arXiv research

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

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 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.

Study asymptotics of unitary matrix elements in quantum mechanics.

Geometric proof of contractibility of unitary group in strong topology.

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…

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

Convexity proven for sums of angles of unitary paths.

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

We will show a theorem of a type of Cheeger and Muller for a noncompact complete hyperbolic threefold of finite vulume. As an application we will compute a special value of Ruelle L-function at the origin for a unitary local system which is cuspidal.

We construct natural Green forms for special cycles in orthogonal and unitary Shimura varieties, in all codimensions, and, for compact Shimura varieties of type O(p,2) and U(p,1), we show that the resulting local archimedean height pairings are related to special values of derivatives of Siegel Eisentein series. A conj…

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

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

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…

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…

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.

New Lagrangian and special Lagrangian examples found in complex space.

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.

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 …

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

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

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…

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…

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

We formulate the unitary rational orbifold conformal field theories in the algebraic quantum field theory framework. Under general conditions, we show that the orbifold of a given unitary rational conformal field theories generates a unitary modular category. Many new unitary modular categories are obtained. We also sh…

Contact group retracts to unitary subgroup.

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

Zeta functions for non-unitary twists are shown to have analytic continuation.

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 , \, \…

Recurrent neural networks are powerful models for processing sequential data, but they are generally plagued by vanishing and exploding gradient problems. Unitary recurrent neural networks (uRNNs), which use unitary recurrence matrices, have recently been proposed as a means to avoid these issues. However, in previous …

We combine recent developments on weakly symmetric pseudo--riemannian nilmanifolds with with geometric methods for construction of unitary representations on square integrable Dolbeault cohomology spaces. This runs parallel to construction of discrete series representations on spaces of square integrable harmonic forms…

Study on determinants of unitary Brownian motion and their asymptotic laws.

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…

Study circle actions on unitary manifolds with discrete fixed points.

A major challenge in the training of recurrent neural networks is the so-called vanishing or exploding gradient problem. The use of a norm-preserving transition operator can address this issue, but parametrization is challenging. In this work we focus on unitary operators and describe a parametrization using the Lie al…

The Riemann sphere of a C*-algebra is a geometric structure derived from a specific projector.

Let I be a symmetrically-normed ideal of the space of bounded operators acting on a Hilbert space H. Let ${p_i}_1 ^w$ $(1\leq w \leq \infty)$ be a family of mutually orthogonal projections on H. The pinching operator associated with the former family of projections is given by P: I --> I, P(x)=\sum_{i=1}^{w} p_i x p_i.…

The study shows ergodicity of unitary frame flows on Kähler manifolds with specific curvature conditions.

A notion of up and down Grover walks on simplicial complexes are proposed and their properties are investigated. These are abstract Szegedy walks, which is a special kind of unitary operators on a Hilbert space. The operators introduced in the present paper are usual Grover walks on graphs defined by using combinatoria…

We consider the Alexander polynomial of a plane algebraic curve twisted by a linear representation. We show that it divides the product of the polynomials of the singularity links, for unitary representations. Moreover, their quotient is given by the determinant of its Blanchfield intersection form. Specializing in the…

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 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…

Study circumcenters in Finsler unitary groups with optimal convexity bounds.

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…

Researchers describe unitary representations of mixed braid groups.

Affirmatively answers Kosniowski conjecture for unitary S^1-manifolds.