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The Dolbeault resolution of the sheaf of holomorphic vector fields $Lie$ on a complex manifold $M$ relates $Lie$ to a sheaf of differential graded Lie algebras, known as the Fr\"olicher-Nijenhuis algebra $g$. We establish - following B. L.…

We extend the results of Riemannian geometry over finite groups and provide a full classification of all linear connections for the minimal noncommutative differential calculus over a finite cyclic group. We solve the torsion-free and…

On an oriented Riemannian manifold, the Gauss-Bonnet-Chern formula asserts that the Pfaffian of the metric represents, in de Rham cohomology, the Euler class of the tangent bundle. Hence, if the underlying manifold is compact, the integral…

Let $X$ be a non-compact geometrically finite hyperbolic 3-manifold without cusps of rank 1. The deformation space $\mc{H}$ of $X$ can be identified with the Teichm\"uller space $\mc{T}$ of the conformal boundary of $X$ as the graph of a…

In this work we use the Rankin-Selberg method to obtain results on the analytic properties of the standard $L$-function attached to vector valued Siegel modular forms. In particular we provide a detailed description of its possible poles…

We approach the question of complexification of the diffeomorphism group of the circle by considering real-analytic maps from the circle into the punctured complex plane with winding number +1. Such complex deformations form an…

In a recent paper, Belishev and Sharafutdinov consider a compact Riemannian manifold $M$ with boundary $\partial M$. They define a generalized Dirichlet to Neumann (DN) operator $\Lambda$ on all forms on the boundary and they prove that the…

We show that every finite-dimensional pointed Hopf algebra over a finite simple Chevalley group, different from $PSL_2(q)$ with q= 3 mod 4 (and from $PSL_3(2)\simeq PSL_2(7)$), is isomorphic to the corresponding group algebra. To do this,…

For a simply connected (non-nilpotent) solvable Lie group $G$ with a lattice $\Gamma$ the de Rham and Dolbeault cohomologies of the solvmanifold $G/\Gamma$ are not in general isomorphic to the cohomologies of the Lie algebra $\mathfrak g$…

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 revisit and generalize our previous algebraic construction of the chiral effective action for Conformal Field Theory on higher genus Riemann surfaces. We show that the action functional can be obtained by evaluating a certain Deligne…

We study sections of the relative Picard bundle of a family of curves of genus $g \geq 2$ through the rank of the associated normal function. Using Griffiths' formula for the infinitesimal invariant and higher Schiffer variations, we…

Isoparametric submanifolds and hypersurfaces in space forms are geometric objects that have been studied since E. Cartan. Another important class of geometric objects is the orbits of a polar action on a Riemannian manifold,e.g., the orbits…

For smooth families of projective algebraic curves, we extend the notion of intersection pairing of metrized line bundles to a pairing on line bundles with flat relative connections. In this setting, we prove the existence of a canonical…

Within a framework of noncommutative geometry, we develop an analogue of (pseudo) Riemannian geometry on finite and discrete sets. On a finite set, there is a counterpart of the continuum metric tensor with a simple geometric…

This paper unites the gauge-theoretic and hyperbolic-geometric perspectives on the asymptotic geometry of the character variety of SL(2,C) representations of a surface group. Specifically, we find an asymptotic correspondence between the…

In the setting of a variety $X$ admitting a tilting bundle $T$ we consider the problem of constructing $X$ as a quiver GIT quotient of the endomorphism algebra $A=\textrm{End}_X(T)^{\textrm{op}}$ corresponding to the tilting bundle. We…

Let $\Omega \subseteq \mathbb C^m$ be a bounded connected open set and $\mathcal H \subseteq \mathcal O(\Omega)$ be an analytic Hilbert module, i.e., the Hilbert space $\mathcal H$ possesses a reproducing kernel $K$, the polynomial ring…

In this paper, we mainly build up the theory of sheaf-correspondence filtered spaces and stratified de Rham complexes for studying singular spaces. We prove the finiteness of a stratified de Rham cohomology and obtain its isomorphism to…

We prove that the intersection cohomology (together with the perverse and the Hodge filtrations) for the moduli space of one-dimensional semistable sheaves supported in an ample curve class on a toric del Pezzo surface is independent of the…