arXiv research

Curious structure of special orthogonal, unitary, and symplectic groups as products of Grassmannians discovered.

We construct new proper biharmonic functions defined on open and dense subsets of the special unitary group SU(2). Then we employ a duality principle to obtain new proper biharmonic functions from the non-compact 3-dimensional hyperbolic space H^3.

We develope a new scheme for the construction of explicit complex-valued proper biharmonic functions on Riemannian Lie groups. We exploit this and manufacture many infinite series of uncountable families of new solutions on the special unitary group $SU(n)$. We then show that the special orthogonal group $SO(n)$ and th…

Study asymptotics of unitary matrix elements in quantum mechanics.

Geometric proof of contractibility of unitary group in strong topology.

The paper uses singularity theory to find normal forms for sub-Riemannian exponential maps.

Study on random representations of surface groups into SU(n), focusing on asymptotic expansions.

Convexity proven for sums of angles of unitary paths.

We construct a new family of toric manifolds generating the unitary bordism ring. Each manifold in the family is the complex projectivisation of the sum of a line bundle and a trivial bundle over a complex projective space. We also construct a family of special unitary quasitoric manifolds which contains polynomial gen…

We give a new method for manufacturing complete minimal submanifolds of compact Lie groups and their homogeneous quotient spaces. For this we make use of harmonic morphisms and basic representation theory of Lie groups. We then apply our method to construct many examples of compact minimal submanifolds of the special u…

The study finds invariant Einstein metrics on complex Stiefel manifolds and special unitary groups.

Improved bounds on moments of word measures on unitary groups.

We introduce a novel approach to perform first-order optimization with orthogonal and unitary constraints. This approach is based on a parametrization stemming from Lie group theory through the exponential map. The parametrization transforms the constrained optimization problem into an unconstrained one over a Euclidea…

Study extends JB-algebra structure group results to infinite dimensions.

The paper studies a special Grassmannian space and shows it's an orbit of a unitary group.

New Lagrangian and special Lagrangian examples found in complex space.

Contact group retracts to unitary subgroup.

New structure on unitary group of Hilbert space.

There are five unimodular simply connected three dimensional unimodular non abelian Lie groups: the nilpotent Lie group $\mathrm{Nil}$, the special unitary group $\mathrm{SU}(2)$, the universal covering group $\widetilde{\mathrm{PSL}}(2,\mathbb{R})$ of the special linear group, the solvable Lie group $\mathrm{Sol}$ and…

Efficient diffusion model for symmetric manifolds reduces training and computation costs.

If the conformal holonomy group $Hol(\mathcal{T})$ of a simply connected space with conformal structure of signature $(2p-1,2q-1)$ is reduced to $\U(p,q)$ then the conformal holonomy is already contained in the special unitary group $\SU(p,q)$. We present two different proofs of this statement, one using conformal trac…

Researchers study injectivity of magnetic and thermostatic nonabelian ray transforms on compact surfaces.

We classify all harmonic maps of finite uniton number from a Riemann surface into SU(n) in terms of certain pieces of the Bruhat decomposition of the subgroup of algebraic loops in SU(n). We give a description of the "Frenet frame data" for such harmonic maps in a given class.

For a unitary local system of rank one on a complete hyperbolic threefold of finite volume which has only one cusp, the Ruelle L-function is defined. We will show that if the first cohomology group of the local system vanishes its value at s=0 is equal to the square of the Franz-Reidemeister torsion.

For a given Hilbert space $\mathcal H$, consider the space of self-adjoint projections $\mathcal P(\mathcal H)$. In this paper we study the differentiable structure of a canonical sphere bundle over $\mathcal P(\mathcal H)$ given by $$ \mathcal R=\{\, (P,f)\in \mathcal P(\mathcal H)\times \mathcal H \, : \, Pf=f , \, \…

We explore the class of triples (M, nabla, P) where M is a manifold, nabla is an affine connection in M and P is a G-structure in M. Inside this class there are infinitesimally homogeneous manifolds, characterized by having G-constant curvature, torsion and inner torsion. For each matrix Lie group G subgroup of GL(Rn) …

Researchers describe unitary representations of mixed braid groups.

The Hardy space H^2(R) for the upper half plane together with a unimodular function group representation u(λ) = \exp(i(λ_1ψ_1 + ... + λ_nψ_n)) for λin R^n, gives rise to a manifold M of orthogonal projections for the subspaces u(λ)H^2(R) of L^2(R). For classes of admissible functions ψ_i the strong operator topology cl…

Study circumcenters in Finsler unitary groups with optimal convexity bounds.

The dimension of the space of SU(n) and translation invariant continuous valuations on $\mathbb{C}^n, n \geq 2$ is computed. For even $n$, this dimension equals $(n^2+3n+10)/2$; for odd $n$ it equals $(n^2+3n+6)/2$. An explicit geometric basis of this space is constructed. The kinematic formulas for SU(n) are obtained …

We use filtrations of the Grassmannian model to produce explicit algebraic formulae for all harmonic maps of finite uniton number from a Riemann surface, and so all harmonic maps from the 2-sphere, to the unitary group for a general class of factorizations by unitons. We show how these specialize to give explicit formu…

Semi-direct products of finite groups have permutation representations that are constructed from the permutation representations of their constituents. One can envision these in a metaphoric sense in which a rope is made from a bundle of threads. In this way, subgroups and quotients are easily visualized. The general i…

Let $U_c(H)={u: u is unitary and u-1 is compact}$ stand for the unitary Fredholm group. We prove the following convexity result. Denote by $d_\infty$ the rectifiable distance induced by the Finsler metric given by the operator norm in $U_c(H)$. If $u_0,u_1,u\in U_c(H)$ and the geodesic $β$ joining $u_0$ and $u_1$ in $U…

Study of logarithms in SVD-closed subgroups of unitary group.

We propose unitary group convolutions (UGConvs), a building block for CNNs which compose a group convolution with unitary transforms in feature space to learn a richer set of representations than group convolution alone. UGConvs generalize two disparate ideas in CNN architecture, channel shuffling (i.e. ShuffleNet) and…

The Hull-Strominger system for supersymmetric vacua of the heterotic string allows general unitary Hermitian connections with torsion and not just the Chern unitary connection. Solutions on unimodular Lie groups exploiting this flexibility were found by T. Fei and S.T. Yau. The Anomaly flow is a flow whose stationary p…

Study nonrigid dynamics of unitary groups on Lie groups via kinetic energy metrics.

Let F be a finite group with a Sylow 2-subgroup S that is normal and abelian. Using hyperelementary induction and cartesian squares, we prove that Cappell's unitary nilpotent groups UNil_*(Z[F];Z[F],Z[F]) have an induced isomorphism to the quotient of UNil_*(Z[S];Z[S],Z[S]) by the action of the group F/S. In particular…

Classifies matrices in the quaternionic hyperbolic unitary group.

Minimal dimensions found for flag manifolds embeddings.

Random flat bundles on surfaces have least eigenvalues at least 1/4.

With the usual definition of a super Hilbert space and a super unitary representation, it is easy to show that there are lots of super Lie groups for which the left-regular representation is not super unitary. I will argue that weakening the definition of a super Hilbert space (by allowing the super scalar product to b…

We consider the construction of the basic bundle gerbe on SU(n) introduced by Meinrenken and show that it extends to a range of groups with unitary actions on a Hilbert space including U(n), diagonal tori and the Banach Lie group of unitary operators differing from the identity by an element of a Schatten ideal. In all…

We describe explicit presentations of all stable and the first nonstable homotopy groups of the unitary groups. In particular, for each n >= 2 we supply n homotopic maps that each represent the (n-1)!-th power of a suitable generator of pi_2n(U(n)) = Z_{n!}. The product of these n commuting maps is the constant map to …

The existence of kinematic formulas for area measures with respect to any connected, closed subgroup of the orthogonal group acting transitively on the unit sphere is established. In particular, the kinematic operator for area measures is shown to have the structure of a co-product. In the case of the unitary group the…

Solved a specific case of Salter's question on Burau representation.

Holomorphic Higgs bundles on Teichmüller space for surface groups.

This paper introduces a submanifold of the moduli space of unitary representations of the fundamental group of a punctured sphere with fixed local monodromy. The submanifold is defined via products of involutions through Lagrangian subspaces. We show that the moduli space of Lagrangian representations is a Lagrangian s…