Defines a map linking quandle homology to Schur multiplier.
problem Understanding the relationship between quandle homology and Schur multiplier.
method Defined a map from second quandle homology to Schur multiplier and expressed second homology of Alexander quandles in terms of exterior algebras.
result Expressed the second homology of Alexander quandles in terms of exterior algebras.
New unoriented versions of Schur and Bogomolov multipliers for finite groups.
problem Defining and analyzing unoriented versions of Schur and Bogomolov multipliers.
method Using cohomology groups and quotient groups to define unoriented multipliers.
result Triviality of unoriented Bogomolov multiplier for certain groups, nontriviality for others.
We show that the Schur multiplier of Sp(2g,Z/DZ) is Z/2Z, when D is divisible by 4.
Study invariants for branched G-covers of surfaces, focusing on stable cases.
problem Classifying equivalence of homomorphisms on surfaces with punctures.
method Classifying space for (framed) C-branched G-covers and branched Schur invariants.
result Provide a stable answer to equivalence of homomorphisms for surfaces with enough genus and punctures.
The study extends group actions from surfaces to 3-manifolds using G-cobordisms.
problem When a finite group action on a surface extends to a 3-manifold.
method Using Schur multiplier and G-cobordisms, the study examines principal actions. result Affirmative extension for abelian, dihedral, symmetric, and alternating groups.
Study on Milnor fibrations of arrangements with trivial algebraic monodromy.
problem Explicit formulas for Milnor fiber Betti numbers in complex hyperplane arrangements.
method Analysis of cohomology jump loci and lower central series quotients of π1(F).
result Found arrangements with same Betti numbers but different fundamental groups.
Paper extends Schur's theorem to spherical curves via monotonicity.
problem Comparing chord lengths of convex and spherical curves.
method Monotonicity and expansion module approach.
result Schur's Theorem extended to spherical curves.
Paper proves almost Schur Lemma on smooth metric measure spaces.
problem Proving a specific lemma on metric measure spaces.
method Proves almost Schur Lemma using closed smooth metric measure spaces.
result Implications of the lemma for X. Cheng's and De Lellis-Topping's results.
Paper proves a new volume comparison theorem for Riemannian manifolds.
problem Comparing volumes of boundaries in Riemannian manifolds.
method Inspired by Schur's theorem, applies to Riemannian manifolds with Ricci curvature.
result Provides a new Schur's type volume comparison theorem.
The paper defines new types of positivity and proves properties of Schur forms for vector bundles.
problem Defining and characterizing new types of positivity for vector bundles.
method Introducing and characterizing two types of strongly decomposable positivity, proving properties of Schur forms.
result Schur forms of strongly decomposable positive vector bundles are positive or weakly positive, answering a question of Griffiths.
In this paper, we study a general almost Schur Lemma on pseudo-Hermitian (2n+1)-manifolds (M,J,θ) for n≥2. When the equality of almost Schur inequality holds, we derive the contact form θ is pseudo-Einstein and the pseudo-Hermitian scalar curvature is constant.
The paper proves a new theorem in Riemannian geometry and offers a new proof for Toponogov's theorem in Alexandrov geometry.
problem Proving new theorems in Riemannian and Alexandrov geometries.
method Inspired by the proof of the Schur-Toponogov theorem, a new proof of Toponogov's theorem is provided.
result A new theorem in Riemannian geometry and a new proof of Toponogov's theorem in Alexandrov geometry.
The paper defines new Schur-constant models for non-negative variables.
problem Modeling equilibrium distributions for non-negative variables.
method Introduces Schur-constant equilibrium distribution models for arithmetic non-negative random variables.
result Derived properties include implicit correlation and sum distribution.
New CR almost Schur Lemma estimates curvature on compact manifolds.
problem Estimating curvature on compact pseudohermitian manifolds.
method Established a new CR almost Schur Lemma with specific positivity conditions.
result Estimates pseudohermitian scalar curvature as a constant.
Proves a formula for Kontsevich-Witten tau-function using Schur Q-polynomials.
problem Proving the Kontsevich-Witten tau-function formula.
method Directly shows Q-polynomial expansion satisfies Virasoro constraints.
result Direct proof of the formula without matrix model.
Quantifies Schur's theorem for curves in CAT(k) spaces.
problem Quantifying Schur's comparison theorem for curves in CAT(k) spaces.
method Comparison formula for curves in model planes, curvature measures, moment arm, and Reshetnyak's theorem.
result Sharpens and extends classical arm and bow lemmas and Riemannian analogues.
Computes cohomology of rank 2 bundles, linking to skew Schur polynomials.
problem Cohomology of rank 2 stable bundles with odd determinant.
method Integral cohomology ring computations using Zagier's methods.
result Computes nilpotency degree of a generator in mod two cohomology.
New formula for twist knots using skew Schur polynomials.
problem Understanding and generalizing knot polynomials for twist knots.
method Reformulated prescription for twist knots in double-column representations using skew Schur polynomials.
result Mysterious shift from standard topological locus complicates generalization.
Let X=G/P be cominuscule rational homogeneous variety. (Equivalently, X admits the structure of a compact Hermitian symmetric space.) We say a Schubert class [S] is Schur rigid if the only irreducible subvarieties Y of X with homology class [Y] = r [S], for an integer r, are Schubert varieties. Robles and The identifie…
Efficiently computes embeddings for large graphs using coarsening.
problem Inefficient computation of graph embeddings for large-scale graphs.
method Graph coarsening based on Schur complements and Gaussian elimination.
result Efficiently computed embeddings on coarsened graph match Schur complement embeddings in expectation.
Schur theorem proven for weakly Landsberg Finsler metrics.
problem Proving the Schur theorem for a specific class of Finsler metrics.
method Using the Ricci curvature and properties of the mean Landsberg tensor, the theorem is proven for weakly Landsberg metrics.
result For weakly Landsberg Finsler metrics, the Ricci scalar must be constant.
New basis and Schur-Weyl duality for loop Hecke algebra defined.
problem Define a new basis for the loop Hecke algebra.
method Use higher linear rewriting theory and combinatorics of Dyck paths.
result Yields a conjecture of Damiani-Martin-Rowell and provides a representation theoretic interpretation.
New method connects neural networks to diagrammatic algebra.
problem Constructing permutation equivariant neural networks.
method Schur-Weyl duality between symmetric group and partition algebra.
result Simple diagrammatic method for calculating weight matrices.
The paper proves new curvature estimates in quaternionic contact geometry.
problem Estimating the curvature of compact qc manifolds.
method Establishing quaternionic contact versions of the Almost Schur Lemma.
result Curvature is a constant in terms of specific norms and components.
Study framizations of algebras using Schur--Weyl duality and tied braids.
problem Understanding framizations of algebras and their connections to quantum groups.
method Developing a general setting for framizations of algebras, including Yokonuma--Hecke and tied braids.
result Obtained Schur--Weyl duality for various algebras, including new framizations.
Let L be a link in a thickened annulus. We show that its sutured annular Khovanov homology carries an action of the exterior current algebra of the Lie algebra sl_2. When L is an m-framed n-cable of a knot K in the three-sphere, its sutured annular Khovanov homology carries a commuting action of the symmetric group S_n…
Researchers prove a stability result for a 3-sphere inequality, extending previous work.
problem Quantitative stability of nonlinear Yamabe-type inequalities on the 3-sphere.
method Proved a two-term refinement of the Schur lemma inequality in the conformal class of the 3-sphere.
result Deduced quantitative stability of an entire family of nonlinear Yamabe-type inequalities.
A new method for efficient portfolio optimization using graph structures.
problem Optimizing portfolio weights while reducing computational complexity.
method Hierarchical graph structures and Schur complement method.
result Optimal portfolio weights can be computed efficiently by inverting small submatrices.
The paper studies connectivity of Schur-Horn map images in real Grassmannians.
problem Connectivity of Schur-Horn map images in real Grassmannians.
method Criterion for pre-images of vectors in \(\mathbb{R}^n\) to be connected.
result Criterion for pre-images of vectors in \(\mathbb{R}^n\) to be connected.
The Schur's theorem of antiholomorphic type is proved for arbitrary almost Hermitian manifolds, namely: If a connected almost Hermitian manifold of dimension greater or equal to 6 is of pointwise constant antiholomorphic sectional curvature, then this curvature is a global constant.
Defines odd Khovanov homology via categorification of q-Schur algebra.
problem Constructing odd Khovanov homology via representation theory.
method Supercategorification of q-Schur algebra, odd foams, tensor product on chain complexes.
result Odd Khovanov homology defined via categorification.
New gauge invariants from framed 3-manifolds match Hopf algebra indicators.
problem Computing indicators for Hopf algebras.
method Using Kuperberg invariants from framed 3-manifolds.
result Kuperberg invariants match higher Frobenius-Schur indicators of Hopf algebras.
Constructs new topological theories in 2D not fitting standard axioms.
problem Developing new topological theories in 2D that don't conform to traditional axioms.
method Universal construction by Blanchet et al., Kronecker's characterization, field extension, Hankel matrices, Schur polynomials, and foam evaluation.
result Introduction of non-multiplicative theories and classification over finite-dimensional state spaces.
In this short note we establish an integral geometric inequality in a smooth metric measure space of the nonnegative Bakry-Émery Ricci curvature. This result can be regarded as a mild generalization of the almost Schur theorem due to De Lellis and Topping (Calc. Var., DOI: 10.1007/s00526-011-0413-z).
Method for conditional sampling with pre-trained normalizing flows.
problem Conditional sampling for incomplete observations.
method Variational Schur conditional sampling with normalizing flows.
result Successfully applied to invertible residual networks for inference and classification.
New algorithm learns ReLU networks efficiently using Schur polynomials.
problem PAC learning a linear combination of ReLU activations under Gaussian distribution.
method Uses tensor decomposition and Schur polynomials to identify and analyze higher-order moments.
result Near-optimal sample and computational complexity for learning ReLU networks.
Schur's lemma states that every Einstein manifold of dimension n≥3 has constant scalar curvature. Here (M,g) is defined to be Einstein if its traceless Ricci tensor $$\Rico:=\Ric-\frac{R}{n}g$$ is identically zero. In this short note we ask to what extent the scalar curvature is constant if the traceless Ricci …
The paper finds new inequalities for convex polygons.
problem Finding precise inequalities for convex polygons.
method Analytic isoperimetric inequalities based on Schur convex functions, followed by Bonnesen-style and inverse Bonnesen-style inequalities.
result Sharp discrete isoperimetric inequalities for planar convex polygons.
Spatial statisticians and quantitative investors use the same mathematical object: a Schur complement, damped by one parameter.
problem The Schur complement is used in both spatial modeling and portfolio allocation, but the parameters are different.
method The Schur complement is interpreted as reliability shrinkage of a conditional Gaussian.
result The Schur complement is the same in both applications.
In our previous paper in \cite{C}, we generalized the almost-Schur lemma of De Lellis and Topping for closed manifolds with nonnegative Rcci curvature to any closed manifolds. In this paper, we generalize the above results to symmetric (2,0)-tensors and give the applications including rth mean curvatures of closed …
The paper proves positivity of characteristic forms for certain vector bundles.
problem Characterizing positivity conditions for vector bundles.
method Operator theory, pushforward identities, and differential forms.
result Schur polynomials in Chern forms of Nakano and Griffiths positive vector bundles are positive as differential forms.
Study of zonal spherical functions on partial flag manifolds using Jacobi polynomials.
problem Understanding zonal spherical functions on partial flag manifolds.
method Matrix variate version of Koornwinder's method for constructing orthogonal polynomials, using Hermitian Jacobi polynomials and multivariate Schur polynomials.
result Conjecture that Hermitian Jacobi polynomials are elementary zonal spherical functions.
Preprint proves quantum coideal Schur-Weyl duality and generalizes Jones-Wenzl projectors.
problem Quantum coideal Schur-Weyl duality and Jones-Wenzl projectors in type B/D.
method Combinatorial proofs and functional analytic arguments.
result Explicit proof of quantum coideal Schur-Weyl duality and generalization of Jones-Wenzl projectors.
The paper optimizes policies constrained to Schur stabilizing controllers using a Newton-type algorithm.
problem Optimizing policies under linear constraints in control systems.
method Newton-type algorithm on a manifold of Schur stabilizing controllers with a Riemannian metric.
result Local convergence guarantees for the Newton-type algorithm without relying on exponential mapping or retractions.
Paper improves performance guarantees for Rademacher projections.
problem Improving statistical guarantees for Rademacher random projections.
method Algebraic framework for proving Schur-concavity properties.
result Novel Schur-concavity property of Rademacher projections with improved performance.
Deep neural networks are proven universally powerful using Koopman operator.
problem Proving the universality of deep neural networks.
method Formal deep network as a dual voice transform with Koopman operator, using group actions and Schur's lemma.
result Simple proof of the universality of DNNs.
Study describes deformation of topological locus for Hopf link.
problem Character description of Hopf link invariant.
method Deformation of topological locus, conjugate and composite representations.
result Definition of adjoint Schur functions.
After shortly recalling the construction of the Khovanov-Kuperberg algebras, we give a characterisation of indecomposable web-modules. It says that a web-module is indecomposable if and only if one can deduce it directly from the Kuperberg bracket (via a Schur lemma argument). The proofs relies on the construction of i…