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Group theory, functional analysis and ergodic theory – three distinct areas of mathematics that meet within the theory of von Neumann algebras. The RIGIDITY project, funded by the ERC, aims to classify families of von Neumann algebras.
Understanding complex structures means separating irrelevant information to get to something simpler and easier to understand. When you look at something from a distance – although you don’t see all the details, you can still describe what you see. ERC grantee Balázs Szegedy has developed several mathematical tools for providing a compressed yet useful view of complex structures.
How does one infer the dynamics of a DNA minicircle in solution? How does one align the neuronal firing patterns of several neurons across individuals? These questions are intrinsically statistical, but nevertheless escape the traditional tools of statistics. The ComplexData project investigated such questions from a mathematical and an applied context.
Once limited to modelling physical problems in engineering, today Partial Differential Equations (PDEs) are used by a diverse array of fields, from natural resources to meteorology, aeronautics, oil and gas and biomedicine – to name only a few. But key mathematical issues remain unsolved, particularly when addressing their control, a must in technological transfer. The ERC-funded DYCON project aims to find answers.
Funded by the ERC, Prof. Anna Wienhard studies several new geometric forms that have been discovered over the past 20 years. These structures are closely related to the generalisation of Teichmüller space, which describes how the surface of a pretzel can be endowed with the geometry of an Escher painting.
How do you study arithmetic objects like integer points using the theory of dynamical systems? The answer is homogenous dynamics, and this connection goes both ways. The GMODGAMMADYNAMICS project, funded by an ERC grant, took a broad approach towards studying this rich interplay.
Chaotic systems are everywhere: the weather, molecules in a gas, the stock market. Small variations in initial conditions can lead to a drastically different time evolution, a phenomenon known as the butterfly effect. Systems can be classified according to how fast different, nearby initial conditions diverge in time. Supported by the ERC, Prof. Corinna Ulcigrai is investigating systems for which nearby initial conditions diverge slowly in time, to uncover mechanisms which explain their complex behaviour.
Artur Avila is a franco-brazilian leading mathematician and an ERC grantee since 2010. At the age of 16, he won the International Mathematical Olympic gold medal and before finishing high school, he received a scholarship for the Instituto Nacional de Matemática Pura e Aplicada (IMPA) of Rio de Janeiro. He is now senior researcher both at the National Center for Scientific Research - CNRS and IMPA. In this interview, Prof. Avila tells us about his international career and the research he conducts both in Brazil and France.
Prof. Ben Green is all about pure mathematics. Asked what is at the heart of modern society, he would probably insist on mankind’s capacity to solve problems and pass its knowledge on to new generations. The 37-year old mathematician can actually boast about his contribution to both: the International Congress of Mathematicians (ICM) which takes place this year in Seoul (South Korea) will see him give a plenary session which he proudly qualifies as the pinnacle of his career. With his ERC grant, he is now providing young mathematicians with an opportunity to shine.
Mathematicians are similar to historians in that they are devoted to finding and interpreting patterns. Like historians, they have to deal with criticism that some theories are of little practical benefit. This is a futile debate, says mathematician Dr Francis Brown who is attending the International Congress of Mathematicians (ICM) taking place in Seoul this summer (South Korea). Through an ERC-funded project, he has developed an algorithm of immense importance to particle physics, using numbers first developed for their aesthetic appeal over 300 years ago. Sometimes the significance of mathematics – as with history – takes time to be revealed.