Weierstrass lecture focusses on the distribution of zeros of L-functions
Professor Dr Peter Clive Sarnak, the keynote speaker at this year's Weierstrass Lecture at Paderborn University, is one of the world's leading number theorists. His work has led to fundamental developments, for example in the context of the so-called Ramanujan conjecture or the "Golden Gates". Professor Dr Tobias Weich, Head of the Spectral Analysis Group at Paderborn University, puts the results in context: "Peter Sarnak has demonstrated extraordinarily profound results - for example, the best known approximations to the famous Ramanujan conjecture, a seemingly simple but in reality incredibly difficult statement about module and measure forms. Such results are almost impossible for outsiders to grasp. They take place in the most abstract areas of pure Mathematics."
Professor Sarnak, who will be giving insights into his research in Paderborn on 12 June, worked at Stanford University, USA, until 1991. Since 1991, he has been a professor of Mathematics at Princeton University, also in the USA. Since 2007, he has also been one of eight permanent members of the prestigious School of Mathematics at Princeton. His honours include the George Pólya Prize of the Society for Industrial and Applied Mathematics (SIAM), the Ostrowski Prize and the Levi L. Constant Prize. He was honoured with the Frank Nelson Cole Prize from the American Mathematical Society for his work on number theory.
Professor Weich explains: "The really impressive thing is that Sarnak repeatedly obtains very concrete, application-relevant results from highly abstract theorems." The Institute for Advanced Study in Princeton has therefore dedicated the portrait "A Connector's Connector" to him - a "connector" who builds bridges between seemingly unrelated disciplines. Two examples illustrate this particularly well. These are the Ramanujan graphs and the Golden Gates.
"The most amazing gift that pure mathematicians have ever given to theoretical Computer Science"
Many technical systems can be described as a graph: as a network of points and connections. The World Wide Web is such a graph - every website is a node, every hyperlink a connection. A paradoxical combination is desired for such networks: economical (few connections per node, which saves costs) and yet highly networked (short paths between every two points, without bottlenecks). Professor Sarnak constructed such optimal networks in 1988 with Alexander Lubotzky and Ralph Phillips using pure number theory. These Ramanujan graphs are demonstrably the best possible of their kind and can be used directly in practice. Today they are used in algorithms, data storage and cryptography. The Yale computer scientist Daniel Spielman aptly described how surprising this gift was: "the most amazing gift that pure mathematicians have ever given to theoretical Computer Science."
"Golden Gates" - building blocks for quantum computers
A second, highly topical example leads directly to quantum technology, one of the major focal points of research here in Paderborn. A quantum computer calculates by manipulating quantum states using gates - the quantum mechanical counterparts to the building blocks of conventional chips. The crucial question is: which gates compose arbitrary arithmetic operations with as few steps as possible and as cost-effectively as possible? "Sarnak has shown that this question is also essentially number-theoretical. His 'golden gates' have been proven to deliver particularly efficient quantum gates. A recent further development (2025) manages with around ten times fewer of the particularly 'expensive' T-gates for certain operations than the standard methods - and it is precisely these that are complex to realise in practice," says Professor Weich.
Mathematics and quantum research in Paderborn
The decomposition of gates into smaller building blocks is a topic that Mathematics at Paderborn is actively involved in as part of the Institute for Photonic Quantum Research (PhoQS). The work takes place at the interface between Mathematics, Physics and Computer Science. PhoQS brings together expertise in the fields of photonics and quantum research at Paderborn University. Thanks to an interdisziplinär team of leading experts from Physics, Mathematics, Computer Science and Electrical Engineering, PhoQS conducts excellent research in the fields of quantum simulation, communication, metrology and computing. In 2024, for example, Germany's first light-based quantum computer (PaQS) was able to start work in Paderborn.
Paderborn and Bielefeld scientists are also investigating integer structures in a Collaborative Research Centre (CRC) funded by the German Research Foundation (DFG) since 2023. In geometry, for example, such structures appear as pavements - repeating partial surfaces, arranged symmetrically and without gaps, which can result in complex, multidimensional patterns. However, mathematicians encounter integer structures not only in geometry, but also in algebra, analysis, number theory and other mathematical subfields. Mathematics research at the interface of these sub-areas characterises the scientific profile of the Paderborn Institute of Mathematics.
Professor Weich states: "Both examples, i.e. the Ramanujan graphs and the Golden Gates, are based on the same mathematical foundation - arithmetic groups, quadratic forms, modular forms - i.e. precisely those 'integer structures' that are at the centre of our special research area. Sarnak's virtuosity, with which he links various sub-disciplines, is an outstanding source of inspiration for many of the CRC's projects. He epitomises in the most impressive way what our discipline can achieve: turning seemingly abstract basic research into relevant applications that nobody could have foreseen."
All interested parties are cordially invited to attend the Weierstrass lecture on Friday, 12 June, from 4 to 6 p.m. in lecture theatre O1.
[This text was translated automatically]
