Related papers: Matrix Formulae for Decorated Super Teichm\"uller …
We construct homotopy formulae $f=\overline\partial\mathcal H_qf+\mathcal H_{q+1}\overline\partial f$ for $(0,q)$ forms on the product domain $\Omega_1\times\dots\times\Omega_m$, where each $\Omega_j$ is either a bounded Lipschitz domain in…
We study the moment space corresponding to matrix measures on the unit circle. Moment points are characterized by non-negative definiteness of block Toeplitz matrices. This characterization is used to derive an explicit representation of…
We compare some natural triangulations of the Teichm\"uller space of hyperbolic surfaces with geodesic boundary and of some bordifications. We adapt Scannell-Wolf's proof to show that grafting semi-infinite cylinders at the ends of…
Let $M$ be a hyperkahler manifold, $\Gamma$ its mapping class group, and $Teich$ the Teichmuller space of complex structures of hyperkahler type. After we glue together birationally equivalent points, we obtain the so-called birational…
For arbitrary F-algebra, in which the operation of addition is defined, I explore biring of matrices of mappings. The sum of matrices is determined by the sum in F-algebra, and the product of matrices is determined by the product of…
Hierarchically hyperbolic spaces provide a common framework for studying mapping class groups of finite type surfaces, Teichm\"uller space, right-angled Artin groups, and many other cubical groups. Given such a space $\mathcal X$, we build…
We give sufficient conditions which ensure that a functor of finite length from an additive category to finite-dimensional vector spaces has a projective resolution whose terms are finitely generated. For polynomial functors, we study also…
Let $A$ be an associative simple (central) superalgebra over ${\mathbb C}$ and $L$ an invariant linear functional on it (trace). Let $a\mapsto a^t$ be an antiautomorphism of $A$ such that $(a^t)^ t=(-1)^{p(a)}a$, where $p(a)$ is the parity…
In this PhD thesis, we give a new geometric approach to higher Teichm\"uller theory. In particular we construct a geometric structure on surfaces, generalizing the complex structure, and we explore its link to Hitchin components. The…
We introduce a wide category of superspaces, called locally finitely generated, which properly includes supermanifolds but enjoys much stronger permanence properties, as are prompted by applications. Namely, it is closed under taking finite…
We study a particular class of infinite-dimensional representations of $\mathfrak{osp}(1|2n)$. These representations $L_n(p)$ are characterized by a positive integer $p$, and are the lowest component in the $p$-fold tensor product of the…
The Euler characteristic of a very affine variety encodes the number of critical points of the likelihood equation on this variety. In this paper, we study the Euler characteristic of the complement of a hypersurface arrangement with…
In this paper, we revisit our paper "Matrix Riccati Equations, Kaluza-Klein, Finsler Spaces, and Mapping Among Manifolds"[1]. We will build mapping among generalized quadratic Hamiltonians and we construct Calabi's Riemmannian Line Elements…
We introduce a refined Sobolev scale on a vector bundle over a closed infinitely smooth manifold. This scale consists of inner product H\"ormander spaces parametrized with a real number and a function varying slowly at infinity in the sense…
We present an effective algorithm for computing the standard cohomology spaces of finitely generated Lie (super) algebras over a commutative field K of characteristic zero. In order to reach explicit representatives of some generators of…
There are two frameworks for mating Kleinian groups with rational maps on the Riemann sphere: the algebraic correspondence framework due to Bullett-Penrose-Lomonaco \cite{BP94,BL20} and the simultaneous uniformization mating framework of…
We generalize the super period matrix of a super Riemann surface to the case that Ramond punctures are present. For a super Riemann surface of genus g with 2r Ramond punctures, we define, modulo certain choices that generalize those in the…
We announce a generalization of Zimmer's cocycle superrigidity theorem proven using harmonic map techniques. This allows us to generalize many results concerning higher rank lattices to all lattices in semisimple groups with property $(T)$.…
Let $S$ be a closed, orientable surface of genus at least 2. The cotangent bundle of the "hyperbolic'' Teichm\"uller space of $S$ can be identified with the space $\CP$ of complex projective structures on $S$ through measured laminations,…
The combinatorial Mandelbrot set is a continuum in the plane, whose boundary can be defined, up to a homeomorphism, as the quotient space of the unit circle by an explicit equivalence relation. This equivalence relation was described by…