Program

09:15 Dennis The (Univ. of Tromsø)
Simply-transitive CR real hypersurfaces in C^3

Abstract: Holomorphically (locally) homogeneous CR real hypersurfaces M^3 in C^2 were classified by Elie Cartan in 1932. A folklore legend tells that an unpublished manuscript of Cartan also treated the next dimension M^5 in C^3 (in conjunction with his study of bounded homogeneous domains), but no paper or electronic document currently circulates.

Over the past 20 years, significant progress has been made on the 5-dimensional classification problem. Recently, only the simply-transitive, Levi non-degenerate case remained. Kossovskiy-Loboda settled the Levi definite case in 2019, and Loboda announced a recent solution to the Levi indefinite case in June 2020, both implementing normal form methods.

In my talk, I will describe joint work with Doubrov & Merker in which we use an independent approach to settle the simply-transitive, Levi non-degenerate classification.

10:00 Alexander Schmeding (Univ. of Bergen)
Control theory and stochastic analysis on Lie groups modeled on Hilbert and Fréchet spaces

Abstract: In recent years, several previously though unconnected areas of mathematics and engineering have been connected. The discovery of a common underlying structure of character groups of Hopf algebras brought together fields such as geometric control theory, numerical analysis, rough path theory and free probability. Complementing this algebraic picture is an emerging geometric theory of infinite-dimensional Lie groups. In the talk we will present a short introduction to this theory. We will then report on some new developments to apply differential geometry to locally convergent series arising in control theory applications.

11:25 Torstein Nilssen (Univ. of Agder)
Rough path variational principles for fluid equations

Abstract: In this presentation we will first recall the insight of Arnold that Euler's equation can be understood as a geodesic equation on an infinite dimensional manifold. Using Lagrange multipliers, this yields a natural framework for understanding the structure of random/irregular perturbations of fluid equations. In this presentation we will consider rough path perturbations and the corresponding variational principles.

12:10 Stine Marie Berge (NTNU Trondheim)
Convexity Properties for Harmonic Functions on Riemannian Manifolds

Abstract: In the 70's Almgren noticed that for a harmonic real-valued function defined on a ball, its L^2-norm over a subsphere will have an increasing logarithmic derivative with respect to the radius of mentioned sphere. We examined similar integrals over a more general class of parameterized surfaces by studying harmonic functions defined on compact subdomains of Riemannian manifolds. The integrals over spheres are also generalized to level sets of a given function satisfying certain conditions. If we consider the L^2 norms over these level sets parametrized by a generalization of the radius, we again reproduce Almgren’s convexity property. We will sketch the proof of this result and illustrate the usefulness of the convexity result by examining some explicit parameterized families of surfaces, e.g. geodesic spheres and ellipses.

13:50 Nicolas Gilliers (NTNU Trondheim)
Rough equations in C^⋆ algebras

Abstract: In this talk, I will introduce a certain class C of controlled differential equations with state space a C^⋆ algebra driven by irregular noises. In the realm of free probability theory, Biane and Speicher defined stochastic integration against free brownian motion, paralleling the classical Ito theory. This lead to existence of solutions for equations in C driven by this process. Later, Deya and Schott discussed refinement of rough path theory tailored to equations in C driven by Hölder noises with exponent greater than 31 , defining in particular the notion of product Levy area.

I will address the question of importing rough path principles to study equations in C driven by Hölder noises with arbitrary exponent. This is a work in progress.

14:35 Charles Curry (NTNU Gjøvik)Representation and Simulation of Diffusions

Abstract: We discuss differing representations of diffusions, through variously Kolmogorov’s PDEs, Ito’s SDEs and Stroock-Varadhan martingale problems. We then turn our attention to simulation of diffusions, with a focus on Ito representations. Central to our discussion will be a certain ‘gauge invariance’, namely extra degrees of freedom in the representation of a diffusion by an SDE. We aim to exploit this extra freedom through geometric integration methods to improve the efficiency of numerical simulations.

15:40 Arnaud Eychenne (Univ. of Bergen)
Construction of multi-solitary waves for the dispersion generalized Benjamin-Ono equation. (short talk)

Abstract: During this presentation I will talk about construction of multi-solitary waves for the equation $\partial_t u -D^{\alpha}\partial_xu + \partial_x(u^2)$, for $\alpha\in]\frac{1}{2},2]$. When $\alpha\neq2$ the operator $D^{\alpha}$ is non local, I will explain briefly the main difference in the proof between the non local case ($\alpha\neq2$) and the KDV equation ($\alpha=2$).

16:00 Eirik Berge (NTNU Trondheim)
Wavelet Spaces as Reproducing Kernel Hilbert Spaces (short talk)

Abstract: In this talk, I will introduce the wavelet spaces arising from a square integrable representation. This notion generalizes the Garbor spaces in time-frequency analysis. The emphasis will be on how one can use machinery from reproducing kernel Hilbert space theory to deduce interesting properties of the wavelet spaces. If time permits, we will also discuss the relationship with the HRT-conjecture.