arXiv research

Complete classification of quaternionic skew-Hermitian symmetric spaces found.

Study various submanifolds in quaternionic skew-Hermitian spaces.

The paper characterizes curvature of quaternionic skew-Hermitian manifolds and constructs related geometric structures.

Paper proves injectivity of non-abelian X-ray transform on certain spaces.

In this paper we extend DDVV-type inequalities involving the Frobenius norm of commutators from real symmetric and skew-symmetric matrices to Hermitian and skew-Hermitian matrices.

This paper studies geometric structures on manifolds with specific symplectic properties.

We prove that there is no faithful finite-dimensional representation by skew-hermitian matrices of a ``basic algebra of observables'' B on a noncompact symplectic manifold M. Consequently there exists no finite-dimensional quantization of any Lie subalgebra of the Poisson algebra C^\infty(M) containing B.

For a two-dimensional simple magnetic system, we study the attenuated magnetic ray transform $I_{A,Φ}$, with attenuation given by a unitary connection $A$ and a skew-Hermitian Higgs field $Φ$. We give a description for the range of $I_{A,Φ}$ acting on $\mathbb C^n$-valued tensor fields.

Study of geometric structures on manifolds, focusing on integrability conditions.

In this paper we generalize the known DDVV-type inequalities for real (skew-)symmetric and complex (skew-)Hermitian matrices to arbitrary real, complex and quaternionic matrices. Inspired by the Erdős-Mordell inequality, we establish the DDVV-type inequalities for matrices in the subspaces spanned by a Clifford system …

We show that for a simple surface with boundary the attenuated ray transform in the presence of a unitary connection and a skew-Hermitian Higgs field is injective modulo the natural obstruction for functions and vector fields. We also show that the connection and the Higgs field are uniquely determined by the scatterin…

We consider integral geometry inverse problems for unitary connections and skew-Hermitian Higgs fields on manifolds with negative sectional curvature. The results apply to manifolds in any dimension, with or without boundary, and also in the presence of trapped geodesics. In the boundary case, we show injectivity of th…

The purpose of this paper is to establish a Lagrangian potential theory, analogous to the classical pluripotential theory, and to define and study a Lagrangian differential operator of Monge-Ampere type. This development is new even in ${\bf C}^n$. However, it applies quite generally -- perhaps most importantly to symp…

To every $Q$-irreducible representation $r$ of a finite group $H$, there corresponds a simple factor $A$ of $Q[H]$ with an involution $τ$. To this pair $(A,τ)$, we associate an arithmetic group $Ω$ consisting of all $(2g-2)\times (2g-2)$ matrices over a natural order of $A^{op}$ which preserve a natural skew-Hermitian …

Convexity proven for sums of angles of unitary paths.

Let $R$ be an integral domain and $G$ be a subgroup of its group of units. We consider the category $\mathbf{\mathsf{Cob}}_G$ of 3-dimensional cobordisms between oriented surfaces with connected boundary, equipped with a representation of their fundamental group in $G$. Under some mild conditions on $R$, we construct a…

Let $p$ be an even positive integer and $U_p(H)$ be the Banach-Lie group of unitary operators $u$ which verify that $u-1$ belongs to the $p$-Schatten ideal $B_p(H)$. Let ${\cal O}$ be a smooth manifold on which $U_p(H)$ acts transitively and smoothly. Then one can endow ${\cal O}$ with a natural Finsler metric in terms…

The paper studies totally nonnegative parts of flag varieties and their topologies.