The Alfréd Rényi Institute of Mathematics is a significant center of international mathematical life. It provides an ideal working environment for mathematical research; since its foundation in 1950 it has played a prominent role in shaping Hungarian mathematics. The institute has a lot of experience in handling larger scale grants; in the past decade it hosted 7 Lendület (Momentum) grants and 9 ERC grants (one of them is a Synergy Grant). In addition to its world class ranking in research, it is engaged in scientific publishing and networking. The institute regularly runs summer schools for young researchers, seminars, named lecture series, and major conferences, sometimes with hundreds of participants.
The Renyi Institute is a research institute in mathematics, focusing mainly on theoretical research, but also open for collaborations related to applications that lead to fundamental mathematical questions. Realizing new trends in mathematics and other sciences, the institution has started new research directions in the subject of mathematical foundations of AI (mainly concerning neural networks and deep learning). The institute has built up an AI team that has a careful balance of theorists and deep learning / machine learning practitioners. The practitioners in the team are well-versed and up-to-date in the extremely fast-paced world of deep learning, but at the same time, they can to contribute to foundational research. The theorists of the team are world-class experts in the highly abstract theoretical machinery, but at the same time, they do not shy away from running simulations. This balance creates an optimal environment for a free flow of ideas between theory and practice, thus, underpins pursuing the general goal to bridge the gap between mathematical theory and deep learning practice.
Project highlights
Dynamics and Structure of Networks (DYNASNET)
Start date: 1 September 2019
End date: 31 August 2025
Overall budget: €9 315 424 (Rényi budget: €3 507 573)
Funded under: H2020-EU.1.1.
Networks define our life. They are essential to biology, communications, social and economic systems, they influence virtually all areas of science and technology. But their workings are not fully understood. László Lovász from Alfréd Rényi Institute of Mathematics and Jaroslav Nešetřil from Charles University in Prague, renowned mathematicians specialising in graph theory, and Albert-László Barabási, a leading expert in network science based at the Central European University in Budapest, aim to build a mathematically sound theory of dynamical networks. They want to transform our understanding of complex systems and prepare ground for applications in multiple disciplines.
Both graph theory in mathematics and the study of networks have made major conceptual advances in the past decade. However, the research communities working in these two disciplines had little conversation between each other, and that limited our insight. The research funded with an ERC Synergy Grant can potentially change it, constructing a coherent theory of dynamical networks, and exploiting its applications and predictive power for various real systems. To enhance the wider impact of the proposed mathematical advances, the principal investigators plan to establish steady links with experts from different domains that encounter and explore networks, from cell biology to brain science and transportation and communication networks, inspiring with novel questions and helping the application of our advances in these domains.
Limits of Discrete Structures (StrucLim)
Start date: 1 February 2014.
End date: 31 July 2019.
ERC call details: ERC-2013-CoG, PE1
Max ERC Funding:1 175 200 €
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. When it comes to understanding complex systems, the challenge 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 single molecule, a course, large-scale view is often enough to predict the behaviour of the system. “To understand complex systems, we usually don’t need a complete description of all the parts,” says Prof. Szegedy. “The challenge is to separate the relevant information, known as the structure, from the irrelevant noise".
To help researchers make this separation, the StrucLim project developed a variety of mathematical tools for compressing data describing large, complex systems into a simpler, more practical form. Many of these tools are based on the emerging subject of graph limit theory. “An important achievement of the project is that we were able to connect random matrix theory with graph limit theory through our results on random regular graph,” explains Prof. Szegedy. “It is always exciting to find connections between different fields".