Related papers: Problems for the problem session of Workshop "Affi…
We first establish several general properties of modality of algebraic group actions. In particular, we introduce the notion of a modality-regular action and prove that every visible action is modality-regular. Then, using these results, we…
The article is a contribution to the local theory of geometric Langlands correspondence. The main result is a categorification of the isomorphism between the (extended) affine Hecke algebra, thought of as an algebra of Iwahori bi-invariant…
The algebra of exterior differential forms on a regular 3-Sasakian 7-manifold is investigated, with special reference to nearly-parallel $G_2$ 3-forms. This is applied to the study of 3-forms invariant under cohomogeneity-one actions by…
This is the text of the Hermann Weyl Prize lecture given by the author at the XXIV Colloquium on Group Theoretical Methods in Physics, Paris, July 2002 (to appear in the Proceedings of the Colloquium).
This work introduces author's approach to harmonic analysis on algebraic groups over functional two-dimensional local fields. For a two-dimensional local field a Hecke algebra which is formed by operators which integrate…
We investigate Lie bialgebra structures on simple Lie algebras of non-split type $A$. It turns out that there are several classes of such Lie bialgebra structures, and it is possible to classify some of them. The classification is obtained…
This is an extended abstract of my talk at the Oberwolfach-Workshop "Representation Theory of Finite-Dimensional Algebras" (19-24 January 2020). It introduces a geometric orbifold model for the bounded derived category of a skew-gentle…
The analytic and formal solutions to a family of singularly perturbed partial differential equations in the complex domain involving two complex time variables are considered. The analytic continuation properties of the solution of an…
We present some new and recent algorithmic results concerning polynomial system solving over various rings. In particular, we present some of the best recent bounds on: (a) the complexity of calculating the complex dimension of an algebraic…
In this article we survey and describe various aspects of the geometry and arithmetic of Kleinian groups - discrete nonelementary groups of isometries of hyperbolic $3$-space. In particular we make a detailed study of two-generator groups…
We review Bacry and Levy-Leblond's work on possible kinematics as applied to 2-dimensional spacetimes, as well as the nine types of 2-dimensional Cayley-Klein geometries, illustrating how the Cayley-Klein geometries give homogeneous…
This paper agrees basically with the talk of the author at the workshop "Homological Mirror Symmetry and Applications", Institute for Advanced Study, Princeton, March 2007.
We give an explicit affine algebraic variety whose coordinate ring is isomorphic (as an algebra with the action of the Weyl group) with the equivariant cohomology of some Springer fibers.
We consider a family of linear singularly perturbed Cauchy problems which combines partial differential operators and linear fractional transforms. We construct a collection of holomorphic solutions on a full covering by sectors of a…
We showcase applications of nonlinear algebra in the sciences and engineering. Our review is organized into eight themes: polynomial optimization, partial differential equations, algebraic statistics, integrable systems, configuration…
For a class of K3 surfaces, the action of a Lie algebra which is a certain affinization of a Kac-Moody algebra is given on the cohomology of the moduli spaces of rank 1 torsion free sheaves on the surface. This action is generated by…
This paper is devoted to a systematic study of the geometry of nondegenerate $\bbR^n$-actions on $n$-manifolds. The motivations for this study come from both dynamics, where these actions form a special class of integrable dynamical systems…
Making new methods for quantum problems often relies on using basic operations in linear algebra. Often these routines are hidden behind well-known libraries that have been optimized over decades. Attempting to improve on those basic…
Fix a finite group $G$. We seek to classify varieties with $G$-action equivariantly birational to a representation of $G$ on affine or projective space. Our focus is odd-dimensional smooth complete intersections of two quadrics, relating…
Fractional action-like variational problems have recently gained importance in studying dynamics of nonconservative systems. In this note we address multi-dimensional fractional action-like problems of the calculus of variations.