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Recent seminars in Analysis and PDE

List of the most recent semesters of seminars

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2022 (Fall)

August

  • August 25, 2022: Louis Emerald, Rennes
    • Title: On the rigorous justification of full dispersion models in coastal oceanography
    • Abstract: The first full dispersion model was introduced by G. Whitham in 1967 to study the Stokes waves of maximal amplitude and the wavebreaking phenomenon. It is a modification of the Korteweg-de Vries equations which have the same dispersion relation as the general water-waves model. Afterward, numerous unidirectional and bidirectional full dispersion models were introduced in the literature. In the first part of the talk, we use classical techniques on free surface elliptic equations to derive rigorously the Whitham-Boussinesq systems. In the second part, we justify rigorously Whitham's model using two different methods. One is adapted to the propagation of unidirectional waves and uses pseudo-differential calculus. The other one is adapted to the propagation of bidirectional waves. It is based on a generalisation of Birkhoff's normal form algorithm.

 

October

  • October 6, 2022: Gianmarco Vega-Molino, Bergen
    • Title: Local Invariants and Geometry of the sub-Laplacian on H-type Foliations
    • Abstract: H-type foliations are studied in the framework of sub-Riemannian geometry with bracket generating distribution defined as the bundle transversal to the fibers. Equipping the manifold with the Bott connection we consider the scalar horizontal curvature as well as a new local invariant induced from the vertical distribution. We extend recent results on the small-time asymptotics of the sub-Riemannanian heat kernel on quaternion-contact (qc-)manifolds due to A. Laaroussi and we express the second heat invariant in sub-Riemannian geometry as a linear combination of the mentioned invariant. The use of an analog to normal coordinates in Riemannian geometry that are well-adapted to the geometric structure of H-type foliations allows us to consider the pull-back of Korányi balls. We explicitly obtain the first three terms in the asymptotic expansion of their Popp volume for small radii. Finally, we address the question of when our manifold is locally isometric as a sub-Riemannian manifold to its H-type tangent group.This talk is based on a joint work with Irina Markina (Bergen), Wolfram Bauer (Hannover) and Abdellah Laaroussi (Hannover)
  • October 13, 2022: Didier Pilod, Bergen
    • Title: Unique continuation properties for the fractional Laplacian
    • Abstract: In the first part of the talk, I will introduce the unique continuation property, and describe a classical method of proof based on Carleman estimates. 

      In the second part of the talk, I will review some unique continuation properties for the fractional Laplacian obtained by Rüland (CPDE 2015) and Ghosh, Salo, Uhlmann (Analysis&PDE 2020).  These properties rely on the Cafarelli-Silvestre extension, allowing to describe the fractional Laplacian through a singular elliptic problem in the upper-half space, and a Carleman estimate for the extension problem obtained by Rüland. 

 

November

  • November 10, 2022: Gunhee Cho, UC Santa Barbra

    • Title: Stochastic differential geometry and applications of probabilistic coupling methods

    • Abstract: The first part discusses how stochastic differential geometry, especially coupling methods, are used in relation to the long-standing open problems of hyperbolic complex geometry, and the recent results including the stochastic Schwarz Lemma and applications.The second part examines further why the Kendall-Cranston probabilistic coupling method is potentially important and useful in studying differential geometry, and discusses the first Dirichlet eigenvalue estimate on K\"ahler manifolds, which is also a very recent result. The first part is relevant to the joint work with M. Gordina, M. Chae, and G. Yang. Also, the second part is related to the joint work with F. Baudoin, and G. Yang.

  • November 10, 2022: Kenro Furutani, Osaka Metropolitan University, Osaka Central Advanced Mathematical Institute
    • Title: Calabi-Yau structure and Bargmann type transformation on the Cayley projective plane
  • November 17, 2022: Adrien Laurent, Bergen
    • Title: The aromatic bicomplex for the study of volume-preserving integrators
    • Abstract: Aromatic B-series were introduced as an extension of standard Butcher-series for the study of volume-preserving integrators. It was proven with their help that no B-series method can preserve volume in general, except for the exact solution of the differential equation. The question was raised whether there exists a volume-preserving integrator that can be expanded as an aromatic B-series. In this talk, we provide a complete characterization of the divergence-free aromatic forests by introducing a similar object to the variational bicomplex from differential geometry in the context of aromatic forests. The analysis provides a handful of new tools to study aromatic B-series, gives insights on the process of integration by parts of trees, and allows to describe explicitly the aromatic B-series of a volume-preserving integrator.

 

December

  • December 8, 2022: Giacomo Gavelli, Bologna
    • Title: Various notions of hyperbolicity and their relationship
    • Abstract: The aim of the talk is to show a comparison between three different notions of hyperbolic spaces: Riemannian manifolds of constant negative curvature, Gromov-hyperbolic metric spaces and Kobayashi-hyperbolic complex spaces. 

      In the first part of the talk I will introduce three classical hyperbolic spaces as Riemannian manifolds, which result to be isometric: the hyperboloid, the Poincaré ball and the Poincaré half-space. On these I will introduce the notions of hyperbolic segments, angles, triangles, geodesics, circles and isometries.

      In the second part I will introduce Gromov and Kobayashi-hyperbolicity. According to Gromov, hyperbolic spaces are characterized by having, in a sense, slim-triangles. The classical hyperbolic spaces have this property. Kobayashi, in 1967, defined a natural semi-distance on any complex space, which is said to be Kobayashi-hyperbolic when such semi-distance is a distance. I  will state a result, proven by Bonk and Balogh in 2000, which grants that bounded strictly pseudoconvex domains with C2-smooth boundary, when equipped with the Kobayashi distance, are Gromov hyperbolic.
       

2022 (Spring)

February

  • February 1, 2022: Erlend Grong, University of Bergen,
    • Title: The Gauss-Bonnet theorem in sub-Riemannian geometry
    • Abstract: The Gauss-Bonnet theorem connects a global invariant, the Euler characteristic, with a local invariants, the Gaussian curvature of the surface and the geodeisc curvature of the boundary. It also explains why we can find the curvature of a space by checking if the angles of a triangle add up to more or less than 180 degrees. We will give a general, low-level introduction to this result.Next, we are going to look at what happens when the inner product on the ambient space goes to infinity outside a given subbundle. Surprisingly, this shows that the Euler characteristic is only dependent on local invariants near to what is known as the characteristic set.
  • February 8, 2022: Ilia Zlotnikov, University of Stavanger (Zoom)
    • Title: On extreme points of the unit balls of the spaces of analytic function.
  • February 15, 2022: Irina Markina, University of Bergen
    • Title: On the module of measure.
    • Abstract: In the talk, I want to explain one tool used in the geometric measure theory. It originated in the theory of functions of complex variables to show completeness of some functional classes and it got the name extremal lengths or module of a family of curves. It was extended to a notion in n-dimensional Euclidean spaces with generalization to the module of measure supported on k-dimensional Lipschitz manifolds.For some specific situations, the module of curves coincides with the capacity widely used to describe the exceptional sets in the Sobolev spaces. If time allows, I will touch on the recent results related to the study of modulus in some non-Euclidean geometric measure theory.
  • February 22, 2022: Arnaud Eychenne, University of Bergen
    • Title: Asymptotic N-soliton-like solutions of the fractional Korteweg-de Vries equation
    • Abstract: In 1834, John Scott Russell discovered a soliton, which is a wave moving without deformation. Since this discovery, solitons have been studied extensively. Nowadays, solitons are natural objects in physics or biology. We will talk about the existence of N-soliton-like solutions, which are solutions that behave at infinity like a sum of N decoupled solitons, for the fractional Korteweg-de Vries equation (fKdV) $\partial_t u +\partial_x (|D|^{\alpha}u - u^2)$. The first proof of the existence of N-soliton-like solutions that is not based on the complete integrability was done around 20 years ago for local generalizations of the KdV equation. In this talk, we will explain briefly the construction of the N-soliton-like solutions and the difficulties caused by the non-locality of the operator $|D|^{\alpha}$ that we had to overcome in the case of fKdV. 

 

March

  • March 1, 2022: Yuri Nikonorov, Russian Academy of Sciences (Zoom)
    • Title: Geometry of Riemannian Homogeneous spaces, Lecture 1: Homogeneous Riemannian manifolds and their description
  • March 8, 2022: Didier Pilod, University of Bergen
    • Title: Finite point blowup for the critical generalized Korteweg-de Vries equation.
    • Abstract: In the last twenty years, there have been significant advances in the study of the blow-up phenomenon for the critical generalized Korteweg-de Vries (gKdV) equation, including the determination of sufficient conditions for blowup, the stability of blowup in a refined topology and the classification of minimal mass blowup. Exotic blow-up solutions with a continuum of blow-up rates and multi-point blow-up solutions were also constructed. However, all these results, as well as numerical simulations, involve the bubbling of a solitary wave going at infinity at the blow-up time, which means that the blow-up dynamics and the residue are eventually uncoupled. Even at the formal level, there was no indication whether blowup at a finite point could occur for this equation.After reviewing the theory of blow-up for the critical gKdV equation in the first part of the talk, we will answer this question by constructing solutions that blow up in finite time under the form of a single-bubble concentrating the ground state at a finite point with an unforeseen blow-up rate.Finding a blow-up rate intermediate between the self-similar rate and other rates previously known also reopens the question of which blow-up rates are actually possible for this equation.This talk is based on a joint work with Yvan Martel (École Polytechnique/France).

  • March 15, 2022: Yuri Nikonorov, Russian Academy of Sciences (Zoom)
    • Title: Geometry of Riemannian Homogeneous spaces, Lecture 2: Some special classes of Riemannian homogeneous spaces
  • March 22, 2022: Douglas Svensson Seth, NTNU
    • Title: The three-dimensional steady water wave problem with vorticity
    • Abstract: The water wave problem concerns solving a free boundary problem. Specifically, the equations of motion for a fluid in a two- or three-dimensional domain where the shape of the upper boundary is unknown. The problem becomes steady through the assumption that the waves travel with uniform and constant speed. The two-dimensional theory is generally more developed than the three-dimensional due to being the older of the two. Already in the mid 1800s Stokes had begun investigating the two-dimensional problem and the first rigorous existence results are due to Nekrasov and Levi-Cività (independently) in the 1920s. On the other hand, the first corresponding rigorous existence result in three dimensions took until 1981 and is due to Reeder and Shinbrot.The first part of the talk will be dedicated to a brief overview of the water wave problem in both two and three dimensions. Here we also highlight some of the differences between the two- and three-dimensional problems. The second part of the talk will be dedicated to two more recent existence results for the three-dimensional problem where the vorticity (curl of the velocity) is nonzero. In the first we assume that the velocity field of the water is a Beltrami field. In the other, the vorticity is instead given by an assumption that stems from magnetohydrodynamics. This talk is based on joint work with Erik Wahlén (Lund University), Evgeniy Lokharu (Lund University) and Kristoffer Varholm (NTNU).

  • March 29, 2022: Yuri Nikonorov, Russian Academy of Sciences (Zoom)
    • Title: Geometry of Riemannian Homogeneous spaces, Lecture 3: Geodesic orbit Riemannian spaces and some their subclasses

 

April

  • April 4, 2022: Yuri Nikonorov, Russian Academy of Sciences (Zoom)
    • Title: Geometry of Riemannian Homogeneous spaces, Lecture 4: The Ricci curvature and related problems
  • Date and place: Tuesday April 12: Easter break
  • April 19, 2022: Gianmarco Vega-Molino, University of Bergen
    • Title: Morse Theory, Studying Geometry and Topology through Critical Points
    • Abstract: Elementary results in the calculus of functions of the real line show that we can understand the geometry of curves through the study of zeros of derivatives.  Extrema of differentiable functions always occur at these critical points, and the nature of the critical points can be understood by considering the behavior of the second derivative.   These notions are extended to multivariate functions by analogously considering the gradient and Hessian.  In the 1920s Marston Morse initiated the study of surfaces (and more generally manifolds) by considering the critical points of functions on them, from which it is possible to determine both local and global properties; in particular, one can recover topological information such as the Euler characteristic.  In this talk, I will present an introduction to Morse theory.  Familiarity with differential geometry (in particular, smooth manifolds) is not a prerequisite.
  • April 26, 2022: Fátima Silva Leite,  Coimbra - Portugal
    • Title: Interpolation on Riemannian Manifolds
    • Abstract: The problem of finding a smooth curve that interpolates a set of points on a Riemannian manifold, and satisfies some boundary conditions, is particularly important in applied areas such as robotics, computer vision and medical imaging.

      In this seminar we start with some motivating examples and then will revisit several methods to generate interpolating Riemannian splines. Our main focus will be on a variational approach to generate splines that minimize the intrinsic acceleration, and on a method based on rolling motions that emerged to overcome difficulties in finding explicit solutions for the Euler-Lagrange equation of the previous optimization problem.

 

May

  • May 3, 2022: Felipe Linares, IMPA Brazil
    • Title: The Cauchy problem for the L2−critical generalized Zakharov-Kuznetsov equation in dimension 3.
  • May 10, 2022: Martin Oen Paulsen, University of Bergen
    • Title: Justification of a full dispersion model from the water wave system​
    • Abstract: A fundamental question in the derivation of an asymptotic model is whether its solution converges to the solution of the original physical system. In particular, we say that an asymptotic model is a valid approximation of the Euler equations with a free surface if we can answer the following points in the affirmative:rThe solutions of the water wave equations exist on the relevant scale.The solutions of the asymptotic model exist (at least) on the same time scale.Lastly, we must establish the consistency between the asymptotic model and the water wave equations and show that the error is "small" when comparing the two solutions.In the first part of the talk, the aim is to introduce the governing equations in water wave theory. Then discuss the three points above and explain the rigorous justification of some famous asymptotic models. The second part of the talk will focus on 2., explaining a new 'long time' well-posedness result for a Whitham-Boussinesq system.
  • May 17, 2022: National Holiday
  • May 24: Francesco Ballerin, University of Bergen,
    • Tile: Sub-Riemannian Geometry and its applications to Image Processing
    • Abstract: A 2D image is perceived by the human brain through the visual cortex V1, a part of the occipital lobe which is sensitive to orientation. This sensitivity intrinsically fills-in gaps in the perceived image depending on the gradient of the image in a neighborhood, restoring corrupted portions of the field of view. The visual cortex V1 can be mathematically modeled as SE(2), a sub-Riemannian geometry which can be exploited to produce image restoration algorithms. In this talk the current state-of-the-art regarding image restoration models based on SE(2) is presented and a proposed new algorithm is introduced.
  • May 31: No seminar. Abel prize lecture by Lázió Lovász.

 

June

  • June 20: René Langøen, University of Bergen
    • Title: The direct monodromy problem and isomonodromic deformations for the Rabi model.
    • Abstract: We discuss the local and global solutions of the Rabi model in Garnier form, a linear system of first order differential equations, with complex rational coefficients. The analytic continuation of the local solutions are described by a monodromy group, which gives a matrix representation of the fundamental group of the punctured Riemann sphere. A detailed geometric description of linear systems of first order differential equations is given, in terms of a local family of connection forms on a principal bundle. The geometric description reveals the Frobenius integrability conditions, which are used to obtain necessary and sufficient conditions for an isomonodromic deformation of the Rabi model.

 

 

2021 (Fall)

Date and place: Wednesday November 17, 2021, Hjørnet and Zoom, 12:15.

Speaker: Jorge Hidalgo Calderon, University of Bergen

Title: The Geometric Maximum Principle and the Alexandrov Theorem

Abstract:

I will explain the geometric maximum principle and use it to prove one of the milestones in the global theory of constant mean curvature hypersurfaces embedded in the euclidean space: the celebrated Alexandrov Theorem. The technique employed in this proof, known in the literature as Alexandrov reflection method, has proved itself quite useful in several branches of mathematics, particularly in PDE's and in Differential Geometry.

 

 

Date and place: Wednesday November 10: No seminar

 

 

Date and place: Wednesday November 3, 2021, Hjørnet and Zoom, 12:15.

Speaker: Stein Andreas Bethuelsen, University of Bergen

Title: Random walks in dynamic random environment

Abstract:

The classical random walk model is a central object within mathematics that e.g. models the propagation of a particle through a medium. In this talk we will focus on certain extensions of the random walk model allowing for random irregularities in the medium. This theory is closely linked to what is known as stochastic homogenization theory.

In the first part of the talk we will review classical results about such random walks in random environment which reveal a rich phenomenology concerning their asymptotic behavior. At the same time we will see that some of the most fundamental questions remain mathematically unsolved.

In the second half we will focus on the case where the environment evolves dynamically with time. In this case, the mixing properties of the dynamics will play an important role, both concerning the phenomenology and the available mathematical theory. This latter part will partly be based on joint work with Florian Völlering (University of Leipzig).

 

 

Date and place: Wednesday October 27, 2021, Hjørnet and Zoom, 12:15.

Speaker: René Langøen, University of Bergen

Title: Complex structures on manifolds and holomorphic maps

Abstract:

I will explain complex holomorphic manifolds and give the definitions of holomorphic maps on them. We shall define the tangent space of a complex holomorphic manifold at a point p, as the vector space of C-linear derivations of holomorphic function germs at p. However, it is not immediately clear how to describe this vector space in more detail. We will thus follow a different approach, taking the real 2n dimensional vector space obtained from the smooth structure on the manifold, impose an almost complex structure on it and then complexify it. We will obtain several linear (real and complex) isomorphisms relating the different constructions.

 

Date and place: Wednesday October 20, 2021, Hjørnet and Zoom, 12:15.

Speaker: Jean-Claude Saut, Université Paris-Saclay

Title: Old and new on the intermediate long wave equation

Abstract:

The Intermediate Long Wave equation (ILW) is a classical asymptotic weakly nonlinear model of internal waves in stratified fluids. It also turns out to be completely integrable.We will first recall the rigorous derivation of the ILW and related models and survey the already known results on the Cauchy problem, mainly obtained by "PDE" techniques. On the other hand there is so far norigorous results on the Cauchy problem, even for small initial data, using Inverse Scatering techniques. In particular the soliton resolution, clearly shown by numerical simulations, is not yet proven.We will then present new results on the qualitative properties of solutions obtained in a joint work with Claudio Munoz and Gustavo Ponce.

 

 

Date and place: Wednesday October 13, 2021, Hjørnet and Zoom, 12:15.

Contact the seminar organizer for link.

Speaker: Irina Markina, University of Bergen

Title: H-type Lie algebras

Abstract:

I will describe 2-step nilpotent Lie algebras, closely related to the Clifford algebras. The H(eisenberg)-type Lie algebras, introduced by Aroldo Kaplan at 1980 for the study of hypoelliptic operators. I will try to explain the relation of H-type Lie algebras to the Clifford algebras and maybe to the composition of quadratic forms. For that, I will briefly revise the definition of Clifford algebras and their representations. I will introduce modules of Clifford algebras versus admissible modules.If time allows I will sketch such questions as the existence of rational structure constants, classification up to isomorphism, Atiyah-Bott periodicity inherited from the Clifford algebras.

 

Date and place: Wednesday October 6: No seminar

 

Date and place: Wednesday September 29, 2021, Hjørnet and Zoom, 12:15.

Contact the seminar organizer for link.

Speaker: Gianmarco Vega-Molino, University of Bergen

Title: H-type Foliations

Abstract:

We discuss H-type foliations, a topic jointly introduced with Fabrice Baudoin, Erlend Grong, and Luca Rizzi. Arising as the sub-Riemannian geometries transverse to Riemannian foliations they are ideally suited to study at the intersection of "extrinsic" and "intrinsic" approaches to sub-Riemannian geometry. We begin with a broad introduction to sub-Riemannian geometry suitable to non-experts and will cover several topics, including forthcoming work on the holonomy of these spaces.

 

Date and place: Wednesday September 22, 2021, Hjørnet and Zoom, 12:15.

Contact the seminar organizer for link.

Speaker: Erlend Grong, University of Bergen

Title: Statistics of manifolds and most probable paths

Abstract:

For a collection of data on a nonlinear space, we cannot use pluss to discuss things such as mean and covariance. In our seminar, we will discuss how to use stochastic processes to model normal distribiutions with a given covariance on a Riemannian manifold. For actual computation, a good tool to use are most probable path; the path a random time dependent variable is most likely to follow. These can be descibed using sub-Riemannian geometry. We will give some explicit formulas for such curves and applications. These results are from a joint work with Stefan Sommer.

 

Date and place: Wednesday September 15: No seminar.

 

Date and place: Wednesday September 8, 2021, Zoom, 15:15. Contact the seminar organizer for link.

Speaker: Vitali Vougalter

Title: On the solvability of some systems of integro-differential equations with anomalous diffusion in higher dimensions

Abstract:

The work deals with the studies of the existence of solutions of a system of integro-differential equations in the case of theanomalous diffusion with the negative Laplace operator in a fractional power in R^d, d=4,5. The proof of the existence of solutions is based on a fixed point technique. Solvability conditions for non Fredholm elliptic operators in unbounded domains are used.

 

Date and place: Wednesday September 1, 2021, Delta + Lunchroom and Zoom, 12:15. Contact the seminar organizer for link.Joint session with the Algebra seminar.

Speaker: Adrien Laurent

Title: Exotic aromatic B-series and multiscale methods for the integration of ergodic and stiff stochastic dynamics in R^d or on manifolds

Abstract:

After a brief summary of the standard results for the weak and the long-time numerical integration of stochastic problems, we propose a new multirevolution method for the weak integration of SDEs with a fast stochastic oscillation. Then, we present a new formalism of Butcher-series, called exotic aromatic B-series, for creating high order integrators for sampling the invariant measure of ergodic stochastic differential equations in R^d and on manifolds. In the particular case of overdamped Langevin dynamics, we obtain the order conditions for a class of Runge-Kutta methods, extend the results to postprocessors and partitioned problems in the context of R^d, and introduce an integrator of order two in the manifold case. We conclude with a new method whose accuracy remains the same in R^d and on a manifold for the integration of penalized Langevin dynamics.

Of course, depending on the knowledge of the audience and on the interactions during the talk, I can adapt my presentation.

 

2021 (Spring)

 

Date and place: May 10, 2021, Auditorium 3 and Zoom, 14:15.

Contact the seminar organizer for link.

Speaker: Hans Z. Munthe-Kaas, University of Bergen

Title: Lie–Butcher series for geodesic flows

Abstract:

We introduce series developments similar to Butcher’s B-series for geodesic flows on manifolds equipped with a general affine connection. This includes the Levi–Civita connection on Riemannian metric spaces as a special case. This novel theory paths new ways for analysing numerical integration schemes based on geodesic flows as well as the study of rough paths in affine geometries.

(joint work with Kurusch Ebrahimi-Fard and Dominique Manchon)

 

Date and place: May 3, 2021, Auditorium 3 and Zoom, 14:15.

Contact the seminar organizer for link.

Speaker: Eirik Berge, NTNU Trondheim

Title: The Feichtinger Algebra - Too Good to be True?

Abstract:

In this talk, I will motivate and explain the Feichtinger algebra. The algebra was invented in the '80s by Hans Georg Feichtinger and has, since the turn of the century, been a central player in time-frequency analysis. Interestingly, the Feichtinger algebra can be defined in a multitude of ways, emphasizing e.g. representation theory, geometric decompositions, or classical analysis. If time permits, I will talk briefly about my own research in the area towards the end of the talk. The goal of the talk is to convince you that the Feichtinger algebra is a beautiful piece of modern mathematics that should be more well-known. I've tried to make the talk accessible for master students. 

 

Date and place: April 19, 2021, Auditorium 3 and Zoom, 14:15.

Contact the seminar organizer for link.

Speaker: Frédéric Valet, University of Bergen

Title: Growth of Sobolev norms for solutions of the Zakharov-Kuznetsov equation in 2D

Abstract:

This is a joint work with Raphaël Côte. For a non dispersive equation like the wave equation, a wave packet moves at a fixed velocity. For a dispersive equation like the Zakharov-Kuznetsov equation (ZK) in 2D, a wave packet can be influenced by the dispersive effects, which means that the energy moves from a low wavenumber to a higher wavenumber along the time. This displacement of frequencies is known in physics like the energy cascade phenomenon. In other words, the influence of the dispersive effect is determined by the evolution of Sobolev norms along the time. In this talk, I will detail the cascade phenomenon, explain how to obtain first an exponential bound of the growth of Sobolev norms and how to improve it into a polynomial bound.

 

Date and place: March 15, 2021, Auditorium 3 and Zoom, 14:15.

Contact the seminar organizer for link.

Speaker: Erlend Grong, University of Bergen

Title: Geometry of sub-Riemannian (2,3,5) manifolds.

Abstract:

We will give details of finding a canonical choice of grading and connection on a sub-Riemannian manifold with growth vector (2,3,5).

The first part of the talk will be to explain what the previous words mean. Then we will give our result and discuss how it relates to the sub-Riemannian equivalence problem. In particular, we will give a flatness theorem.

The talk focuses on a special case of results found in

https://arxiv.org/abs/2010.05366