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We calculate the additive and multiplicative structure (together with the grading) of the Hochschild homology and cohomology and the cyclic homology of preprojective algebras of types T. We also compute the calculus structure which is…

We explain some fundamental differences between the theories of mixed Hodge modules and mixed twistor modules (including the difference in weight system on the nearby cycle functor) which do not seem to be clarified explicitly in the…

For a simplicial complex X on {1,2, ..., n} we define enriched homology and cohomology modules. They are graded modules over k[x_1, ..., x_n] whose ranks are equal to the dimensions of the reduced homology and cohomology groups. We…

It's well known that multiple polylogarithms give rise to good unipotent variations of mixed Hodge-Tate structures. In this paper we shall {\em explicitly} determine these structures related to multiple logarithms and some other multiple…

We consider Holder continuous cocycles over hyperbolic dynamical systems with values in the group of invertible bounded linear operators on a Banach space. We show that two fiber bunched cocycles are Holder continuously cohomologous if and…

This paper explores the cohomology of linear cycle sets, focusing on extensions of a specific linear cycle set H by an abelian group I. We derive explicit formulas for the second cohomology group, which classifies these extensions, and…

We compute relations of rational equivalence among special codimensional two cycles on families of abelian surfaces using elements of higher chow groups.

We discuss variations of mixed Hodge structure arising from projective morphisms of complex analytic spaces. Then we treat generalizations of Koll\'ar's torsion-free theorem, vanishing theorem, and so on, for reducible complex analytic…

We introduce the singular cohomology ring of a matroid which extends the Chow ring of a matroid. This is defined as the singular cohomology ring of a certain quasi-projective toric variety associated to the matroid. Using the matroidal…

We introduce several highness notions on degrees related to the problem of computing isomorphisms between structures, provided that isomorphisms exist. We consider variants along axes of uniformity, inclusion of negative information, and…

We compare the context of Hodge structures with that of vertex algebras of conformal field theory. Vertex algebras appear as the highest weight representations of infinite dimensional Lie algebras. A correspondence between Higgs bundles and…

Given a complex variety $X$, a linear algebraic group $G$ and a representation $\rho$ of the fundamental group $\pi\_1(X,x)$ into $G$, we develop a framework for constructing a functorial mixed Hodge structure on the formal local ring of…

Given a derived equivalence of orbifolds associated to projective varieties with (not necessarily Gorenstein) quotient singularities, we deduce consequences related to the behavior of orbifold Hodge numbers and the Picard variety, extending…

This text can be considered as a non-technical and arithmetically motivated introduction to the definition of the limiting mixed Hodge structure. We state several assertions in terms natural to the classical theory of ordinary differential…

Quantum symmetric algebras (or noncommutative polynomial rings) arise in many places in mathematics. In this article we find the multiplicative structure of their Hochschild cohomology when the coefficients are in an arbitrary bimodule…

We prove that the cohomology algebra of elliptic arrangements depends only on the poset of layers. In the particular case of braid elliptic arrangements, we study the cohomology as representation and we compute some Hodge numbers. Finally,…

We explain the theory of refined cycle maps associated to arithmetic mixed sheaves. This includes the case of arithmetic mixed Hodge structures, and is closely related to work of Asakura, Beilinson, Bloch, Green, Griffiths, Mueller-Stach,…

We show that a weighted homogeneous complex surface singularity is metrically conical (i.e., bi-Lipschitz equivalent to a metric cone) only if its two lowest weights are equal. We also give an example of a pair of weighted homogeneous…

We determine the Jordan-Holder decomposition multiplicities of projective and cell modules over periplectic Brauer algebras in characteristic zero. These are obtained by developing the combinatorics of certain skew Young diagrams. We also…

Soergel bimodules are certain bimodules over polynomial algebras, associated with Coxeter groups, and introduced by Soergel in the 1990's while studying the category O of complex semisimple Lie algebras. Even though their definition is…