Paper connects Turaev cobracket to Kashiwara--Vergne problem via Hodge theory.
problem Solving the Kashiwara--Vergne problem in algebraic geometry.
method Combining Turaev cobracket and Goldman bracket results, constructing torsors of solutions.
result Found a torsor of solutions to the Kashiwara--Vergne problem that depends only on topology.
This paper characterizes Kashiwara-Vergne groups using algebraic structures of knotted tubes.
problem Characterizing Kashiwara-Vergne groups using algebraic structures.
method Using algebraic structures of welded foams and arrow diagrams, the paper describes the Kashiwara-Vergne groups and their associated graded circuit algebras.
result The paper provides a description of the graded Grothendieck-Teichmüller group as automorphisms of arrow diagrams.
By introducing a refinement of the Goldman-Turaev Lie bialgebra, we interpret the divergence cocycle in the Kashiwara-Vergne problem and the Enomoto-Satoh obstructions for the surjectivity of the Johnson homomorphisms as some part of a regular homotopy version of the Turaev cobracket.
Two elliptic Kashiwara-Vergne Lie algebras match.
problem Matching two elliptic Kashiwara-Vergne Lie algebras.
method Using mould theoretic techniques and graded formality isomorphisms.
result The two elliptic Kashiwara-Vergne Lie algebras coincide.
Proof of injection from double shuffle to Kashiwara-Vergne Lie algebra.
problem Injecting double shuffle Lie algebra into Kashiwara-Vergne Lie algebra.
method Inclusion of brunnian braids group on different genus 0 surfaces, using lower central series of brunnian Lie algebras, and explicit links between maps.
result Injection of double shuffle Lie algebra into symmetric Kashiwara-Vergne Lie algebra.
We define a family KV(g,n) of Kashiwara-Vergne problems associated with compact connected oriented 2-manifolds of genus g with n+1 boundary components. The problem KV(0,3) is the classical Kashiwara-Vergne problem from Lie theory. We show the existence of solutions of KV(g,n) for ar…
New proof connects Kashiwara-Vergne equations to Goldman-Turaev Lie bialgebra.
problem Proving Kashiwara-Vergne equations from isomorphism of Lie bialgebras.
method Novel characterization of conjugacy classes in free Lie algebra via cyclic words.
result Automorphisms inducing isomorphisms in Goldman-Turaev Lie bialgebra satisfy Kashiwara-Vergne equations.
We introduce a simplified version of Drinfeld's equations that still solves Kashiwara-Vergne equations.
problem Solving Kashiwara-Vergne equations using Drinfeld's associator equations.
method Introducing a weak version of Drinfeld's associator equations and showing its solutions lead to solutions of the Kashiwara-Vergne equations.
result Solutions to emergent Drinfeld equations still solve the Kashiwara-Vergne equations.
Survey on algebraic aspects of Turaev cobracket and related problems.
problem Measuring self-intersection of loops on surfaces.
method Algebraic review of cobracket and its variants, mapping class group applications, Kashiwara-Vergne problem relation.
result Homological description of the cobracket by Hain.
Study formalities on closed surfaces using connections.
problem Formalities of Goldman-Turaev Lie bialgebra on closed surfaces.
method Reformulated Kashiwara-Vergne groups and associators in higher genera using non-commutative connections.
result Determined pro-unipotent automorphism group of associated graded.
Functorial approach connects operads to Lie bialgebras.
problem Establishing a natural way to derive Lie bialgebras from operads.
method Functorial construction of bracket and cobracket operations.
result Reproduced Alekseev-Torossian injection for higher genus groups.
This is the second in a series of papers dedicated to studying w-knots, and more generally, w-knotted objects (w-braids, w-tangles, etc.). These are classes of knotted objects that are wider but weaker than their "usual" counterparts. To get (say) w-knots from usual knots (or u-knots), one has to allow non-planar "virt…
For an oriented 2-dimensional manifold Σ of genus g with n boundary components the space Cπ1(Σ)/[Cπ1(Σ),Cπ1(Σ)] carries the Goldman-Turaev Lie bialgebra structure defined in terms of intersections and self-intersections of curves. Its associated graded (under the natural filtratio…
In this paper, we describe a surprising link between the theory of the Goldman-Turaev Lie bialgebra on surfaces of genus zero and the Kashiwara-Vergne (KV) problem in Lie theory. Let Σ be an oriented 2-dimensional manifold with non-empty boundary and K a field of characteristic zero. The Goldman-Turaev Lie…
For a compact oriented surface Σ of genus g with n+1 boundary components, the space g(Σ) spanned by free homotopy classes of loops in Σ carries the structure of a Lie bialgebra equipped with a natural decreasing filtration, whose structure morphisms are called the Goldman bracket and the (framed) T…
In this paper we show that, after completion in the I-adic topology, the Goldman bracket on the space spanned by homotopy classes of loops on a smooth, complex algebraic curve is a morphism of mixed Hodge structure. We prove similar statements for the natural action (defined by Kawazumi and Kuno) of the loops in X on p…
This is the first in a series of papers studying w-knotted objects (w-knots, w-braids, w-tangles, etc.), which make a class of knotted objects which is {w}ider but {w}eaker than their usual counterparts. The group of w-braids was studied (as "{w}elded braids") by Fenn-Rimanyi-Rourke and was shown to be isomorphic to th…
We survey the status of some decision problems for 3-manifolds and their fundamental groups. This includes the classical decision problems for finitely presented groups (Word Problem, Conjugacy Problem, Isomorphism Problem), and also the Homeomorphism Problem for 3-manifolds and the Membership Problem for 3-manifold gr…
Optimal transport reformulates multiple quantile hedging problem.
problem Multiple quantile hedging problem in incomplete markets.
method Reformulated as Monge optimal transport problem, introduced Kantorovitch version, proved no duality gap.
result Multiple quantile hedging problem can be seen as semi-discrete optimal transport problem.
Solves four problems related to circle families in the plane.
problem Four basic problems of circle families in the plane.
method Solves all four basic problems of circle families in the plane.
result All four basic problems are solved.
Solves four problems related to sphere families in 3D space.
problem Four basic problems of sphere families in Euclidean 3-space.
method Solves all four basic problems of sphere families in Euclidean 3-space.
result All four basic problems are solved.
The paper solves optimal control problems for various convex sets using convex trigonometry.
problem Optimal control problems with 2D convex compact sets.
method Using convex trigonometry to derive extremals for various problems.
result Geodesics in multiple sub-Finsler problems are derived.
Explains eigenvalue and generalized eigenvalue problems with examples.
problem Eigenvalue and generalized eigenvalue problems.
method Introduction and examples from machine learning.
result Solutions to eigenvalue and generalized eigenvalue problems.
This paper solves the Christoffel problem in hyperbolic space and its equivalent on spheres.
problem Prescribing curvatures for convex hypersurfaces in hyperbolic space.
method Proving a full rank theorem to establish the existence of solutions.
result Existence of solutions to the Christoffel problem and its equivalent Nirenberg-Kazdan-Warner problem on spheres.
In the present paper, the primal-dual problem consisting of the investment risk minimization problem and the expected return maximization problem in the mean-variance model is discussed using replica analysis. As a natural extension of the investment risk minimization problem under only a budget constraint that we anal…
Study proves only origin-centered spheres solve certain curvature problems.
problem Proving uniqueness of solutions to curvature problems.
method Using the Heintze-Karcher inequality, the study proves the uniqueness of smooth, strictly convex solutions to a class of Minkowski type problems.
result Only origin-centered spheres solve isotropic and Lp-Gaussian-Minkowski problems. MathChat uses LLM agents to solve challenging math problems through conversational problem-solving.
problem Solving math problems expressed in natural language.
method MathChat is a conversational framework combining an LLM agent and a user proxy agent for collaborative problem-solving.
result MathChat improves tool-using prompting methods by 6% on difficult math problems.
New algorithm solves non-convex min-max problems in signal processing.
problem Non-convex min-max problems in signal processing and communication.
method Hybrid Block Successive Approximation (HiBSA) algorithm alternating gradient descent and ascent steps.
result HiBSA converges to first-order stationary solutions with global rates.
This article reviews ranking problems and their solutions.
problem Ranking problems in statistical learning.
method Systematic review of ranking problems and optimization techniques.
result Unified notation for optimization problems and identification of strengths and limitations of algorithms.
The paper solves a generalized Christoffel-Minkowski problem using a curvature flow.
problem Solving the (p,q)-Christoffel-Minkowski problem.
method Investigating the problem via an expanding curvature flow.
result Existence and uniqueness of smooth solutions to the (p,q)-Christoffel-Minkowski problem.
Paper solves Gromov-Wasserstein for point clouds efficiently.
problem Quantifying similarity between two formations or shapes.
method Reformulates QAP as low-rank concave quadratic optimization problem.
result Global solution for large-scale problems with thousands of points.
The paper explains how microlocal analysis solves geometric inverse problems.
problem Recovering geometric information from boundary measurements.
method Microlocal analysis applied to three inverse problems.
result Microlocal techniques solve specific inverse problems in Riemannian geometry.
Proves NP and co-NP status for knot core recognition in solid torus.
problem Determining if a knot is the core of a solid torus.
method Alternate proof and corollary of Hopf link recognition problem.
result Proves NP and co-NP status for solid torus core recognition problem.
A new method solves complex control problems with random coefficients.
problem Solving LQ McKean-Vlasov control problems with random coefficients.
method Decomposes the problem into two decoupled stochastic optimal control problems.
result The sum of optimal controls of auxiliary problems equals the original problem's optimal control.
This is a survey of some problems in geometric group theory which I find interesting. The problems are from different areas of group theory. Each section is devoted to problems in one area. It contains an introduction where I give some necessary definitions and motivations, problems and some discussions of them. For ea…
We present updates to the problems on Hirzebruch's 1954 problem list focussing on open problems, and on those where substantial progress has been made in recent years. We discuss some purely topological problems, as well as geometric problems about (almost) complex structures, both algebraic and non-algebraic, about co…
27 problems identified in automating movie/TV subtitle translation.
problem Challenges in translating movie/TV subtitles.
method Categorized problems into three categories and evaluated translation quality.
result Frontier NLP systems struggle with subtitles and require post-processing.
New method solves generalized Minkowski problem for torsional rigidity.
problem Generalized Minkowski problem for torsional rigidity.
method Flow method
result Existence of solutions for general measures.
Solves Brezis' first open problem on ball solutions.
problem Existence of solutions to Brezis-Nirenberg problem on a 3D ball.
method Building on sign-changing solutions to the Yamabe problem.
result Infinitely many sign-changing, nonradial solutions found.
Classical knot recognition problem solved in NP with exponential time algorithm.
problem Determining if a virtual knot is classical.
method Proved NP membership and provided an exponential time algorithm.
result Classical knot recognition problem is in NP.
Paper solves four problems of pseudo-circle envelopes in Minkowski plane.
problem Four problems of pseudo-circle envelopes in Minkowski plane.
method Solutions to four basic problems.
result Solved four problems of pseudo-circle envelopes in Minkowski plane.
In this paper, we address the inverse problem, or the statistical machine learning problem, in Markov random fields with a non-parametric pair-wise energy function with continuous variables. The inverse problem is formulated by maximum likelihood estimation. The exact treatment of maximum likelihood estimation is intra…
Describes state variables in sequential decision problems, linking them to Markovian and non-Markovian models.
problem Sequential decision problems, especially in active learning and POMDPs, where decisions affect what is observed and learned.
method Canonical framework and novel two-agent perspective of POMDPs, defining state variables to claim Markovian or non-Markovian models.
result Properly modeled sequential decision problems are Markovian, while real decision problems are often non-Markovian.
Conference compiles problems on foliations and diffeomorphisms.
problem Challenges in foliations and diffeomorphism groups.
method Compilation of problems from conference participants.
result Compilation of 20+ problems on foliations and diffeomorphisms.
Solves double coset problem for braid group H_n.
problem Double coset problem in braid group B_n modulo H_n.
method Uses Garside's decomposition of braids in B_n.
result Demonstrates stable equivalence of Link Problem to solvable algebraic problem.
This paper solves the dual Minkowski problem for q-torsional rigidity.
problem The dual Minkowski problem for q-torsional rigidity.
method Introduced the p-th dual q-torsional measure and solved the p-th dual Minkowski problem for q-torsional rigidity using a Gauss curvature flow.
result Existence of smooth even and non-even solutions to the p-th dual Minkowski problem for q-torsional rigidity.
In this paper, we discuss the uniqueness in an integral geometry problem in a strongly convex domain. Our problem is related to the problem of finding a Riemannian metric by the distances between all pairs of the boundary points. For the proof, the problem is reduced to an inverse source problem for a kinetic equation …
Study on geometric variational problems for existence, regularity, and uniqueness of solutions.
problem Geometric variational problems, focusing on existence, regularity, and uniqueness of solutions.
method Formulated in Federer and Fleming's theory of currents, discussed the existence theory, and presented core ideas of the (interior) regularity theory for area-minimizing currents and optimal transport paths. Two original results on generic uniqueness of solutions were presented.
result Generic uniqueness of solutions for both Plateau's problem and optimal branched transport problem.