Optimizes quantum channel mapping between Hilbert spaces.
problem Optimal mapping between Hilbert spaces based on wavefunction measurements.
method Maximizes total fidelity subject to partial unitarity constraints using an iterative algorithm.
result Developed an algorithm for finding the global maximum of the optimization problem.
We give a 3-page description of the Gassner invariant / representation of braids / pure braids, along with a description and a proof of its unitarity property.
New method preserves unitarity for Schrödinger equation learning, reducing errors and improving time generalization.
problem Learning the evolution operator for time-dependent Schrödinger equation with varying Hamiltonians.
method Linear estimator preserving weak unitarity, with theoretical error bounds and time generalization.
result Achieves up to two orders of magnitude smaller relative errors than existing methods.
Study non-semisimple TQFT for Burau representation density and unitarity.
problem Density and unitarity of the Burau representation from a non-semisimple TQFT perspective.
method TQFT construction of Squier's Hermitian form on the Burau representation.
result Density of the image of braid group in unitary representations.
New method shows unitarity in quantization for toric manifolds.
problem Unitarity in quantization commutes with reduction for toric manifolds.
method Generalized coherent state transform (gCST) and geodesic rays of toric Kähler polarizations.
result Quantization commutes unitarily with reduction for the new mixed polarization.
We prove the existence of a new class of constant mean curvature cylinders with an arbitrary number of umbilics by unitarizing the monodromy of Hill's equation.
This is the first in a series of papers devoted to an analogue of the metaplectic representation, namely, the minimal unitary representation of an indefinite orthogonal group; this representation corresponds to the minimal nilpotent coadjoint orbit in the philosophy of Kirillov-Kostant. We begin by applying methods fro…
We describe the unitary globalization of cohomologically induced modules $A_{\fq}(λ)$. The purpose of the paper is to give a geometric realization of the unitarizable modules. Our results do not constitute a proof of unitarity.
Paper connects algebraic and analytic methods for braid group representations.
problem Constructing representations of braid groups using algebraic and analytic approaches.
method Katz-Long-Moody construction and multiplicative middle convolution for KZ-type equations.
result Multiplicative middle convolution preserves unitarity and provides an algorithm to determine the signature of a Hermitian matrix.
A natural one-parameter family of Kähler quantizations of the cotangent bundle T∗K of a compact Lie group K, taking into account the half-form correction, was studied in \cite{FMMN}. In the present paper, it is shown that the associated Blattner-Kostant-Sternberg (BKS) pairing map is unitary and coincides with the…
We provide a geometric construction of the unitary structure which is projectively preserved by the Hitchin connection. We analyze the asymptotic behavior of it and we establish that it is uniformly in the level equivalent to the Hermitian structure induced by the L2 inner product on smooth sections.
We decompose the de Rham Laplacian on Sasaki-Einstein manifolds as a sum over mostly positive definite terms. An immediate consequence are lower bounds on its spectrum. These bounds constitute a supergravity equivalent of the unitarity bounds in dual superconformal field theories. The proof uses a generalization of Kah…
We present a theorem on the unitarizability of loop group valued monodromy representations and apply this to show the existence of new families of constant mean curvature surfaces homeomorphic to a thrice-punctured sphere in the simply-connected 3-dimensional space forms R3, $\bbS^3 $ and $\bbH^3$. Additionally, we…
We use some Lie group theory and Budney's unitarization of the Lawrence-Krammer representation, to prove that for generic parameters of definite form the image of the representation (also on certain types of subgroups) is dense in the unitary group. This implies that, except possibly for closures of full-twist braids, …
We extend the coherent state transform (CST) of Hall to the context of the moduli spaces of semistable holomorphic vector bundles with fixed determinant over elliptic curves. We show that by applying the CST to appropriate distributions, we obtain the space of level k, rank n and genus one non-abelian theta functions w…
We describe two simple obstructions to the existence of Ricci-flat Kahler cone metrics on isolated Gorenstein singularities or, equivalently, to the existence of Sasaki-Einstein metrics on the links of these singularities. In particular, this also leads to new obstructions for Kahler-Einstein metrics on Fano orbifolds.…
We establish various results on the large level limit of projective quantum representations of surface mapping class groups obtained by quantizing moduli spaces of flat SU(n)-bundle. Working with the metaplectic correction, we proved that these projective representations lift to asymptotic representations. We show that…
Factorization of the differential expansion coefficients for HOMFLY-PT polynomials of double braids, discovered in arXiv:1606.06015 in the case of rectangular representations R, is extended to the first non-rectangular representations R=[2,1] and R=[3,1]. This increases chances that such factorization will take p…
RP-GFRFT unifies fractional order and rotation control for graph signals.
problem Lack of rotation-based spectral control in GFRFT and zero-angle degeneracy in AGFT.
method Rotation-parameterized graph fractional Fourier transform (RP-GFRFT) with degeneracy preserving rotation matrix.
result RP-GFRFT improves spectral filtering performance over existing methods.
Hybrid QML model improves recovery rate prediction accuracy.
problem Complex nonlinear dependencies, high-dimensional feature spaces, and limited sample sizes in recovery rate forecasting.
method Hybrid Quantum Machine Learning (QML) with Amplitude Encoding, leveraging PQC and qubit data compression.
result Significantly lower RMSE (0.228) compared to classical models.
This work discovers algebraic structures from data using a differentiable measure.
problem Discovering discrete algebraic rules from data.
method Formalizes the problem through Cayley-table completion and uses HyperCube operator-valued tensor factorization.
result Derives an absolute lower bound for the differentiable measure of algebraic complexity, proving it is attained only for group structures.
An differential field (F;∂1,...,∂m) of characteristic zero, a subgroup H of affine group GL(n,C)∝Cn with respect to its identical representation in Fn and the following two fields of differential rational functions in x=(x1,x2,...,xn)-column vector, $$C< x, \partial >^H=\{f^{\part…
The L2-∂∂-Lemma is extended to complete Kähler manifolds with a gap in the spectrum.
problem Extending the L2-∂∂-Lemma to non-compact Kähler manifolds. method Proving the L2-∂∂-Lemma on complete Kähler manifolds with a gap in the spectrum. result The L2-∂∂-Lemma is generalized to complete Kähler manifolds. Sharp Hölder regularity found for complex Frobenius theorem coordinates.
problem Finding optimal Hölder-Zygmund regularity for complex Frobenius theorem coordinates.
method Analyzing necessary and sufficient conditions for coordinate charts achieving the theorem's structure.
result The optimal Hölder-Zygmund regularity for coordinate charts is shown to be α. In a recent paper~\cite{DDL10} we studied basic properties of partial immersions and partially free maps, a generalization of free maps introduced first by Gromov in~\cite{Gro70}. In this short note we show how to build partially free maps out of partial immersions and use this fact to prove that the partially free map…
Two-dimensional Riemannian manifolds uniquely determined by boundary data.
problem Determining a 2D Riemannian manifold from boundary data.
method Calderón problem approach using Dirichlet-to-Neumann operator.
result A 2D compact connected Riemannian manifold is uniquely determined up to conformal equivalence.
PRNet registers partial 3D shapes using deep learning.
problem Partial-to-partial point cloud registration.
method Self-supervised deep learning network for non-convex alignment and partial correspondence.
result Outperforms existing methods on synthetic data.
Study cohomologies of complex manifolds with symplectic forms and their stability.
problem Analyzing cohomologies of complex manifolds with symplectic forms.
method Investigate the Hard Lefschetz Condition on Dolbeault cohomology groups using a double complex.
result Stability of the ∂∂Λ-Lemma under small deformations of ω but not under complex structure. Let M be a surface sum of 3-manifolds M1 and M2 along a bounded connected surface F and ∂i be the component of ∂Mi containing F. If Mi has a high distance Heegaard splitting, then any minimal Heegaard splitting of M is the amalgamation of those of M1,M2 and M∗, where $M^i=M…
The paper studies maps from pseudo-Hermitian to Kähler manifolds, proving harmonic map properties.
problem Analyzing maps between pseudo-Hermitian and Kähler manifolds.
method Investigates partial energy functionals and critical maps, proving foliated results for ∂b- and ∂b-harmonic maps. result Generalizes Siu's holomorphicity result to ∂b- and ∂b-harmonic maps. The paper proves Hodge decompositions and partial bar partial lemmas for G2 and Calabi-Yau manifolds.
problem Proving Hodge decompositions and partial bar partial lemmas for G2 and Calabi-Yau manifolds.
method Defining cohomology spaces analogous to Bott-Chern cohomology and relating them to harmonic forms on the manifolds.
result Geometric interpretation of cohomology classes in terms of submanifolds and gerbes for G2 manifolds.
The paper studies holomorphic Poisson manifolds and their deformations under specific cohomology conditions.
problem Understanding deformations of holomorphic Poisson manifolds under certain cohomological assumptions.
method Investigates properties of Koszul-Brylinski homology and Dolbeault cohomology, proving formality of a DGLA.
result The DGLA is shown to be formal, and Maurer-Cartan elements induce complex structure deformations.
The paper characterizes when the ∂∂-lemma holds for twistor spaces.
problem Characterizing the ∂∂-lemma for twistor spaces. method Study Bott-Chern and Aeppli cohomologies of twistor spaces.
result Explicit computation of Dolbeault cohomology for flat torus twistor space.
Let M be a compact hypersurface with boundary ∂M=∂D1∪∂D2, ∂D1⊂Π1, ∂D2⊂Π2, Π1 and Π2 two parallel hyperplanes in Rn+1 (n≥2). Suppose that M is contained in the slab determined by these hyperplanes and that the mean cu…
The paper studies deformations of Calabi-Yau manifolds using Gauduchon metrics.
problem Deformations of Calabi-Yau manifolds under co-polarised conditions.
method Analyzes local deformations of Calabi-Yau ∂∂ˉ-manifolds using Gauduchon metrics and constructs a new hp-HS form. result Proves the p-SKT h-∂∂ˉ-property is deformation open. Paper tackles distribution matching by partially matching distributions, achieving robust results.
problem Robustly aligning two probability distributions.
method Developed a partial Wasserstein adversarial network (PWAN) to efficiently approximate the partial Wasserstein-1 (PW) discrepancy.
result The PWAN effectively produces highly robust matching results, outperforming state-of-the-art methods.
We give a simple proof of a result on the ∂∂ˉ-lemma property under a blow-up transformation by Deligne--Griffiths--Morgan--Sullivan's criterion. Here, we use an explicit blow-up formula for Dolbeault cohomology given in our previous work, which can be induced by a morphism expressed on the level of…
Differential structure on partial isometries over Grassmannian constructed.
problem No specific problem stated; abstract focuses on method and result.
method Construction of differential structure on partial isometries over restricted Grassmannian.
result Set of partial isometries over restricted Grassmannian becomes a Banach Lie groupoid.
Ribbon cobordism forms a partial order in 3-manifolds.
problem Understanding partial orders in 3-manifolds.
method Utilizing recent methods from Ian Agol's work on knot concordance.
result Ribbon rational homology cobordism forms a partial order.
On a compact complex manifold X, we prove a Frölicher-type inequality for Bott-Chern cohomology and we show that the equality holds if and only if X satisfies the ∂∂-Lemma.
Abstract study of HKT manifolds, proving Hodge theory and formality properties.
problem Analyzing Hodge theory and formality in HKT manifolds.
method Study of Dolbeault operators and Laplacians, proving relations and properties.
result Formality of differential graded algebra for certain HKT manifolds.
Study on 4-dimensional almost-Hermitian manifolds, proving ∂-harmonic forms invariant under certain metrics.
problem Proving ∂-harmonic forms are topological invariants for specific metrics on 4-dimensional almost-Hermitian manifolds. method Analyzing ∂-Laplacian and using globally conformally Kähler and strictly locally conformally Kähler metrics. result Dimension of ∂-harmonic (1,1)-forms is a topological invariant, answering Kodaira and Spencer's problem. We consider partial matchings, which are finite graphs consisting of edges and vertices of degree zero or one. We consider transformations between two states of partial matchings. We introduce a method of presenting a transformation between partial matchings. We introduce the notion of the lattice presentation of a par…
It is shown that any smooth strictly convex global solution of det(∂ξi∂ξj∂2u)=exp{−∑i=1ndi∂ξi∂u−d0}, where d0, d1,...,dn are constants, must be a quadratic polynomial. This extends a well-known theorem of Jö…
New theorems on Hodge numbers and Kähler structures derived from complex differential forms.
problem Deformation invariance and local stability of Hodge numbers and Kähler structures.
method Using the exponential operator and power series method, the approach focuses on d-closed extensions and foliated cases. result Local stabilities of transversely p-Kähler structures and new theorems on Hodge numbers. 3-manifold curvature comparison with rotationally symmetric bodies.
problem Comparing scalar curvature of 3-manifolds with rotationally symmetric boundaries.
method Inspired by Gromov, comparing mean curvatures and induced metrics.
result Flatness of 3-manifolds under certain curvature conditions.
Study shows partial hyperbolicity leads to Anosov dynamics in 3-manifolds.
problem Understanding dynamics in hyperbolic 3-manifolds and Seifert manifolds.
method Classification of partially hyperbolic diffeomorphisms and pseudo-Anosov dynamics.
result Complete classification of partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds and Seifert manifolds.
The paper introduces a new class of manifolds based on the Hodge decomposition and spectral sequences.
problem Characterizing and understanding new classes of compact complex manifolds.
method Introducing a new class of page-r-∂∂ˉ-manifolds and using spectral sequences. result Characterized and provided examples of the new class of manifolds.