The Domast student seminar is an informal seminar for doctoral students of mathematics and statistics. The aim is to give students an opportunity to develop science communication and presentation skills and to get a peak into other fields than their own. Students of Domast may get study credits for presenting in or organising the seminar. After every speaker there will be a relaxed discussion as well as a feedback session, so be prepared to take part in these if you attend. The seminar is organised for the first time fall 2020 and will be developed along the way.
Everyone is welcome and we hope to have speakers from all Domast fields!
If you wish to present in the seminar or have other questions, contact the seminar organisers Saara Sarsa or Anna Suomenrinne-Nordvik (firstname.lastname@example.org).
The seminar is held bi-weekly on Fridays at 15-17, currently on Zoom.
|4 September 2020||Eugenia Franco
One dimensional reduction of a Renewal Equation with a continuum of states at birth
Structured population models aim at studying phenomena at the population level, starting from mechanisms at the individual level. Individuals are characterized by their state (age, size, immunity level, …) and appear (for example through birth) in the population with a certain state, which is called state at birth.
If the set of the states at birth is finite it is possible to formulate the model as a Renewal Equation; the asymptotic behviour of its solution can be studied with the Renewal Theorem.
When, instead, we deal with a continuum of states at birth we can still easily formulate the model as a Renewal Equation, but its analysis will typically be difficult.
The focus of this talk is on the second case and on linear Renewal Equations.
Since a Renewal Equation is an integral equation it is chracterized by an integral kernel, which is the bridge between the mathematical formalization and the biological interpretation.
In the first part of the talk I will focus on the modelling aspect, presenting also some examples.
In the second part of the talk I will explain how, under a factorization assumption on this kernel, it is possible to deduce the asymptotic behviour of the solution of the Renewal Equation.
|18 September 2020||Akseli Haarala
On the electrostatic Born-Infeld equations and the Lorentz mean curvature operator
In 1930’s Born and Infeld proposed a new model of nonlinear electrodynamics. In the electrostatic case the Born-Infeld equations lead to the study of a certain quasilinear, non-uniformly elliptic operator that comes with a natural gradient constraint. The same operator appears also as the mean curvature operator of spacelike surfaces in the Lorentz-Minkowski space, the setting of special relativity. We will explain both of these contexts to motivate the mathematical study of said operator.Our main focus will be on the regularity of the solutions of the electrostatic Born-Infeld equations. We will talk about some now classical results as well as some recent developments. We hope to give some ideas on the problems and methods involved without going into details.
|2 October 2020||Tommi Heikkilä
Shearlet regularization in applied dynamic tomography
Tomography is a common example of an ill-posed inverse problem where in order to gain information on the object of interest, the incomplete, indirect and noisy measurements need to be complemented with additional information. This process is known as regularization and a wide variety of options have been developed over the years. In particular wavelets and their extensions are a robust choice in many situations.
The aim of this talk is to give a brief introduction to tomography and motivate the regularization process using shearlets – which are quite a modern representation system for multivariate data such as images or animations.
|16 October 2020||Kalle Koskinen
Convergence of local observables in a statistical ensemble model of a simple paramagnet
In physics literature, ensemble equivalence is usually evoked to swap between less computationally tractable models to more computationally tractable models. Although ensemble equivalence is effectively a very elaborate parameter matching scheme, it is then tacitly assumed that any relevant information of the model is carried over. In this presentation, we will go through the basics of ensemble equivalence and apply these results to prove convergence of local observables between two distinct statistical ensemble models of a simple paramagnet. In doing so, we will have given a completely rigorous example of the interchange between statistical ensembles at the level of local observables.
|30 October 2020||Brecht Donvil (Time exceptionally 13-15)
A short introduction to open quantum systems
The goal of open quantum systems is to study the impact of the environment on the dynamics of a quantum system. The study of such systems today is perhaps more relevant than ever due to the development of genuine quantum devices such as quantum computers. Generally it can be said that the influence of the environment is detrimental to the working of these devices. In my talk I will focus on one of the simplest examples of an open quantum system: a qubit, i.e. a quantum bit, in contact with a thermal environment. Using this example I will sketch the scene of open quantum systems and present some of the main results. Finally, I will tie this in with some results I derived working on my PhD.
|13 November 2020||Emil Airta
Introduction to singular integrals and commutators
In the field of harmonic analysis, the most central objects are the singular integrals. One of the most studied ones are the Calderón-Zygmund operators; the history of classical studies dates to the mid-1900s. Recently, commutator estimates of these operators have drawn more attention. In this talk, I aim to present some reasons for this.The focus of the talk is to introduce the most basic examples of singular integrals and the related commutator estimates. Moreover, we briefly consider singular integrals on a general form and present some classical questions.
|27 November 2020||Stefanos Lappas
The dyadic representation theorem using smooth wavelets with compact support
The representation of a general Calderón–Zygmund operator in terms of dyadic Haar shift operators first appeared as a tool to prove the $A_2$ theorem, and it has found a number of other applications. In this talk we present a new dyadic representation theorem by using smooth compactly supported wavelets in place of Haar functions. A key advantage of this is that we achieve a faster decay of the expansion when the kernel of the general Calderón–Zygmund operator has additional smoothness.
|11 December 2020||Tuomo Nieminen