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We consider geometric invariant theory for \emph{graded additive groups}, groups of the form $\mathbb{G}_a^r\rtimes_w\mathbb{G}_m$ such that the $\mathbb{G}_m$-action on $\mathbb{G}_a^r$ is a scalar multiplication with weight…
After we have given a survey on the Burnside ring of a finite group, we discuss and analyze various extensions of this notion to infinite (discrete) groups. The first three are the finite-G-set-version, the inverse-limit-version and the…
Recent progress in holographic correspondence uncovered remarkable relations between key characteristics of the theories on both sides of duality and certain integrable models. In this note we revisit the problem of the role of certain…
We study some graded geometric constructions appearing naturally in the context of gauge theories. Inspired by a known relation of gauging with equivariant cohomology we generalize the latter notion to the case of arbitrary Q-manifolds…
For a complex analytic variety with an action of a finite group and for an invariant 1-form on it, we give an equivariant version (with values in the Burnside ring of the group) of the local Euler obstruction of the 1-form and describe its…
We study relations between the classical modular symbols associated with congruence subgroups and Kontsevich-Pestun-Tschinkel groups $\mathcal{M}_n(G)$ associated with finite abelian groups $G$.
We present a practical application of parallel symbolic computation in General Relativity: the calculation of curvature invariants for large dimension. We discuss the structure of the calculations, an implementation of the technique and…
Many moduli spaces are constructed as quotients of group actions; this paper surveys the classical theory, as well as recent progress and applications. We review geometric invariant theory for reductive groups and how it is used to…
This work deals with relations between a bounded cohomological invariant and the geometry of Hermitian symmetric spaces of noncompact type. The invariant, obtained from the K\"ahler class, is used to define and characterize a special class…
A survey on recent developments in (algebraic) integral geometry is given. The main focus lies on algebraic structures on the space of translation invariant valuations and applications in integral geometry.
A surface in a three-dimensional metric Lie group $G$ is said invariant if it is invariant with respect to a one-dimensional subgroup $\Gamma$ of the isometry group of $G$. Is this work we focus on unimodular metric Lie groups $G$ that can…
We study differential geometric properties of cuspidal edges with boundary. There are several differential geometric invariants which are related with the behavior of the boundary in addition to usual differential geometric invariants of…
In this paper we give a geometric construction of the Borel equivariant (co)homology for spaces with a $G$-action, where $G$ is a compact Lie group with the property that the adjoint representation is orientable. A nice feature of these…
In this paper we introduce a projective invarinat measure on the special unitary group. It is directly related to transition probabilities. It has some interesting connection with convex geometry. Applications to approximation of quantum…
We motivate and survey the theory of BPS invariants of categories and BPS cohomology of stacks, indicating applications in enumerative geometry and representation theory, as well as recent advances.
This is a brief overview of our work on the theory of group invariant solutions to differential equations. The motivations and applications of this work stem from problems in differential geometry and relativistic field theory. The key…
We provide a geometric characterisation of binary sextics with vanishing quadratic invariant.
We extend projectively equivariant quantization and symbol calculus to symbols of pseudo-differential operators. An explicit expression in terms of hypergeometric functions with noncommutative arguments is given. Some examples are worked…
In theories like SM or MSSM with a complex gauge group structure the complete set of Feynman diagrams contributed to a particular physics process can be splited to exact gauge invariant subsets. Arguments and examples given in the review…
An algorithm is given for computing explicit formulas for the generators of relations among the invariant rational functions for vector-valued bilinear forms. These formulas have applications in the geometry of Riemannian submanifolds and…