We study a prototypical geometric wave equation, given by wave maps from the Minkowski space R 1+d into the sphere S d , under the influence of additive stochastic forcing, in all energy-supercritical dimensions d ≥ 3. In the deterministic setting, self-similar finite-time blowup is expected for large data, but remains open beyond perturbative regimes. We show that adding a non-degenerate Gaussian noise provokes finite-time blowup with positive probability for arbitrary initial data. Moreover, the blowup is governed by the explicit self-similar profile originally identified in the deterministic theory. Our approach combines local well-posedness for stochastic wave equations, a Da Prato-Debussche decomposition, and a stability analysis in self-similar variables. The result corroborates the conjecture that the self-similar blowup mechanism is robust and represents the generic large-data behavior in the deterministic problem.

This is joint work with M. Hofmanova and E. Luongo (Bielefeld)

We study a very general class of first-order linear hyperbolic
systems that both become weakly hyperbolic and contain singular
lower-order coefficients at a single time t = 0. In "critical" weakly
hyperbolic settings, it is well-known that solutions lose a finite
amount of regularity at t = 0. Here, we both improve upon the analysis
in the weakly hyperbolic setting, and we extend this analysis to systems
containing critically singular coefficients, which may also exhibit
modified asymptotics and regularity loss at t = 0.

In particular, we give precise quantifications for (1) the asymptotics
of solutions as t approaches 0, (2) the scattering problem of solving
the system with asymptotic data at t = 0, and (3) the loss of regularity
due to the degeneracies at t = 0. Finally, we discuss a wide range of
applications for these results, including weakly hyperbolic wave
equations (and equations of higher order), as well as equations arising
from relativity and cosmology (e.g. at big bang singularities).

This is joint work with Bolys Sabitbek (Ghent).

As temperatures are increasing, so is the presence of meltwater lakes sitting on the surface of the Greenland Ice Sheet. Such lakes have the possibility of draining through cracks in the ice to the bedrock. Observed discharge rates have found that these lakes can drain at three times the flow rate of Niagara Falls. Current models of subglacial drainage systems are unable to cope with such a large and sudden volume of water. This motivates the idea of a 'subglacial blister' which propagates and slowly dissipates underneath the ice sheet. We present a basic hydrofracture model for understanding this process, before carrying out a number of extensions to observe the effects of turbulence, topography, leak-off and finite ice thickness.

In this talk, I will examine the dynamics of the fermion–rotor system, originally introduced by Polchinski as a toy model for monopole–fermion scattering. Despite its simplicity, the system is surprisingly subtle, with ingoing and outgoing fermion fields carrying different quantum numbers. I will show that the rotor acts as a twist operator in the low-energy theory, changing the quantum numbers of excitations that have previously passed through the origin to ensure scattering consistent with all symmetries, thereby resolving the long-standing Unitarity puzzle. I will then discuss generalizations of this setup with multiple rotors and unequal charges, and demonstrate how the system can be viewed as a UV-completion of boundary states for chiral theories, establishing a connection to the proposed resolution of the puzzle using boundary conformal field theory.

Let G be a connected reductive group defined over a non-Archimedean local field k. Endow G with a k-involution and take H to be the fixed-point subgroup in G of that involution. In this talk I will report on some of my recent results regarding Chabauty limits of H(k) inside G(k). Although the results are similar to the real and complex cases, the techniques are totally different and with a strong geometric flavor. Some of the main actors are the Bruhat—Tits building associated with G(k) and basic methods from CAT(0) geometry.

Persistent homology (PH) is an operation which, loosely speaking, describes the different holes in a point cloud via a collection of intervals called a barcode. The two most frequently used variants of persistent homology for point clouds are called Čech PH and Vietoris-Rips PH. How much information is lost when we apply these kinds of PH to a point cloud? We investigate this question by studying the subspace of point clouds with the same barcodes under these operations. We establish upper and lower bounds on the dimension of this space, and find that the question of when the persistence map is identifiable has close ties to rigidity theory. For example, we show that a generic point cloud being locally identifiable under Vietoris-Rips persistence is equivalent to a certain graph being rigid on the same point cloud.

For a tree $T$ whose bipartition classes have sizes $t_1 \ge t_2$, two simple constructions shows that the Ramsey number of $T$ is at least $\max\{t_1+2t_2,2t_1\}-1$. In 1974, Burr conjectured that equality holds for every tree. It turns out that Burr’s conjecture is false for certain trees called the double stars, though all of the known counterexamples have large maximum degrees. In 2002, Haxell, Łuczak, and Tingley showed that Burr’s conjecture is approximately true if one imposes a maximum degree condition.

We show that Burr’s conjecture holds for all trees with up to small linear maximum degrees. That is, there exists $c>0$ such that for every $n$-vertex tree $T$ with maximum degree at most $cn$ and bipartition class sizes $t_1\ge t_2$, its Ramsey number $R(T)$ is exactly $\max\{t_1+2t_2,2t_1\}-1$. We also generalise this result to determine the exact asymmetric Ramsey number $R(T,S)$ of two trees $T$ and $S$ under certain additional conditions, and construct examples showing that these conditions are necessary. 

This talk is based on joint work with Richard Montgomery and Matías Pavez-Signé.

In this talk, I will introduce the nilpotent cohomological Hall algebra COHA(S, Z) of coherent sheaves on a smooth quasi-projective complex surface S that are set-theoretically supported on a closed subscheme Z. This algebra can be viewed as the "largest" algebra of cohomological Hecke operators associated with modifications along a subscheme Z of S. When S is the minimal resolution of an ADE singularity and Z is the exceptional divisor, I will describe how to characterize COHA(S, Z) in terms of the Yangian of the corresponding affine ADE quiver Q (based on joint work with Emanuel Diaconescu, Mauro Porta, Oliver Schiffmann, and Eric Vasserot, arXiv:2502.19445). More generally, I will discuss nilpotent COHAs arising from Bridgeland stability conditions on the bounded derived category of nilpotent representations of the preprojective algebra of Q, following joint work with Olivier Schiffmann and Parth Shimpi (arXiv:2511.08576).

The capacity of a set is a classical notion in potential theory and it is a measure of the size of a set as seen by a random walk or Brownian motion. Recently Zhu defined the notion of branching capacity as the analogue of capacity in the context of a branching random walk. In this talk I will describe joint work with Amine Asselah and Bruno Schapira where we introduce a notion of capacity of a set for critical bond percolation and I will explain how it shares similar properties as in the case of branching random walks. 

In this talk from Alexander Ravnanger, he provides a dynamical description of the largest AF-ideal in certain crossed products by the integers. In the case of the uniform Roe algebra of the integers, this reveals an interesting connection to a well-studied object in topological semigroup theory. On the way, he gives an overview of what is known about the abundance of projections in such crossed products, the structure of the simple quotients, and concepts of low-dimensionality for uniform Roe algebras.

Join us to discuss x+y: A Mathematician’s Manifesto for Rethinking Gender by Eugenia Cheng.

I will discuss old and new results about the distribution of zeros of modular forms, and relation to Quantum Unique Ergodicity. It is known that a modular form of weight k has about k/12 zeros in the fundamental domain . A classical question in the analytic theory of modular forms is “can we locate the zeros of a distinguished family of modular forms?”. In 1970, F. Rankin and Swinnerton-Dyer proved that the zeros of the Eisenstein series all lie on the circular part of the boundary of the fundamental domain. In the beginning of this century, I discovered that for cuspidal Hecke eigenforms, the picture is very different - the zeros are not localized, and in fact become uniformly distributed in the fundamental domain. Very recently, we have investigated other families of modular forms, such as the Miller basis (ZR 2024, Roei Raveh 2025, Adi Zilka 2026), Poincare series (RA Rankin 1982, Noam Kimmel 2025) and theta functions (Roei Raveh 2026),  finding a variety of possible distributions of the zeroes.

(Joint seminar with Random Matrix Theory)

I will discuss old and new results about the distribution of zeros of modular forms, and relation to Quantum Unique Ergodicity. It is known that a modular form of weight k has about k/12 zeros in the fundamental domain . A classical question in the analytic theory of modular forms is “can we locate the zeros of a distinguished family of modular forms?”. In 1970, F. Rankin and Swinnerton-Dyer proved that the zeros of the Eisenstein series all lie on the circular part of the boundary of the fundamental domain. In the beginning of this century, I discovered that for cuspidal Hecke eigenforms, the picture is very different - the zeros are not localized, and in fact become uniformly distributed in the fundamental domain. Very recently, we have investigated other families of modular forms, such as the Miller basis (ZR 2024, Roei Raveh 2025, Adi Zilka 2026), Poincare series (RA Rankin 1982, Noam Kimmel 2025) and theta functions (Roei Raveh 2026),  finding a variety of possible distributions of the zeroes.

The mitigation of bacterial adhesion to surfaces and subsequent biofilm formation is a key challenge in healthcare and manufacturing processes. To accurately predict biofilm formation you must determine how changes to bacteria behaviours and dynamics alter their ability to adhere to surfaces. In this talk, I will present a framework for incorporating microscale behaviour into continuum models using techniques from statistical mechanics at the microscale combined with boundary-layer theory at the macroscale.

We will examine the flow of a dilute suspension of motile bacteria over a flat absorbing surface, developing an effective model for the bacteria density near the boundary inspired by the classical Lévêque boundary layer problem. We use our effective model to derive analytical solutions for the bacterial adhesion rate as a function of fluid shear rate and individual motility parameters of the bacteria, validating against stochastic numerical simulations of individual bacteria. We find that bacterial adhesion is greatest at intermediate flow rates, since at higher flow rates shear-induced upstream swimming limits adhesion.

Subspace methods are commonly used for finding approximate eigenvalues and singular values of large-scale matrices. Once a subspace is found, the Rayleigh-Ritz method (for symmetric eigenvalue problems) and Petrov-Galerkin projection (for singular values) are the de facto method for extraction of eigenvalues and singular values. In this work we derive error bounds for approximate eigenvalues obtained via the Rayleigh-Ritz process. Our bounds are quadratic in the residual corresponding to each Ritz value while also being robust to clustered Ritz values, which is a key improvement over existing results. We apply these bounds to several methods for computing eigenvalues and singular values, including Krylov methods and randomized algorithms.

The canonical example of AdS/CFT relates N=4 SYM in 4d to supergravity on AdS5 x S5 by considering a stack of D3-branes. A natural question then emerges: what about considering other Dp-branes? The worldvolume theory is again SYM but is not conformal anymore, while the supergravity dual is now only conformally AdS. Despite these differences, some control remains, and some inspiration from the p=3 case can be sought. In this talk, I will review this setup and discuss the recent results of [2503.18770] and [2503.14685] regarding the computation of correlation functions.

Professor Ana Djurdjevac will talk about; 'The Dean–Kawasaki Equation: Theory, Numerics, and Applications'

The Dean–Kawasaki equation provides a stochastic partial differential equation description of interacting particle systems at the level of empirical densities and has attracted considerable interest in statistical physics, stochastic analysis, and applied modeling. In this work, we study analytical and numerical aspects of the Dean–Kawasaki equation, with a particular focus on well-posedness, structure preservation, and possible discretization strategies. In addition, we extend the framework to the Dean–Kawasaki equation posed on smooth hypersurfaces. We discuss applications of the Dean–Kawasaki framework to particle-based models arising in biological systems and modeling social dynamics.

We study optimal investment and consumption in an incomplete stochastic factor model for a power utility investor on the infinite horizon. When the state space of the stochastic factor is finite, we give a complete characterisation of the well-posedness of the problem and provide an efficient numerical algorithm for computing the value function. When the state space is a (possibly infinite) open interval and the stochastic factor is represented by an Ito diffusion, we develop a general theory of sub- and supersolutions for second-order ordinary differential equations on open domains without boundary values to prove existence of the solution to the Hamilton-Jacobi-Bellman (HJB) equation along with explicit bounds for the solution. By characterising the asymptotic behaviour of the solution, we are also able to provide rigorous verification arguments for various models, including the Heston model. Finally, we link the discrete and continuous setting and show that that the value function in the diffusion setting can be approximated very efficiently through a fast discretisation scheme.

Specialist species thrive under specific environmental conditions in narrow geographic ranges and are widely recognized as heavily threatened by climate deregulation. Many might rely on both their potential to adapt and to disperse towards a refugium to avoid extinction. It is thus crucial to understand the influence of environmental conditions on the unfolding process of adaptation. I will present a PDE model of the eco-evolutionary dynamics of a specialist species in a two-patch environment with moving optima. The transmission of the adaptive trait across generations is modelled by a non-linear, non-local operator of sexual reproduction. In an asymptotic regime of small variance, I justify that the local trait distributions are well approximatted by Gaussian distributions with fixed variances, which allows to report the analysis on the closed system of moments. Thanks to a separation of time scales between ecology and evolution, I next derive a limit system of moments and analyse its stationary states. In particular, I identify the critical environmental speed for persistence, which reflects how both the existence of a refugium and the cost of dispersal impact extinction patterns. Additionally, the analysis provides key insights regarding the path towards this refugium. I show that there exists a critical environmental speed above which the species crosses a tipping point, resulting into an abrupt habitat switch from its native patch to the refugium. When selection for local adaptation is strong, this habitat switch passes through an evolutionary ‘‘death valley’’ that can promote extinction for lower environmental speeds than the critical one.

A (saturated) fusion system on a p-group P contains data about conjugacy within P, the typical case being the system induced by a group on its Sylow p-subgroup. Fusion systems are completely determined by looking at their essential subgroups, which must admit an automorphism of order coprime to p. For p=2, we describe two new methods that address the question: given an essential subgroup $E<P$ of a fusion system on P, what can we say about P? In particular, one method gives us sufficient conditions to deduce that $E\triangleleft P$, while the other explores cases where we have strong control over the normaliser tower of E in P.

Stability is a key property of topological invariants used in data analysis and motivates the fundamental role of metrics in persistence theory. This talk reviews noise systems, a framework for constructing and analysing metrics on persistence modules, and shows how a rich family of metrics enables the definition of metric-dependent stable invariants. Focusing on one-parameter persistence, we discuss algebraic Wasserstein distances and the associated Wasserstein stable ranks, invariants that can be computed and compared efficiently. These invariants depend on interpretable parameters that can be optimised within machine-learning pipelines. We illustrate the use of Wasserstein stable ranks through experiments on synthetic and real datasets, showing how different metric choices highlight specific structural features of the data.

Hamilton’s Ricci flow is a widely studied tool of geometric analysis, with a variety of applications. It is sometimes possible to obtain existence results for Ricci flows coming out of singular spaces, which leads to the question of uniqueness in these cases. In this talk, we will discuss a new result on uniqueness of Ricci flows coming out of Reifenberg Alexandrov spaces, and give some indication of the methods used in the proof.

I will discuss an ongoing joint work with Luigi Appolloni and Andrea Malchiodi concerning the above objects. Minimal surfaces are critical points of the area functional, which is analytic in this case, so we should expect critical points (minimal surfaces) to be either isolated or to belong to smooth nearby minimal foliations. On the other hand, the flat plane of multiplicity two in $\mathbb{R}^3$ can be (in compact regions) approximated by a blown-down catenoid, which will converge back to the plane with multiplicity two in the limit. Hence a plane of multiplicity two cannot be thought of as being isolated, or belonging solely to a smooth family, because there are “nearby” minimal surfaces of distinct topology weakly converging to it. We will nevertheless prove that, when the ambient manifold is closed and analytic, this type of local degeneration is impossible amongst closed and embedded minimal surfaces of bounded topology: such surfaces, even with multiplicity are either isolated or belong to smooth families of nearby minimal surfaces.  

I will introduce the Polchinski dynamics, a general framework to study asymptotic properties of statistical mechanics and field theory models inspired by renormalisation group ideas. The Polchinski dynamics has appeared recently under different names, such as stochastic localisation, and in very different contexts (Markov chain mixing, optimal transport, functional inequalities...) Here I will motivate its construction from a physics point of view and mention a few applications. In particular, I will explain how the Polchinski dynamics can be used to generalise Bakry and Emery’s Γ2 calculus to obtain functional inequalities (e.g. Poincaré, log-Sobolev) in physics models which are typically high-dimensional and non-convex. 

The aim of this talk is to discuss a finite-volume scheme for the aggregation-diffusion family of equations with non-linear mobility
∂tρ = ∇ · (m(ρ)∇(U′(ρ) + V + W ∗ ρ)) in bounded domains with no-flux conditions. We will present basic properties of the scheme: existence, decay of a free, and comparison principle (where applicable); and a convergence-by-compactness result for the saturation case where m(0) = m(1) = 0, under general assumptions on m,U, V , and W. The results are joint works published in [1, 2]. At the end of the talk, we will discuss an extension to the Porous-Medium Equation with non-local pressure that corresponds to m(ρ) = ρm, U, V = 0 and W(x) = c|x|^−d−2s.

This project is joint work with Jose Carrillo (University of Oxford). 
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to follow

I discuss two separate projects which evoke/strengthen connections between combinatorics and ideas from statistical physics.

The first concerns the minimum number of independent sets in triangle-free graphs of a given edge-density. We present a lower bound using a generalisation of the inductive method of Shearer (1983) for the sharpest-to-date off-diagonal Ramsey upper bound. This result is matched remarkably closely by the count in binomial random graphs.

The second sets out a qualitative generalisation of a well-known sharp result of Haxell (2001) for independent transversals in vertex-partitioned graphs of given maximum degree. That is, we consider the space of independent transversals under one-vertex modifications. We show it is connected if the parts are strictly larger than twice the maximum degree, and if the requirement is only at least twice the maximum degree we find an interesting sufficient condition for connectivity.

These constitute joint works with Pjotr Buys, Jan van den Heuvel, and Kenta Ozeki.

If time permits, I sketch some thoughts about a systematic pursuit of more connections of this flavour.

Convex geometry has long been influenced by the study of dualities and extremal inequalities, with origins in classical affine geometry and functional analysis. In this talk, Kasia Wyczesany will explore an abstract concept of duality, focusing on the classical idea of the polar set, which captures the duality of finite-dimensional normed spaces. This notion leads to fundamental questions about volume products, inspiring some of the most famous inequalities in the field. Whilst Mahler’s influential 1939 conjecture regarding the minimiser of the volume product will be mentioned, the emphasis will be on the Blaschke–Santaló inequality, which identifies the maximiser, along with its modern extensions. Main new results are joint work with S. Artstein-Avidan and S. Sadovsky, and S. Artstein-Avidan and M. Fradelizi. 

Given a quantum Hamiltonian, represented as an $N \times N$ Hermitian matrix $H$, we derive an expression for the largest Lyapunov exponent of the classical trajectories in the phase space appropriate for the dynamics induced by $H$. To this end we associate to $H$ a graph with $N$ vertices and derive a quantum map on functions defined on the directed edges of the graph. Using the semiclassical approach in the reverse direction we obtain the corresponding classical evolution (Liouvillian) operator. Using ergodic theory methods (Sinai, Ruelle, Bowen, Pollicott\ldots) we obtain closed expressions for the Lyapunov exponent, as well as for its variance. Applications for random matrix models will be presented.

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Speaker Yongho Park will talk about 'Subspace Correction Methods for Convex Optimization: Algorithms, Theory, and Applications'

This talk considers a framework of subspace correction methods for convex optimization, which provides a unified perspective for the design and analysis of a wide range of iterative methods, including advanced domain decomposition and multigrid methods. We first develop a convergence theory for parallel subspace correction methods based on the observation that these methods can be interpreted as nonlinearly preconditioned gradient descent methods. This viewpoint leads to a simpler and sharper analysis compared with existing approaches. We further show how the theory can be extended to semicoercive and nearly semicoercive problems. In addition, we explore connections between subspace correction methods and other classes of iterative algorithms, such as alternating projection methods, through the lens of convex duality, thereby enabling a unified treatment. Several applications are presented, including nonlinear partial differential equations, variational inequalities, and mathematical imaging problems. The talk concludes with a discussion of relevant and emerging research directions.

Conventional data feeds from exchanges, even L3 feeds, generally only tell one what happened: accepted submissions of maker and taker orders,  cancellations, and the evolution of the order book and the best bid and ask prices. However, by analyzing a dataset derived from the blockchain of the highly liquid cryptocurrency exchange Hyperliquid, we are able to see all messages (4.5 bn in our one-month sample), including rejections. Unexpectedly, almost 60% of message traffic is generated by submission and subsequent rejection of a single order type: post-only limit orders sent to the 'wrong' (aggressive) side of the book, for example a buy limit order at a price at or above the best ask. Such orders are automatically rejected on arrival except in the (rare) case that the price moves up while the order is in transit. Nearly 30% of message traffic relates to cancellations, leaving a small fraction for all other messages.

I shall describe this order flow in detail, then address the question of why message traffic is dominated by rejected submissions which, by their nature, do not influence the order book in any way at all, and are invisible to all traders except the submitter. We propose that the reason lies in a market-making strategy whose aim is to gain queue priority immediately after any price change, and I shall show how the evidence supports this hypothesis. I shall also discuss the risk/return characteristics of the strategy, and finally discuss its pivotal role in replenishing liquidity following a price move.

Joint work with Jakob Albers, Mihai Cucuringu and Alex Shestopaloff.