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The triad refers to embedding of two systems of polynomials, symmetric ones and those of the Baker-Akhiezer type into a power series of the Noumi-Shiraishi type. It provides an alternative definition of Macdonald theory and its extensions.…
We construct a conformally invariant random family of closed curves in the plane by welding of random homeomorphisms of the unit circle given in terms of the exponential of Gaussian Free Field. We conjecture that our curves are locally…
We survey structure-preserving discretizations of minimal surfaces in Euclidean space. Our focus is on a discretization defined via parallel face offsets of polyhedral surfaces, which naturally leads to a notion of vanishing mean curvature…
We have established a 1-1 correspondence between a solution of the universal Whitham hierarchy and a twistor space. The twistor space consists of a complex surface and a family of complex curves together with a meromorphic 2-form. The…
Using variational considerations, we establish that there exists a new symmetric trace-free tensor conformal invariant of hypersurfaces embeddings in even dimensional conformal manifolds. This conformal invariant completes the family of…
The Weyl principle is extended from the Riemannian to the pseudo-Riemannian setting, and subsequently to manifolds equipped with generic symmetric $(0,2)$-tensors. More precisely, we construct a family of generalized curvature measures…
We introduce the $\mathbb{Z}_N$-twisted trigonometric sigma models, a new class of integrable deformations of the principal chiral model. Starting from 4d Chern-Simons theory on a cylinder, the models are constructed by introducing a…
We show a closed Bach-flat Riemannian manifold with a fixed positive constant scalar curvature has to be locally spherical if its Weyl and traceless Ricci tensors are small in the sense of either $L^\infty$ or $L^{\frac{n}{2}}$-norm.…
We first give an exposition of how the Polyakov path integral for the bosonic string produces a natural mapping class group invariant measure, $d(Poly)$, on the Teichm\"uller space of Riemann surfaces of each fixed genus. The description of…
Recently, a remarkable new class of spacetimes describing black holes immersed in a non-aligned electromagnetic field has been found. While still of type D, this class goes beyond the famous Pleba\'nski--Demia\'nski family. Here we…
We gave a new very simple proof that the completion of the space of the diffeomorphism of the circle modulo conformal maps with respect to the Weil-Petersson Metric is a complex analytic manifold modeled on the Hilbert space with 3/2…
We develop a degree theory for compact immersed hypersurfaces of prescribed $K$-curvature immersed in a compact, orientable Riemannian manifold, where $K$ is any elliptic curvature function. We apply this theory to count the (algebraic)…
We prove that the universal Teichmuller space T(1) carries a new structure of a complex Hilbert manifold. We show that the connected component of the identity of T(1), the Hilbert submanifold T_{0}(1), is a topological group. We define a…
This paper studies novel four-dimensional integrable field theories that are deformations of self-dual Yang-Mills. They are engineered by considering holomorphic Chern-Simons and BF type theories on covers of twistor space obtained by…
We introduce the specialization map in Scholzes theory of diamonds. We consider v-sheaves that behave like formal schemes and call them kimberlites. We attach to them: a reduced special fiber, an analytic locus, a specialization map, a…
Given a semi-Riemannian manifold, we give necessary and sufficient conditions for a Riemannian submanifold of arbitrary co-dimension to be umbilical along normal directions. We do that by using the so-called \emph{total shear tensor}, i.e.,…
This is a commentary on Teichm\"ullers' paper "Ver\"anderliche Riemannsche Fl\"achen" (Variable Riemann Surfaces), published in 1944. This paper is the last one that Teichm\"uller wrote on the problem of moduli. At most places the paper…
The space of topological decompositions into triangulations of a surface has a natural graph structure where two triangulations share an edge if they are related by a so-called flip. This space is a sort of combinatorial Teichm\"uller space…
A spring-mass model arranged in a diamond structure --- mechanical diamond --- is analyzed in terms of topology in detail. We find that, additional springs connecting the next-nearest-neighbor pairs of mass points and the modulation of the…
We interpret the combinatorial Mandelbrot set in terms of \it{quadratic laminations} (equivalence relations $\sim$ on the unit circle invariant under $\sigma_2$). To each lamination we associate a particular {\em geolamination} (the…