Tags: TAG News Forschung Mathematik, TAG News DMI]]>

As part of the ERC project **Advances in effective evolution equations for classical and quantum systems (AEQUA)**, Saffirio is addressing the question of how, within the framework of the kinetic theory of gases, rigorous derivation of effective macroscopic equations can be derived from microscopic laws of classical and quantum mechanics. "If you look at plasmas and gases at microscopic scale, you can see that they are made of a huge number of particles. In the transition from the microscopic to the macroscopic level, they appear to exhibit a collective behavior that can be described by simpler models called effective equations. I investigate in a rigorous way the step from the microscopic to the macroscopic description to detect the regimes of applicability of certain physical models."

Addressing the aforementioned question could contribute to the resolution of longstanding open problems, such as the emergence of irreversibility from microscopic time-reversible dynamics. "In many microscopic systems, you can go back and forth in time. This is no longer the case at the macroscopic scale where we experience the emergence of the so-called arrow of time." Furthermore, addressing this question will lead to the development of innovative mathematical methods originating from bridging different branches of mathematical physics and analysis, namely classical and quantum many-body systems, kinetic theories, and PDEs.

The AEQUA project has four main goals: The derivation of the Vlasov-Poisson equation with Coulomb and gravitational interactions from many-particle quantum dynamics, the derivation of the Vlasov-Poisson equation from Newtonian mechanics in the mean-field regime, the derivation of the quantum Boltzmann equation from a system of many interacting fermions in the weak coupling limit and the derivation of the classical Boltzmann equation from a system of many classical particles considering the boundary conditions.

In her everyday life, Saffirio often feels reminded of her research: “If you take, for example, the formation of political opinions in the run-up to an election, you look at an entire nation of voters - millions of people, informing themselves, talking to each other, reevaluating their opinions and so on. Basically, this is a highly complex system – I can trace back the interaction between two people but looking at the entire population at the macroscopic level, I can no longer unravel who spoke to whom, when an interaction took place and what impact it had." Macroscopic effective equations approximating in some sense these complex systems are the key to understand these phenomena and make predictions that turn out to be useful for applications. The same applies to market research or the prediction of disease progression. The search for reduction of complexity in highly complex systems is neither new nor rare. "Of course, these are simply some practical examples that I don't actively work on as a researcher, but they show some of the many reference points between mathematics and our day-to-day lives." The methods developed within the framework of AEQUA will offer a new perspective on the derivation of kinetic equations with special focus on the derivation of the Boltzmann equation. They have long been of great importance for mathematics, theoretical physics, and the philosophy of science and have served as prototype models for applications, not only in physics, but also in biology, economy and social sciences.

The ERC Starting Grants are among the most renowned grants for young researchers in Europe. The ERC awards these grants for innovative basic research and promotes the independent work of talented young researchers.

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Isabel Wagner, born in 1979, studied computer science at the University of Erlangen, where she earned her doctorate in 2010 with a thesis in the area of wireless networks. Following a period as a JSPS Postdoctoral Fellow at Osaka University in Japan, she worked as a lecturer at the University of Hull, before moving to De Montfort University in Leicester in 2015. There she was appointed senior lecturer in 2016 and promoted to associate professor in computer science (cybersecurity) in 2019.

Wagner’s research activities are focused on privacy protection and privacy-enhancing technologies, particularly metrics to quantify the effectiveness of privacy protection mechanisms. Potential fields of application include smart cities, vehicular networks, smart grids and genomics.

She is also interested in bio-inspired mechanisms for privacy protection and methods to ensure greater transparency in the use of surveillance systems by technology companies. In her 2022 book *Auditing Corporate Surveillance Systems*, Wagner describes how users are monitored by big tech companies, how this surveillance can be investigated experimentally, and what the latest studies have revealed.

The professorship for cybersecurity is based in the subject area of computer science with its focal areas of distributed systems and machine intelligence, and has numerous points of contact with research in other faculties, for example in relation to blockchain and smart contracts or data protection in personalized medicine.

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After completing his doctorate at the University of Basel, Philipp Habegger worked as a postdoctoral researcher at ETH in Zurich, where he proved a then-unproven hypothesis by Enrico Bombieri, Umberto Zannier, and his doctoral supervisor, David Masser, in 2009. Based on this result, he then introduced innovative methods based on complex techniques from algebraic geometry into the study of heights. In 2013, he expanded these to produce the first version of the result still known as the *Höhenungleichung* (‘height inequation’) among experts. His findings showed a deep relationship between the complexity of points in the parameter space of Abelian varieties and the complexity of group theory objects, thus generalizing earlier works from the 1980s by Joseph Silverman of Brown University. Abelian varieties are named after the Norwegian mathematician Niels Henrik Abel and are relevant to various fields of mathematics and theoretical physics. Having returned to Basel as a professor, Habegger continued to develop his ideas together with his co-authors, ultimately proving a uniform version of the Mordell conjecture. The Mordell conjecture represents a central connection between number theory and geometry and was first proven by Gerd Faltings in 1983, who received the Fields Medal for his work.

Habegger’s research has been highly influential internationally as well as within the University of Basel. Lars Kühne, who was an Ambizione fellow in Basel until 2020, combined the *Höhenungleichung* with concepts from the theory of uniform distribution in 2021 in order to prove a then-unsolved conjecture by Barry Mazur of Harvard University. Another version of the Mazur conjecture was proven in a positive-characteristic setting by Robert Wilms, today a postdoctoral researcher at the University of Basel.

Philipp Habegger will give his lecture at the ICM on July 9 in slot 9b. He will also speak at a satellite event at ETH Zurich on July 12. Both events will be streamed online and/or published on YouTube afterwards.

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