Lagrangian Topology Research Articles

A Lagrangian pictionary

We describe a dictionary of geometry ⟷ algebra in Lagrangian topology. As a by-product, we obtain a tautological (in a particular sense which we explain) proof of a folklore conjecture (sometimes attributed to Kontsevich) claiming that the objects and structure of the derived Fukaya category can be represented through immersed Lagrangians. Our construction is based on certain Lagrangian cobordism categories endowed with a structure called surgery models.

Finite-time circulation changes from topological rearrangement of distinguished curves and non-advective fluxes

A vorticity budget diagnoses the growth or decay of a vortex from advective transport, or non-advective fluxes such as those due to friction or vortex tilting. However, when a budget calculation is performed with respect to a fixed coordinate, errors may result from time-dependence of the flow, leading to disagreement between the vorticity tendency and the observed vorticity field. An adaptive Lagrangian coordinate resolves this problem, provided that the resulting Lagrangian structure does not become too complicated.In this study, a numerical simulation of Hurricane Nate (2011), the vorticity tendency is evaluated along distinguished material curves. There can be no net advective flux along a closed material curve, therefore, the total circulation tendency for a material region includes only the non-advective uxes acting along its boundary. A distinguished set of material curves (DMCs) associated with a distinguished hyperbolic trajectory (DHT) form a Lagrangian topology similar to that of a cat’s eye flow or “pouch” at each Eulerian snapshot. The time-dependence of velocities allows additional regions called lobes, which are formed by the intersections of DMCs, to exchange fluid across the vortex boundary by redefining the boundary.Because the vortex boundary changes, we refer to this redefinition of material boundary as “topological rearrangement”. The method is useful for unsteady cat’s-eye flows and more complex interactions of multiple waves, vortices and background shear. All advective changes of the vortex circulation are identified by exchanges of the lobes, and all non-advective uxes act between the vortex and either the lobes or environmental flow. The Lagrangian topology and combination of advective and non-advective uxes relative to the topology is used to describe the evolution of the circulation of Nate during its time of formation.

Power Systems Topology and State Estimation by Graph Blind Source Separation

In this paper, we consider the problem of blind estimation of states and topology (BEST) in power systems. We use the linearized DC model of real power measurements with unknown voltage phases (i.e. states) and an unknown admittance matrix (i.e. topology) and show that the BEST problem can be formulated as a blind source separation (BSS) problem with a weighted Laplacian mixing matrix. We develop the constrained maximum likelihood (ML) estimator of the Laplacian matrix for this graph BSS (GBSS) problem with Gaussian-distributed states. The ML-BEST is shown to be only a function of the states second-order statistics. Since the topology recovery stage of the ML-BEST approach results in a high-complexity optimization problem, we propose two low-complexity methods to implement it 1) Two- phase topology recovery, which is based on solving the relaxed convex optimization and then finding the closest Laplacian matrix, and 2) Augmented Lagrangian topology recovery. We derive a closed-form expression for the associated Cramer-Rao bound (CRB) on the topology matrix estimation. The performance of the proposed methods is evaluated for the IEEE-14 bus test-case system and for a random network. It is shown that, asymptotically, the state estimation performance of the proposed ML-BEST methods coincides with the oracles minimum mean-squared-error (MSE) state estimator, and the MSE of the topology estimation achieves the proposed CRB.

Bifurcations and degenerate periodic points in a three dimensional chaotic fluid flow.

Analysis of the periodic points of a conservative periodic dynamical system uncovers the basic kinematic structure of the transport dynamics and identifies regions of local stability or chaos. While elliptic and hyperbolic points typically govern such behaviour in 3D systems, degenerate (parabolic) points also play an important role. These points represent a bifurcation in local stability and Lagrangian topology. In this study, we consider the ramifications of the two types of degenerate periodic points that occur in a model 3D fluid flow. (1) Period-tripling bifurcations occur when the local rotation angle associated with elliptic points is reversed, creating a reversal in the orientation of associated Lagrangian structures. Even though a single unstable point is created, the bifurcation in local stability has a large influence on local transport and the global arrangement of manifolds as the unstable degenerate point has three stable and three unstable directions, similar to hyperbolic points, and occurs at the intersection of three hyperbolic periodic lines. The presence of period-tripling bifurcation points indicates regions of both chaos and confinement, with the extent of each depending on the nature of the associated manifold intersections. (2) The second type of bifurcation occurs when periodic lines become tangent to local or global invariant surfaces. This bifurcation creates both saddle-centre bifurcations which can create both chaotic and stable regions, and period-doubling bifurcations which are a common route to chaos in 2D systems. We provide conditions for the occurrence of these tangent bifurcations in 3D conservative systems, as well as constraints on the possible types of tangent bifurcation that can occur based on topological considerations.

Lagrangian topology and enumerative geometry

We use the "pearl" machinery in our previous work to study certain enumerative invariants associated to monotone Lagrangian submanifolds.

Scalar dispersion in a periodically reoriented potential flow: Acceleration via Lagrangian chaos

Although potential flows are irrotational, Lagrangian chaos can occur when these are unsteady, with rapid global mixing observed upon flow parameter optimization. What is unknown is whether Lagrangian chaos in potential flows results in accelerated scalar dispersion, to what magnitude, how robustly, and via what mechanisms. We consider scalar dispersion in a model unsteady potential flow, the Lagrangian topology of which is well understood. The asymptotic scalar dispersion rate q and corresponding scalar distribution (strange eigenmode) are calculated over the flow parameter space Q for Peclét numbers Pe=10{1}-10{4}. The richness of solutions over Q increases with Pe, with pattern mode locking, symmetry breaking transitions to chaos and fractally distributed maxima observed. Such behavior suggests detailed global resolution of Q is necessary for robust optimization, however localization of local optima to bifurcations between periodic and subharmonic eigenmodes suggests novel efficient means of optimization. Acceleration rates of 150 fold at Pe=10{4} are observed; significantly greater than corresponding values for chaotic Stokes flows, suggesting significant scope for dispersion acceleration in potential flows in general.

Lagrangian topology of a periodically reoriented potential flow: Symmetry, optimization, and mixing

Scalar transport in closed potential flows is investigated for the specific case of a periodically reoriented dipole flow. Despite the irrotational nature of the flow, the periodic reorientations effectively create heteroclinic and/or homoclinic points arising from the joining of stable and unstable manifolds. For scalar advection, Lagrangian chaos can be achieved with breakdown of the regular Hamiltonian structure, which is governed by symmetry conditions imposed by the dipole flow. Instability envelopes associated with period-doubling bifurcations of fixed points govern which regions of the flow control parameter space admit global chaos. These regions are further refined via calculation of Lyapunov exponents. These results suggest significant scalar transport enhancement is possible within potential flows, given appropriate programming of stirring protocols.

Two applications of prequantization in Lagrangian topology

The main theorem characterizes the Lagrangian homology classes of a compact symplectic 2n-manifold with integral symplectic form w. An integral homology n-class a is Lagrangian (i.e., can be represented by a Lagrangian n-cycle) if and only if α ∩ [w] = 0.