Study mixed commutator lengths in wreath products and their relation to general ranks.
problem Understanding mixed commutator lengths in wreath products and their relation to general ranks.
method Analyzing wreath products (G,N)=(Z≀Γ,⨁ΓZ) and determining mixed commutator lengths in terms of general rank. result Mixed commutator lengths and ordinary commutator lengths coincide under certain conditions.
The fundamental ideas of aplicability of Levi-Malcev Theorem for Bol algebras, which plays a basic role in structural theory are outlined
New spectral theory for non-associative algebras with applications to Moufang dynamics.
problem Spectral theory of non-associative algebras and their applications.
method Introducing almost periodic Banach--Malcev algebras and analyzing their spectral properties.
result Spectral characterization and continuous functional calculus for almost periodic derivations.
Explicit BCH series radii found for special Banach-Malcev shift algebras.
problem Finding convergence radii for BCH series in specific algebraic structures.
method Established explicit convergence radii using continuity estimates and algebraic properties.
result Explicit formula for convergence radii derived and validated for various shift algebras.
Infinitesimal calculations link fundamental groups to Lie algebras.
problem Calculating logarithm maps in fundamental groups.
method Hopf invariants defined by Harrison cohomology of commutative cochains.
result Zeroth Harrison cohomology is a universal dual to Malcev Lie algebra.
Proves properties of Torelli Lie algebra for surfaces.
problem Properties of Torelli Lie algebra for surfaces.
method Proves two theorems about Malcev Lie algebra associated to Torelli group.
result Stable Koszul property and trivial Johnson homomorphism kernel.
Extends foliation results to singular cases.
problem Understanding foliations near singular leaves.
method Proves semi-local Levi-Malcev theorem for holonomy Lie algebroid.
result Formal semi-local triviality for all 2-connected and a wide class of 1-connected leaves.
A Levi-Malcev type decomposition for 2-step solvable Lie algebras with a complex structure
problem Decomposition of 2-step solvable Lie algebras with a complex structure method Proving a Levi-Malcev type decomposition
result Fino-Vezzoni conjecture holds for 2-step solvable unimodular Lie algebras We represent the coordinate ring of algebraic hulls (which are generalizations of the Malcev completions of nilpotent groups for solvable groups) of solvmanifolds G/Γ by using Miller's exponential iterated integrals (which are extensions of Chen's iterated integrals) of invariant differential forms.
We introduce a notion of a Fox pairing in a group algebra and use Fox pairings to define automorphisms of the Malcev completions of groups. These automorphisms generalize to the algebraic setting the action of the Dehn twists in the group algebras of the fundamental groups of surfaces. This work is inspired by the Kawa…
We show that the Malcev Lie algebra of the fundamental group of a compact 2n+1-dimensional Sasakian manifold with n≥2 admits a quadratic presentation by using Morgan's bigradings of minimal models of mixed-Hodge diagrams. By using bigradings of minimal models, we also simplify the proof of the result of Cappelle…
Extends Floquet-Bloch theory to nilpotent groups for geometric applications.
problem Asymptotic problems on nilpotent covers of negatively curved manifolds.
method Generalized Floquet-Bloch theory using Malcev completions.
result Branching formula relating finite and infinite-dimensional representations.
Given a smooth manifold M equipped with a properly and discontinuous smooth action of a discrete group G, the nerve M∙G is a simplicial manifold and its vector space of differential forms TotN(ADR(M∙G)) carry a C∞-algebra structure m∙. We sh…
Paper proves non-triviality of Johnson kernel torsion subgroup.
problem Non-triviality of the torsion subgroup of the abelianized Johnson kernel.
method Action of mapping class group on Malcev Lie algebra, diagrammatic techniques.
result Proves non-triviality of the torsion subgroup with a purely 2-dimensional proof.
Let S be a compact connected oriented surface with one boundary component. We extend each of Johnson's and Morita's homomorphisms to the Ptolemy groupoid of S. Our extensions are canonical and take values into finitely generated free abelian groups. The constructions are based on the 3-dimensional interpretation of the…
In 1964, John Stallings established an important relationship between the low-dimensional homology of a group and its lower central series. We establish a similar relationship between the low-dimensional homology of a group and its derived series. We also define a torsion-free-solvable completion of a group that is ana…
Study reveals vanishing Massey products on compact complex surfaces, impacting their fundamental group structure.
problem Understanding the real homotopy type of compact complex surfaces.
method Analyzes Massey products and fundamental group presentations, providing explicit presentations in non-Kähler cases.
result Explicit presentations of fundamental groups based on first Betti numbers, vanishing Massey products beyond certain lengths.
In this note, we address the following question: Which 1-formal groups occur as fundamental groups of both quasi-Kähler manifolds and closed, connected, orientable 3-manifolds. We classify all such groups, at the level of Malcev completions, and compute their coranks. Dropping the assumption on realizability by 3-manif…
Study prolongations of nilpotent Lie algebras with specific structural subalgebras.
problem Understanding prolongations of nilpotent Lie algebras with specific structural subalgebras.
method Analyzing finite dimensional almost and quasi-effective prolongations of nilpotent Z-graded Lie algebras, focusing on those with decomposable reductive structural subalgebras.
result Obtained Levi-Malčev and Levi-Chevalley decompositions and precise properties of prolongations.
In this paper we prove that finite index subgroups of genus 3 mapping class and Torelli groups that contain the group generated by Dehn twists on bounding simple closed curves are not Kahler. These results are deduced from explicit presentations of the unipotent (aka, Malcev) completion of genus 3 Torelli groups and of…
The generalized Dehn twist along a closed curve in an oriented surface is an algebraic construction which involves intersections of loops in the surface. It is defined as an automorphism of the Malcev completion of the fundamental group of the surface. As the name suggests, for the case where the curve has no self-inte…
Let X and Y be finite-type CW-complexes (X connected, Y simply connected), such that the rational cohomology ring of Y is a k-rescaling of the rational cohomology ring of X. Assume H^*(X,Q) is a Koszul algebra. Then, the homotopy Lie algebra pi_*(Omega Y) tensor Q equals, up to k-rescaling, the graded rational Lie alge…
We explore the graded and filtered formality properties of finitely generated groups by studying the various Lie algebras over a field of characteristic 0 attached to such groups, including the Malcev Lie algebra, the associated graded Lie algebra, the holonomy Lie algebra, and the Chen Lie algebra. We explain how thes…
We analyze the degree-two part of the Torelli group's associated graded.
problem Understanding the structure of the degree-two part of the Torelli group's associated graded.
method We use algebraic topology and group theory to analyze the structure of the degree-two part of the Torelli group's associated graded.
result The abelian group Γ2I/Γ3I is torsion-free and described as a lattice in a rational vector space. Let Σ be a compact oriented surface. The Dehn twist along every simple closed curve γ⊂Σ induces an automorphism of the fundamental group π of Σ. There are two possible ways to generalize such automorphisms if the curve γ is allowed to have self-intersections. One way is to consider the `generalized Deh…
Let S be a compact connected oriented surface with one boundary component, and let P be the fundamental group of S. The Johnson filtration is a decreasing sequence of subgroups of the Torelli group of S, whose k-th term consists of the self-homeomorphisms of S that act trivially at the level of the k-th nilpotent quoti…
This paper protects rankings from differential privacy breaches.
problem Leakage of personal information in rankings.
method Develops ε-ranking differential privacy and a multistage ranking algorithm.
result Establishes the connection between Mallows model and ε-ranking differential privacy.
The paper addresses privacy in rank aggregation using randomized responses.
problem Preserving privacy while aggregating pairwise rankings.
method Adaptive debiasing method for randomized response rankings.
result Established minimax rates for estimation errors and optimal privacy guarantees.
This paper addresses the problem of rank aggregation, which aims to find a consensus ranking among multiple ranking inputs. Traditional rank aggregation methods are deterministic, and can be categorized into explicit and implicit methods depending on whether rank information is explicitly or implicitly utilized. Surpri…
We study the problem of rank aggregation: given a set of ranked lists, we want to form a consensus ranking. Furthermore, we consider the case of extreme lists: i.e., only the rank of the best or worst elements are known. We impute missing ranks by the average value and generalise Spearman's ρto extreme ranks. Our main …
Develops a method to infer partial rankings from sparse comparisons.
problem Challenges in ranking items with limited and noisy comparisons.
method Nonparametric Bayesian approach for learning partial rankings.
result Finds partial rankings that distinguish meaningful differences only when data supports it.
Research characterizes learnability of multilabel ranking problems.
problem Learnability of multilabel ranking problems with relevance-score feedback.
method Characterizes learnability in batch and online settings for a large family of ranking losses.
result Characterizes two equivalence classes of ranking losses based on learnability.
This paper compares rank aggregation methods for partial label ranking.
problem Handling partial label ranking with ties.
method Scoring-based and non-parametric probabilistic-based rank aggregation methods.
result Scoring-based variants consistently outperform the state-of-the-art method.
We study the problem of learning to rank from multiple information sources. Though multi-view learning and learning to rank have been studied extensively leading to a wide range of applications, multi-view learning to rank as a synergy of both topics has received little attention. The aim of the paper is to propose a c…
Paper tackles underranking in group-fair ranking systems, proving a trade-off and presenting an algorithm.
problem Underranking in group-fair ranking systems can worsen social and economic inequalities.
method Formulated underranking as a new problem, proved a lower bound, and presented a fair ranking algorithm.
result Algorithm achieves best of underranking and group fairness, confirming theoretical trade-off.
New ranking system balances fairness and user utility.
problem Achieving group fairness in ranking systems.
method Formulated a minimax game between a ranking player and an adversary.
result Better utility for highly fair rankings.
Introduces nondecreasing rank for matrices and tensors, developing methods and applications.
problem Finding low-rank approximations for matrices and tensors with monotonic constraints.
method Developed a variant of hierarchical alternating least squares algorithm for finding low ND rank approximations.
result Low ND rank factorizations can be found and interpreted for real-world datasets.
Analytic proof for minimal rank Sard conjecture.
problem Proving the minimal rank Sard conjecture in the analytic category.
method Using subanalytic abnormal distribution from [4], we establish a proof.
result The set of points accessible through singular horizontal curves of minimal rank has Lebesgue measure zero.
Rank-one measurements limit feasible sets for low-rank PSD matrices.
problem Feasibility of PSD matrices under rank-one measurements.
method Characterization of feasible sets for PSD matrices given rank-one projections.
result Radius of feasible sets determines singleton solution sets for low-rank matrices.
Sparse reduced-rank regression selects variables and ranks via manifold optimization.
problem Traditional rank selection fails when true rank is high.
method Sparse regularization and manifold optimization for rank and variable selection.
result Accurate estimation of coefficient parameter with high true rank.
We consider the problem of statistical inference for ranking data, specifically rank aggregation, under the assumption that samples are incomplete in the sense of not comprising all choice alternatives. In contrast to most existing methods, we explicitly model the process of turning a full ranking into an incomplete on…
The paper tackles learning true rankings from noisy, incomplete data.
problem Learning true rankings from incomplete and noisy data.
method Introduces a selective Mallows model for noisy rankings and derives upper and lower bounds on sample complexity.
result Strong asymptotically tight bounds on sample complexity for learning complete rankings and top-k rankings.
Paper introduces GAMs for interpretable learning-to-rank models.
problem Need for transparent ranking models in legal or policy scenarios.
method Developed generalized additive models (GAMs) for ranking tasks using neural networks.
result Neural ranking GAMs achieve better performance than traditional GAMs while maintaining interpretability.
Proposes tensor Q-rank for better tensor rank recovery in complex data.
problem Improving tensor rank recovery for complex data with low sampling rate.
method Introduces tensor Q-rank and two selection methods for Q, proposing VMTQN and MOTQN models. result Demonstrates superior performance in tensor completion problems compared to TNN-based methods.
Low-rank framework for task-specific LLM ranking from sparse comparisons.
problem Challenges in reliable task-specific ranking of LLMs under sparse, imbalanced comparisons.
method Low-rank modeling of task-by-model ability matrix, max-norm accurate estimator, task-wise top-K recovery guarantees, uncertainty quantification framework.
result Improves sample efficiency and produces tighter, better-calibrated ranking certificates.
New method solves nonsmooth low-rank matrix optimization problems efficiently.
problem Nonsmooth and low-rank matrix optimization problems in statistics and machine learning.
method Low-rank Extragradient Method with warm-start initialization.
result The extragradient method converges to an optimal solution with rate O(1/t) and requires only two low-rank SVDs per iteration. New methods rank players using covariates and comparisons, outperforming existing algorithms.
problem Ranking players based on incomplete and noisy pairwise comparisons.
method Three spectral ranking methods incorporating player covariates.
result Proposed methods outperform existing algorithms in simulations.
Proposes a cross entropy loss for better ranking algorithms.
problem Improving the theoretical understanding and performance of ranking algorithms.
method Introduces a cross entropy-based loss function that is a convex bound on NDCG and consistent with NDCG.
result Empirically, the proposed method outperforms existing algorithms in quality and robustness.