# Analysis and PDE seminar: Spring 2024

This is the page for the current semester of the seminars in Analysis and PDE at the University of Bergen. This semester seminars are held on Thursdays in the room Sigma at 14.15 until 16.00.

## Main content

### Detailed entries with abstracts

## January 25, Irina Markina

**Date and time**: Thursday, January 25, at 14.15

**Place**: Aud. Sigma

**Speaker**: Irina Markina, Professor, Department of Mathematics, UiB

**Title**: KdV equation as a minimising L^2-energy equation.

**Abstract**: This talk is oriented to two parts of the Analysis and PDE group: the group with an interest in differential geometry and the group that is interested in non-linear partial differential equations. The motion of a rigid body in 3-D space is successfully described as a motion in the group of Euclidean transformations (rotations and translations) by making use of the Euler angles. V. Arnold proposed to describe fluid motion by replacing the finite-dimensional group of Euclidean transformations with the infinite-dimensional group of diffeomorphic transformations of a suitable space. We will consider the simplest infinite dimensional group, which is the group of diffeomorphism Diff(S) of the unit circle S, which corresponds to the description of periodic solutions of one variable. I will define the group (slightly different from Diff(S)), its Lie algebra, the metric on it (or the energy) and finally show that the equation describing the geodesics (the curves minimizing the energy) is the famous KdV equation in the fluid mechanics.

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## February 1, René Langøen

**Date and time**: Thursday, February 1, at 14.15

**Place**: Aud. Sigma

**Speaker**: René Langøen, PhD. student, Department of Mathematics, UiB

**Title**: Stokes graphs of a quadratic differential related to a Rabi model

**Abstract**: To study the behaviour of solutions to a second-order linear differential equation 𝑦″+𝑄(𝑧,𝑡)𝑦=0 one can associate the quadratic differential 𝑄(𝑧)𝑑𝑧^{2} on the punctured Riemann sphere and consider its Stokes graph. We consider an ODE related to a Rabi problem describing a light-atom interaction in physics. The associated quadratic differential is meromorphic with two finite poles. The integrability condition for this type of ODE under isomonodromic deformations is related to a non-linear second-order differential equation, known as Painlevé V. In my talk, I will explain a classification of the Stokes graphs according to the nature of the zeros of the meromorphic quadratic differential originated in the Rabi model. This is a joint work with I. Markina (University of Bergen) and A. Solynin (Texas Tech, USA).

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## February 08, Martin Oen Paulsen

**Date and time**: Thursday, February 8, at 14.15

**Place**: Aud. Sigma

**Speaker**: Martin Oen Paulsen, PhD. student, Department of Mathematics, UiB

**Title**: On the square root of a Laplace-Beltrami operator

**Abstract**: A central part of the study of free boundary problems is related to the properties of the Dirichlet to Neumann operator (DNO). In particular, it is known that DNO is related to the Beltrami-Laplace operator associated with the surface. An important application of this observation was given in the celebrated paper by Lannes published in the Journal of the AMS in 2005, where he proved the well-posedness of the water waves equations.

In this talk, I will present this key observation that relates the DNO with the square root of the Laplace-Beltrami operator.

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## February 15, Erlend Grong

**Date and time**: Thursday, February 15, at 14.15

**Place**: Aud. Sigma

**Speaker**: Erlend Grong, Researcher, Department of Mathematics, UiB

**Title**: Infinite dimensional control on manifolds with boundary

**Abstract**: The talk will relate to controllability; the ability to reach a certain point in a space given a permissible set of movements.Though there are several standard results for finite-dimensional manifolds, controllability in infinite dimensional spaces are less understood. We are going to focus on the special case of diffeomorphism groups. In 2009, Agrachev and Caponigro described how we could look at how flows of vector fields move points around on a manifold and determine from such observations which diffeomorphisms can be generated by similar flows. We want to consider the same problem, but now allow the manifold to have boundary, and even corners in some special cases. We will both describe the Lie group structure for such groups as well as controllability result. The results presented are based on joint work with Alexander Schmeding (NTNU).

## March 7, Jonatan Stava

**Date and time**: Thursday, March 7, at 14.15

**Place**: Aud. Sigma

**Speaker**: Jonatan Stava, PhD.-student, Department of Mathematics, UiB

**Title**: What is the best way to understand the Bianchi Identities?

**Abstract**: The Bianchi identities are relations involving the curvature and torsion that will always be satisfied for a smooth manifold with an affine connection. These identities are equivalent to the Jacobi identity with respect to the skew-symmetrization of the tensor product in the free tensor algebra over the smooth vector fields. To get this equivalence we must include the connection to make the tensor algebra into what is called a D-algebra. It is also known that the Bianchi identities can be derived from the canonical principal connection and the canonical solder form of the frame bundle.

In this presentation we will be in two parts:

- First we look at the frame bundle. What structure is induced on the frame bundle from the connection on the underlying manifold? The goal is to get a good understanding of the curvature, torsion and the Bianchi identities on the frame bundle.

- Secondly we will see what the D-algebra is and how the Bianchi identities can be represented here. We might speculate on the relation between the frame bundle and the D-algebra.

This talk is based on early stage of a potential research project.

**Keywords:** Bianchi identities, connection, frame bundle, D-algebra

## March 21, Martin Oen Paulsen

**Date and time**: Thursday, March 21, at 14.15

**Place**: Aud. 4

**Speaker**: Martin Oen Paulsen, PhD.-student, Department of Mathematics, UiB

**Title**: The Nash-Moser Theorem and Applications

## April 4, Francesco Ballerin

**Date and time**: Thursday, April 4, at 14.15

**Place**: Aud. Sigma

**Speaker**: Francesco Ballerin, PhD Candidate, Department of Mathematics, UiB

**Title: **SO(3)-equivariant neural networks for meteorological predictions

**Abstract**: Geometric deep learning is a field that extends traditional deep learning methods to data with an underlying geometric structure, such as graphs and manifolds. It equips neural networks with mechanisms to handle non-Euclidean data and force either invariance or equivariance depending on the type of problem. This means that the result of a prediction does not depend on the symmetries of the underlining geometrical space: rotating a picture of a cat does not produce a picture of a dog and changing atlas for a manifold does not change the manifold. This tackles some of the problems of the black-box nature of neural networks forcing some useful mathematical constraints. We explore the ideas behind geometric deep learning, and we see an application to meteorological data prediction.

## April 11, Torunn Stavland Jensen

**Date and time**: Thursday, April 11, at 14.15

**Place**: Aud. Sigma

**Speaker**: Torunn Stavland Jensen, Master student, Department of Mathematics, UiB

**Title: **Unique Continuation for the Schrödinger Equation

**Abstract**: The first part of the talk will be a short history on unique continuation, Carleman estimates and how Hardy’s Uncertainty Principle is related to a unique continuation result for the free Schrödinger equation. In a series of work by Escauriaza, Kenig, Ponce and Vega this result was extended to the Schrödinger equation with potential and to the NLS. I will present the main steps of the proof for this result for the Schrödinger equation with potential. The proof is based on Carleman estimates, which formally is based on calculus and convexity arguments. However, going from a formal level to a rigorous one is not straight forward, and if we do not justify the computations rigorously, we can prove wrong results. In particular I will present an example of a formal Carleman argument for which the corresponding inequalities leads to a false statement.

## April 18, Sylvie Vega-Molino

**Date and time**: Thursday, April 18, at 14.15

**Place**: Aud. Sigma

**Speaker**: Sylvie Vega-Molino, Postdoc, Department of Mathematics, UiB

**Title: **Controllability of Shapes through Landmark Manifolds

**Abstract**: Landmark manifolds consist of distinct points that are often used to describe shapes. We show that in the Euclidean space, we can preselect two vector fields such that their flows will be able to take any n-landmark to another, regardless of the number of points n. This is a joint work with Erlend Grong.

## April 25, Mihaela Ifrim, University of Wisconsin Madison

**Date and time**: Thursday, April 25, at 14.15

**Place**: Aud. Sigma

**Speaker**: Mihaela Ifrim, Associate Professor, from University of Wisconsin Madison.

**Title: **The small data global well-posedness conjecture for 1D defocusing dispersive flows

**Abstract**: I will present a very recent conjecture which broadly asserts that small data should yield global solutions for 1D defocusing dispersive flows with cubic nonlinearities, in both semilinear and quasilinear settings. This conjecture was recently proved in several settings in joint work with Daniel Tataru.

## May 10, Marcos Salvai, National University of Cordoba, Argentina

**Date and time**: Friday, May 10, at 10.15

**Place**: Aud. Sigma

**Speaker**: Marcos Salvai, Professor from National University of Cordoba, Argentina.

**Title: **The sub-Riemannian geometry of screw motions with constant pitch

**Abstract**: Let M be an oriented three-dimensional Riemannian manifold of constant sectional curvature k = 0 , 1, -1 and let SO(M) be its direct orthonormal frame bundle (direct refers to positive orientation), which has dimension six and may be thought of as the set of all positions of a small body in M. Given λ ∈ R, there is a three-dimensional distribution D^λ on SO(M) accounting for infinitesimal rototranslations of constant pitch λ. When λ ≠ k^2, there is a canonical sub-Riemannian structure on D^λ. We describe its geodesics. For k = 0, -1, we compute the lengths of all periodic geodesics of ( SO(M) , D^λ ) in terms of the lengths and the holonomies of the periodic geodesics of M, when M has positive injectivity radius.

It turns out that the notion of rototranslating with constant pitch makes sense for some higher dimensional Riemannian manifolds, for instance, for R^7 via the octonionic cross product, or for compact Lie groups. We define sub-Riemannian structures analogous to the above and find some of their geodesis.

## May 16, Hans Z. Munthe-Kaas

**Date and time**: Thursday, May 16, at 14.15

**Place**: Aud. Sigma

**Speaker**: Hans Z. Munthe-Kaas, Professor, Department of Mathematics, UiB.

**Title: **Rough paths on manifolds

**Abstract**: This work studies rough differential equations (RDEs) on homogeneous spaces. We provide an explicit expansion of the solution at each point of the real line using decorated planar forests. The notion of planarly branched rough path is developed, following Gubinelli's branched rough paths. The main difference being the replacement of the Butcher–Connes–Kreimer Hopf algebra of non-planar rooted trees by the Munthe-Kaas–Wright Hopf algebra of planar rooted forests. The latter underlies the extension of Butcher's *B*-series to Lie–Butcher series known in Lie group integration theory. Planarly branched rough paths admit the study of RDEs on homogeneous spaces, the same way Gubinelli's branched rough paths are used for RDEs on finite-dimensional vector spaces. An analogue of Lyons' extension theorem is proven. Under analyticity assumptions on the coefficients and when the Hölder index of the driving path is one, we show convergence of the planar forest expansion in a small time interval.

## June 3, Gutavo Ponce

**Date and time**: Monday, June 3, at 10.00

**Place**: Aud. Sigma

**Speaker**: Gustavo Ponce, Professor, Department of Mathematics, University of California, Santa Barbara, USA

**Title: **Unique Continuation Properties of Nonlocal Nonlinear Dispersive Models.

**Abstract**: We shall study unique continuation properties for a class of non-local “dispersive" models.

We shall consider two type of unique continuation problems: local and asymptotic to infinity.

We shall review several results for several equations including the Benjamin-Ono equation, the ILW equation., the Camassa-Holm equation, and the Benjamin-Bona-Mahony equation.

## June 3, Luis Vega

**Date and time**: Monday, June 3, at 11.00

**Place**: Aud. Sigma

**Speaker**: Luis Vega, Professor, BCAM, University of the Basque Country, Spain

**Title: **The binormal flow and the evolution of viscous vortex filaments.

**Abstract**: I will present the so called Localized Induction Approximation that describes the dynamics of a vortex filament according to the Binormal Curvature Flow (BF). I’ll give a result about the desingularization of the Biot-Savart integral proved with Marco A. Fontelos within the framework of Navier-Stokes equations. Some particular examples regarding BF obtained with Valeria Banica will be also considered. These examples allow to connect BF with the so-called Riemann non-differentiable function and the Frisch-Parisi approach to turbulence.

## June 7, Jonatan Stava

**Date and time**: Friday, June 7, at 14.15

**Place**: Delta (4A9f), Realfagbygget

**Speaker**: Jonatan Stava, PhD. candidate @ the University of Bergen

**Title: **Nomizu's canonical connection of the second kind

**Abstract**: Runge-Kutta methods is a common class of methods for solving initial value problems in R^n. These methods approximate the exact flow by combining curves with constant velocity, or just "straight lines". If we want to general these methods to manifolds other than R^n we must use the equivalent of a constant velocity curve which is called a geodesic, but these curves are dependent on the choice of connection on the manifold. A connection is a way of taking the derivative of one vector field in the direction of another. In general, there exists an infinite number of possible connections on a manifold. We say that a connection is canonical if it is the unique connection which satisfies certain conditions which are natural for the manifold in question. For instance, on a Riemannian manifold, the Levi-Civita connection is the canonical connection which is torsion free and preserves the metric.

Homogeneous spaces are an important type of manifolds that can be represented as quotients of Lie groups. Is there a canonical connection on homogeneous spaces? This was answered by Nomizu in an article from 1954. The goals of this talk:

Give a good introduction to the topic of connections.

Prove Nomizu's theorem which relates invariant connections on a reductive homogeneous space G/H with products on a vector subspace of the Lie algebra of G.

Use this theorem to express some particularly nice connections, ending with the one that Nomizu names the canonical connection of the second kind.