Related papers: Geometric Hodge structures with prescribed Hodge n…

We conjecture the existence of a simple geometric structure underlying questions of reducibility of parabolically induced representations of reductive p-adic groups.

We compute the integral homology and cohomology groups of configuration spaces of two distinct points on a given real projective space. The explicit answer is related to the (known multiplicative structure in the) integral cohomology---with…

We introduce and study Hodge-de Rham numbers for compact almost complex 4-manifolds, generalizing the Hodge numbers of a complex surface. The main properties of these numbers in the case of complex surfaces are extended to this more general…

Traditionally, Hodge structures are associated with complex projective varieties. In my expository lectures I discussed a non-commutative generalization of Hodge structures in deformation quantization and in derived algebraic geometry.

This is a survey of our research on geometric structures of projective embeddings and includes some topics of our talks in several symposia during 1990-99. We clarify our main problem, which is to construct a kind of geometric composition…

Hodge theory is a beautiful synthesis of geometry, topology, and analysis, which has been developed in the setting of Riemannian manifolds. On the other hand, spaces of images, which are important in the mathematical foundations of vision…

These are the notes for the talk "Hodge numbers of a hypothetical complex structure on $S^6$" given by the author at the MAM1 "(Non)-existence of complex structures on $S^6$" held in Marburg in March 2017. They are based on [A. Gray, A…

Let $\mathcal{A}$ be a smooth proper C-linear triangulated category Calabi-Yau of dimension 3 endowed with a (non-trivial) rank function. Using the homological unit of $\mathcal{A}$ with respect to the given rank function, we define Hodge…

''Positive geometries'' are a class of semi-algebraic domains which admit a unique ''canonical form'': a logarithmic form whose residues match the boundary structure of the domain. The study of such geometries is motivated by recent…

Let $Y$ be a projective submanifold of the total space of the inverse of a very ample line bundle $\pi:L^{-1}\rightarrow B$ over a projective manifold $B$. Any section of $L^{-1}\rightarrow B$ is isomorphic to $B$ and the Hodge numbers of…

Skeletal polyhedra and polygonal complexes are finite or infinite periodic structures in 3-space with interesting geometric, combinatorial, and algebraic properties. These structures can be viewed as finite or infinite periodic graphs…

We define Hodge correlators for a compact Kahler manifold X. They are complex numbers which can be obtained by perturbative series expansion of a certain Feynman integral which we assign to X. We show that they define a functorial real…

We begin by introducing the concept of a Hodge structure and give some of its basic properties, including the Hodge and Lefschetz decompositions. We then define the period map, which relates families of Kahler manifolds to the families of…

This is primarily an expository piece and the first sentence of the introduction pretty much sums it up: This article is aimed at people who already know what mixed Hodge structures are and what they are good for, but who are not sure how…

This paper studies the possible Hodge groups of simple polarizable $\mathbb{Q}$-Hodge structures with Hodge numbers $(n,0,\ldots,0,n)$. In particular, it generalizes earlier work of Ribet and Moonen-Zarhin to completely determine the…

This manuscript develops a geometric approach to ordinary cohomology of smooth manifolds, constructing a cochain complex model based on co-oriented smooth maps from manifolds with corners. Special attention is given to the pull-back product…

We identify the category of real mixed Hodge structures with the category of vector bundles with connections (not necessarily flat) on C, equivariant with respect to C^*. Here C is the complex plane considered as a 2-dimensional real…

Polyhedral surfaces are fundamental objects in architectural geometry and industrial design. Whereas closeness of a given mesh to a smooth reference surface and its suitability for numerical simulations were already studied extensively, the…

In this paper, we explore a notion of nonabelian Hodge structure on the fundamental group of an algebraic variety. This is approach is compared to some alternative approaches due to Morgan, Hain and others. We also give criteria for a…

Geometric number systems, obtained by extending the real number system to include new anticommuting square roots of +1 and -1, provide a royal road to higher mathematics by largely sidestepping the tedious languages of tensor analysis and…