Rényi's Two Researchers in a New International Simons Collaboration

The New Simons Collaboration will explore discrete subgroups of Lie Groups, which play a fundamental role in several areas of modern mathematics and are closely connected to theoretical physics. The Simons Foundation does more than simply fund large research grants: its flagship programmes also include long-term, major international collaborations. One of the programme’s key features, which has attracted sustained international attention, is its support for exploratory, or basic, research – that is, research pursuing long-term scientific questions – as well as scientific collaborations. Two senior research fellows at Rényi Institute, Miklós Abért and Gábor Pete, participate in this major international research collaboration. The project will launch in January.

The Simons Foundation recently announced the launch of the Simons Collaboration on Discrete Subgroups of Lie Groups. Lie groups are mathematical structures used to describe symmetries. They make it possible to capture mathematically, for example, the rotation of a sphere, movements in space, or certain symmetries of physical systems. Miklós Abért, a senior research fellow in the Rényi Institute’s Algebra Research Group and one of the Hungarian Principal Investigators (PI-s) in the successful Simons grant, explains the significance of Lie groups: “If we take three-dimensional space or a two-dimensional plane, it has congruences, that is, isometries, such as translations, rotations or reflections. If we reflect across an axis, or across a plane in three dimensions, this is a symmetry of the space, because applying it gives us the same space back. But as a movement, it is not the identity, because it sends many points to different points. A symmetry is a movement that leaves the entire shape unchanged. If we take a circle, for example, it can also be rotated or reflected. The symmetries of a given space form a group. Here, ‘group’ means that I can apply two symmetries one after the other – in other words, multiply them – and obtain another symmetry. For example, applying two reflections across axes one after the other results in a rotation by twice the angle between the two axes. The first interesting observation is that the order in which I do this matters. If you have two axes that intersect, say, at an angle of 10 degrees, then the order in which you use the axes matters when performing the resulting rotation. One gives a rotation of plus 20 degrees, while the other gives minus 20 degrees. In other words, the group is non-commutative: the order in which I multiply the elements matters.” Lie groups can be thought of as the continuous symmetry groups of so-called symmetric geometric spaces. For example, the isometries of a sphere, a plane or hyperbolic space form such groups. Lie groups are named after the Norwegian mathematician Sophus Lie.

The study of Lie groups is closely intertwined with the study of their discrete subgroups. These contain only certain, separated elements among the possible symmetries described by a Lie group. For example, if we inscribe a cube in a sphere, the symmetry group of the cube is a discrete subgroup of the symmetry group of the sphere. “Discrete subgroups have important connections and applications, among other areas, in mathematical physics. Within mathematics itself, there are also many fields where it is essential to understand them,” Abért explains.

Lie groups are important tools in modern mathematics and physics, playing a role in areas including quantum physics, general relativity and particle physics. The new collaboration explores this field from three closely interconnected perspectives: randomness, representation theory, and geometry and dynamics.

The collaboration brings together 16 PI-s from five countries (by affiliation, nine are based in the United States, three in Israel, two in Hungary, one in Poland and one in Germany), all outstanding researchers from leading universities and research institutions around the world. The collaboration is led by Yair Minsky, a mathematician at Yale University. Explaining the choice of the three research areas, he said: “These three areas, which are all quite classical, have each developed their own culture and their own ways of asking questions. It seems to us that there are ripe signs of connections that we want to try to get to by learning from each other.”
As mathematical fields begin to influence one another, it is like building a bridge between two islands,” Abért explains. “Then another bridge is built to a different field, and, following the idea that a friend of my friend is also my friend, these fields begin to influence one another as well. Topics in mathematics that have an impact on many different areas are particularly interesting because they also allow those areas to influence each other. It is enormously valuable when mathematical energy can flow between different fields. Discrete subgroups of Lie groups are one such topic: they touch on five or six different areas of mathematics.”

For Rényi Institute, the collaboration represents more than professional recognition and significant funding. The two Hungarian researchers are becoming part of a joint effort that, as Abért puts it, could lead to “new mathematical understanding.” Returning to his earlier metaphor, the 16 leading researchers demonstrated in their successful proposal that these bridges can indeed be built: “We knew about each other, but many of us had never had direct mathematical collaborations with one another before. Now we have shown that this can change, and also how it can change,” he adds.

Abért also describes how the proposal itself was prepared. “It was not easy, since there were 16 of us writing it,” he says. “A smaller group coordinated the process under Yair Minsky’s leadership, but we all read each other’s texts. We started asking questions of one another, and there was a fairly long period of learning. There were discoveries, but also failures as we searched for common directions. But that is a natural part of our work and of science itself. After a while, it became clear where we were heading. This was a useful process in its own right, apart from the grant application,” he adds. “The reason two researchers from Rényi Institute are involved is that we can represent more than one of the main areas. Gábor Pete (senior research fellow at Rényi Institute’s and head of Probability and Statistics Research Department, ed.) represents probability theory, while I represent geometry and dynamics. We already had the professional communication between us that was needed for this project. In other words, we had already built several bridges between our respective fields, and this allowed us to mediate between the others as well, taking on a kind of mathematical translator role. The strong emphasis on probability among the research areas was partly due to the fact that the two of us, representing two different mathematical fields, understand each other so well. The probabilistic perspective is perhaps the newest element in this network of connections, so we hope it will open up particularly many new directions.”
 

Under the current programme, the Simons Foundation plans to support no more than three new MPS Collaborations each year. According to our information, around 150 groups applied in this round. To our knowledge, this is the first time that researchers based at a Hungarian institution have participated as PI-s in a Simons Collaboration.

“There are around 25 ongoing Simons Collaborations. The funding is for four years, and if the programme proves highly successful, it can be extended for another three years. The fields covered are mathematics, physics, their interfaces, and computer science. The grant provides up to 8 million dollars in total over four years,” Abért explains.

In response to a question from renyi.hu, he also shared that securing this kind of funding is “extremely difficult.” “Per researcher, it provides fewer resources than a large individual ERC grant, but it is a completely different type of support, and it sends a different message as well. We can be particularly proud that two PI-s from  Rényi Institute participate at the same time. Rényi has a strong international reputation, and this collaboration can strengthen it even further.”

“What matters about this funding is not only its size, but also the idea behind it. The Simons Foundation brings together researchers working in different fields because it starts from the premise that fundamental new results often emerge precisely at the intersection of different areas. It provides the time, freedom and genuine collaboration needed to make this possible. I find it particularly valuable that a private foundation is doing all this from its own resources because it considers basic research important. There is very little tradition of this kind of support in Hungary. To me, this culture of science funding is at least as noteworthy as the funding itself.” 

An important feature of the Simons model is that it supports long-term, higher-risk basic research that can be difficult to fund through traditional grant systems. Its aim is to accelerate research into major problems in mathematics, theoretical physics and theoretical computer science whose solutions could potentially redirect the development of an entire field. The Simons model builds an entire research community around a particular important question. Leading researchers from several countries and institutions work together, regularly sharing ideas while they are still taking shape. This creates opportunities for connections between seemingly distant areas of mathematics or theoretical physics that are less likely to emerge through a conventional conference or grant system. In these fields, a researcher may spend years working on a very narrowly defined problem. Meanwhile, a mathematician or physicist working in another area may already have a method that could provide the key to solving it. The Simons model brings together the best researchers while ensuring that they approach the same problem from different perspectives. The collaboration announced now is an ideal example of how the Simons model works. The project brings together geometry and dynamics, representation theory and probability theory, while its results may have connections to fields such as quantum physics and robotics.

A key element of the Simons Foundation’s philosophy is to involve young researchers in the collaboration. Postdocs and PhD students will be closely engaged in the work, with intensive interaction taking place not only online. The research leaders are also expected to meet in smaller subgroups, as well as once a year in a plenary meeting. The Hungarian team’s commitment also includes two semesters of activities at Renyi's Erdős Center. “Of the two semesters at Erdős, one will be organised under the umbrella of the collaboration, while the other will be closely connected to it. We expect that many members of the collaboration will also attend the second one. Postdoctoral researchers from the group will come, and there will be schools for young researchers with around 100 participants. This is another reason why it is so valuable that we have Erdős Center. During the application process, they also look at what we are able to offer: whether there is a(n infra)structure behind the beauty of the mathematics, and whether we can put it to use. The heads of the institutions involved were also formally asked to confirm their support. András Stipsicz, Director General of Rényi Institute, formally committed in writing as part of the application that, if we won the grant, Erdős Center would organise these two semesters. Erdős Center represents both infrastructure and quality assurance, making it a valuable addition in the eyes of decision-makers. This may also have played a role in our success,” Abért says, emphasising that Erdős Center, which has operated at Rényi Institute since 2019 and has already gained international reputation, has previously received a Simons grant. That grant, however, was specifically for organising international scientific events.(Click HERE for our interview with Károly Böröczky, director of Erdős Center.)

Results are to be delivered within four years. An important part of the Simons Foundation’s policy is that it does not focus on what can be measured quickly, but on creating the opportunity to ask big questions and work on them over a long period of time. In basic research, it is impossible to predict in advance when and how a mathematical discovery will lead to a practical application. The history of mathematics offers countless examples of theories that initially appeared completely abstract but became technologically significant decades later.“The real, most powerful impact often does not happen within those four years, even if our collaboration leads to significant results,” Abért emphasises. “The same principle applies here: a new generation of researchers will emerge who are already at home in all of these fields, and they may achieve the biggest breakthrough, perhaps ten years from now. Basic research is always a long-term investment, even when it has specific benefits in the shorter term. But if theoretical mathematicians were forced to look for and produce direct, practical applications, that would come at the expense of our theoretical work. You don’t use a camera to hammer a nail into a wall. The people evaluating these major grants think of such projects as investments whose benefits will not necessarily be theirs directly, but will ultimately belong to humanity as a whole. Of course, they also need to make sure that the people entrusted with these substantial resources are themselves thinking responsibly.”

 “In theoretical mathematics, there is a kind of collective, more developed way of thinking – we have no choice but to work for the long term, because shorter time scales can only be interpreted in an artificial way. This does not mean that I do not plan. I need to know what I am thinking about today, or even what I will be thinking about next week. Otherwise, I am just staring at something. There has to be intention, awareness and discipline. But if I start from the assumption that I have to produce something this year and write a certain number of papers, I will end up doing bad mathematics. And in the same way, at the level of science, you get bad agriculture or bad sociology if all you have in mind is short-term planning. So this is not only true of mathematics, although mathematics is an extreme example: those who want to produce papers on a quantitative scale will inevitably end up writing a lot of bad papers.”

“For mathematicians, it goes without saying that ‘we work for eternity’, and this is one of the reasons why we invest an extraordinary amount of energy in the next generation. In Hungary, there is a culture of mathematics, and of research in general, including basic research. There is still genuine respect for intellectual achievement, which matters greatly when young people are deciding what they want to devote their professional lives to. People can spend 10 or 20 years thinking about a major mathematical question.” 

Miklós Abért totally agrees with the suggestion that this could also serve as an example for humanity as a whole. Among other things, for instance, climate thinking could benefit from such a long-term perspective. “Certainly. It would be good if humanity could learn from this, although I have serious doubts that it will. But deep down, most people know that thinking only about tomorrow is both problematic and ominous.”