Paper reviews algebraic research in machine learning theory.
problem Understanding phase transitions in machine learning models.
method Algebraic approaches in statistical mechanics.
result Algebraic methods are essential for analyzing machine learning models with singularities.
Research connects Lie algebras to configuration space (co)homology.
problem Understanding the (co)homology of configuration spaces.
method Identifying Lie algebra (co)homology as a counterpart to configuration space (co)homology.
result Lie algebras and configuration spaces have a deep mathematical relationship.
Research on dualities in geometric stereotypes.
problem Understanding dualities in geometric stereotypes.
method Continuation of previous research on stereotype spaces and algebras.
result New insights into geometric stereotype dualities.
Monograph explores algebraic structures related to Yang-Baxter equation.
problem Yang-Baxter equation and its solutions in algebra.
method Investigation of skew braces, quandles, racks, and Rota-Baxter groups.
result Interrelations and applications of these structures to knot theory.
Research on knots, braids, and their invariants.
problem Understanding and computing knot invariants.
method Definition of knot equivalence, use of braid groups, Hecke algebras, and HOMFLY polynomial.
result General formula for HOMFLY polynomial of looped Coxeter braids.
Research examines curves of degree 8 with specific singularities.
problem Existence of curves with prescribed singularities.
method Algebraic and symplectic approaches.
result Characterization of curves with specific singularities.
This is a survey article, based on the author's lectures in the 2015 AMS Summer Research Institute in Algebraic Geometry, and to appear in the Proceedings.
Research explores Lie algebras in Riemannian manifolds.
problem Understanding Lie algebras in Riemannian manifolds.
method Analyzes the Lie algebra of infinitesimal isometries.
result Identifies two commutative ideals in the Lie algebra.
Non-associtive algebras is a research direction gaining much attention these days. New developments show that associative algebras and some not-associative structures can be unified at the level of Yang-Baxter structures. In this paper, we present a unification for associative algebras, Jordan algebras and Lie algebras…
In this short survey we give a background and explain some recent developments in algebraic minimal cones and nonassociative algebras. A good deal of this paper is recollections of my collaboration with my teacher, PhD supervisor and a colleague, Vladimir Miklyukov on minimal surface theory that motivated the present r…
This is a survey work on Lie algebras with ad-invariant metrics. We summarize main features, notions and constructions, in the aim of bringing into consideration the main research on the topic. We also give some list of examples in low dimensions.
This overview paper is intended as a quick introduction to Lie algebras of vector fields. Originally introduced in the late 19th century by Sophus Lie to capture symmetries of ordinary differential equations, these algebras, or infinitesimal groups, are a recurring theme in 20th-century research on Lie algebras. I will…
Research decouples Lie algebroids using bicocycle double cross product theory.
problem Understanding decoupling and coupling phenomena in Lie algebroids.
method Bicocycle double cross product realization method.
result Unified product, double cross product, semi-direct product, and cocycle extension frameworks are instances of the general method.
Researchers develop neural networks for approximating functions in Banach spaces.
problem Approximating Banach space valued continuous functions.
method Quasi-interpolation Banach space valued neural network operators using algebraic sigmoid functions.
result Jackson type inequalities for function approximation.
New algebraic framework for Jacobi manifolds connects geometric mechanics and dimensional analysis.
problem Lack of clear algebraic interpretation for Jacobi manifolds.
method Developed a dimensioned algebra approach to capture algebraic counterparts of Jacobi manifolds.
result Poly-Jacobi manifolds provide a new connection between geometric mechanics and dimensional analysis.
Researchers solve a 25-year-old conjecture about vector fields.
problem Proving a 25-year-old conjecture about divergence-free vector fields.
method Analysis of a Leibniz algebra underlying these vector fields.
result Construction of the universal central extension for divergence-free vector fields and diffeomorphisms.
Researchers present and compare different representations of dissipative Hamiltonian DAE systems.
problem Understanding and transforming dissipative Hamiltonian DAE systems.
method Global geometric and algebraic points of view, translations between representations, characterizations, and numerical methods for computing structural information.
result A general DAE system can be transformed into a dissipative Hamiltonian or port-Hamiltonian DAE system.
Researchers prove an equivariant index theorem on Euclidean space.
problem Calculating the equivariant index of the Bott-Dirac operator on R2n. method Continuous field of C∗-algebras and equivariant index theorem. result Explicit calculation of the equivariant index of the Bott-Dirac operator on R2n. Researchers compute c-projective symmetry algebras for Kähler surfaces.
problem Understanding symmetries in Kähler surfaces.
method Defined and analyzed c-projective vector fields and computed their symmetries.
result Computed c-projective symmetry algebras for Kähler surfaces with essential c-projective vector fields.
Researchers found abnormal extremals on specific Lie groups.
problem Identifying abnormal extremals on four-dimensional Lie groups.
method Using left-invariant sub-Finsler quasimetrics and seminorms on Lie algebra.
result Established a criterion for strict abnormality of extremals.
Researchers prove Lie algebras of differential operators and Grothendieck constructions coincide.
problem Understanding derivations and Lie algebras of vector bundles.
method Proving Lie algebras coincide through differential operators and Grothendieck constructions.
result Lie algebras coincide up to an isomorphism.
A new category of Lie algebras, called generalized Lie algebras, is presented such that classical Lie algebras and Lie-Rinehart algebras are objects of this new category. A new philosophy over generalized Lie algebroids theory is presented using the notion of generalized Lie algebra and examples of objects of the categ…
New C∗-algebra approach unifies machine learning strategies.
problem Lack of diverse and information-rich data models in machine learning.
method Integrates C∗-algebra into machine learning frameworks. result Unified learning strategies and new data models.
Research explores Kähler and semi-para-Kähler structures on specific Lie groups.
problem Existence of Kähler and semi-para-Kähler structures on six-dimensional unsolvable Lie groups.
method Examines four specific Lie algebras and their structures.
result One Lie algebra admits Kähler metrics, others admit semi-para-Kähler and semi-Kähler structures.
Researchers address the generation of differential invariants for geometric structures.
problem Finite generation of differential algebra of relative differential invariants.
method Investigation of algebraic and differential properties, localization, weight analysis.
result Localization on a finite set of relative invariants makes the differential algebra finitely generated.
This research introduces Lie brackets on spaces of biderivations in Lie algebras.
problem Understanding higher-order infinitesimal symmetries in Lie algebras.
method Study of right biderivations and Lie brackets on their spaces.
result New Lie algebra framework for biderivations with applications in deformation theory.
Researchers explore valuations on polyhedra and topological arrangements without imposing algebraic structures.
problem Understanding valuations on polyhedra and their connections to topological arrangements.
method Generalizes the setting of valuations on convex polyhedra to collections of defining hyperplanes without imposing algebraic structures.
result Uncovered a close relationship between scissors congruence problems and finite hyperplane arrangements.
Authors discuss complex and non-Archimedean geometry, proving a conjecture.
problem Proving a version of the Yau--Tian--Donaldson conjecture for Kähler metrics.
method Relation between complex, analytic, and non-Archimedean geometry.
result Sketch of proof for Yau--Tian--Donaldson conjecture.
Researchers use Mellin-Barnes integrals to study trinomial equations and their braids.
problem Analyzing the roots of trinomial algebraic equations.
method Global analytic continuation and Mellin-Barnes integral representations.
result Precise description of the Galois group of trinomial equations.
Researchers link vertex algebras to non-Kähler solutions of the Hull-Strominger system.
problem Constructing representations of vertex algebras from non-Kähler solutions of the Hull-Strominger system.
method Embedding the N=2 superconformal vertex algebra in the chiral de Rham complex of a string Courant algebroid, with a condition on the Hermitian-Yang-Mills connection.
result Any solution of the Hull-Strominger system satisfying the Hermitian-Yang-Mills condition has an associated N=2 embedding.
Researchers extend pseudodifferential calculus on filtered manifolds using fixed point algebras.
problem Defining operators with varying orders on filtered manifolds.
method Using generalized fixed point algebras and nilpotent Lie groups, they construct a new calculus.
result They establish a new calculus that reflects the behavior of differential operators on filtered manifolds.
In~\cite{Kim} the author generalized the Conway algebra and constructed the invariant valued in the generalized Conway algebra defined by applying two skein relations to crossings, which is called a generalized Conway type invariant. The generalized Conway type invariant is a generalization of Homflypt polynomial. In t…
Researchers create explicit representations for skein algebras of small surfaces, revealing their Azumaya loci.
problem Understanding the structure and properties of skein algebras of small surfaces.
method Constructed finite-dimensional representations at all roots of unity, using explicit formulas and analyzing reducibility.
result Azumaya loci of the surfaces contain the smooth loci of classical shadow varieties, with equality for the one-punctured torus and proper containment for the four-punctured sphere.
Researchers prove positivity of skein algebra structure constants for specific surfaces.
problem Positivity of structure constants in skein algebras of specific surfaces.
method Mirror symmetry construction based on higher genus Gromov-Witten theory applied to a complex cubic surface.
result Proved positivity of structure constants for skein algebras of the 4-punctured sphere and 1-punctured torus.
Research on refined algebraic domains respecting differential geometry.
problem Understanding shapes and regions of real algebraic curves.
method Investigates points in two curves, singular points, inflection points, and points of double tangent lines, considering differential geometry.
result Proves fundamental properties and investigates examples of refined algebraic domains.
Research explores real algebraic realization of round fold maps of codimension -1.
problem Real algebraic realization of round fold maps of codimension -1.
method Generalizes canonical projections of unit spheres to round fold maps and discusses their real algebraic realization.
result Developed new studies in real algebraic geometry focusing on round fold maps of codimension -1.
Researchers describe a Ceresa class for tropical and topological curves, linking algebraic and cohomological perspectives.
problem Explicitly describe the Ceresa class for non-hyperelliptic curves.
method Combining algebraic, tropical, and topological perspectives, defining the Ceresa class for curves and surfaces.
result The Ceresa class is torsion in all settings: tropical curves, topological surfaces, and smooth algebraic curves over C((t)). Researchers prove smoothings for surfaces with triple points.
problem Smoothings of surfaces with triple points.
method Differential geometric proof.
result Proves existence of smoothings for surfaces satisfying suitable conditions.
AIDN uses deep learning to represent algebraic structures.
problem Building learning systems to uncover algebraic laws from data.
method AIDN is a deep learning algorithm that represents algebraic objects using neural networks.
result AIDN can robustly compute representations of various algebraic structures.
Researchers reconstruct algebraic maps onto curves based on prescribed Reeb graphs.
problem Reconstructing smooth real algebraic maps onto curves with specific Reeb graphs.
method Developed a method to reconstruct functions from general finite graphs, focusing on curves.
result Reconstructed functions from prescribed Reeb graphs, providing a new approach in real algebraic geometry.
Researchers prove formulas for flag area measures, extending previous work.
problem Proving additive kinematic formulas for flag area measures.
method Introducing an algebraic framework to compute these formulas explicitly.
result Existence and explicit computation of additive kinematic formulas for flag area measures.
Research examines octonionic slice regular functions and their automorphisms and invariants.
problem Analyzing slice regular functions in the octonionic algebra.
method Investigates automorphisms and invariants of octonionic slice regular functions.
result Characterizes the automorphisms and invariants of octonionic slice regular functions.
Researchers confirm integral formulas for G2-structures in detail.
problem Verifying integral formulas for G2-structures on Riemannian manifolds. method Detailed analysis and explicit expressions of intrinsic torsion using exterior algebra.
result Agreement of integral formulas with intrinsic torsion components.
This article was submitted to a volume under preparation, with Benson Farb as the editor, on the topic of open problems in surface mapping class groups. The braid group B_n is the mapping class group of an n-times punctured disk. The Iwahori-Hecke algebra H_n is a quotient of the braid group algebra of B_n by a quadrat…
Researchers establish a connection between knot homology and Lie algebra actions.
problem Understanding the HOMFLY-PT homology of (n,n+1) torus knots. method Constructing an explicit isomorphism and computing tautological class actions.
result The tautological class action extends to Hamiltonian vector fields and differentials in spectral sequences.
This work explores algebraic structures from curvature and torsion in affine connections.
problem Understanding algebraic structures from curvature and torsion in affine connections.
method Post-Lie algebra, D-algebra, and special polynomials.
result A particular class of geometrically special polynomials is generated by torsion and curvature.
The paper mostly collects material on generic rank of A--modules with respect to differential geometric applications. Our research was motivated by geometry of A--structures. In particular, we discuss the case where A is an unitary associative algebra not necessary with inversion. Some of the examples are studied…
Researchers solve the realization of Jordan-Kronecker invariants in Lie algebras.
problem Identifying which Jordan-Kronecker invariants can be realized by Lie algebras.
method Analyzing the Kronecker and Jordan cases, proving impossibility for certain invariants, and describing realizability for others.
result Complete solution for Jordan and Kronecker cases, partial answers for others.