Two national Centres of Excellence of the Academy of Finland operate at the department, The Finnish Centre of Excellence in Inverse Problems Research and The Finnish Centre of Excellence in Analysis and Dynamics Research.

# Centres of Excellence

#### INVERSE PROBLEMS

Inverse problems appear in several fields, including medical imaging, image processing, mathematical finance, astronomy, geophysics, non-destructive material testing and sub-surface prospecting. Typical inverse problems arise from asking simple questions "backwards". For instance, the simple question might be "If we know precisely the structure of the inner organs of a patient, what kind of X-ray images would we get from her?" The same question backwards is "Given a set of X-ray images of a patient, what is the three-dimensional structure of her inner organs?" This is the inverse problem of Computerized Tomography, or CT imaging.

Usually the inverse problem is more difficult than the simple question that it reverses. For example, even though the Earth's gravitational field is governed by Newton's law of gravitation, the inverse problem of finding sub-surface structures from minor variations of the gravitational field on the surface is extremely hard. Successful solution of inverse problems requires specially designed algorithms that can tolerate errors in measured data.

Inverse problems research concentrates on the mathematical theory and practical interpretation of indirect measurements. The study of inverse problems is an active area of modern applied mathematics and one of the most interdisciplinary field of science.

**The Centre of Excellence (CoE) in Analysis and Dynamics Research** belongs to the Centres of Excellence in Research Programme 2014–2019 of the Academy of Finland. Its goals are

- To develop a new culture in Finnish mathematics that encourages collaboration in pure mathematics and connects the highest level of pure mathematics with applications.
- To renew researcher training in our fields aiming to counter early specialization of PhD students by offering them a broad education, including applications, thereby also improving their job opportunities.

The research covers a wide spectrum of mathematical analysis and its applications: dynamical systems, fractal geometry, random geometry, partial differential equations and turbulence, statistical mechanics and mathematical models of biological evolution.

#### Analysis and Dynamics

The modern field of dynamics emerged in the second half of the 20th century when computer simulations started to reveal the extraordinary rich phenomena present in simple dynamical models. It thoroughly transformed *mathematical biology*and *mathematical physics* as well as *complex analysis*. In addition, *geometric measure theory* was invigorated by the study of fractals. The dynamical origin of pattern formation in natural systems became the object of study of *extended dynamics* which employed as mathematical tools *nonlinear partial differential equations*. Today one of the most important applications of dynamics occurs in dynamical models of the atmosphere and climate change.

The past decade has seen a remarkable influx of probabilistic ideas in analysis and its applications. A new active field of *random geometry* combines ideas from mathematical physics and complex analysis in the study of random fractal curves and surfaces having their origin in physical models of magnetization, percolation and quantum gravity. Martingales and other notions from stochastics have become standard tools in *harmonic analysis*, a classical field which is prominent in applications. Probabilistic methods are central in the study of *non-equilibrium systems* where stochastic differential equations, large deviation principles and random matrix theory have played prominent roles. One of the most interesting phenomena combining both chaos and randomness is fully developed *turbulence*.