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

# Analysis and PDE Seminar: semester plan

Analysis and PDE seminar takes place weekly, on Tuesdays, at 14.15. Location for Spring 2022: Delta, 4th floor, Realfagbygget, Allégaten 41 and Zoom.

Seminars the semester Spring 2022 will be Tuesdays 14:15 in Delta and Zoom

### 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:
r
The 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.