UGConvs improve CNN accuracy with unitary transforms.
problem Improving CNN accuracy with richer representations.
method UGConvs combine group convolutions with unitary transforms.
result HadaNets achieve similar accuracy to circulant networks with lower complexity.
Contact group retracts to unitary subgroup.
problem Understanding contact structures on 3-sphere.
method Proving deformation retraction to unitary subgroup.
result Group of contactomorphisms retracts to U(2).
New structure on unitary group of Hilbert space.
problem No specific problem stated; constructing a new structure.
method Constructing a Banach Poisson-Lie group structure.
result Banach Poisson-Lie group structure on unitary group of Hilbert space.
Consider a compact locally symmetric space M of rank r, with fundamental group Γ. The von Neumann algebra $\vn(Γ)$ is the convolution algebra of functions f∈ℓ2(Γ) which act by left convolution on ℓ2(Γ). Let Tr be a totally geodesic flat torus of dimension r in M and let $Γ_0\cong\bb Z^r$ be t…
New biharmonic functions created on Lie groups.
problem Constructing explicit biharmonic functions on Lie groups.
method Developed a new scheme for constructing complex-valued biharmonic functions on Riemannian Lie groups.
result Manufactured infinite series of new solutions on SU(n) and showed applicability to SO(n) and Sp(n). Researchers describe unitary representations of mixed braid groups.
problem Understanding unitary representations of mixed braid groups.
method Explicitly describe unitary representations on cohomology of Abelian branched covers.
result Image of the representation is generated by complex reflections and related to the multivariate Burau representation.
Study circumcenters in Finsler unitary groups with optimal convexity bounds.
problem Existence and convexity of circumcenters in Finsler unitary groups.
method Analysis of distance functions and p-Schatten norm on Lie algebra.
result Existence of circumcenters for sets with radius < π/2 in several metrics.
New unitary representations for mapping class groups without almost invariant vectors.
problem Understanding unitary representations of mapping class groups.
method Space of measured foliations and Teichmüller space.
result None of the representations has almost invariant vectors.
Geometric proof of contractibility of unitary group in strong topology.
problem Contractibility of unitary group in strong operator topology.
method Direct geometric proof and construction of special subspaces and operators.
result Direct geometric proof of contractibility theorem.
Curious structure of special orthogonal, unitary, and symplectic groups as products of Grassmannians discovered.
problem Understanding the structure of special orthogonal, unitary, and symplectic groups.
method Expressing these groups as products of Grassmannians realized as involution matrices.
result Special orthogonal, special unitary, and symplectic groups can be expressed as products of their corresponding Grassmannians.
Study asymptotics of unitary matrix elements in quantum mechanics.
problem Asymptotic behavior of unitary matrix elements in quantum mechanics.
method Uses Berezin-Toeplitz quantization and symplectic geometry.
result Recover asymptotics of Wigner's d-matrix elements for spin representations.
Let Uc(H)=u:uisunitaryandu−1iscompact stand for the unitary Fredholm group. We prove the following convexity result. Denote by d∞ the rectifiable distance induced by the Finsler metric given by the operator norm in Uc(H). If u0,u1,u∈Uc(H) and the geodesic β joining u0 and u1 in $U…
Develops quantum circuits for faster learning with symmetry considerations.
problem Speeding up learning quantum states with symmetry considerations.
method Utilizes Okounkov-Vershik approach and Young-Jucys-Murphy elements to develop Sn-equivariant convolutional quantum circuits. result Proves Sn-CQA generates any unitary in any given Sn irrep sector, universal for SU(d) symmetry. QCNNs avoid barren plateaus, making them trainable.
problem Exponentially vanishing gradients in QNNs.
method Graph-based method to analyze Haar-distributed unitaries.
result QCNNs do not exhibit barren plateaus, implying trainability.
Study of logarithms in SVD-closed subgroups of unitary group.
problem Understanding logarithms in SVD-closed subgroups of unitary groups.
method Analysis of generalized principal logarithms and minimizing geodesics.
result Set of generalized principal logarithms is a disjoint union of diffeomorphic subsets.
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.
problem Understanding the dynamics of unitary groups on Lie groups using kinetic energy metrics.
method Least action principle applied to geodesics of the kinetic energy metric on G. result Kinetic energy metric on G is not complete and not invariant. Convexity proven for sums of angles of unitary paths.
problem Proving convexity of sums of eigenvalues of unitary matrices.
method Analyzing paths of unitary matrices and their angles, using operator norms.
result Sum of first m angles of unitary path is convex.
Classifies matrices in the quaternionic hyperbolic unitary group.
problem Understanding the structure of matrices in the quaternionic hyperbolic unitary group.
method Used complex representation and characteristic polynomial to study matrices.
result Computed the characteristic polynomial and studied its sign.
Minimal dimensions found for flag manifolds embeddings.
problem Finding the smallest dimensions for flag manifolds embeddings.
method Equivariant embeddings of orthogonal and unitary groups acting on real and complex flag manifolds.
result Minimal dimensions achieved at isospectral models.
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 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 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.
problem Under what conditions are matrices in the image of the Burau representation of B3. method Algorithmically constructed a counterexample to Salter's specific question.
result The central quotient of the Burau image group is not the central quotient of a certain subgroup of the unitary group.
Holomorphic Higgs bundles on Teichmüller space for surface groups.
problem Characterizing representations of surface groups admitting holomorphic Higgs data.
method Non-abelian Hodge correspondence, unitarity conditions, and existence proofs for higher ranks.
result Holomorphic dependency of Higgs data is equivalent to unitarity for SL(2,C) representations but fails for higher ranks. A new method for optimizing neural networks with orthogonal constraints.
problem Optimizing neural networks with orthogonal constraints.
method Parametrization using the exponential map to transform constrained optimization into unconstrained.
result Faster, more accurate, and stable convergence in RNNs with orthogonal recurrent weights.
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…
We construct projective unitary representations of the smooth Deligne cohomology group of a compact oriented Riemannian manifold of dimension 4k+1, generalizing positive energy representations of the loop group of the circle. We also classify such representations under a certain condition. The number of the equivalence…
Compactifies Minkowski space using unitary matrices.
problem Compactifying Minkowski space for quantum field theories.
method Using Cayley transform and unitary group $\U(2)$.
result Defines interesting backgrounds for quantum field theories.
When looking at Bott's original proof of his periodicity theorem for the stable homotopy groups of the orthogonal and unitary groups, one sees in the background a differential geometric periodicity phenomenon. We show that this geometric phenomenon extends to the standard inclusion of the orthogonal group into the unit…
New mechanics on non-associative octonions discovered.
problem Discrete mechanics on non-associative groups.
method Generalized Lagrangian and Hamiltonian mechanics to non-associative objects.
result Discrete mechanics on unitary octonions achieved.
Develops a framework for designing quantum neural networks that respect symmetries.
problem Trainability and generalization issues in quantum neural networks.
method Equivariant quantum neural networks (EQNN) for any symmetry group.
result Efficient construction of equivariant layers for EQNNs, including QCNNs.
New unitary representations for Thompson's group V and virtual links.
problem Constructing unitary representations for Thompson's group V and virtual links.
method Defining a surjection from V to virtual equivalence classes of CC links, constructing unitary representations from kei and operator quandle colorings.
result Realizes all oriented almost classical virtual links.
Improved bounds on moments of word measures on unitary groups.
problem Analyzing the moments of word measures on unitary groups.
method Using Haar-uniform sampling and algebraic invariants to study asymptotic behavior.
result Found a new algebraic invariant related to the moments of word measures.
In this second part of the `essay on the completion of quantum theory' we define the {\em unitary setting of completed quantum mechanics}, by adding as intrinsic data to those from Part I (arXiv:1711.08643) the choice of a north pole N and south pole S in the geometric space. Then we explain that, in the unitary settin…
Random representations of surface groups approach asymptotic freeness in large n limit.
problem Asymptotic freeness of Haar unitary matrices for surface groups.
method Interplay between Dehn's work and classical invariant theory.
result Expected value of trace of a fixed non-identity element is bounded as no∞. We give in explicit form the principal kinematic formula for the action of the affine unitary group on $\C^n$, together with a straightforward algebraic method for computing the full array of unitary kinematic formulas, expressed in terms of certain convex valuations introduced, essentially, by H. Tasaki. We introduce …
Study SKK groups of manifolds to classify non-unitary TQFTs.
problem Classify non-unitary invertible topological quantum field theories.
method Apply Galatius-Madsen-Tillman-Weiss and Genauer-Schommer-Pries results to compute SKK groups.
result Complete classification of non-unitary invertible TQFTs in dimensions 1-5.
Study on random representations of surface groups into SU(n), focusing on asymptotic expansions.
problem Understanding random representations of surface groups into special unitary groups.
method Use of a symplectic form on moduli space, establishing asymptotic expansions for trace values.
result Existence of large n asymptotic expansions for expected values of trace of elements under random representations.
A Hermitian TQFT from non-semisimple quantum sl(2) modules.
problem Constructing a Hermitian TQFT from a non-semisimple category.
method Endowed a non-semisimple category of quantum sl(2) modules with a Hermitian structure and proved the resulting TQFT is Hermitian.
result Projective representations of the mapping class group in indefinite unitary matrices.
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 paper studies a special Grassmannian space and shows it's an orbit of a unitary group.
problem Investigating a specific Grassmannian space of infinite-dimensional subspaces.
method Analyzing the restricted p-Schatten class Grassmannian and showing it's an affine coadjoint orbit of a unitary group. result The restricted p-Schatten class Grassmannian is shown to be an affine coadjoint orbit of an infinite-dimensional restricted unitary group. We give a completely explicit formula for all harmonic maps of finite uniton number from a Riemann surface to the unitary group U(n) in any dimension, and so all harmonic maps from the 2-sphere, in terms of freely chosen meromorphic functions on the surface and their derivatives, using only combinations of projections …
An orbifold is a Morita equivalence class of a proper {\' e}tale Lie groupoid. A unitary equivalence class of spectral triples over the algebra of smooth invariant functions are associated with any compact spin orbifold. In the case of an effective spin orbifold we construct a collection of spectral triples over the sm…
In a "naive" attempt to create algebraic quantum field theories on the circle, we obtain a family of unitary representations of Thompson's groups T and F for any subfactor. The Thompson group elements are the "local scale transformations" of the theory. In a simple case the coefficients of the representations are polyn…
Private algorithms approximate matrices with private data.
problem Approximate matrices with same spectrum using private data.
method Differential privacy algorithms for unitary orbit optimization.
result Upper and lower bounds on approximation error.