for this workshop

## Geometric partial differential equations from unified string theories

at the

American Institute of Mathematics, Pasadena, California

organized by

Tristan Collins, Anna Fino, and Duong H Phong

This workshop, sponsored by AIM and the NSF, will be devoted to developing a systematic PDE approach to the geometric equations arising from string theories. String theories are at this moment the only known viable candidate for a unified theory of all physical interactions, and their equations can be expected to be a great source of inspiration for mathematics. Indeed, the first solutions of string theories were found to be given by Calabi-Yau manifolds and Hermitian-Einstein metrics, two fundamental notions of canonical metrics in Kaehler geometry whose existence had been established by Yau and Donaldson-Uhlenbeck-Yau very recently at that time. But phenomenological considerations as well as subsequent developments such as mirror symmetry, dualities, and non-perturbative effects, all suggested that a fuller understanding of these equations is necessary, and such a fuller understanding would lead to suitable notions of canonical metrics in non-Kaehler geometry and/or symplectic geometry. The corresponding equations found so far have all been novel, and of considerable interest in their own right for PDE theory, complex geometry, symplectic geometry, or symplectic geometry. While many examples have been studied in the physics literature, no systematic PDE approach for them has been developed as yet, notably in any of the above fields in the mathematics literature, which would have been their natural realm.

The present workshop aims at rectifying this situation. In particular, some of the main (intertwining) themes to be discussed are

- New special geometries in relation to string theory
- Canonical metrics in non-Kaehler geometry
- Geometric flows beyond the Ricci flow, particularly in symplectic geometry and non-integrable complex structures

This event will be run as an AIM-style workshop. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

Space and funding is available for a few more participants. If you would like to participate, please apply by filling out the on-line form no later than September 10, 2025. Applications are open to all, and we especially encourage women, underrepresented minorities, junior mathematicians, and researchers from primarily undergraduate institutions to apply.

Before submitting an application, please read the description of the AIM style of workshop.

For more information email *workshops@aimath.org*