Higher structures have emerged as a new eld in mathematics, as well as in mathematical and theoretical physics. Covering many di erent topics, this eld encompasses the following areas: moduli spaces, defor mation theory, derived geometry, enumerative geometry, representation theory, homotopy algebras, Poisson geometry and quantization, mirror symmetry, string theory, and quantum eld theory.
Higher structures refer to the structures introduced into various elds of mathematics by treating the objects of study as objects in higher categories. In many elds, this point of view has led to the discovery of hidden structures, often yielding deep new insights into the research area. Over the years, homotopy theory has developed powerful tools for dealing with higher categories, and as a result, the study of higher structures often involves introducing techniques from homotopy theory into other elds of mathematics.
In deformation theory, this means replacing Lie algebras with L or homotopy Lie algebras. In algebraic geometry, the ring of holomorphic functions on a variety is replaced by a di erential graded algebra. In representation theory, representations on vector spaces are replaced by representations on categories. In enumerative geometry, counting invariants (which answer questions such as how many twisted cubic curves lie on a Calabi-Yau threefold? ) are replaced by the dimensions of certain vector spaces (such as Gopakumar Vafa invariants). Quantum eld theories are de ned as functors from higher cobordism categories to higher linear categories, to name just a few examples.
Even though they are investigating similar fundamental questions and adopting the same general higher structural perspective, di erent communities have mostly worked independently of each other. Indeed, the techniques developed by the di erent schools are quite distinct. These topics are extremely active and are related to many areas of mathematics and physics. One objective of this project is to bring together the various higher structure communities active in di erent areas and originating from diverse backgrounds such as algebraic geometers, di erential geometers, and mathematical physicists to encourage more interaction and cross-fertilization between the elds, as well as identify new emerging directions. The project will bring together leading experts and young researchers in these subjects to promote interaction and collaboration.
To this purpose, we plan to include a week-long school, where a series of mini-courses will be delivered by world-renowned leading experts, aiming to facilitate communication between the elds. We will also have workshops presenting recent research. In addition, we plan to run a weekly seminar during April-June 2026. Furthermore, there will be a number of informal series of lectures and mini-courses on related topics. In particular, the project will focus on lectures in these areas:
･ Higher structures in Moduli spaces and Enumerative Geometry
･ Homotopy algebras, deformation theory, and quantization
･ DG manifolds in geometry and physics
　RIMS Symposium I (WS1)
　Theme: School on Moduli Spaces in Various Flavors of Geometry , April 13-17, 2026
　Organizers: Kai Behrend, Hsuan-Yi Liao, Kaoru Ono, Mathieu Stienon, Ping Xu
　Mini-course speakers
･ Alexey Bondal
･ Barbara Fantechi
･ Kenji Fukaya
･ Bernhard Keller *
･ Hiraku Nakajima
(*) to be con rmed
RIMS Symposium II (WS2)
Theme: Workshop on Moduli Spaces in Various Flavors of Geometry , April 20-24, 2026
Organizers: Kai Behrend, Hsuan-Yi Liao, Kaoru Ono, Mathieu Stienon, Ping Xu
RIMS Symposium III (WS3)
Theme: Workshop on DG Manifolds in Geometry and Physics , March 8-12, 2027
Organizers: Hsuan-Yi Liao, Kaoru Ono, Mathieu Stienon, Ping Xu

The RIMS research project 2026 "The Mathematical Roads to QFT" will bring together worldwide leading mathematicians interested in the mathematical aspects of Quantum Field Theory, and their wide-ranging applications from high-energy physics to quantum simulation. It will take place at RIMS from April to July, 2026. The organizing committee is composed by C. Brennecke (IAM Bonn), S. Cenatiempo (GSSI L'Aquila), M. Falconi (Politecnico di Milano), F. Hiroshima (Kyushu University) and H. Ochiai (Kyushu University).

This project aims at stimulating scientific interactions between the different mathematical approaches to QFT, that range from probability theory and stochastic PDEs to operator theory and operator algebras, as well as statistical mechanics and microlocal analysis. The goal is to further advance our knowledge of QFT and its applications, with the ultimate goal of tackling the outstanding open problems in QFT such as the rigorous renormalization of physical theories, spontaneous symmetry breaking and critical phenomena.

The RIMS will host international guests both for the whole duration of the research project and for shorter visits. To foster an active and engaging participation of both experienced and young researchers, and stimulate interactions with the local community, some key events will be organized throughout the project, including an opening workshop in the second half of April, and a conference at the end of June. These events will offer a possibility, especially to young researchers, both to come in contact with the recent groundbreaking ideas, and to showcase and discuss the most recent advances in the field.

It is the commitment and hope of the organizing committee for this project to be successful in being a building block of new and exciting developments in the mathematics of quantum field theory.

Arithmetic homotopy geometry (AHG) exploits arithmetic and geometric invariants via homotopy theory and especially the theory of fundamental groups (in its étale, Galois, motivic, and topological versions). It thrives and revisits leading questions and constructions on both ends of the number theory-geometry spectrum, as can be seen in the following 3 contemporary structuring topics:

• A. Homotopy, Rationality, and Geometry. Homotopic methods have deepened our understanding of rationality phenomena, from rational obstructions (non-abelian Chabauty, section conjecture, Brauer–Manin) to estimates in dimension growth and height estimates (IUT, Heath-Brown-Serre conjecture, Batyrev-Manin conjecture, Malle conjecture) and non-rationality (Hilbert specialization property).
  Moduli situations (Hurwitz spaces and SL2-torsors) provide new contexts for geometry and arithmetic to interact. Anabelian algorithms reveal new structures (e.g., monoids and quasi-tripods) and new models for central objects of number theory (e.g., BGT subgroups wrt the absolute Galois group of rational numbers).
• B. The Homology-Homotopy Frontier, The two arithmetic universalities frameworks, (linear) motivic theory and (non-linear) anabelian homotopy theory, once pursued separately, now increasingly overlap: (a) nearly-abelian reconstructions (m-step reconstruction, abelian-by-central, section conjecture), (b) non-abelian Chabauty. (pro-unipotent, m-step unipotent quotient, Selmer sections), and (c) monodromy method in local systems illustrate this new interface.
  In these settings, derived perverse formalism, mixed Hodge structures, and p-adic Hodge techniques (esp. period maps) indicate special loci of interest for finer control of arithmetic properties; étale topological type provides an anabelian-motivic bridge for progress on both sides of the frontier.
• C. Combinatorial Arithmetic Geometry.Original combinatorial methods (braid groups, Lie algebras, DM-compactification) have matured into categorical tools for refinement of structuring questions of arithmetic homotopy theory (e.g., Ihara program, Oda’s question): (a) combinatorial anabelian geometry attacks higher dimensional arithmetic, (b) operads and Quillen model categories link Drinfeld-Kohno Lie algebra, hairy graph complex, and Galois-Teichmüller theory, all together for (c) further number theory-topology interactions (e.g., graph homology, Johnson homomorphisms, knots and primes, and associators).
  These (anabelian) arithmetic and topological combinatorics must now be confronted with each other for exploiting decisive progress on both sides (e.g., in arithmetic of moduli of higher genus curves, and for Goodwillie-Weiss manifold calculus).

The objectives of the AHG RIMS Research Project are to gather a broad international mathematical audience/actors: (a) to report and present the most recent techniques, principles, and results of these topics, (b) to stimulate and crystallize the shaping of new leading questions and conjectures, and (c) to support the emergence of the next-generation of arithmetic homotopy geometers.
This year-long AHG Research Project is structured in 3 A-B-C seasons, each with a main international conference, a series of introductive mini-courses, some expert workshops, and some satellite events for exploring new interfaces. A biweekly seminar links all the seasons. The invitation of world-class international and domestic experts for mid-term stay will promote research exchanges locally and in the long term.
Mid-career researchers will be provided the opportunity of workshop organization and promoting the participation of early-career researchers. Participation of female researchers will be strongly encouraged to promote gender diversity in research mathematics in Japan.

Recent Developments in Fluid Mechanics PDEs

Fluid mechanics has numerous real-world applications, e.g. weather prediction and agriculture. Modeling and analyzing the motion of fluids through partial differential equations (PDEs) have been studied for centuries by physicists and engineers. Mathematical analysis on such nonlinear PDEs is a notoriously difficult research direction. However, it has recently seen significant developments due to multiple breakthroughs in the areas of nonuniqueness, mixing, boundary layer, blowups, and stochasticity through interdisciplinary approaches, e.g. the technique of convex integration that originated in geometry and led to results on nonuniqueness of solutions to PDEs of fluid mechanics and recently singular stochastic PDEs driven by random noise, for which even the existence of solutions had long been open.

The objective of this series of workshops ``Recent Developments in Fluid Mechanics PDEs'' is to gather a large group of active, influential, and internationally renowned researchers in fluid mechanics PDEs and organize the following three workshops.

A) 05/11/2027-05/14/2027 at International Institute for Advanced Studies:
on convex integration: Convex integration technique has its roots in Nash's work on isometric embedding in geometry. In the past two decades, it has developed rapidly as a breakthrough technique to prove various non-uniqueness results that led to resolutions of Onsager's conjecture, Taylor's conjecture, and more.
B) 5/26/2027-5/28/2027 at Research in Mathematical Sciences (RIMS)
on mixing and boundary layers: In fluid dynamics, mixing generated by fluid motion is an important research topic, both theoretically and in terms of applications. Most substances experience molecular diffusion, but mixing due to fluid motion can induce energy dissipation on a timescale much shorter than that of the trivial energy dissipation from molecular diffusion. This phenomenon is known as enhanced dissipation, and its mathematical theory to achieve the quantitative estimates for the rate of dissipation has seen significant development in recent years. Another notable example where this interaction is prominent is the boundary layer of viscous fluid. Indeed, when a fluid satisfies the no-slip boundary condition on a solid wall, a boundary layer forms in the vicinity of the wall, strongly reflecting the effects of fluid viscosity.
C) 5/31/2027-06/02/2027 at RIMS
on fluid mechanics in general such as global regularity and blowup: One of the outstanding open Millennium Prize Problems asks whether or not, in case the spatial domain is a three-dimensional whole space or torus, starting from a sufficiently smooth divergence-free initial data, unforced Navier-Stokes equations admit a unique solution for all time with bounded energy. The past decade has witnessed a remarkable progress towards ill-posedness direction, e.g. finite-time blow-up, norm inflation, nonuniqueness, and non-existence of solutions.

Our workshops will allow ample time for interactions, enabling in-depth discussions that can lead to future collaborations. The first day of the workshop on convex integration will be devoted to introductory lectures that are accessible for non-experts for training young scholars.