# Mathematics Spotlight on ERC projects - Europa EU

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Clusters of most used keywords in ERC projects related to mathematics research The European Research Council Set up in 2007, the European Research Council (ERC) is the first and the only evaluation criterion being excellence. To date, pan-European funding body designed to support investigator- the ERC has awarded over 8 000 long-term grants to individual driven frontier research and to stimulate scientific excellence researchers of any nationality and age who wish to carry across Europe. It aims to support the best and most creative out their research projects in Europe. With a budget of over scientists to identify and explore new directions in any field of EUR 13 billion from 2014 to 2020, the ERC is part of the EU research (Physical Sciences and Engineering, Life Sciences and Research and Innovation framework programme, Horizon Social Sciences and Humanities) with no thematic priorities 2020.

Introduction Unlike the common and extended idea that mathematics is an Similarly, mathematics has kept its traditional relationships with isolated and somehow lonely subject, research in this field, over other fields such as computer science, engineering, econometrics the years, especially in recent ones, is continuously disproving this or physics, since nowadays science builds highly on computation statement. Mathematics is probably much closer to real life than and simulation where mathematics has proven to be of great help. many other disciplines. It is playing an important role in countries’ Lately, mathematics is more and more showing its transversal economies and is key in the current data science era. value, especially in the field of life sciences. Mathematics inspires other sciences but also gets inspired by them, some very exciting Within the field itself and in addition to the classical interactions mathematical topics have originated from concepts in, for example, among different mathematical disciplines, mathematics is theoretical physics, where mathematicians have found subjects of experiencing lately an increasing collaboration between subfields, study that hide highly rich mathematical structures. where interaction was somehow not expected in the past; some examples being probability and combinatorics, commutative Mathematics is growing more and more interconnected with algebra and statistics or topology, geometry and analysis. The research at the interface of multiple mathematical subfields and joint forces and the application of well-known tools in one subfield is at the same time increasing its connections with other sciences. to another has opened up Pandora’s box, giving birth in some cases to new interesting areas of research and solving/advancing To date, the European Research Council (ERC) has supported 377 in problems that were open for years. More and more, the ERC projects in mathematics, representing more than EUR 600 million grantees build cross-disciplinary teams with very heterogeneous of investment in this research area. A sample of these projects backgrounds where their expertise complements that of other is highlighted in this brochure published on the occasion of the members, being able to attack problems that would resist International Congress of Mathematicians 2018 (ICM 2018), taking otherwise. Cross-disciplinarity is playing a crucial role in the recent place from 1 to 9 August in Rio de Janeiro (Brazil). advances in mathematics.

Classifying von Neumann algebras Group theory, functional analysis and ergodic theory – three As to group actions, whenever a group acts on a measured distinct areas of mathematics that meet within the theory of von space there is an associated crossed product von Neumann Neumann algebras. The RIGIDITY project, funded by the ERC, algebra. “The first key result obtained in the RIGIDITY project was the aims to classify families of von Neumann algebras. construction of von Neumann algebras with precisely two crossed product decompositions,” explains Prof. Vaes. In mathematics, groups encode symmetry. Functional analysis, on the other hand, provides a solid mathematical framework for Von Neumann algebras arise naturally in free probability theory. particle physics, since quantum mechanical observables behave Very little is known about the classification of the so-called free like matrices of infinite size. Last but not least, ergodic theory Araki-Woods factors. This, however, is changing as Prof. Vaes and provides us with the tools needed to understand the long-term his team have established definitive classification theorems for behaviour of a dynamical system. these factors. These three areas meet in the theory of von Neumann algebras. A third focus of the RIGIDITY project is on the quantum But how do we classify these algebras? Finding the answer to symmetries of von Neumann algebras. “Our work here led us to this fundamental question is exactly what the RIGIDITY project the first truly quantum instance of Kazhdan’s property (T), which is aims to do. “Using Popa’s deformation/rigidity theory, we classify a rigidity property used in the construction of expander graphs and von Neumann algebras that arise from group actions and free thus widely used in the theoretical aspects of computer science,” probability theory,” explains Prof. Stefaan Vaes, who leads the adds Prof. Vaes. project. “The ultimate goal is to understand which aspects of a symmetry group can be recovered from the much coarser structure given by the ambient von Neumann algebra.” Researcher: Stefaan Vaes, KU Leuven (Belgium) ERC project: Rigidity and classification of von Neumann algebras (RIGIDITY) © KU Leuven, Rob Stevens ERC funding: Consolidator Grant 2013, EUR 1.4 million (2014-2019) Stefaan Vaes is a mathematician working in operator algebras and functional analysis. He leads the Functional Analysis research group at KU Leuven (Belgium). The focus of his work is on von Neumann algebras and their connections to ergodic theory, group theory and subfactors. In 2015, he was awarded the Francqui prize, the main Belgian scientific award.

New approaches to controlling dynamics Once limited to modelling physical problems in engineering, “Although there are a range of methods for solving and controlling today Partial Differential Equations (PDEs) are used by a PDEs, some of the key mathematical issues remain unsolved, thus diverse array of fields, from natural resources to meteorology, limiting their applicability to real-life problems”. aeronautics, oil and gas and biomedicine – to name only a few. But key mathematical issues remain unsolved, particularly when Prof. Zuazua and his team of international researchers aim to addressing their control, a must in technological transfer. The make a ground-breaking contribution to the broad area of ERC-funded DYCON project aims to find answers. Control of PDEs. To do this, their work focuses on addressing the key unsolved analytical and computational issues in the A PDE is a model used to describe motion: elastic bodies, fluids, field. “The coordinated and focused effort that we aim to develop crowds, opinions, etc. It is expressed as an algebraic system is timely and much needed to solve these issues and will help bridge involving the partial derivatives of an unknown function and the gap between PDE control and real applications via computer state (depending on space and time variables). PDEs are used simulations” says Prof. Zuazua. to mathematically formulate and describe the motion of the relevant entities in nature and industry, such as heat and sound, The numerous algorithms, tutorials, sample codes, software and fluid flow, elasticity, infections and diseases. simulations developed in the course of the project will all be made available via the DYCON Computational Laboratory. Freely PDEs must be solved and, often times, in practical applications, available via the project webpage, these methods and tools have also controlled, designed and tuned. Prof. Enrique Zuazua explains: also been released via Zenodo, GitHub and MathWorks. Researcher: Enrique Zuazua, Fundación Deusto (Spain) ERC project: Dynamic Control and Numerics of Partial Differential Equations (DYCON) ERC funding: Advanced Grant 2015, EUR 2 million (2016-2021) Enrique Zuazua is the Director of the Chair in Computational Mathematics at DeustoTech Laboratory at the University of Deusto, Bilbao and a Professor of Applied Mathematics at the Universidad Autónoma de Madrid – UAM (Spain). His fields of expertise, in the broad area of Applied Mathematics, cover topics related with Partial Differential Equations, Systems Control and Numerical Analysis.

Getting to grips with (slow) chaos Chaotic systems are everywhere: the weather, molecules in a gas, tends to display different chaotic features, so it’s hard to identify the stock market. Small variations in initial conditions can lead common phenomena and mechanisms for chaos.” to a drastically different time evolution, a phenomenon known as the butterfly effect. Systems can be classified according to how Many of the parabolic systems we know best have a form of fast different, nearby initial conditions diverge in time. Supported ‘homogeneity’, and in some sense have a lot of internal structure. by the ERC, Prof. Corinna Ulcigrai is investigating systems for One of the project components is to study ‘perturbations’ of this which nearby initial conditions diverge slowly in time, to uncover system, where this homogeneity and structure breaks down. For mechanisms which explain their complex behaviour. these ‘inhomogeneous’ systems, one cannot use algebraic tools, but some geometric features still persist and can be exploited. The ChaParDyn project is focussing on so-called parabolic dynamical systems: mathematical models for the many While fundamental research, such as that carried out by Prof. phenomena which display a “slow” form of chaotic evolution. Ulcigrai, is not driven by potential applications of results, virtually Examples of parabolic systems span from famous physics all of the systems that impact our everyday lives are chaotic, so models, like the Novikov model describing electrons in metals, ‘understanding chaos’ mathematically is essential. The project’s to fundamental mathematical objects, such as flows on surfaces. research highlights the central role played by geometric phenomena in explaining the features of ‘slow chaos’. The Prof. Ulcigrai, who leads the project, explains, “When it comes to project also demonstrated that when homogeneity is broken, parabolic dynamical systems, the problem is that only few examples typical features do indeed seem to appear, which gives hope for are understood well and have been studied in depth. Each example a unifying description. Researcher: Corinna Ulcigrai, University of Bristol (United Kingdom) ERC project: Chaos in Parabolic Dynamics: Mixing, Rigidity, Spectra (ChaParDyn) ERC funding: Starting Grant 2013, EUR 1.2 million (2014-2019) Corinna Ulcigrai studied at the Scuola Normale in Pisa (Italy) and received her PhD from Princeton University under Y. Sinai in 2007. She became Full Professor at the University of Bristol in 2015 and at the University of Zurich in 2018. Her awards include a European Mathematical Society Prize (2012), a Whitehead Prize (2013) and a Wolfson Research Merit award (2017).

New tools to understand the large-scale behaviour of complex systems Understanding complex structures means separating To help researchers make this separation, the StrucLim project irrelevant information to get to something simpler and easier developed a variety of mathematical tools for compressing data to understand. When you look at something from a distance – describing large, complex systems into a simpler, more practical although you don’t see all the details, you can still describe what form. Many of these tools are based on the emerging subject you see. ERC grantee Balázs Szegedy has developed several of graph limit theory. “An important achievement of the project mathematical tools for providing a compressed yet useful view is that we were able to connect random matrix theory with graph of complex structures. limit theory through our results on random regular graph,” explains Prof. Szegedy. “It is always exciting to find connections between When it comes to understanding complex systems, the challenge different fields.” is to separate the signal from the noise. For example, to describe how water flows, you don’t need to ‘see’ the position of every In graph limit theory, the project developed methods for studying single molecule, a course, large-scale view is often enough to the large-scale behaviour of bounded degree random networks predict the behaviour of the system. and many other sparse networks. In higher order Fourier analysis, the StrucLim team researchers obtained a unified treatment of “To understand complex systems, we usually don’t need a complete the so-called Gowers norms and Host-Kra semi-norms. “These description of all the parts,” says Prof. Szegedy. “The challenge is to results have various applications. For example, we obtained a separate the relevant information, known as the structure, from the structure theory for the characteristic factors of the Host-Kra semi- irrelevant noise.” norms for nilpotent actions.” says Prof. Szegedy. Researcher: Balázs Szegedy, Hungarian Academy of Science - Alfréd Rényi Institute of Mathematics (Hungary) ERC project: Limits of discrete structures (StrucLim) ERC funding: Consolidator Grant 2013, EUR 1.2 million (2014-2019) Balázs Szegedy is a Hungarian mathematician. After finishing his PhD in 2003 he held short term positions at Microsoft Research and the Institute for Advanced Study (USA). In 2006 he joined the University of Toronto (Canada) and he returned to Hungary in 2013. Currently he works at the Alfréd Rényi Institute of Mathematics. His research focuses mainly on combinatorics and group theory.

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Understanding the deformation spaces of geometric structures Funded by the ERC, Prof. Anna Wienhard studies several new Higher Teichmüller spaces are connected components of the variety geometric forms that have been discovered over the past of representations of the fundamental group of a surface into a 20 years. These structures are closely related to the generalisation semi-simple Lie group, which consist entirely of discrete and faithful of Teichmüller space, which describes how the surface of a pretzel representations. There are two families of higher Teichmüller spaces: can be endowed with the geometry of an Escher painting. Hitchin components, defined for split real Lie groups; and maximal representations, defined Lie groups of Hermitian type. Topologically, a surface is classified by its genus, which is determined by the number of holes in it. A sphere with no holes Researchers recently discovered the underlying structure, which has genus zero, whereas a donut is genus one and a pretzel explains why there are Teichmüller spaces for some groups but genus three. If the genus is greater than one, the uniformisation not others. “This underlying structure is a new notion of positivity in theorem states that the surface be endowed with a hyperbolic semi-simple real Lie groups, which at the same time generalises the structure. concept of total positivity and the Lie semi-groups of Lie groups of Hermitian type,” says Prof. Wienhard. There are an infinite number of ways of endowing such a surface with a hyperbolic structure. Known as the Teichmüller space, According to her, this new notion of positivity raises additional this is a prime example of a ‘deformation space’ of geometric questions. “Total positivity plays a role in many areas of mathematics structures. In general, a deformation space describes all possible and has applications in, for example, statistical mechanics,” she ways in which a given topological manifold can be endowed says. “I think the new positivity will be a big part of this project’s with a given geometric structure. Prof. Wienhard is investigating legacy, and I want to explore if our notion provides new approaches such deformation spaces, in particular, those related to higher for applications.” Teichmüller spaces. Researcher: Anna Wienhard, Heidelberg University (Germany) ERC project: Deformation Spaces of Geometric Structures (GEOMETRICSTRUCTURES) © HITS (Heidelberg Institut für ERC funding: Consolidator Grant 2013, EUR 1.6 million (2014-2018) Theoretische Studien) Anna Wienhard is a Full Professor at the Mathematical Institute of Heidelberg University, and a group leader at the Heidelberg Institute of Theoretical Studies (Germany). She is a Fellow of the American Mathematical Society. Her research is at the interface between geometry, topology and algebra. She investigates moduli spaces of geometric structures.

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Statistical inference for random functions and measures How does one infer the dynamics of a DNA minicircle in solution? “While our work here was motivated by the dynamics of DNA in How does one align the neuronal firing patterns of several solution (seen as a curve moving in space, hence giving rise to a neurons across individuals? These questions are intrinsically time series of curves), one can imagine a plethora of other situations statistical, but nevertheless escape the traditional tools of where data fall in this framework,” says Prof. Victor Panaretos. statistics. The ComplexData project investigated such questions from a mathematical and an applied context. Another of the project’s key contributions was a novel framework to analyse physiological processes, such as neuronal firing patters, The data revolution produces not only big data, but often among several individuals where each individual has their own, complex data too: objects whose intrinsic structure requires and unknown, time scale. This corresponded to viewing the data a more sophisticated mathematical formalism than usual to as random elements of the so-called Wasserstein space, and analyse statistically. For instance, such objects may lie in spaces using tools from optimal measure transport to define appropriate that are infinite dimensional and/or curved. The ComplexData notions of means and common time-scales. project, funded by the ERC, developed theory and methodology for such data arising in contexts from biophysics to neuroscience. The project extended existing techniques, but also generated and applied new ones. “We were able, for instance, to provide Among its main contributions was a novel spectral framework to rigorous statistical evidence on the nature and degree of association analyse time series of curves or surfaces – time series in Hilbert between the base-pair composition and mechanical behaviour of spaces - where the observation at each time point is a function DNA minicircles at persistence length” Prof. Panaretos explains. itself. This was applied to provide the first rigorous statistical evidence that the base-pair composition of DNA strands influences their dynamical behaviour. Researcher: Victor Panaretos, École Polytechnique Fédérale de Lausanne - EPFL (Switzerland) ERC project: Statistics for Complex Data: Understanding Randomness, Geometry and Complexity with a view Towards Biophysics (COMPLEXDATA) © Alain Herzog, EPFL ERC funding: Starting Grant 2010, EUR 0.7 million (2011-2016) Victor Panaretos is Associate Professor of Mathematical Statistics at EPFL (Switzerland). He obtained his PhD at UC Berkeley (USA) in 2007, receiving the Lehman Award for his thesis. His research focusses on functional and geometrical statistics. An elected fellow of the International Statistical Institute, he serves on the board of several journals, and has been named the 2019 Bernoulli Society Forum Lecturer.

Spectral density kernel amplitudes © Annals of Statistics, Institute of Mathematical Statistics

Unravelling the mysteries of homogeneous dynamics How do you study arithmetic objects like integer points using A key achievement of this work was the creation of a very general the theory of dynamical systems? The answer is homogenous joining classification theorem for higher rank diagonalisable dynamics, and this connection goes both ways. The groups. “In some sense, this tool is a definitive result: one that has GMODGAMMADYNAMICS project, funded by an ERC grant, took already found interesting applications,” says Prof. Lindenstrauss. a broad approach towards studying this rich interplay. For example, this joining result was used by other researchers to establish the joint distribution of integer vectors on Eucleadian Homogeneous dynamics is the study of the asymptotic properties spheres and the shape of the lattice orthogonal to this vector. of the action of Lie groups on homogenous spaces such as the space of lattices in n dimensional space. “This project looked at Another important outcome was the solving of a 50-year-old the connection these actions have to number theory, to the spectral conjecture on local limit theorem for random walks on the theory of homogenous spaces and to arithmetic combinatorics,” isometry group of Euclidean spaces by a team member, Dr. says Prof. Elon Lindenstrauss. “It’s a beautiful area that interacts Peter Varju, which gave way to a very precise local-central limit with a lot of mathematics.” theorem for such random walks. One of the mysteries that the project team addressed was the Despite these significant breakthroughs, Prof. Lindenstrauss says subtle rigidity properties of higher rank abelian actions on that this is just the beginning: “There are ideas developed in this homogenous spaces. They also focused on the combinatorial project that we are eager to take even further forward. Furstenberg’s properties of arithmetic groups, equidistribution, spectral gaps, Conjecture on x2 x3 invariant measures and Littlewood’s Conjecture random walks and quantum ergodicity. are both still open, so there’s still plenty more to think about!” Researcher: Elon Lindenstrauss, Hebrew University of Jerusalem (Israel) ERC project: Dynamics on homogeneous spaces, spectra and arithmetic (GMODGAMMADYNAMICS) ERC funding: Advanced Grant 2010, EUR 1.2 million (2011-2016) Elon Lindenstrauss obtained his PhD at the Hebrew University of Jerusalem (Israel) under the guidance of Benjamin Weiss in 1999. He held positions at the IAS in Princeton, Stanford University and Princeton University (USA) before returning to the Hebrew University in 2008 where he is currently Professor of Mathematics and since 2016 the Chair of the Einstein Institute of Mathematics. He received several prizes for his mathematical achievements, including a Fields Medal. He is a member of the Israel Academy of Sciences and Humanities and the Academia Europaea.

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JZ-04-18-565-EN-C “The European Research Council has, in a short time, achieved world-class status as a funding body for excellent curiosity-driven frontier research. With its special emphasis on allowing top young talent to thrive, the ERC Scientific Council is committed to keeping to this course. The ERC will continue to help make Europe a power house for science and a place where innovation is fuelled by a new generation.” Jean-Pierre Bourguignon ERC President and Chair of its Scientific Council The European Research Council Executive Agency may not be held responsible for the use to which this information may be put, or for any possible errors. © European Research Council Executive Agency, 2018• Reproduction of the text is permitted provided the source is acknowledged. Reproduction of the http://erc.europa.eu photographs is prohibited. DOI 10.2828/24832 - ISBN 978-92-9215-070-9

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