# Research

The Master’s programme in mathematics and statistics is based on top research. Research within the disciplines of the degree programme is of high international standard and is highly regarded. The research focus within the disciplines in the degree programme are e.g. geometric analysis and measure theory, analysis in metric spaces, partial differential equations, functional analysis, harmonic analysis, mathematical physics, stochastics, inverse problems, mathematical logic and set theory, biomathematics, time series analysis, biometry, econometry, psychometrics and social statistics. The programme is part of the Analysis and dynamics and the Inverse problems centres of excellence.

Analysis is one of the main research areas of mathematics and statistics at the University of Helsinki. Analysis has deep traditions in Helsinki, and during the last years the most valuable results have been achieved in nonlinear potential theory, nonlinear partial differential equations, mathematical physics, conformal geometry and quasiregular mappings. The group has strong international and national contacts. Most of the papers are joint papers with foreign researchers. Visits to and from the group are frequent at all levels; senior researchers, post docs and graduate students. The group has a well established international reputation as evidenced, for example, by many invited talks at high level conferences. On the national level co-operation with Jyväskylä is very intense. There are also good co-operation with the Aalto University and Universities of Joensuu, Oulu and Turku. Co-operation with other research groups inside the department is active, in particular with mathematical physics and inverse problems.

Inverse problems research lies at the intersection of pure and applied mathematics. The forward problem corresponding an inverse problem is usually a well defined problem in mathematical physics. What is inverted in inverse problem is the causality: Whereas in a forward problem we start from the causes and end up with the results, in an inverse problem we start with partial knowledge of the causes and the result and infer more about the causes. For example, in direct scattering we know the incoming wave and the scatterer and calculate the scattered wave. In inverse scattering we know some amount of incoming waves and the corresponding scattered waves and deduce some properties of the scatterer (e.g. the shape or number of components).

Mathematical physics research group works on mathematically rigorous analysis of a wide variety of problems including quantum and statistical field theories, transport and kinetic theory of phonons, waves and quantum evolution, open quantum systems, turbulence and stochastic evolution equations, such as Schramm–Loewner evolution.

Mathematical logic uses exact mathematical methods, originally developed in algebra, topology, measure theory, analysis, and combinatorics to study the two thousand year old subject of logic. During the 20th century, thanks to the revolutionary results of Gödel, but also of Skolem, Gentzen, Church, Turing and Cohen, mathematical logic developed into a deep research area with applications to philosophy, computer science, linguistics and, indeed, mathematics itself. In mathematical logic the Helsinki Logic group focuses on set theory, set-theoretic model theory, model theory, finite model theory, dependence and independence logic, second order logic, philosophical logic, as well as the history of logic and foundations and philosophy of mathematics. The group has developed methods in infinitary logic involving transfinite games and trees to investigate the structure of uncountable models, with connections to stability theory. The group is also known for its work in generalized quantifiers: their hierarchies, their applications in linguistics and computer science, and their set-theoretical properties, as well as for its work in the theory of abstract elementary classes and metric model theory. A recent topic of interest is dependence logic, a project to develop the mathematics and logic of dependence and independence concepts, as they are used in mathematics, computer science and elsewhere.

Research is done both in Bayesian and classical statistics. The research is applied in bioinformatics, economics and image processing. Statistical research, handling uncertaintly which is an inherent part of all scientific activity, by the use of probabilistic models. Computational inference methods for complex statistical models in a wide focus of applications ranging from evolution to modeling ecosystems. Development of fast and robust generic algorithmic tools for inference in statistical models. Statistical machine learning for handling Big Data. Enabling better and faster solutions to scientific inference problems and aiding development of future technologies targeting several practical questions, such as human health and resource management.

Social Statistics supports evidence-based decision making for the government, organizations and private companies. Applications range from pensions and health care to finance and social media. Questions of inequality between genders, regions or generations typically involve statistical analyses. Dynamic probabilistic modelling is a primary tool in analyses involving forecasting and risk management. Finland has an extensive system of administrative registers that provide high quality individual level data, dating back to as early as 18th century. These are complemented by survey data. New sources of data include the internet and social media. Social Statisticians at the University of Helsinki are distinguished in the handling of such data and as developers of new methods for their statistical analysis.

The Biomathematics Research Group focuses on mathematical modelling and analysis of biological phenomena and processes. Our main interest is in mathematical population dynamics, applied to ecology and evolution. Focal areas include the dynamics of structured populations, including metapopulations, as well as adaptive dynamics, a mathematical framework for modelling evolution by natural selection in complex ecological systems. Next to developing the basic theory of these areas, we pursue diverse applications such as the evolution of dispersal or the evolution of pathogens. We use a variety of modelling approaches, most importantly ordinary and stochastic differential equations, renewal equations and stochastic processes, and develop also numerical methods. A successful analysis of the models requires research in the theory of (finite or infinite dimensional) dynamical systems and such research is also carried out in the group.