Graduate student seminar

The seminar takes place at room MaD380 at 14:15–15:15, unless otherwise stated. Everyone is welcome.

If you are willing to give a talk or have any questions you can contact Timo Schultz or Ville Kivioja. See math department personnel for contact information.

Talks in Autumn 2018

21.9. The story of 99 Lie algebras

Ville Kivioja

I will consider the classification of Lie groups by metric geometry. Such a classification means introducing relations among Lie groups using metric geometry notions and asking which Lie groups can be related. A famous open problem of this kind asks if it is true that two nilpotent simply connected Lie groups are related by a quasi-isometry only if they are the same group. While not solving this problem, I shall treat some easier and related problems in low dimensions. I will first recall the basics and clarify what kind of metric geometry problems make sense for Lie groups. Then I will go through the classification results we have in dimensions 3 (complete) and 4 and 5 (partial). This will explain the title of the presentation. The talk is an expansion of the 20min presentation I gave in the Summer School of Global Analysis and Applications in Brasov, Romania this year.

28.9. Pointwise characterization of Sobolev functions and Sobolev space on metric measure space

Zheng Zhu

I will give a pointwise characterization of Sobolev functions, and prove the equivalence between this characterization and the definition of Sobolev space. Then using this characterization to define the Sobolev function on metric measure space. The talk is mainly base on ''P. Hajlasz, Sobolev Spaces on an Arbitrary Metric Space, Potential Analysis 5: 403-415, 1996.''

5.10. Rectifiability of Sets of Finite Perimeter in Carnot groups

Terhi Moisala

Sets of finite perimeter (or Caccioppoli sets) are a classical object of study in geometric measure theory. In the early fifties, they were deeply studied by Federer and De Giorgi, who proved that sets of finite perimeter in R^n have (n-1)-rectifiable measure theoretical boundaries. I will give a short introduction to rectifiability and sets of finite perimeter in the Euclidean setting and present some ideas of De Giorgi and Federer on this subject. I will also introduce the notion of Carnot groups that are Ahlfors-regular metric measure spaces with many vector-space-like properties. Finally, I will describe how the above mentioned concepts can be transmitted into this more general setting and where the difficulties arise.

12.10. Relations between several Sobolev spaces on metric measure space

Zhuang Wang

I will talk about three different Sobolev spaces on metric measure space and discuss the relations between them.

19.10. Inverse kinematic problem and X-ray tomography

Keijo Mönkkönen

Can we determine the inner structure of the Earth by measuring seismic waves on the surface? This problem is known as kinematic problem in seismology and is over 100 years old. I present the problem in the modern language of differential geometry and prove that we can recover the speed of sound from boundary measurements under suitable symmetry assumptions. The same kind of question can be asked about functions: can we recover a function from its integrals over all geodesics? This is known as X-ray tomography and it is kind of linearization of the kinematic problem. Here I will only sketch a proof for the result, that the (attenuated) X-ray transform is injective on L2-functions. If time allows, I will discuss a little bit about the difficulties to generalize these problems to the case, where the sound speed is not only radial but depends also on the direction of motion.

26.10. Schoenflies theorem and Kirszbraun theorem for bi-Lipschitz mappings

Haiqing Xu

This is a summary presentation. We start from the classical Schoenflies theorem and Kirszbraun theorem. Afterwards we show these two theorems for bi-Lipschitz mappings.

2.11. On the equivalence of viscosity and weak solutions

Jarkko Siltakoski

A classical solution to a partial differential equation is a smooth function that satisfies the equation at every point of a domain. However, equations that appear in applications do not always admit a smooth solution. Hence the class of solutions needs to be expanded. One such class are the usual Sobolev weak solutions based on integration by parts. Another is the so called viscosity solutions based on generalized pointwise derivatives. When both classes of solutions can be meaningfully defined, it is natural to ask if they are the same. I shall recall the relevant definitions and discuss this question using the standard Laplace equation as a simple example.

9.11. Regularity from quantified flatness

Eero Hakavuori

A differentiable function is infinitesimally linear. This is a qualitative notion as it is only concerned with the existence of limits of difference quotients. Replacing the qualitative existence by some quantified notion often immediately implies more regularity, e.g. (Hölder) continuity of the derivatives. In this talk I will focus on two notions of quantified flatness. The first is sublinearly bilipschitz equivalences (Cornulier 2011), which can quantify how far a map is from an isometry. The second is the beta-numbers (Jones 1990), which quantify how well a map is approximated by an affine map on any given scale.

16.11. Inverse problems for a fractional conductivity equation

Giovanni Covi

We will show how a fractional version of the conductivity equation can be derived from considering a long jump random walk with random weights. We will then state a pair of inverse problems for such equation and solve them by using recent results about the fractional Schrödinger equation.

23.11. Timo Schultz

TBA

Autumn 2017

Webpage for autumn 2017 seminar can be found here.

Spring 2017

Webpage for spring 2017 seminar can be found here.

Spring 2016

Webpage for spring 2016 seminar can be found here.