The seminar takes place at room MaD355 at 12:15–13:15, 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.
Back to seminar 2022.
Atte Lohvansuu
It is common knowledge that certain functions with rather simple closed form expressions, such as exp(x^2) or sin(x)/x, do not have elementary antiderivatives. But how does one prove such statements? What do "closed form" and "elementary" even mean? In this talk I will give some partial answers to these questions, and prove the nonexistence of the aforementioned elementary antiderivatives.
Timo Schultz
I will talk about (algebraic) topology of finite topological spaces and its connections to studying topological spaces of more interest such as CW- or simplicial complexes, or manifolds. The correspondence between homotopy types of compact CW-complexes and homotopy types of finite topological spaces will be touched upon.
Ville Kivioja
I discuss the meaning of negative curvature in some generality, after which I will present two points of view to visual boundaries of negatively curved spaces. One is the Gromov-boundary with Bourdon-distance and the other is its punctured version with Hamenstädt-distance. While we want to understand the constructions for quite general spaces, I will discuss the example of the real hyperbolic plane in some detail, since there one can understand everything from pictures and some Calculus 1. Finally I will point out what is the link of all of this to the research of nilpotent Lie groups and their quasi-conformal maps.
Toni Ikonen
I will talk about Circle packings. The packings have many applications in the field of complex analysis, image recognition, and metric geometry among other things. I will introduce a specific application of circle packings: the discrete Riemann mapping theorem. The discrete version of the theorem is deeply related to the classical one. I will introduce both versions of the theorem, some ideas behind the proof of the discrete version and its connection to the classical theorem. I will also show some concrete examples of the discrete mapping theorem (pictures will be involved).
Anna Kausamo
I will guide you along the path between measure theory and geometry. In the end, you will find the isoperimetric inequality. There are obstacles on the way, though, like the Brunn-Minkowski inequality. Fortunately, there is a way through - the optimal transport.
Keijo Mönkkönen
I will discuss about unique continuation results of Riesz potentials on distribution spaces and give some applications to inverse problems, especially to integral geometry. In addition, I also mention about the connection to fractional Laplacians and their role in fractional (non-local) inverse problems.
Enrico Pasqualetto
Choquet theory plays a fundamental role in functional analysis, convex geometry, and measure theory. The main result in this field is the celebrated Choquet-Bishop-de Leeuw theorem, which states that any point of a compact convex subset of a normed space (or, more generally, of a locally convex topological vector space) can be obtained as a superposition of the extreme points of the convex set itself. The focus of the seminar will be on some of the countless applications of this result - among which, I just mention the Krein-Milman theorem and the Riesz representation theorem. Time permitting, I will also explain how it can be used to study the extreme points of the unit ball in the space of BV functions.
Matthew Romney
The uniformization theorem, one of the fundamental results of complex analysis, states that every simply connected Riemann surface is conformally equivalent to either the unit disk, the plane, or the sphere. In this talk, we discuss the problem of generalizing this result to metric spaces with minimal assumptions on their geometry. This research direction is motivated by connections to geometric group theory and complex dynamics, where such spaces arise naturally. In the metric space setting, the class of conformal mappings is quite restrictive and it is natural to consider instead some notion of quasiconformal mapping. We give an overview of recent and ongoing work in this area. This talk is intended for a broad mathematical audience.
Nguyen Ngoc Khanh
Let M be a 2-dimensional simply connected Riemann surface which is conformally equivalent to the hyperbolic plane. In this talk, we discuss characterizations of M in terms of Capacities, Green functions, Super-harmonic functions, Schrödinger equations, Volume growth.
Martti Rasimus
In 1993 He and Schramm proved that Koebe's conjecture known also as Kreisnormierungsproblem holds in the countably connected case: Every planar domain with countably many boundary components is conformally equivalent to a circle domain. The aim of this talk is to give an outline for a proof of this theorem starting from the finitely connected case proven by Koebe.