Domast PhD seminar

 

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 Tommi Heikkilä or Janne Siipola (firstname.lastname@helsinki.fi). The available seminar dates for spring 2021 are below. If you wish to be one of the seminar organisers autumn 2021, contact current organisers.

The seminar is held bi-weekly on Fridays at 15-17, currently on Zoom. Meeting ID: 627 4848 5384

 

 

Spring 2021 Speaker
22 January 2021
Jorma Kilpi

Statistical analysis of data that has heavy-tailed characteristics
First, I will introduce the concept of heavy-tailed probability distributions. Then, I will discuss some practical consequences for the statistical analysis that will follow if the distribution of the data is heavy-tailed. Next, I turn the focus on the data and present methods to detect whether an observed data has heavy-tailed characteristics. I will discuss some estimation problems related to the data that has heavy-tailed characteristics. Finally, I will discuss some consequences that will follow if the data exhibit heavy-tailed features.

5 February 2021 Anna Suomenrinne-Nordvik
Mathematical and practical challenges in modeling COVID-19
Use of mathematical modeling to understand infectious disease dynamics as well as predicting future outcomes has played a significant role in management of the current pandemic. The coronavirus modeling group at the Finnish institute of health and welfare uses an age-structured SEIR model, and while this model is well studied, determining model parameters is an ill-posed inverse problem. The novelty of the virus also presents issues as there is on one hand poor prior knowledge and on the other hand an extensive amount of new research published in recent months. Rapid changes in the population contact structure due to restrictions and individuals adapting their behavior pose additional challenges. In this talk I will present parts of the coronavirus modeling work done at the Finnish institute of health and welfare as well as some mathematical and practical challenges in the work.
19 February 2021 Riina Virta
Gamma emission tomography of spent nuclear fuel
Disposal of spent nuclear fuel is a timely topic in Finland, where a geological repository for such disposal is currently being built. Before the spent nuclear fuel can be buried in the bedrock, the contents of the fuel need to be verified to avoid possible diversion of nuclear material. For this safeguards purpose, a non-destructive assay method Passive Gamma Emission Tomography (PGET) is used. With the help of modern inverse computing, a cross-sectional image of the fuel can be reconstructed. Both the attenuation and activity of the spent fuel can be seen and even a single missing fuel rod can be detected.
5 March 2021 Antti Mikkonen
Radiative transfer in atmospheric remote sensing
Remote sensing of atmospheric constituents vastly expands the available data compared to in situ measurements. One atmospheric remote sensing method is measuring the scattered and reflected solar radiation using satellite-based spectrometers. Determining the state of the atmosphere from a radiation spectrum is an ill-posed inverse problem. The full retrieval process of atmospheric carbon dioxide will be examined with the focus on the forward modeling. In general case, the forward model is the radiative transfer equation (RTE), but in practical situations numerical methods and approximations are necessary. Two radiative transfer simulators, Siro and RaySca, will be presented.
19 March 2021 Tuomas Oikari
Singular integral and applications
We outline some motivating examples from partial differential equations and rectifiability of sets for the study of singular integral operators and their commutators. The boundedness theory for both classes of operators is also discussed.
26 March 2021 Mikko Heikkilä
Differentially private partitioned variational inference
Variational inference is a common method in Bayesian learning for approximating intractable posterior distributions.
Partitioned variational inference (PVI) is a recent formulation that unifies many previous variational inference algorithms within a single framework. PVI is especially well suited for federated learning scenarios, where the data are distributed over many participants. In this talk I will discuss our work on combining PVI with differential privacy, a strict mathematical definition of privacy, to enable learning variational approximations in the federated setting while guaranteeing privacy for the individual data subjects.
9 April 2021 Joona Oikarinen
On the Origin of Spin
The aim of this presentation is to explain how spin appears in Quantum Mechanics when implementing rotational symmetry. On the mathematical side this leads us to study projective representations of symmetry groups on projective Hilbert spaces and their lifts to the universal cover of the group. Then we study how the projective representation on the universal group can be lifted to a unitary representation on the Hilbert space. On the physical side we study how the mathematics leads to the appearance of the spin observables for the electron.
16 April 2021 Stefanos Lappas
Extrapolation of compactness on weighted spaces
The extrapolation theorem of Rubio de Francia is one of the most powerful tools in the theory of weighted norm inequalities: it allows one to deduce an inequality (often but not necessarily: the bounded of an operator) on all weighted L^p spaces with a range of p, by checking it just for one exponent p (but all relevant weights). My topic is an analogous method for extrapolation of compactness. In a relatively soft way, it recovers several recent results about compactness of operators on weighted spaces and also gives some new ones.
30 April 2021 TBA

Here are the abstracts for the talks which were held in autumn 2020.

Fall 2020 Speaker
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
Estimation of the time-dependent ratio of SARS-CoV-2 infections to registered disease cases, during the first wave of the 2020 COVID-19 outbreak in Helsinki and Uusimaa region, FinlandIn Finland, the first wave of a COVID-19 epidemic, the illness caused by new severe acute respiratory syndrom coronavirus 2 (SARS-CoV-2), occurred during the spring of 2020, with most of the disease cases in the extended capital region of Helsinki and Uusimaa (HUS area). The clinical manifestations of a SARS-CoV-2 infection ranges from asymptotic to severe, potentially fatal disease. The likelihood of a SARS-CoV-2 infection being registered as a disease case depends on the clinical manifestation as well as the testing policy and capacity at the time, which may miss mild and asymptomatic cases.

COVID-19 disease cases are registered in the Finnish National Infectious Diseases Register (FNIDR). A large proportion of the infections were likely not registered as cases during the first wave of the epidemic in Finland due to testing policy and capacity as well as infections not developing into disease cases. Quantifying the relative difference in the numbers of registered cases and true infections is important, as assessing the severity of the disease caused by SARS-CoV-2 requires knowledge of the number of infections occurring in the population.

Infections usually leave a mark in the form of antibodies, and so knowledge of the number of infections can be attained from serology studies. A population serology survey by the Finnish Institute for Health and Welfare (THL) obtains information on how large a proportion of the population have developed antibodies to SARS-CoV-2 in different regions in Finland (serology survey). The serology survey targets most areas in Finland and individuals aged 18-69.

It is not straightforward to compare the results of the serology survey to the registered cases. The two sources of observations are not directly comparable during any time point as there may be a significant delay from the disease onset date to developing antibodies. To solve this issue, we utilise previously published information of the time to develop antibodies to project the population proportion of infected individuals with antibodies (i.e those who have seroconverted), when the registered infections in the FNIDR are the only infections. We compare the projections to the proportion of seroconverted observed in the serology survey and estimate the time-dependent ratio of infections to registered cases.