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

During their time on the project, the mathematicians around Pierre Le Boudec were not concerned with finding integer solutions for all polynomials. They decided to adopt a statistical perspective on this problem: Provided that a given polynomial equation has the so-called local solutions in all completions of the field of rational numbers, it can be predicted with high probability that it also has an integer solution. These completions were classified by Alexander Ostrowski, a mathematician

working in Basel. It is known that checking the existence of these local solutions can be done algorithmically. Prof. Le Boudec and his co-authors were able to prove a conjecture of Poonen and Voloch, which had been open for almost 20 years, except for cubic surfaces.

So the algorithm Hilbert longed for exists, at least statistically, in most cases. Thus, the researchers may have come as close as possible to solving Hilbert's tenth problem.

The results of Pierre Le Boudec, Tim Browning and Will Sawin have contributed significantly to the progress of basic research in the field of number theory and have extended the structural understanding of polynomial equations.

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For the study, the researchers analyzed data from over 1.1 million inpatient cases at 102 Swiss hospitals in order to investigate the relationship between bed occupancy and the 14-day mortality rate in hospitals. In other words, the cases were observed until the 14-day mark unless they were discharged earlier.

The relationship between bed occupancy and mortality is a complex one. In addition to the occupancy of beds, it is important to consider other factors such as patient transfers in the individual hospitals, the average severity of the cases admitted to the hospital on the respective day, and the patients’ individual risk of dying. Further variables include comorbidities and the age and gender of patients. Account was also taken of the difference between weekdays and weekends and the type of hospital.

The threshold of capacity utilization above which the mortality risk increases is different in each hospital. If a patient is exposed to the bed occupancy above this value, the risk of death increases by around 2% per day. In the event of two or three additional days with excessive capacity utilization, there is a 3.2% or 4.9% increase, respectively, in the probability of 14-day mortality at the hospital. The threshold for the individual facilities ranged between 42.1% and 95.9% bed occupancy.

These considerable differences have a decisive impact on the threshold: the higher the average bed occupancy at a hospital, the higher the threshold. In the case of small hospitals, the average occupancy is around 60%, while this figure rises to 90% at large hospitals. With a lower average occupancy, larger variations can occur – and these variations in bed occupancy result in a lower threshold for increased mortality. Accordingly, this threshold is also reached more quickly.

The reasons for rising mortality in the event of higher occupancy include the fact that certain treatments can no longer be performed or are delayed. Moreover, the number of physicians and nursing staff remains relatively stable despite these wide variations.

According to Simon, the problem can be addressed by reducing the variations in occupancy and ensuring that the hospitals are adequately staffed. Above all, he believes the solution lies at the political level: “It’s difficult to operate lots of small units efficiently. Pooling hospitals or ensuring closer collaboration between them leads to less variation and therefore less risk.”

**Original publication**

Narayan Sharma, Giusi Moffa, René Schwendimann, Olga Endrich, Dietmar Ausserhofer, Michael Simon

The effect of time-varying capacity utilization on 14-day in-hospital mortality: a retrospective longitudinal study in Swiss general hospitals

BMC Health Services Research (2022), doi: 10.1186/s12913-022-08950-y

Applications are welcome and must be submitted via email togianluca.crippa@*clutter*unibas.ch. The review of the applications will start in January 2023 and will continue until the positions are filled.

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Hanspeter Kraft: The modern possibilities of a digital edition with extensive search options and internal and external links give researchers and interested laypeople a tool far exceeding the power of the printed page. Of course, the challenges involved are enormous, especially because of the mix of languages (Latin, old German and French) and the many formulas.

**What is the purpose of a digital edition?**

In the long run, the digital platform OBE-digital (which stands for Opera-Bernoulli-Euler) will contain all the works, correspondence and notebooks of not only Euler, but also Bernoulli and their circle. This includes originals, catalogs, secondary literature and more. Everything will be available as open access and free of charge. This means that interested laypeople, not just experts, can engage with the material.

**Leonhard Euler is still well-known more than 300 years after his birth. Was he really that remarkable in his own time?**

It is accurate to characterize Euler as the greatest scholar of his time, and most mathematicians today consider him the greatest and most productive mathematician of all time. It is truly impressive how many completely new things Euler was able to create in his time – ideas and theories no one had even come close to before.

**Is Euler’s work mostly of historical importance or does it remain relevant today?**

In contrast to a natural science like biology, where the historic interest is largely concerned with the creation and origins of modern ideas, Euler’s mathematics is thoroughly modern and remains relevant today. Mathematical results never become wrong! We may understand them better over time thanks to new insights, or they may turn out to be special cases of more general results, but they remain correct. A common occurrence is that classic results later have incredible applications that no one could have dreamed of at the time. Euler’s ideas and formulas are used in transportation problems (GPS), in simulations of the circulatory system, and in e-banking, as we explain in an accessible way in three short videos.

**What are your hopes for the future of the Euler Edition?**

I very much hope that young scientists will be inspired by this modern form of publishing and get fully on board. The requirements are very high, however: along with very solid historical knowledge, this project also requires good mathematical foundations and a willingness to use modern digital methods. But let’s stay optimistic!

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This is where mathematics comes in, and in particular, partial differential equations (PDEs). “A PDE is a mathematical equation that you get when you take some physical quantity – such as the temperature of water or the density of a pollutant – and link together its variations in space and in time,” explains FLIRT project coordinator Gianluca Crippa from the University of Basel in Switzerland. “The main challenge is that interactions are happening at very different scales. Pollution introduced to a lake for example will affect different parts of the lake differently, at different times.” Mathematics therefore enables us to understand these things analytically, even if we cannot fully observe them or simulate them with a computer. The FLIRT project, supported by the European Research Council, enabled Crippa to delve deeper into this world. “Mathematics is of course a theoretical science in its own right, but I also think it’s cool when we can help to answer questions in the real world,” he adds.

Through the FLIRT project, Crippa sought to develop mathematical models and techniques that can more accurately capture certain behaviours in fluid flows. “This might include adding and stirring milk into your coffee, or adding a pollutant into a lake,” he says. “Mathematical models can help to quantify how these substances will mix and homogenise, and how fast this happens.” Crippa was also interested in mathematically describing turbulence, the motion of fluid characterised by chaotic changes. “This is a fascinating process that is not quite understood mathematically,” he remarks. “Nonetheless, we can find universal behaviours. The exact shape of the water flow after a rock in a river is difficult to predict in detail, but in nature we always observe very similar patterns.”

The FLIRT project has addressed a number of questions, and also provided the first mathematical approach to some physical theories that have lacked a theoretical justification for over half a century. Around 1950, the physicists Obukhov and Corrsin made some predictions about how temperature should behave in fluid turbulence. Together with some collaborators, Crippa was able to show the accuracy of their predictions, using mathematical models. “This fits into the project’s general focus of developing structures and strategies to solve problems in fluid dynamics,” he notes. “We want to find patterns for irregular phenomena, and make sense of them. With mathematics, we can ‘see’ more than what we can observe experimentally.” Crippa also found mathematically universal bounds that govern the mixing of fluids. If you were to add dye to water, the mixing process will be governed by certain parameters such as speed and energy. Crippa was able to place limitations on how quickly the mixing of the dye into water can occur. “I’m really happy that the mathematical work we’ve done here will be of interest to colleagues from theoretical and experimental physics,” says Crippa. “I’m still very much a mathematician – I don’t have a lab and often just work with chalk and a blackboard – but I’m excited to be working at the interface with other fields. Such dialogue is very important.”

**This article is a publication of the Community Research and Development Information Service (CORDIS).**

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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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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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During the subsequent tour of several floors of the building, visitors had the opportunity to take a look at the library and marvel at a few architectural features that still remind us of the fact that the building at Spiegelgasse 1 once belonged to the Basler Kantonalbank. Furthermore, the MicroCluster of the High Performance Computing (HPC) research group was shown. In two VR deployments by the Databases and Information Systems (DBIS) research group, visitors also had the opportunity to play a room escape and view exhibits from the Basel Historical Museum (HMB) in a virtual museum.

During the aperitif that followed, the visitors were able to ask some final questions and engage with professors from both the Mathematics and the Computer Science section. One feature surely did not go unnoticed: «There's even a blackboard in the kitchen - that's when you realize people around here are doing math.» The evening ended with a piece of chalk in one hand, breadsticks in the other, and the search for answers to the question of what distinguishes an «elegant mathematical proof» from an «ugly proof».

Numerous departments, museums, and institutions covering a wide range of topics are brought together under the umbrella of the University of Basel or are associated with it. The «Uni-Einblicke» series of events aims to introduce these institutions to employees and students and allow them to get to know the university, its partners, and topics better.

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