# Recent seminars in Analysis and PDE

List of the most recent semesters of seminars

## Main content

**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