Defines solitons and flows for generalized Ricci flow on nilpotent Lie groups.
problem Generalizing geometric flows on nilpotent Lie groups.
method Defines solitons and flows for generalized Ricci flow on exact Courant algebroids over nilpotent Lie groups.
result Explicit examples provided for the Heisenberg group.
Study shows limitations of Lie bracket commutation for nonsmooth vector fields.
problem Limitations of Lie bracket commutation for nonsmooth vector fields.
method Analysis of nonsmooth vector fields, focusing on commutation of flows and Lie bracket conditions.
result Lie bracket commutation cannot be extended to general a.e. differentiable vector fields, but holds for certain Sobolev regular fields.
A new Poisson bracket defined on Poisson structures with applications to fixed points and cohomology.
problem Defining a Poisson bracket on the space of Poisson structures.
method Constructing a Poisson bracket on P(M) depending on a volume form, and defining invariant of Poisson structures. result Invariant of Poisson structures detects unimodularity and related Poisson bracket for symplectic structures.
In this paper, we study the Ricci flow of solvmanifolds whose Lie algebra has an abelian ideal of codimension one, by using the bracket flow. We prove that solutions to the Ricci flow are immortal, the omega-limit of bracket flow solutions is a single point, and that for any sequence of times there exists a subsequence…
Investigates metric degeneracies on symplectic leaves using a generalized gradient flow.
problem Degeneracies in metrics on symplectic leaves of Poisson manifolds.
method Introduces the generalized double bracket (GDB) vector field to generalize gradient dynamics.
result Identifies admissible regions where the double bracket metric remains non-degenerate on symplectic leaves, enabling GDB as a gradient flow.
Toda flow explained as a porous medium equation.
problem Understanding the Toda flow through the lens of porous medium equations.
method Analyzing the geometry and dynamics of the porous medium equation and comparing it to the Toda flow.
result The Toda flow can be represented as a specific porous medium equation, revealing its gradient and Hamiltonian nature.
In the context of generalized geometry we first show how the Courant bracket helps to define connections with skew torsion and then investigate a five-dimensional invariant functional and its associated geometry. A Hamiltonian flow arising from this corresponds to a version of the Nahm equations using the Courant brack…
Study of Laplacian flow and solitons for G2-structures on homogeneous spaces.
problem Characterizing and understanding Laplacian solitons in homogeneous G2-structures. method Bracket flow/algebraic soliton approach, semi-algebraic solitons characterization, long time existence analysis.
result Found an example of a left-invariant closed semi-algebraic soliton that is not equivalent to any algebraic soliton.
We study curvature flows in the locally homogeneous case (e.g. compact quotients of Lie groups, solvmanifolds, nilmanifolds) in a unified way, by considering a generic flow under just a few natural conditions on the broad class of almost-hermitian structures. As a main tool, we use an ODE system defined on the variety …
Study how large-scale flows align small-scale vortices in 3D Euler equations.
problem Understanding how large-scale flows align small-scale vortices in 3D Euler equations.
method Constructing a Lagrangian coordinate to identify when the Lie bracket is zero and investigating the locality of the pressure term.
result Clarified conditions under which small-scale vortices are aligned by large-scale flows.
We study the evolution of homogeneous Ricci solitons under the bracket flow, a dynamical system on the space of all homogeneous spaces of dimension n with a q-dimensional isotropy, which is equivalent to the Ricci flow for homogeneous manifolds. We prove that algebraic solitons (i.e. the Ricci operator is a multiple of…
The main purpose of this note is to prove that any basis of a nilpotent Lie algebra for which all diagonal left-invariant metrics have diagonal Ricci tensor necessarily produce quite a simple set of structural constants; namely, the bracket of any pair of elements of the basis must be a multiple of some of them and onl…
Steady solitons found in renormalization group flow on Lie groups.
problem Analyzing two-loop renormalization group flow on 3D unimodular Lie groups.
method Using induced bracket flow and finding steady solitons.
result Found various steady solitons, some related to Ricci flow, others not.
Kontsevich flow simplified for 2D Poisson structures.
problem Formality conjecture for 2D Poisson structures.
method Universal flow construction and Poisson-cohomology triviality proof.
result For n=2, the flow Γ1 is Poisson-cohomology trivial. Derives equations for interacting Lie-Poisson systems using 2-cocycle extensions.
problem Understanding collective motion of interacting Lie-Poisson systems.
method Derives equations on dual space of extended structure, including 2-cocycle terms.
result Provides most general realization of Lie-Poisson system coupling.
We study evolution of (strong Kähler with torsion) SKT structures via the pluriclosed flow on complex nilmanifolds, i.e. on compact quotients of simply connected nilpotent Lie groups by discrete subgroups endowed with an invariant complex structure. Adapting to our case the techniques introduced by Jorge Lauret for stu…
Study infinite dimensional Lie algebras and their Poisson structures from Lie group actions.
problem Investigate Poisson structures from Lie group actions on manifolds.
method Computational approach, focusing on Lie algebroids and central extensions.
result Discover a broader class of Poisson brackets than standard Lie Poisson brackets.
New maximal families of compatible Poisson structures derived from geodesically equivalent metrics.
problem Constructing maximal families of compatible Poisson structures.
method Connecting geodesically equivalent metrics and compatible Poisson structures of hydrodynamic type.
result Maximal families of compatible Poisson structures of dimension (n+1)(n+2)/2 are constructed. Study left-invariant pseudo-Riemannian metrics on Lie groups focusing on null cone Lie algebras.
problem Characterize left-invariant pseudo-Riemannian metrics on Lie groups in the null cone.
method Use bracket flow on Lie algebra to study metrics on Lie groups.
result Classify all cases of null cone Lie algebras in signatures (1,q) and (2,q).
Flows on (or variations of) discrete curves in R2 give rise to flows on a subalgebra of functions on that curve. For a special choice of flows and a certain subalgebra this is described by the Toda lattice hierachy. In the paper it is shown that the canonical symplectic structure on R2N, which can be interpre…
We present a new formulation of some basic differential geometric notions on a smooth manifold M, in the setting of nonstandard analysis. In place of classical vector fields, for which one needs to construct the tangent bundle of M, we define a prevector field, which is an internal map from *M to itself, implementing t…
We study the existence of left invariant closed G2-structures defining a Ricci soliton metric on simply connected nonabelian nilpotent Lie groups. For each one of these G2-structures, we show long time existence and uniqueness of solution for the Laplacian flow on the noncompact manifold. Moreover, considering th…
The symmetric product of vector fields on a manifold arises when one studies the controllability of certain classes of mechanical control systems. A geometric description of the symmetric product is provided using parallel transport, along the lines of the flow interpretation of the Lie bracket. This geometric interpre…
Develops a new framework for generalized Ricci flow on Lie groups.
problem Global existence and geometric properties of Ricci flow on Lie groups.
method Inspired by Lauret's bracket flow, studies generalized Ricci flow on discrete quotients of Lie groups.
result Establishes global existence on solvmanifolds in arbitrary dimensions.
We prove that the Ricci flow g(t) starting at any metric on the euclidean space that is invariant by a transitive nilpotent Lie group N, can be obtained by solving an ODE for a curve of nilpotent Lie brackets. By using that this ODE is the negative gradient flow of a homogeneous polynomial, we obtain that g(t) is type-…
Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. Generically, the singular set is an embedded one dimensional manifold and there are three type of point…
Develops a method to study geometric flows on homogeneous spaces.
problem Analyzing geometric flows on homogeneous spaces.
method Bracket flow on Lie algebras to study geometric flows.
result Found a closed G2-structure on a nilpotent Lie group that is an expanding soliton for the Laplacian flow.
New bracket unifies nonholonomic dynamics and Hamilton-Jacobi theory.
problem Unified description of nonholonomic dynamics and Hamilton-Jacobi theory.
method Defined and proved coincidence of three nonholonomic brackets.
result Three nonholonomic brackets coincide.
New bracket theory connects three nonholonomic dynamics models.
problem Nonholonomic dynamics and their bracket formulations.
method Definition and proof of equivalence of three nonholonomic brackets.
result Three nonholonomic brackets are equivalent.
This note proves equivalence between Dorfman brackets and lifts, showing universality of the Courant-Dorfman bracket.
problem Characterizing twistings and symmetries of transitive Dorfman brackets.
method Proving equivalence between Dorfman brackets and lifts, intertwining with Courant-Dorfman bracket.
result Universality of the Courant-Dorfman bracket and characterization of Dorfman brackets via lifts.
Flat connections derived from Poisson brackets on loop spaces.
problem Understanding the structure of Poisson brackets on loop spaces.
method Defined connections by explicit linear combinations of standard connections associated with the Poisson bracket.
result Connections are shown to be flat.
In this paper, we investigate the geometry of a general class of gradient flows with multiple local maxima. we decompose the underlying space into disjoint regions of attraction and establish the adjacency criterion. The criterion states a necessary and sufficient condition for two regions of attraction of stable equil…
It is shown that two definitions for an exterior differential in superspace, giving the same exterior calculus, yet lead to different results when applied to the Poisson bracket. A prescription for the transition with the help of these exterior differentials from the given Poisson bracket of definite Grassmann parity t…
The paper constructs compatible Poisson brackets on gl(N).
problem Constructing compatible Poisson brackets on gl(N).
method Using constant tensors and Schouten brackets, the paper explicitly constructs quadratic Poisson brackets compatible with the standard Lie-Poisson bracket.
result Explicit construction of quadratic Poisson brackets compatible with the standard Lie-Poisson bracket on gl(N).
We survey the many instances of derived bracket construction in differential geometry, Lie algebroid and Courant algebroid theories, and their properties. We recall and compare the constructions of Buttin and Vinogradov, and we prove that the Vinogradov bracket is the skew-symmetrization of a derived bracket. Odd (resp…
Researchers attempt to categorify biquandle brackets using Khovanov homology methods.
problem Categorify biquandle brackets using Khovanov homology methods.
method Outline a Khovanov homology-style construction for biquandle brackets.
result A canonical biquandle 2-cocycle is defined, but not a true categorification of biquandle brackets.
Extends Nambu-Poisson bracket to superspace R^{n|m}.
problem No new problem introduced.
method Constructs Nambu-Poisson algebras of even degree functions using superdeterminant.
result Proves the n-ary bracket satisfies conditions for Nambu-Poisson bracket in R^{n|1} and R^{n|2}.
Graph complex acts on Poisson bi-vectors, producing universal cocycles.
problem Understanding the action of graph complex on Poisson bi-vectors.
method Using Lie derivatives and graph cocycles, the graph complex acts on Poisson bi-vectors.
result A uniform construction of universal cocycles for homogeneous Poisson bi-vectors.
Explains biquandle brackets and quivers for a topology talk.
problem None explicitly stated; focuses on background information.
method Review of biquandle concepts and related structures.
result Clarifies understanding of biquandle bracket quivers.
New method calculates knot and link biquandle brackets using trace diagrams.
problem Computing biquandle brackets of knots and links efficiently.
method Using trace diagrams to compute biquandle brackets of oriented knots and links.
result Identified algebraic conditions for strand moves and stop conditions.
A new framework describes dissipation using a metriplectic 4-bracket.
problem Describing dissipation in a way that preserves energy and entropy.
method Using a metriplectic 4-bracket, a quantity like the Poisson bracket with symmetries motivated by Riemannian curvature.
result The metriplectic 4-bracket dynamics includes all known previous binary bracket theories for dissipation.
New biquandle bracket invariants are linked to biquandle 2-cocycles.
problem Quantum enhancements and biquandle colored links.
method Proving biquandle bracket invariants are pointwise products of other invariants and biquandle 2-cocycles.
result New biquandle bracket invariants are equivalent to the Jones polynomial on knots.
New geometric definition of Lie bracket for undirected curves.
problem Understanding the Lie bracket of undirected curves on a surface.
method Local geometric definition and proof of three results.
result The TWG bracket counts intersection and suggests disjoint representatives.
New examples show non-trivial parity-biquandle bracket.
problem Constructing non-trivial parity-biquandle bracket examples.
method Slightly changed notation and constructed examples of knots and links.
result Minimality theorem: graphs appear as link invariants.
Complete integrability proved for SR geodesic flow on S^7.
problem Complete integrability of subriemannian geodesic flow on S^7.
method Adapting a method by A. Thimm, constructing functionally independent first integrals.
result Complete integrability in the sense of Liouville proved for SR geodesic flow.
Abstract: Poisson bracket on shear coordinates relates to Fenchel-Nielsen bracket on gluing parameters.
problem Relationship between Fenchel-Nielsen coordinates and shear coordinates on Riemann surfaces.
method Explicitly showed the Poisson bracket on shear coordinates induces the Fenchel-Nielsen bracket on gluing parameters.
result Poisson bracket on shear coordinates relates to Fenchel-Nielsen bracket on gluing parameters.
Kishino's knot is not detected by the fundamental group or the bracket polynomial; these invariants cannot differentiate between Kishino's knot and the unknot. However, we can show that Kishino's knot is not equivalent to unknot by applying either the 3-strand bracket polynomial or the surface bracket polynomial. In th…
Enhances quantum invariants using tribracket brackets.
problem Quantum invariants of tribracket-colored knots and links.
method Introduces tribracket brackets as skein invariants.
result Provides new quantum invariants and examples.