Geometry Meets Physics: Finiteness, Tameness, and Complexity

Dates: November 12–14, 2025

Location: CMSA G10, 20 Garden Street, Cambridge MA 02138 Finiteness is a fundamental property in consistent physical theories. From the earliest days of quantum field theory and string theory, the drive to eliminate unphysical infinities has been a guiding principle. More recently, finiteness has emerged as a key criterion for constraining effective theories that can be embedded in quantum gravity. Formulating and testing these constraints remains a central challenge in current research.

In parallel, mathematics has made remarkable advanced in addressing finiteness questions using tame geometry. Built on the framework of o-minimal structures, tame geometry offers a precise language for describing objects of finite geometric complexity. Recent developments, such as sharp o-minimality, go further by introducing a quantitative notion of complexity, opening new directions for analyzing finiteness in mathematics and physics alike.

This workshop brings together mathematicians and physicists to exchange ideas, explore new perspectives, and spark collaborations at the interface of geometry, logic, and fundamental physics.

Invited Speakers:

Vijay Balasubramanian (UPenn), Gregorio Baldi (IHES, France), Gal Binyamini (Weizmann Institute), Raf Cluckers (Lille, France), Bruno Klingler (Humboldt University, Berlin), Delgado Matilda (Max Planck Institute Munich), Adele Padgett (Vienna), David Prieto (Utrecht) , Washington Taylor (MIT), David Urbanik (IHES, France), Cumrun Vafa (Harvard), Mick van Vliet (Utrecht), Benny Zak (Weizmann),

Organizers: Thomas Grimm, Harvard CMSA & Utrecht University | Gal Binyamini, Weizmann Institute | Bruno Klingler, Humboldt University, Berlin

A workshop introducing researchers with some background in harmonic analysis and dispersive partial differential equations to some open problems in general relativity and quantum information.

About: Recently, there have been some remarkable breakthroughs of harmonic analysis techniques in general relativity and quantum information. As some featured examples, we refer to a result in 2021 of Hintz and Vasy and now in 2025 of Hintz, Petersen, and Vasy, in proving global nonlinear stability of Kerr-de Sitter spacetimes, extending Fourier-harmonic analytic methods into nonlinear geometric PDES, and the solution of the L2 curvature conjecture for the Einstein Equations by Klainerman, Rodnianski, and Szeftel.

Some main themes of this workshop will be to explore:

How harmonic analysis can be used in General Relativity settings, which often involve obstacles like a lack of global Fourier transforms, noncompactness of Lorentzian manifolds and the associated scattering theory, and nonlinear and geometric structure of PDEs.

The analysis of Schrodinger operators that arise in quantum computing (including themes such as Lieb-Robinson bounds).

One of the main goals of the workshop will be to introduce researchers with some background in harmonic analysis and dispersive PDE to some open and accessible problems in general relativity and quantum information. The structure of the workshop will accommodate time for plenary talks, problem sessions, and discussion and collaboration time. Thus the workshop is ideal for researchers in harmonic analysis or dispersive PDE seeking an introduction to how these techniques may be applied in a general relativity or quantum information setting, and to be acquainted with some accessible open problems in those fields, or conversely researchers in General Relativity or Quantum Information seeking to incorporate more harmonic analysis techniques into their research.

Participation is in person only. There will be talks that are delivered by video link.

Speakers:

Di Fang (Duke University) - online talk

Jesse Gell-Redman (University of Melbourne)

Zihua Guo (Monash University)

Andrew Hassell (Australian National University)

Xiao Ma (University of Michigan) - online talk

Todd Oliynyk (Monash University)

David Ou Yang (Ludwig-Maximilians-Universität München)

Volker Schlue (University of Melbourne)

Andras Vasy (Stanford University) - online talk

Jingxuan Zhang (Tsinghua University)

Scientific and Organising Committee:

Alexandria Rose (Australian National University)

Pierre Portal (Australian National University)

Xiaoxu Wu (Australian National University)

Tony Martin (Australian National University)

Gauge theory and string geometry have been at the forefront of mathematical and physical research for decades. Gauge theory, originating from physics, provides a framework for understanding fundamental forces and has inspired a range of mathematical advances. Notably, Donaldson’s work using gauge theory has led to significant progress in 4-manifold topology, and the development of Seiberg-Witten theory further refined our understanding of smooth 4-manifolds.

In parallel, string theory emerged as a candidate for unifying all known fundamental forces, leading to a deep interaction with mathematics. It inspired the concept of mirror symmetry, which reveals a duality between seemingly distinct Calabi- Yau manifolds and has motivated many developments in symplectic geometry and enumerative geometry.

The program will provide ample opportunities for participants to interact with experts, discuss open problems, and potentially initiate collaborations. We aim to foster an inclusive and interactive environment that encourages the exchange of ideas between researchers at different stages of their careers.

Gauge theory and string geometry have been at the forefront of mathematical and physical research for decades. Gauge theory, originating from physics, provides a framework for understanding fundamental forces and has inspired a range of mathematical advances. Notably, Donaldson’s work using gauge theory has led to significant progress in 4-manifold topology, and the development of Seiberg-Witten theory further refined our understanding of smooth 4-manifolds.

In parallel, string theory emerged as a candidate for unifying all known fundamental forces, leading to a deep interaction with mathematics. It inspired the concept of mirror symmetry, which reveals a duality between seemingly distinct Calabi- Yau manifolds and has motivated many developments in symplectic geometry and enumerative geometry.

The program will provide ample opportunities for participants to interact with experts, discuss open problems, and potentially initiate collaborations. We aim to foster an inclusive and interactive environment that encourages the exchange of ideas between researchers at different stages of their careers.

The interaction between topology and curvature is a fundamental theme in modern mathematics. The study of scalar curvature plays an increasingly important role not only in geometry and topology, but also in general relativity and theoretical physics more generally through the theory of Dirac operators. This special session will bring together experts and young researchers who study these topics from many different points of view. The aim of this session is to share viewpoints and progress on understanding the scalar curvature and topology of manifolds, and to establish new connections among the culturally diverse groups spread worldwide, but particularly in the USA.

This workshop, sponsored by AIM and the NSF, will bring together experts from different areas in mathematics related to applications of the Bochner technique, including Riemannian geometry, complex geometry, representation theory, and geometric flows.

The Bochner technique is a foundational tool in differential geometry which provides a deep link to topology. Recent advances include new connections to representation theory with applications to vanishing results for Betti and Hodge numbers, the resolutions of the Nishikawa conjecture, projectivity and rational connectedness results for Kähler manifold, or new Kodaira-Bochner formulae. The aim of the workshop is to both push the boundaries of these areas as well as strengthen the interaction among experts in different areas. Utilizing the versatility of the Bochner technique is a key component of the workshop. The workshop is meant to bring together leading experts as well as aspiring new researchers from all areas related to the Bochner technique.

The main topics for this workshop are

(1) Vanishing results and applications to topology and geometric flows

(2) Representation theoretic aspects and symmetric spaces

(3) The curvature operator of the second kind

(4) Nonlinear Kodaira-Bochner formulae and their applications

This conference is addressed to mathematicians and mathematical physicists interested in contemporary mechanics, physics and biology and associated mathematical questions. The application of differential geometry to find new results on manifolds, relativity, hypersurfaces, N-body problem, gauge fields, geometric quantization, rotational sequences, minimal surfaces, biophysical systems, coherent states, Dirac and Seiberg-Witten monopoles, rigid body dynamics, Toda chains dynamics, geometric algebra, Kähler calculus, thermodynamics, etc. The meeting allows participants coming from different fields to share and to interchange geometrical ideas among them with the leading role of differential geometry. The contributions presented at the conference will be invited to be submitted to the series on Geometry, Integrability and Quantization or to the Journal of Geometry and Symmetry in Physics.

This is the 2nd conference organized by the International Society of Nonlinear Mathematical Physics (ISNMP) which is located in Bad Ems, Germany. The conference aims to provide a platform for experts to report and discuss their research progress on all aspects of nonlinear mathematical physics, with a focus on integrable and exactly solvable systems. This time the conference will be held in memory of Wilhelm I. Fushchich on his 90th anniversary and in honour of Jarmo Hietarinta on the occasion of his 80th birthday.