The trimester program aims to bring together experts, postdocs, and students in computer science and certain areas in mathematics (analysis, probability, and combinatorics) in order to learn about some challenging open problems recently raised in computer science, to use and invent necessary new tools and techniques in mathematics to solve these challenging problems, and vice versa to learn and further extend methods developed in computer science to develop new directions in mathematics motivated by questions in computer science. The core topics of the trimester program would be: learning theory, complexity of classical and quantum algorithms, vector valued functions on the hypercube, complex Hypercontractivity, polynomial inequalities on the hypercube, and discrete approximation theory on the hamming cube.

# Future

In recent years, the development of analytic tools to study geometric questions in metric spaces beyond Riemannian manifolds has seen tremendous progress and led to many structural results which shed a new light even on classical questions. The goal of this HIM program is to further refine such tools and explore their applications, like the study of filling invariants, norms, volumes, energies, in various concrete geometric frameworks. The program aims at bringing together researchers in analysis on metric spaces, geometric group theory, differential geometry, higher Teichmüller theory and low dimensional topology to isolate and enhance common core ideas and tie these strands to a more global theory, geared at making progress towards some well established open problems. The trimester will include an Introductory Winter School "Analysis and geometry on groups and spaces" (January, 27-31), the Felix Klein Lectures (tba), and an International Conference "Differential geometry beyond Riemannian manifolds” (March, 24-28). The online application platform to participate in this trimester program will be accessible from January 19 to June 30, 2024 under https://him-application.uni-bonn.de/index.php?id=5701 For applications of PhD students and postdocs, a letter of recommendation is required. Please make sure this letter is sent before July 1, 2024 to application-metric-analysis@him.uni-bonn.de In case of questions concerning services and administration, please contact events@hcm.uni-bonn.de

Formal Mathematics, the program to formalize, check, and manage mathematical knowledge, statements and proofs with computer support, is about to reach a critical threshold where it can efficiently support mathematical research and teaching. It has the chance to profoundly change practices in pure mathematics, as computer algebra systems have already changed computational and experimental mathematics. Computer-checked formalizations of mathematical theories can be seen as the kind of complete presentations that Euclid or Bourbaki were aiming for. Formalized results are absolutely correct (modulo remote chances of computer failures) and can be verified by simple ``mechanical'' methods independent of high-level mathematical intuitions, abilities or traditions. They constitute a solid canon of results on which further work can be founded.

Quantum field theories (QFTs) have been successfully applied, throughout the last 70 years, to model and analyze diverse physical phenomena; in particular, critical behavior in statistical mechanics, and interactions of fundamental particles. However, a rigorous mathematical framework to construct and understand these theories is still limited. This program will further pursue this direction, building on recent developments towards QFT coming from random geometry and probability theory. In particular, the goal is to bring together researchers with different viewpoints and expertise on this multifaceted topic. With this combined expertise, the program aims at addressing some of the main challenges and key open questions in the field, including in the following areas: advances in quantum gauge theories in 3D and 4D rigorous constructions of exactly solvable quantum field theories, especially in 2D more generally, analysis of probabilistic aspects of quantum field theories the theory of phase transitions, in particular for systems with continuous symmetry, and related phenomena