Schedule for: 21w5505 - Moving Frames and their Modern Applications

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

This talk describes difference moving frames, which are discrete moving frames that incorporate the natural prolongation structure generated by the group of translations on $\mathbb{Z}^N$. They can be modified to cope with finite domains. Difference moving frames produce group-invariant reductions of partial difference equations. In particular, they yield invariant formulations of Euler-Lagrange difference equations and an equivariant version of Noether's Theorem. We discuss these, with application to a Toda-type system that stems from the cross-ratio and discrete potential KdV equations.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Curves in Euclidean space enjoy a natural action of the orthogonal group on its ambient space. We apply the Fels-Olver moving frame method paired with the log-signature transform to construct a set of integral invariants for curves in $\mathbb{R}^d$ under rigid motions and to compare curves up to rigid motions (and tree-like extensions). In particular we show that one can construct a set of invariants that characterize the equivalence class of the truncated iterated-integrals signature under orthogonal transformations.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk I will summarize work with Anna Calini and Jing-Ping Wang on the discretization of $W_n$ algebras. I will then introduce the discrete analogue of the algebra of differential and pseudo-differential operators, and I will show that two natural Poisson brackets defined on this algebra coincides with the brackets that were used to integrate discrete systems associated to the $W_n$ algebras. This is ongoing work with Anton Isozimov.

Given is an (unknown) point configuration in ${\mathbb R}^d$. We obtain noisy measurements of the points, and would like to characterize the shape of the point configuration. More specifically, let $\rho(x)$ be a probability density function from which the noisy point measurements are obtained. We would like to characterize the orbit $\{ \rho(g\cdot x) | g\in E(d) \} $, where $E(d)$ denotes the Euclidean group acting on ${\mathbb R}^d$. I will consider the case where $\rho(x)$ is a mixture of Gaussians, rewrite the problem as an algebraic question, and provide a solution.

Semi-discrete equations not only can be semi-discretisations of partial differential equa- tions or semi-continuum limits of partial difference equations, but also arise as mechanical and physical systems themselves, e.g., the Toda lattice and interconnected systems in mechanics. Symmetries are fundamentally important properties that help us to under- stand the solvability/integrability of equations. In this talk, we will introduce a general treatment for computing continuous symmetries of semi-discrete equations through the linearized symmetry condition and extend Noether’s two theorems. Worked examples will be provided. This is joint work with Peter Hydon (University of Kent).

Innovations in pedagogy and placement can increase the equity of STEM instruction, but identification of a robust portfolio of outcome measurements that are easily interpreted by stakeholders can be challenging. Elementary geometry can facilitate communication between analysts and administrators without suppression of potentially crucial information for the sake of simplicity. Moving frames provide a versatile, powerful tool for decomposing multidimensional outcome data into full cohort trends and deviations of outcomes for sub-cohorts of interest from those trends. If gains in one measure are accompanied by losses in other measures (e.g. average course grades in the first STEM course following a preparatory math course increase because all but the highest scoring students in the prep course immediately leave STEM), characterizing those changes using linear transformations of outcome vectors can potentially reveal patterns that are difficult to recognize in tables of scalar data.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Lax representations of nonlinear PDEs are widely recognized as the key feature of integrable systems. Different phenomena thereof, such as bi-Hamiltonian structures, recursion operators, nonlocal symmetries, etc., can be derived from the Lax representations.  Therefore the problem to determine whether a given PDE admits a Lax representation is of great importance.  In this talk, I will discuss how the structure theory of Lie pseudogroups in combination with the theory of infinite-dimensional Lie algebras can be applied to tackle this problem. Considerations will be based on the moving frame technique.

We give an explicit effective construction for a large new class of partial differential systems in four independent variables that are integrable in the sense of soliton theory, thus showing inter alia that there is significantly more of such systems than it appeared before. This is achieved by employing contact vector fields in dimension three in the construction of associated Lax pairs; please see A. Sergyeyev, Lett. Math. Phys. 108 (2018), no. 2, 359-376 (arXiv:1401.2122) for further details.

In a 2009 paper, Anderson, Fels and Vassiliou showed that, for a class of Darboux-integrable (DI) hyperbolic systems, a canonical integrable extension exists which is constructed using the action of the Vessiot group, and which splits as the product of two simpler differential systems. Moreover, each solution of the DI system arises as a `superposition' of a pair of solutions to the simpler systems. In this preliminary report on joint work with Mark Fels, we outline a conjectural picture for the construction of a canonical integrable extension for elliptic DI systems. In general, the extension does not split, but in several examples the extension is contact-equivalent to a prolongation of the Cauchy-Riemann equations, leading to solution formulas in terms of holomorphic functions. If time permits, we will discuss an application of these ideas to the isometric embedding problem for certain surfaces of revolution.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Equations of Lie type are fundamental in the theory of moving frames and these equations have an interesting and rich history. Equations of Lie type appear in the equations for the reconstruction problem for curves with prescribed differential invariants. By studying equations of Lie type, E. Vessiot discovered a generalization of these equations which remarkably turn out to underlie the integration process for partial differential equations that can be integrated by the method of Darboux. I will explain this relationship and demonstrate it with examples.

We will review the main applications of the method of moving frames to the theory of orthogonal separation of variables in pseudo-Riemannian spaces of constant curvature. In this context, the method of moving frames arises as an indispensable tool in the study of algebraic and geometric properties of Killing tensors (symmetry operators)  that determine the orthogonal separation of variables in problems of classical (quantum) mechanics. 

Generic rank 2 distributions on 5-manifolds, i.e. (2,3,5)-distributions, are interesting geometric structures arising in the study of non-holonomic kinematical systems (e.g. two 2-spheres rolling on each other without twisting or slipping), underdetermined ODE of Monge type, conformal 5-manifolds with special holonomy, etc. The origins of their study date to Élie Cartan's "5-variables" paper of 1910, where he gave a tour-de-force application of his method of equivalence. In particular, he obtained a canonical coframing, discovered a fundamental (tensorial) curvature quantity (the Cartan quartic), and gave a (almost complete) local classification of structures that are multiply-transitive, i.e. (locally) homogeneous with isotropy of positive dimension. In my talk, I'll revisit this homogeneous classification and present it from a modern Cartan-geometric perspective.

First introduced as orthonormal basis vectors in the numerical solution of PDEs on curved surfaces, moving frame algorithms have been proved competitively accurate and stable for various PDEs, particularly with high-order discretization schemes. The PDEs include conservational laws, diffusion equations, shallow water equations, and Maxwell’s equations. High-order discretization schemes mean continuous/discontinuous Galerkin method or spectral/hp methods. The most striking feature of moving frames is that moving frames simplifies the type of medium in PDEs. A simple representation of anisotropy by the adjusted length of the frames in diffusion equations or a general representation of rotation surfaces by moving frames provides significant advantages in numerical algorithms. Beyond the spatial representation of complex domain, moving frames aligned along with wave propagation yields connection and Riemann curvature tensor to help to identify and predict the flow pattern. One application of such an algorithm is to analyze the cardiac electric flow where a large amount of a specific component of the Riemann curvature tensor implies conduction block and consequently the possibility of reentry and fibrillation. Another application is to construct a numerical algorithm to simulate neural spike propagation along with a spreading neural fiber bundle in the brain’s white matter to reveal the geometric structure of the brain connectivity. The most recent research also applies moving frames to a field of ‘time’ to achieve the spacetime analysis of time-dependent propagation in the heart and brain. All these moving frames applications demonstrate the beautiful simplicity of moving frames in the complex propagation phenomena in complex domains. However, a question still hangs in the air about its restrictions and real-world interpretation of connection and curvature.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk I wish to review variations on the constructions of rational invariants and the key role of sections therein. The moving frame  by Fels & Olver (1999) provided a method to compute local invariants. In practice it relies on 1/ making explicit the solution of the application of the implicit function theorem 2/ fighting through symbolic expressions involving radicals. For a fully algorithmic approach [Hubert Kogan FoCM 2007] recasted the problem in algebraic terms and offered a construction of local invariant as algebraic functions given by the Gröbner basis of their defining ideal. On the way we proved that we could also compute a generating set of rational invariants [Hubert Kogan JSC 2007], which are global invariants. The general construction gave the intuition to refined constructions for specific group actions, relevant in applications and offering further connections.  Such is the case of scalings, with a new take on the Buckingham-Pi theorem,  [Hubert Labahn FoCM 2013] and the action of the orthogonal group on homogeneous polynomials in in 3D [Görlach Hubert Papadopoulo FoCM 2019] with a computational take on the  slices  of Seshadri (1962).

Underpinning the equivariant moving frame is a basic theorem on congruence of submanifolds in Lie groups. In this talk we present a recent generalization of this theorem to Lie pseudo-groups and the perspective it provides on the equivariant moving frame for Lie pseudo-groups. From this new point of view Cartan's equivalence method emerges naturally.

We compute generators for the algebra of rational scalar differential invariants of general and degenerate Kundt spacetimes. Special attention is given to dimensions 3 and 4 since in those dimensions the degenerate Kundt metrics are known to be exactly the Lorentzian metrics that can not be distinguished by polynomial curvature invariants constructed from the Riemann tensor and its covariant derivatives. The talk is based on joint work with Boris Kruglikov.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

The Gromov-Wasserstein (GW) distance is a generalization of the standard Wasserstein distance between two probability measures on a given ambient metric space. The GW distance assumes that these two probability measures might live on different ambient metric spaces and therefore implements an actual comparison of pairs of metric measure spaces. A metric-measure space is a triple (X,dX,muX) where (X,dX) is a metric space and muX is a Borel probability measure over X. In practical applications, this distance is estimated either directly via gradient based optimization approaches, or through the computation of lower bounds which arise from distributional invariants of metric-measure spaces. One particular such invariant is the so-called ‘global distance distribution’ which precisely encodes the distribution of pairwise distances between points in a given metric measure space. This invariant has been used in many applications yet its classificatory power is not yet well understood. This talk will overview the construction of the GW distance, the stability of distributional invariants, and will also discuss some results regarding the injectivity of the global distribution of distances for smooth planar curves, hypersurfaces, and metric trees.  Part of this work is joint with Facundo Memoli.

In general relativity, the invariant classification of spacetimes is typically formulated in terms of a pseudo-algorithm proposed by Anders Karlhede in 1980. At first glance, this algorithm and its subsequent refinements do not bear much resemblence to Cartan's method for the equivalence of G-structures. Indeed, even if one limits the scope of the equivalence method to that of Riemannian geometries, it is difficult to perceive the relation between the two approaches. To wit, Karlhede's algorithm does not make use of the bundle of orthogonal frames and relies instead on iterated normalizations of the curvature tensor. My aim will be to explain the relativity approach to an audience familiar with the Cartan formalism and to highlight some computational advantages of this way of doing equivalence problems.

Relative invariants help to understand singularities of group actions on manifolds. Their weights are cocycles modulo coboundaries and thus correspond to the first Gelfand-Fuks cohomology. Relative differential invariants correspond to prolongation of the action to jet-spaces, and are important in the equivalence problem. I will discuss the weight lattice and the finiteness theorem for relative differential invariants.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The field of rational algebraic and differential SO(3)-invariants of spherical harmonics were described and were used for the description of regular SO(3)-orbits of spherical harmonics in an algebraic and differential setting.

Let $\Q_3$ be the $3$-dimensional complex quadric equipped with its holomorphic conformal structure. We use the method of moving frame to study conformal geometry of isotropic holomorphic curves in $\Q_3$ and their interrelations with relevant classes of surfaces in Riemannian and Lorentzian space-forms. This is a joint work with Lorenzo Nicolodi.

There has been active work towards definition of super cluster algebras (Ovsienko, Ovsienko-Shapiro, and Li-Mixco-Ransingh-Srivastava), but the notion is still a mystery. In the talk, we present our construction of “super Pluecker embedding” for Grassmannian of $r|s$-planes in $n|m$-space. (Only a very special case was considered before in the literature, namely, of $2|0$-planes in $4|1$-space, by Cervantes-Fioresi-Lledo.) The straightforward algebraic construction of exterior powers goes through for the Grassmannian $G_{r|0}(n|m)$, i.e. completely even planes in the superspace. For the general case of $r|s$-planes, a more complicated construction is needed. Our super Pluecker map takes the Grassmann supermanifold $G_{r|s}(V)$ to a “weighted projective space” $P_{1,-1}(\Lambda^{r|s}(V)\oplus \Lambda^{s|rs}(\Pi V))$, with weights $+1, −1$. Here $\Lambda^{r|s}(V)$ denotes the $(r|s)$th exterior power of a superspace $V$ and $\Pi$ is the parity reversion functor. We identify the super analog of Pluecker coordinates and show that our map is an embedding. We investigate the super analog of the Pluecker relations. We obtain them for arbitrary $r|s$ and $n|m$. The case $r|0$ is relevant for conjectural super cluster algebras. Also, we consider another type of relations suggested by H. Khudaverdian and show that they are equivalent to (super) Pluecker relations for $r|s = 2|0$ (this is new even in the classical case), but in general are only a consequence of the Pluecker relations. (Based on a joint work with Th. Voronov.)

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The classical, differential topological theory of singularities of differential equations is mainly concerned with scalar ordinary differential equations of first or second order with an emphasis on classifications and normal forms. We present an extension of the basic definitions to arbitrary systems of ordinary or partial differential equations based on Vessiot theory and some of the arising open problems. We also relate these results with the notion of a "regular differential equation" - a standard assumption in the geometric theory of differential equations which is rarely made rigorous. If time permits, we will also discuss the question of how the theory can be effective, i.e. translated into algebraic algorithms for detecting singularities and for analysing the local solution behaviour. (Much of the talk is based on the recent article Lange-Hegermann, Robertz, Seiler, Seiss: Singularities of Algebraic Differential Equations, Adv. Appl. Math. 131 (2021) 102266.)

We discuss the application of the method of moving frames to computing generalized Casimir operators of Lie algebras, i.e., invariants of the coadjoint representations of such algebras. We also review results on using the obtained purely algebraic algorithm for finding generalized Casimir operators of low-dimensional Lie algebras and series of solvable Lie algebras with specific structure of their nilradicals, in particular, of the Lie algebras of triangular and strictly triangular matrices of an arbitrary fixed dimension.

We consider the problem of finding a complete set of invariants for the product action of a Lie group $G$ on multiple copies of a homogeneous space $G/H$, where $H$ is a closed Lie subgroup of $G$ and the action is primitive. In the particular the case when $G$ is not simple and the primitive actions have been classified by Golubitsky. We will present a reduction theorem that simplifies the problem of finding invariants and apply it to finding two point invariants in $SU(2)$, and $SL(2)$.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11am