Workshop theme
Why are all drums round? This can be regarded as one of the earliest motivating questions in spectral geometry. Broadly speaking, spectral geometry explores the interplay between the geometric properties of a space and the spectrum of differential operators defined on it. Its study draws upon deep theories and refined tools from several areas of mathematics, including differential geometry, mathematical physics, partial differential equations, number theory, dynamical systems, and numerical analysis. The field also has numerous real-world applications, ranging from physics and biology to computer science and data analysis.
Although most of the questions about to be addressed during the INI programme on Geometric Spectral Theory and Applications are of a pure mathematics nature, recent years have shown that scientific computing and numerical approximations have become essential tools for gaining insight into deep mathematical phenomena. They not only provide valuable intuitive support for developing new conceptual approaches, but also help in constructing rigorous analytical proofs. A major need within the community is to have accessible tools for computing the spectrum of various differential operators and for exploring its interaction with the underlying geometry, in order to better understand extremal behaviours, singular situations, degenerative phenomena, and so on.
Artificial Intelligence could play a promising role in providing numerical support for computing the spectrum and for searching extremal shapes (or geometries) in spectral optimization problems. Traditionally, identifying optimal shapes for spectral functionals relies on sophisticated numerical techniques. These include computation of the spectrum using finite element methods or fundamental solutions, evaluation of shape and topological derivatives, implementation of level set/phase field/relaxation/homogenisation methods — all combined within tailored optimization algorithms. Currently, only experts in numerical analysis and scientific computing can typically carry out this complex sequence.
However, the spectral geometry community has shown a strong and growing interest in this kind of work.
AI tools offer the potential to introduce a completely new approach. What if we could learn the spectrum — or more broadly, any differential or variational quantity — directly from the geometry? A well-designed learning procedure could bypass the entire chain of traditional numerical approximations of PDEs and replace them with neural networks capable of producing sufficiently accurate estimates. Such computations would not only be significantly faster but could also be integrated into standard optimization algorithms operating in finite-dimensional spaces.
While this approach may not immediately attain the high accuracy of classical methods, it could nonetheless provide spectral geometers with an intuitive, easy-to-use, computationally efficient tool. It may be combined with automated proof algorithms: even though such combination would not directly provide rigorous proofs, this strategy could be efficient for guiding intuition and identifying potential counterexamples that may later be confirmed through rigorous mathematical analysis.
The purpose of the workshop is to bring together specialists from several communities: spectral geometers, numerical analysts, and researchers working in AI geometric data processing and PDEs. The development of AI based spectral analysis requires a deep interaction between them. Our primary objective is to exchange perspectives and gain a deeper understanding of the potential role that artificial intelligence and geometric data processing may play in addressing research questions in spectral geometry, and ultimately to foster the development of new collaborations. We aim to explore how AI might provide new methods, insights, or tools that could help us tackle some of numerical problems related either to the computation of the spectrum or the optimization of the geometry in relationship with spectral functionals.
The workshop will include a series of short presentations followed by an open discussion. The topics of the presentations will cover different aspects from spectral geometry, AI applications in computer graphics, numerical analysis, geometry processing, shape optimization and computer assisted proofs.
Confirmed speakers:
Registrations close on the 22nd March 2026.
