"Automorphic Forms and Arithmetic" (AuForA) - this is the name of the project led by mathematician **Prof. Dr. Valentin Blomer**. It is in the field of basic mathematical research. In this project, Blomer investigates connections between classical number-theoretic objects such as integer matrices or integer solutions of equations on the one hand and complex and highly structured functions, the so-called automorphic forms, on the other.

At the center are three fundamental mathematical conjectures, unsolved for more than 15 years, whose conceptual common feature is the statistical behavior of automorphic forms in certain families. The project aims to help achieve substantial progress and solutions for these three conjectures. The ERC Advanced Grant will provide Blomer with approximately two million euros for his research over the next five years.

After studying mathematics and computer science at the University of Mainz, Valentin Blomer earned his doctorate at the University of Stuttgart in 2002 and habilitated three years later at the University of Göttingen, where he was an assistant professor from 2004 to 2005. He then moved to the University of Toronto, initially as an Assistant Professor, and later received a full professorship there. In 2009, he became a professor at the University of Göttingen, and since 2019 he has been at the University of Bonn. The Advanced Grant is not his first ERC funding: from 2010 to 2015, the specialist in analytic number theory already held a Starting Grant. Valentin Blomer is a member of the Hausdorff Center for Mathematics Cluster of Excellence and the Transdisciplinary Research Area "Modelling" at the University of Bonn.

**Theoretically predicting experiments at particle accelerators**

The project "Loop Corrections from the Theory of Motives" (LoCoMotive) by physicist** Prof. Dr. Claude Duhr** aims to make the most accurate theoretical predictions possible for experiments at particle accelerators - first and foremost at the Large Hadron Collider (LHC) at CERN in Geneva. At particle accelerators, subatomic particles such as protons, the building blocks of the atomic nucleus, are made to collide at very high energies. This creates new subatomic particles, such as the Higgs boson, which was first experimentally detected ten years ago at the LHC at CERN. Accurate theoretical predictions for particle accelerators require the evaluation of so-called loop corrections, the calculation of which is often very complicated.

Claude Duhr's project LoCoMotive investigates to what extent modern mathematical methods, especially from the field of the so-called Theory of Motives, can be applied to efficiently evaluate loop corrections. The project is situated at the frontier of particle physics and mathematics, with the aim of applying abstract mathematics in a concrete way to achieve theoretical predictions of unprecedented accuracy. The ERC Consolidator Grant provides Duhr's project with funding of around two million euros.

After studying at the Université Catholique de Louvain in Belgium, theoretical physicist Claude Duhr received his doctorate from the same university in 2009. In the following two years, he worked as a postdoctoral researcher at Durham University in the UK and then for two years at ETH Zurich in Switzerland. This was followed by research and teaching positions at the universities of Louvain-la-Neuve and Durham, and from 2014 to 2021 Duhr worked at CERN in Geneva. Since 2021, he has been a professor at the Institute of Physics at the University of Bonn. His overarching focus is the mathematics of precision physics.

**New quantum states in open systems**

In his project, physicist **Dr. Julian Schmitt** aims to develop a strategy with which new and stable quantum states can be generated and observed in open systems. His interest is focused on states of matter, which inherit a topological character from their coupling to the environment. Topology, the mathematical study of shapes and their geometric properties, is an important fundamental universal concept for our current understanding of states of matter, both on small and large scales - from atomic systems to astrophysics. Moreover, topological materials exhibit a high degree of robustness, which makes them an interesting resource from a technological point of view. Contrary to what had been assumed so far, openness is not necessarily a limitation for such topological systems, but it may even become a tool to generate new topological states, as suggested by Schmitt.

In his TopoGrand project, Schmitt plans to develop a novel photonic system that traps Bose-Einstein condensates of light particles in arrays of tiny optical microresonators. The goal is to generate and observe novel topological states of light. In particular, doing this at room temperature is groundbreaking. Julian Schmitt's approach and the experiments may become relevant for diverse applications, such as information processing on photonic chips. From a fundamental physics perspective, the TopoGrand project will explore the emerging links between photonics, condensed matter systems and quantum computing, and emulate finite-temperature topological systems, which are at the forefront of research in quantum physics. The ERC Starting Grant for the project is endowed with around 1.5 million euros.

After studying physics and earning his doctorate at the University of Bonn, Julian Schmitt was a postdoctoral researcher at the University of Cambridge, and has been a Junior Principal Investigator at the University of Bonn for more than two years. His area of expertise is experimental research with quantum systems of light and matter. In his research work, he has contributed to important achievements in the physics of low-dimensional quantum gases using experiments with photonic and ultracold atomic systems where quantum effects can be controlled and studied under extreme conditions and with great flexibility. In 2020, he received an Independence Grant from the ML4Q Cluster of Excellence.

**Enumerative Geometry of algebraic surfaces**

Enumerative geometry is a classical area of mathematics that deals with the question of how many objects of a certain type exist on a given geometric space, or more precisely on an algebraic variety. Studying and possibly solving these counting problems helps to understand new aspects of the geometry of these spaces and often leads to interesting new algebraic structures as well as to new connections between geometry and other subfields of mathematics.

In the project "Correspondences in enumerative geometry: Hilbert schemes, K3 surfaces and modular forms" (K3Mod) **Prof. Dr. Georg Oberdieck** investigates the enumerative geometry of algebraic surfaces, in particular the so-called K3-surface. The focus is on proving correspondences between different enumerative theories and thereby gaining new insights into these theories. A central goal is to determine the Gromov-Witten theory of Hilbert schemes of points on algebraic surfaces. Part of Oberdieck's approach is to prove symmetries of generating functions of invariants and thereby establish a connection to modular forms, a classical branch of number theory. This allows complicated structures to be computed by determining a few coefficients.

The K3Mod project is part of algebraic geometry, but also features numerous connections with representation theory, number theory and physics. It is funded by the ERC Starting Grant with about 1.5 million euros.

After studying mathematics at ETH Zurich, Georg Oberdieck earned his PhD there in 2015 under Rahul Pandharipande, one of the leading experts in modern algebraic geometry. He then worked at the Massachusetts Institute of Technology (MIT) in the USA before becoming a Bonn Junior Fellow at the Hausdorff Center for Mathematics Cluster of Excellence at the University of Bonn in September 2018. In 2020, he received the prestigious Heinz Maier-Leibnitz Prize of the German Research Foundation.