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Analysis and PDE
Weekly seminar of the Analysis and PDE group, highlighting internal members and guests alike.

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.

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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)𝑑𝑧2 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).

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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:

  1. 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.
     
  2. 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

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April 18, Hans Z. Munthe-Kaas

Date and time: Thursday, March 7, 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.

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