Analysis and PDE seminar: Fall 2024

Date and time: Thursday, September 5, at 14.15

Place: Aud. Sigma

Speaker: Vegard Hansen, Master student,  Department of Mathematics, UiB

Title: An Introduction to Geometric Measure Theory and Rectifiable Sets

Abstract: Geometric measure theory is rougly speaking the study of geometric objects using the techniques of measure theory. Among the core concepts of this theory is the notion of rectifiable subsets. They provide a generalisation of manifolds to a class with a much less rigid structure. I will present the main ingredients in GMT, namely the hausdorff measures and Lipschitz maps. Using these we will define the recifiable sets, and discuss some of their properties.

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Date and time: Thursday, September 26, at 12.15

Place: Aud. Sigma

Speaker: Erik Broman, Senior Lecturer, Chalmers/University of Gothenburg, Sweden

Title: Phase transitions of semi-scale invariant random fractals

Abstract: 

In all semi-scale invariant random fractal models, there is an 

intensity parameter $\lambda>0$ of the underlying Poisson process which essentially determines 

the nature of the resulting random fractal. As $\lambda$ varies, the models

undergo several phase transitions. One is when the fractal set transitions from containing 

connected components, to the phase where it is almost surely totally disconnected. 

Another is when the fractal transitions from being totally disconnected to disappearing 

completely (i.e. it is empty). As we will explain, this is intimately connected to the classical

problem of covering a fixed set by other random sets (see for example the classical papers

by Dvoretsky or Shepp).

In the talk we will present results concerning both of these phase transitions. In particular, 

the results include determination of the exact value of the parameter $\lambda$ at which 

the second transition mentioned occurs. Furthermore, we are able to determine the behavior of the 

fractal sets at the critical points of both of these phase transitions. 

The talk will be non-technical and is aimed at a broad audience.

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Date and time: Thursday, October 10, at 12.15

Place: Hjørnet

Speaker: René Langøen, Phd. student @ Department of Mathematics, UiB

Title: Curvature in the group of measure-preserving diffeomorphisms of the Klein bottle

Abstract: In this talk I introduce the diffeomorphism group of a manifold, with special focus on the diffeomorphism groups of the torus and the Klein bottle. The Lie algebra of a diffeomorphism group of a manifold is given by the vector fields on the manifold.  The torus is a double orientation cover of the Klein bottle implying a direct relation between vector fields on the torus and the Klein bottle. We use this relation to calculate curvature in the diffeomorphism group of the Klein bottle, in particular, we calculate sectional curvature and an infinite dimensional version of the Ricci curvature. The talk is based on recent work with Boris Khesin (University of Toronto) and Irina Markina (UiB).

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Date and time: Thursday, October 17, at 12.15

Place: Sigma, Realfagbygget, UiB

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

Title:  Local invariants and geometry of the sub-Laplacian on H-type foliations

Abstract: 

Let (M,g) be a smooth, oriented, connected Riemannian manifold equipped with a Riemannian foliation with bundle-like complete metric g and totally geodesic leaves satisfying some additional symmetry conditions. The manifold is studied in the framework of sub-Riemannian geometry with bracket generating distribution transversal to the totally geodesic fibers. Equipping M with the Bott connection we find local invariants by studying the small-time asymptotics of the sub-Riemannanian heat kernel. We obtain the first three terms in the asymptotic expansion of the Popp volume for the pull-back of small sub-Riemannian balls. We address also the question of local isometry of M as a sub-Riemannian manifold and its tangent group.

This is the joint work with W. Bauer, A. Laaroussi (Leibnitz University of Hannover, Germany), S. Vega-Molino (University of Bergen, Norway)

 In spite that the abstract sound very technical, I will not go to details and will present only the main idea of the work, such that it would be accessible by master students.

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Date and time: Thursday, October 17, at 12.15

Place: Sigma, Realfagbygget, UiB

Speaker: Didier Pilod, Associate Professor @ Department of Mathematics, UiB

Title: On the fractional Schrödinger equation with variable coefficients

Abstract: We study the initial value problem (IVP) associated to the semi-linear fractional Schödinger equation with variable coefficients. We deduce several properties of the anisotropic fractional elliptic operator modelling the dispersion relation and use them to establish the local well-posedness for the corresponding IVP. Also, we obtain unique continuation results concerning the solutions of this problem. These are consequences of uniqueness properties that we prove for fractional elliptic operators with variable coefficients.

The talk is based on a joint work with Carlos Kenig (Chicago), Gustavo Ponce (Santa Barbara) and Luis Vega (Bilbao)

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Date and time: Thursday, November 14, at 12.15

Place: Sigma, Realfagbygget, UiB

Speaker: Håkon Lillerødvann Strandjord, Master Student @ Department of Mathematics, UiB

Title: The Hodge Decomposition Theorem and de Rham Cohomology

Abstract: Given a differentiable manifold M one can define the "exterior p-bundle" of M. The space of smooth sections that are smooth p-forms on M, is denoted by E^{p}(M). E^{p}(M) has a certain orthogonal decomposition - known as the Hodge decomposition - related to the Laplace-Beltrami operator. The Hodge decomposition gives a necessary and sufficient condition for the existence of unique solutions to the Poisson equation for smooth p-forms.  If M is a compact oriented Riemannian manifold of dimension n, then the space E^{p}(M) endowed with the L^{2} product gives rise to the de Rham cohomology theory of M, such as Poincaré duality and an isomorphism between the p-th cohomology group of M and harmonic p-forms.

In the talk we give the necessary background to go through the ideas of the proof of the Hodge decomposition theorem. If time allows, we will explain the isomorphism between harmonic p-forms and p-th de Rham cohomology group including the Poincaré duality.

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Date and time: Thursday, November 21, at 12.15

Place: Sigma, Realfagbygget, UiB

Speaker: Torunn Stavland Jensen, Ph.D. Student @ Department of Mathematics, UiB

Title: Fourier uniqueness pairs and discrete unique continuation for the Schrödinger equation

Abstract: 

A pair of sets (A,B) such that if the function f vanishes on A and the Fourier transform of f vanishes on B implies that f is identically zero is called a Fourier uniqueness pair. The problem of finding Fourier uniqueness pairs is connected to uncertainty principles. After a breakthrough in this field by Radchenko and Viazovska in 2019, several new Fourier uniqueness results have been proved.

Applications to PDEs: Since the solution of the free Schrödinger equation can be written in terms of the Fourier transform, it is possible to obtain unique continuation results for the free Schrödinger equation by applying the Fourier uniqueness results. Therefore, it is natural to wonder whether these results also hold for Schrödinger equations with potential and nonlinear Schrödinger equations(NLS).

This talk will focus on works by J.P.G Ramos and M.Sousa and J.P.G Ramos and C. Kehle where they in the first work proved a Fourier uniqueness result, and in the second work adapted these techniques to prove a «discrete unique continuation» result for the cubic NLS and Schrödinger with potential. We will also discuss how we aim to improve these results for the Schrödinger equation and explain the techniques that are used. This is planned to be a collaboration with D. Pilod and J.P.G. Ramos.

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