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: Algebraic properties of orbifold models on arbitrary Riemann surfaces are investigated. The action of mapping class group transformations and of standard geometric operations is given explicitly. An infinite dimensional extension of the…

Algebraically constructible functions connect real algebra with the topology of algebraic sets. In this survey we present some history, definitions, properties, and algebraic characterizations of algebraically constructible functions, and a…

This paper gives an algebraic characterization of expansive actions of countable abelian groups on compact abelian groups. This naturally extends the classification of expansive algebraic $\mathbb{Z}^d$-actions given by Schmidt using…

This thesis studies arithmetic of linear algebraic groups. It involves studying the properties of linear algebraic groups defined over global fields, local fields and finite fields, or more generally the study of the linear algebraic groups…

We define and study complex structures and generalizations on spaces consisting of geodesics or harmonic maps that are compatible with the symmetries of these spaces. The main results are about existence and uniqueness of such structures.

In this article we review the question of constructing geometric quotients of actions of linear algebraic groups on irreducible varieties over algebraically closed fields of characteristic zero, in the spirit of Mumford's geometric…

This is a survey article on real algebra and geometry, and in particular on its recent applications in optimization and convexity. We first introduce basic notions and results from the classical theory. We then explain how these relate to…

In this paper, we study the real and the complex Waring rank of reducible cubic forms. In particular, we compute the complex rank of all reducible cubic forms. In the real case, for all reducible cubics, we either compute or bound the real…

The paper concerns two versions of the notion of real forms of Lie superalgebras. One is the standard approach, where a real form of a complex Lie superalgebra is a real Lie superalgebra such that its complexification is the original…

We study the structure of combinatorial Burnside groups, which receive equivariant birational invariants of actions of finite groups on algebraic varieties.

We survey some results on real rational surfaces focused on their topology and their birational geometry.

This is a preliminary version of the first chapter of a book project on the character theory of finite groups of Lie type. It provides the foundations from the general theory of reductive algebraic groups over a finite field.

We prove a version of the Chevalley Restriction Theorem for the action of a real reductive group G on a topological space X which locally embeds into a holomorphic representation. Assuming that there exists an appropriate quotient X//G for…

This text is a survey of derived algebraic geometry. It covers a variety of general notions and results from the subject with a view on the recent developments at the interface with deformation quantization.

In this paper we construct compact forms associated with a complex Lie supergroup with Lie superalgebra of classical type.

We consider actions of real Lie subgroups G of complex reductive Lie groups on Kaehlerian spaces. Our main result is the openness of the set of semistable points with respect to a momentum map and the action of G.

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…

This paper is, essentially, a survey related to the problem of understanding the combinatorics of the action of the monoidal category of finite dimensional modules over a simple finite dimensional Lie algebra on various categories of Lie…

We systematically produce algebraic varieties with torus action by constructing them as suitably embedded subvarieties of toric varieties. The resulting varieties admit an explicit treatment in terms of toric geometry and graded ring…

Let G be a real or complex linear algebraic reductive group. Let H and F be reductive subgroups. We study the natural H action on G/F. The main theorem of this note shows that generic H orbits are closed. This theorem is then applied to…