an eigenfunction…

Let $\mathbb{R}$ be the field of real numbers. We consider the problem of computing the real isolated points of a real algebraic set in $\mathbb{R}^n$ given as the vanishing set of a polynomial system. This problem plays an important role…

In previous works, we proposed a method to characterize jointly self-similarity and anisotropy properties of a large class of self--similar Gaussian random fields. We provide here a mathematical analysis of our approach, proving that the…

We give an alternative argument for the classification of real bundle pairs over smooth symmetric surfaces and extend this classification to nodal symmetric surfaces. We also classify the homotopy classes of automorphisms of real bundle…

We study how geometric properties of tropical convex sets and polytopes, which are of interest in many application areas, manifest themselves in their algebraic structure as modules over the tropical semiring. Our main results establish a…

We present a reduction of codimension theorem for surfaces with parallel mean curvature in symmetric spaces.

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.

Cone spherical surfaces are orientable Riemannian surfaces with constant curvature one and a finite set of conical singularities. A subset of these surfaces, referred to as dihedral surfaces, is characterized by their monodromy groups,…

We construct examples of flat surfaces in $\mathbb{H}^3$ which are graphs over a two-punctured horosphere and classify complete embedded flat surfaces in $\mathbb{H}^3$ with only one end and at most two isolated singularities.

We introduce a notion of resultant of two meromorphic functions on a compact Riemann surface and demonstrate its usefulness in several respects. For example, we exhibit several integral formulas for the resultant, relate it to potential…

We generalize a result of Freedman and He, concerning the duality of moduli and capacities in solid tori, to sufficiently regular metric spaces. This is a continuation of the work of the author and K. Rajala on the corresponding duality in…

We consider singularities of frontal surfaces of corank one and finite frontal codimension. We look at the classification under left-right-equivalence and introduce the notion of frontalisation for singularities of fold type. We define the…

While several results have pointed to the existence of exactly quasisymmetric fields on a surface (Garren & Boozer 1991a,b; Plunk & Helander 2018), we have obtained the first such solutions using a vacuum surface expansion formalism. We…

We extend a result proved in \cite{Col} for mirror symmetries of planar systems to measure-preserving non-linear reversibilities of $n$-dimensional systems, dropping the analyticity and nondegeneracy conditions.

We study the differential and metric structures of the set of real square roots of a non-singular real matrix, under the assumption that the matrix and its square roots are semi-simple, or symmetric, or orthogonal.

We solve the Bonnet problem for surfaces in the homogeneous 3-manifolds with a 4-dimensional isometry group. More specifically, we show that a simply connected real analytic surface in H^2xR or S^2xR is uniquely determined pointwise by its…

We present a series of examples of pairs of singular semialgebraic surfaces (real semialgebraic sets of dimension two) in ${\mathbb R}^3$ and ${\mathbb R}^4$ which are bi-Lipschitz equivalent with respect to the outer metric, ambient…

Locally analytically, any isolated double point occurs as a double covering of a smooth surface. It can be desingularized via the canonical resolution, as it is well-known. In this paper we explicitly compute the fundamental cycle of both…