### Department of Mathematics and Systems Analysis

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15. September 2022

Opponent is Assistant Professor **Cristiana De Filippis**, University of Parma, Italy

Custos is Professor **Juha Kinnunen, **Aalto University School of Science, Department of Mathematics and Systems Analysis

Contact details of the doctoral student: cintia.pacchiano@aalto.fi, +358 (50) 4149900

The public defence will be organised on campus.

The doctoral thesis is publicly displayed 10 days before the defence in the publication archive Aaltodoc of Aalto University.

Press release:

This dissertation studies existence and regularity properties of functions related to the calculus of variations on metric measure spaces that support a weak Poincaré inequality and doubling measure. We concentrate especially on variational solutions to the total variation flow, and quasiminimizers to the (p,q)-Dirichlet integral. The main interest in this work is to extend some classical results of the calculus of variations to metric measure spaces.

Variational methods appeared as an answer to the problem of finding minima of functionals. It is about giving a necessary and sufficient condition for the existence of the minimum, as well as conditions that allow its calculation and algorithms that let us compute it. Variational calculus is intimately linked with the theory of partial differential equations since the conditions for existence of a solution to the minimization problem normally depend on the fact that said solution satisfies a certain differential equation.

This dissertation focuses on various classes of functions related to a Dirichlet type integral. We first define variational solutions to the total variation flow (TVF) in metric measure spaces. We establish their existence and, using energy estimates and the properties of the underlying metric, we give necessary and sufficient conditions for a variational solution to be continuous at a given point. As far as we know, this is the first time that existence and regularity questions are discussed for parabolic problems with linear growth on metric measure spaces. We then take a purely variational approach to a (p,q)-Dirichlet integral. We define its quasiminimizers, and using the concept of upper gradients together with Newtonian spaces, we develop interior regularity and regularity up to the boundary. Lastly, we prove higher integrability together with stability results in the context of general metric measure spaces.

Analysis on metric spaces is nowadays an active and independent field, bringing together researchers from different parts of the mathematical spectrum. It has applications to disciplines as diverse as geometric group theory, nonlinear PDEs, and even theoretical computer science. This can offer us a better understanding of the phenomena and also lead to new results, even in the classical Euclidean case.

12. August 2022

With the recent developments in the precision of measurement technology and storage capacity, massively large and high dimensional data sets have become commonplace over nearly all fields of science. Functional data – data arising from measuring a generating process of continuous nature over its continuum – has emerged as a prominent type of such big data due to the richness of its structural features. One does not have to search far to find an abundance of great examples of functional data sets: the growth curves of children, measurements of meteorological events such as temperature or precipitation and hourly electricity consumption over a day, are all examples of processes within the realm of functional data.

Detailed analysis of the shape features of functional data is often the key to revealing important modes of variance in functional data. For instance, recognizing structural deviancies from the typical in the growth pattern of school aged children can be one of the earliest markers warning of potential underlying problems in health and well-being. Accurately predicting the hourly electricity consumption is crucial for an electricity company to be able to match the production to the demand. Distinguishing between the silhouettes of a child and a dog can be crucial in computer vision applications for self-driving cars. In short, sensitivity of the developed methodology to variations in shape has become an important topic in the literature. However, precisely defining typicality or atypicality in shape has proven to be a difficult problem. In how fine detail should the variations in local features be considered? What precisely makes a curve ‘too curvy’ in comparison to a set of other observations? Clearly, it is time to leave the classical, location-based considerations in the offside and shift our focus towards the intricacies in shape and structure.

In the dissertation, we develop methods for assessing the shape typicality and similarity of observations and study their properties in theory and in practice. Furthermore, we study the practical implementations of the methods in some prominent, common applications such as supervised learning and outlier detection, and evaluate their performance compared to some popular modern competitors. In particular, we demonstrate the excellent properties of the proposed methods and show that in many commonly encountered settings, they are able to match or even outperform many of the leading competitors.

Opponent is Professor **Thomas Verdebout**, Université Libre de Bruxelles, Belgium

Custos is Professor **Pauliina Ilmonen, **Aalto University School of Science, Department of Mathematics and Systems Analysis

Contact details of the doctoral student: sami.helander@aalto.fi, +358 50 5186136

**The public defence will be organised on campus **(Otakaari 1, lecture hall H304).

The doctoral thesis is publicly displayed 10 days before the defence in the publication archive Aaltodoc of Aalto University.

27. June 2022

Camilla Hollanti is head of the research group “Algebra, Number Theory and Applications” at Aalto University, which has a very striking gender balance. Read the interview (European Women in Mathematics)

27. June 2022

This thesis studies two different prototypes of nonlinear partial differential equations with porous medium type and p-growth structure. These equations can be interpreted as nonlinear generalizations of the heat equation modeling various physical phenomena such as gas flow in porous medium, heat conduction or water movement in soil. The thesis focuses on regularity properties of solutions as well as their gradients to these equations. Boundary regularity for the gradient is shown in terms of higher integrability for porous medium type equations. Moreover, we demonstrate that both solutions and their gradients are stable with respect to the parameter characterizing the equation. If there is an obstacle restricting the behavior of the solution, we show that the solution is continuous provided that the obstacle is sufficiently regular. The obstacle problem is closely connected to the concept of supersolutions, which we define in the thesis as functions obeying a comparison principle. We show that supersolutions according to this definition are divided into two mutually exclusive classes for which we give several characterizations. The results in the thesis show that both solutions and their gradients possess properties that are mathematically relevant. The proofs of our results require new mathematical techniques and their implementations, which could be expected to be useful in other contexts as well.

Opponent is Professor **José Miguel Urbano**, University of Coimbra, Portugal

Custos is Professor **Juha Kinnunen, **Aalto University School of Science, Department of Mathematics and Systems Analysis

Contact details of the doctoral candidate: kristian.moring@aalto.fi

**The public defence will be organised on campus **(Otakaari 4, lecture hall 216).

The doctoral thesis is publicly displayed 10 days before the defence in the publication archive Aaltodoc of Aalto University.

27. June 2022

The foremost international meeting of mathematicians, ICM 2022, was supposed to be held in Saint Petersburg in July 2022, and **Vladimir Putin** was supposed to have opened it. Due to the war in Ukraine, the organisers decided to move the congress online and find another venue for the opening ceremony. The International Mathematical Union (IMU) received many offers of new venues. The Finnish community of mathematicians offered to organise the event in Helsinki on 5th-6th July 2022, and IMU accepted Finland's invitation.

The opening ceremony for the ICM 2022 congress will be held in Helsinki on 5^{th} July 2022, and at the ceremony, distinguished mathematicians will be awarded the Fields medals, the Carl Friedrich Gauss prize, the Chern medal, the Leelavati prize, and the Abacus prize for mathematical computing. These prizes are highly esteemed among mathematicians as proof of remarkable scientific achievements in the field of mathematics. Among the medals awarded by IMU, the Fields medal is best known outside the field of mathematics as a recognition comparable to the Nobel Prize.

The opening ceremony will be held at the Töölö hall of Aalto University, and some 600 attendees are expected along with members of the international press, since the award ceremony is traditionally also a media event. President of the Republic of Finland **Sauli Niinistö** will open the ceremony. The event will be hosted by Professor **Camilla Hollanti **from Aalto University.

On the following day, July 6^{th}, the awardees will give scientific talks on the most current breakthrough research in mathematics at the Töölö auditorium of Aalto University. Both the award ceremony and the lectures of the awardees will be streamed live to the participants of the virtual ICM conference.

The organisers of the event in Finland are the national committee of mathematics, member of the Council of Finnish Academies, chaired by professor in mathematics at the University of Helsinki, Professor **Antti Kupiainen**, and the Finnish Mathematical Society, chaired by professor in mathematics at the University of Helsinki, Professor **Tuomo Kuusi**, as well as a large number of Finnish mathematicians from these two organisations.

Professor Antti Kupiainen and Professor Tuomo Kuusi consider it a great honour for Finland and Finnish mathematics to have the ceremony here. It has been held in Finland only once before, in 1978.

Many get-togethers and satellite conferences will be organised in connection with the event for mathematicians from different fields, such as the conference World Meeting for Women in Mathematics, which will award the prize for mathematical physics, named after **Olga Aleksandrovna Ladyzhenskaya**, who was born a hundred years ago. The Probability and Mathematical Physics Satellite Conference, which is organized in Helsinki between 28 June - 7 July, will feature several previous Fields medalists as speakers.

The IMU Award Ceremony 2022 will be streamed on 5 July

Professor Antti Kupiainen, University of Helsinki, antti.kupiainen@helsinki.fi, phone +358 50 4480305

Professor Tuomo Kuusi, University of Helsinki, tuomo.kuusi@helsinki.fi, phone +358 50 5560814

Professor Camilla Hollanti, Aalto University, camilla.hollanti, phone +358 50 5628987

World Meeting for Women in Mathematics:

Senior University Lecturer Kirsi Peltonen, kirsi.peltonen@aalto.fi, phone: +358 50 5747006

13. May 2022

Risk management and decision-making under uncertainty are common challenges in business and public administration. Often the framework of a decision-making problem consists of various types of factors and variables whose mutual probabilistic dependencies may be difficult to know or perceive exactly. For instance, there might not be suitable historical data available, or the relevant data may be difficult to identify. These problems are typical in situations where risks are novel or unprecedented. Among such instances are, e.g., unique projects, ecological and economical disasters, and governmental conflicts.

Even though there might be a lack of suitable historical data, there is often an abundance of expert insight available, along with diverse information on indirectly related factors. In these situations, analysis of risks and decision-making under uncertainty can effectively be supported by Bayesian networks (BNs). A BN represents a system of linked components both visually and numerically enabling a rigorous quantification of risks and a clear communication of the components’ interaction. BNs can be constructed based on various information sources such as experimental data, historical data, and expert knowledge. The applications of BNs are numerous and cover a wide range of domains, such as medical decision support, risk analysis concerning epidemics, ecosystems, and industry, as well as policy and military planning.

The dissertation elaborates the construction of BNs by expert elicitation which involves subjective assessments of a domain expert and is often required in practical applications. The main contribution is the development of new elicitation approaches that help the expert to establish required numerical dependencies between BN components. The approaches improve an existing elicitation method commonly used in BN applications. They reduce the elicitation effort of the expert and also extend the application scope of the underlying method. Their practical execution is supported by thorough guidelines and online implementations. Consequently, the new approaches facilitate and promote the effective and diverse utilization of BNs in various applications.

Opponent is Professor **Norman Fenton**, Queen Mary University of London, UK

Custos is Professor **Kai Virtanen, **Aalto University School of Science, Department of Mathematics and Systems Analysis

Contact details of the doctoral student: pekka.laitila@aalto.fi

**The public defence will be organised on campus and via Zoom. **Link to the event

5. April 2022

Perhaps the date (April Fool's Day) had an effect on the slightly low amount of the participants..nevertheless Biitsi was super nice, food delicious and we had great fun!

Thanks for the participants! Here are some photos.

5. August 2021

In 2022 we are planning to hold an in-person Bridges conference! Join us for invited and contributed talks, hands-on workshops, art galleries and live performance events, and an excursion.Page content by: webmaster-math [at] list [dot] aalto [dot] fi