Developments in the Mathematical Sciences - DIMS 2025

conference

16.06.25 18.06.25

Developments in the Mathematical Sciences - DIMS 2025

Founded almost 30 years ago, the Max Planck Institute for Mathematics in the Sciences is one of two mathematics institutes within the Max Planck Society, an internationally renowned and leading society for basic research in Germany and beyond. Our fundamental mission is to conduct cutting-edge research in mathematics. Our connection to the sciences includes both the application of mathematical tools and concepts and also the condensation of scientific questions into fruitful mathematical problems.

We see it as our mission to presenting the diversity of fascinating developments in mathematics even beyond the current scientific scope of the institute. To this end, we have established the "Developments in the Mathematical Sciences" conference series, which aims to provide insights into the latest developments.

After 2019 and 2021, this is the third conference in this series. We want to discuss current trends and get an overview of today's mathematical landscape – especially with regard to the label "mathematics in the sciences". Speakers will motivate their work and place it in a general context. The talks will be accessible to mathematicians from a wide range of backgrounds. There will be plenty of time for questions and informal interaction.

Speakers:

Nalini Anantharaman (University of Strasbourg)
Bjoern Bringmann (Institute of Advanced Study)
Elisenda Feliu (University of Copenhagen)
Mikolaj Fraczyk (Jagiellonian University)
Javier Gómez-Serrano (Brown University)
Massimiliano Gubinelli (University of Oxford)
Nina Holden (New York University)
Anton Mellit (University of Vienna)
Emmy Murphy (University of Toronto)
Nicolas Perkowski (Freie Universität Berlin)
Alessandro Pigati (Università Bocconi)
Stefan Steinerberger (University of Washington)
Floris van Doorn (University of Bonn)
Yilin Wang (Institut des Hautes Études Scientifiques)
Melanie Weber (Harvard College)

Program

08:30 - 09:00	Registration and Welcome Coffee
09:00 - 09:15	Welcome Address
09:15 - 10:05	Nalini Anantharaman On the bottom of the spectrum of random hyperbolic surfaces Although there are several ways to ''choose a compact hyperbolic surface at random'', putting the Weil-Petersson probability measure on the moduli space of hyperbolic surfaces of a given topology is certainly the most natural. The work of M. Mirzakhani has made possible the study of this probabilistic model: it is one of the only model of ''random riemannian manifolds'' where some explicit calculations are actually possible. One may thus ask questions about of the geometry and the spectral statistics of the laplacian of a randomly chosen surface – in analogy with what is usually asked for models of random graphs. I will be interested in the spectral gap of the laplacian for a random compact hyperbolic surface, in the limit of large genus (j.w. Laura Monk).
10:15 - 11:05	Bjoern Bringmann Heat, wave, and Schrödinger dynamics of Gibbs measures Gibbs measures were first introduced in statistical mechanics, where they have been used to describe statistical equilibria of physical systems. Over the last decades, Gibbs measures have also become central objects in probability theory, partial differential equations, and mathematical physics. In this talk, we are interested in the heat, wave, and Schrödinger dynamics corresponding to Gibbs measures. In the first part of the talk, we discuss the invariance of the $Φ_{3}^{4}$ -measure under the three-dimensional cubic nonlinear wave equation. In the second part, we then discuss the global well-posedness of the stochastic Abelian-Higgs model, which is the Langevin equation for the Abelian-Higgs measure. The two results above have been obtained in joint work with Y. Deng, A. Nahmod, and H. Yue and S. Cao, respectively.
11:15 - 11:45	Coffee & Tea
11:45 - 12:35	Elisenda Feliu Reaction networks and real algebraic geometry: A fruitful interplay The study of equilibria of (bio)chemical reaction networks with mass-action kinetics dates back to the 1970s, with foundational work by Horn, Jackson, and Feinberg, who established the field of Chemical Reaction Network Theory (CRNT). A central question in CRNT is whether a given reaction network admits multiple equilibria for some choice of parameter values. Since equilibria correspond to the positive solutions of parametric polynomial systems, the field has seen a significant shift over the past two decades with the integration of ideas, tools, and techniques from (real) algebraic geometry and computational algebra. This interaction has not only deepened our understanding of reaction networks but also revealed that many results initially developed within CRNT have far-reaching applications beyond chemistry. In this talk, I will start by introducing key concepts and questions from reaction network theory and its algebraic framework in terms of vertically parametrised systems. After this motivation, I will present selected examples of the interplay between the theory of reaction networks and real algebraic geometry.
12:45 - 13:45	Lunch
13:45 - 14:30	Breakout Session
14:30 - 15:20	Nina Holden Scaling limits of random planar maps Planar maps are graphs embedded in the sphere such that no two edges cross, where we view two planar maps as equivalent if we can get one from the other via a continuous deformation of the sphere. In this talk we will present scaling limit results (i.e., convergence results) for random planar maps and we will focus in particular on a notion of convergence known as convergence under conformal embedding.
15:30 - 16:00	Coffee Break
16:00 - 16:50	Stefan Steinerberger Growth Models in the Plane We will study the growth of (two-dimensional) things. Think about lichen growing on a tree (tends to be sort of round). Another fun example is electricity propagating through wood (tends to be sort of fractal). A celebrated and rather mysterious model is called DLA: it forms the most beautiful fractal patterns, pictures will be provided. Despite being very well known, DLA is actually fairly poorly understood and we will quickly survey the existing ideas. We will then discuss a new type of growth model that behaves very similarly and which can be analyzed using a different and arguably more elementary set of tools.

09:00 - 09:50	Anton Mellit Symmetric functions in geometry There have been several recent developments in which symmetric functions encode answers to sophisticated counting problems: cohomology of spaces of solutions to various matrix problems such as moduli spaces of higgs bundles, character and quiver varieties; commutative algebra problems such as coinvariant modules; knot invariants; various combinatorial problems. Usually Macdonald polynomials come up and hint at mysterious connections between these different subjects. I'll discuss some examples and try to give an overview of this field.
10:00 - 10:50	Yilin Wang The Brownian loop measure on Riemann surfaces and applications to length spectra The goal of this talk is to showcase how we can use stochastic processes to study the geometry of surfaces. More precisely, I will recall the basic facts about surfaces with constant curvature and Brownian motion on them. Then, we use the Brownian loop measure to express the lengths of closed geodesics on a hyperbolic surface and zeta-regularized determinant of the Laplace-Beltrami operator. This gives a tool to study the length spectra of a hyperbolic surface and we obtain a new identity between the length spectrum of a compact surface and that of the same surface with an arbitrary number of additional cusps. The talk is mainly based on a joint work with Yuhao Xue (IHES).
11:00 - 11:30	Coffee Break
11:30 - 12:20	Mikolaj Fraczyk Minimal submanifolds, low dimensional actions and higher expansion phenomena In a recent work with Ben Lowe we studied the behavior of low codimensional minimal submanifolds in locally symmetric spaces. This led us to discover a curious dichotomy, where (under some reasonable conditions on the ambient space) any low codimensional subset of a locally symmetric space has to be “very complicated” or otherwise it can be collapsed to lower dimension. This proved to be surprisingly useful observation, which we can exploit to prove branched cover stability theorems for octonionic hyperbolic manifolds, show new fixed point theorems for low dimensional action of lattices in semisimple Lie groups and build new examples of Riemannian higher expanders in the sense of Gromov. I will explain how minimal submanifolds and geometric measure theory tools enter the picture and advertise several new ways they could be used to study locally symmetric spaces.
12:30 - 13:30	Lunch
13:30 - 14:30	Breakout Session
14:30 - 15:20	Javier Gómez-Serrano Machine Learning in PDE: Discovering new, unstable solutions In this talk I will explain several recent results combining machine learning techniques and more traditional mathematics. The overarching theme is the interplay between modern (ML) and classical methods in order to discover new solutions of certain PDE with low or very low numerical error. I will also outline how to turn the numerical approximate solutions into a rigorous proof via computer-assisted methods, leading in some cases to singularity formation, or to the existence of certain special solutions (e.g. traveling waves). In particular, unstable solutions are now amenable to be discovered. While the motivation is originally coming from fluid mechanics' equations such as Euler or Navier-Stokes some of the methods can be tuned to other PDE.
15:30 - 16:00	Coffee Break
16:00 - 16:50	Emmy Murphy Flexibility in symplectic and Weinstein geometry A Stein manifold is a complex manifold admitting a proper holomorphic embedding into C^n. A number of theorems due to Oka, Cartan, and Grauert (among others) show that Stein manifolds have geometric flexibility properties, compared to their compact/projective complex counterparts. A culmination of this is the work of Cieliebak-Eliashberg, which states that Stein manifolds can be completely classified (up to deformation) via their underlying symplectic geometry (called a Weinstein structure). While this Stein=Weinstein equivalence holds for all Stein manifolds, the relations between Weinstein manifolds and contact geometry gives rise to a "deeper level" of h-principle flexibility, which only holds in complex dimension >2 and furthermore only holds for a subclass of Weinstein manifolds. These techniques allow us to apply techniques from high dimensional smooth topology to prove theorems about symplectic and Stein manifolds, which have been used for applications such as classification theorems and constructions of exotic submanifolds.

09:00 - 09:50	Nicolas Perkowski Recent progress on SPDEs beyond subcriticality Effective models of interacting particle systems or turbulent fluids are often given by singular stochastic partial differential equations (SPDEs). While research on subcritical SPDEs continues to flourish, building on the foundations of Hairer's regularity structures, recent years have seen major breakthroughs in tackling critical and supercritical models. In this survey talk I will explore this new frontier, covering foundational models like the 2D stochastic heat equation and KPZ/Burgers equation, as well as complex systems from fluid dynamics like stochastic Navier-Stokes or surface geostrophic equations. I will discuss the origin of these equations, including the challenging Dean-Kawasaki equation, and survey some novel techniques that now allow for their rigorous analysis.
10:00 - 10:50	Melanie Weber Data and Model Geometry in Deep Learning The utility of encoding geometric structure, such as known symmetries, into machine learning architectures has been demonstrated empirically in domains ranging from biology to computer vision.However, rigorous analysis of its impact on the learning complexity of neural networks remains largely unexplored. Recent advances in learning theory have shown that training shallow, fully connected neural networks, which are agnostic to data geometry, has exponential complexity in the statistical query model, a framework encompassing gradient descent. In this talk, we ask whether knowledge on data and model geometry is sufficient to alleviate the fundamental hardness of learning neural networks. We analyze two geometric settings: First, equivariant neural networks, a class of geometric architectures that explicitly encode symmetries; and second, learning neural networks under the manifold hypothesis, which assumes that high-dimensional input data lies on or near a low-dimensional manifold.
11:00 - 11:30	Coffee Break
11:30 - 12:20	Massimiliano Gubinelli Supersymmetry and elliptic SPDEs I will discuss a theorem about elliptic SPDEs in two dimensions which has a remarkable supersymmetric proof via a dimensional reduction argument first noticed by Parisi and Sourlas in the ’80. The goal is to advertise this not very well known area of probability theory inhabited by strange but interesting objects.
12:30 - 13:30	Lunch
13:30 - 14:30	Breakout Session
14:30 - 15:20	Alessandro Pigati Minimal submanifolds as energy concentration sets: from phase transitions to gauge theoretic models An important problem in calculus of variations and geometric analysis is the construction of minimal submanifolds, namely critical points for the area, due to their role in relating curvature and topology of the ambient. In the last three decades, a fruitful way to approximate the area functional in low codimension is to interpret submanifolds as the nodal sets of maps (or sections of vector bundles), critical for suitable physical energies or well-known lagrangians from gauge theory. Inspired by the well understood Allen-Cahn energy in codimension one, we will survey the situation in codimension two, where the abelian U(1) Higgs model has provided a successful framework (leaving the regularity theory aside), as well as the non-abelian SU(2) Yang-Mills-Higgs as a natural candidate in codimension three, emphasizing analogies and differences, as well as the new key difficulties. This talk is based on collaborations with Stern and Parise-Stern.
15:30 - 16:00	Coffee Break
16:00 - 16:50	Floris van Doorn talk cancelled In recent years, formalization has received more attention from mathematicians because of success stories such as the Liquid Tensor Experiment, which formalized a deep result from condensed mathematics in Lean. I will discuss the Carleson project, an ongoing formalization that verifies a modern generalization of Lennart Carleson's 1966 theorem in harmonic analysis about the pointwise convergence of Fourier series under weak conditions. Carleson's original proof is notoriously difficult, and all subsequent proofs required a lot of intricate details. In particular, I will reflect on the collaborative nature of this formalization.

Organizers

Felix Otto

Max Planck Institute for Mathematics in the Sciences

Bernd Sturmfels

Max Planck Institute for Mathematics in the Sciences

László Székelyhidi

Max Planck Institute for Mathematics in the Sciences

Anna Wienhard

Max Planck Institute for Mathematics in the Sciences

Administrative Contact

Jörg Lehnert

Max Planck Institute for Mathematics in the Sciences Contact via Mail

Katharina Matschke

Max Planck Institute for Mathematics in the Sciences Contact via Mail

Developments in the Mathematical Sciences - DIMS 2025

Developments in the Mathematical Sciences - DIMS 2025

Links

Program

Organizers

Felix Otto

Bernd Sturmfels

László Székelyhidi

Anna Wienhard

Administrative Contact

Jörg Lehnert

Katharina Matschke