Past

The flow of viscous fluid through a soft porous medium exerts drag on the matrix and induces non-uniform deformation. This behaviour can become increasingly complicated when the medium has a complex rheology, such that deformations exhibit elastic (reversible) and plastic (irreversible) behaviour, or when the rheology has a viscous component, making the response of the medium rate dependent. This is perhaps particularly the case when compaction is repeated over many cycles, or when additional forces (e.g. gravity or an external load) act simultaneously with flow to compact the medium, as in many industrial and geophysical applications. Here, we explore the interaction of viscous effects with elastic and plastic media from a theoretical standpoint, focussing on unidirectional compaction. We initially consider how the medium responds to the reversal of flow forcing when some of its initial deformation is non-recoverable. More generally, we explore how spatial variations in stress arising from fluid flow interact with the stress history of the sample when some element of its rheology is plastic and rate-dependent, and characterise the response of the medium depending on the nature of its constitutive laws for effective stress and permeability.

Consider a scalar conservation law with a spatially discontinuous flux at a single point x = 0, and assume that the flux is uniformly convex when x ̸= 0. I will discuss controllability problems for AB-entropy solutions associated to the so-called (A, B)-interface connection. I will first present a characterization of the set of profiles of AB-entropy solutions at a time horizon T > 0, as fixed points of a backward-forward solution operator. Next, I will address the problem of identifying the set of initial data driven by the corresponding AB-entropy solution to a given target profile ω T, at a time horizon T > 0. These results rely on the introduction of proper concepts of AB-backward solution operator, and AB-genuine/interface characteristics associated to an (A, B)-interface connection, and exploit duality properties of backward/forward shocks for AB-entropy solutions.
 

Based on joint works with Luca Talamini (SISSA-ISAS, Trieste)

The classical Leopoldt conjecture predicts that the global units of a number field (tensored with Qp) inject into the local units at p. In this talk, I'll discuss some non-abelian generalisations of this in the setting of Galois representations.

The talk will be about a natural percolation model built from the so-called Brownian loop soup. We will give sense to studying its phase transition in dimension d = 2 + epsilon, with epsilon varying in [0,1], and discuss how to perform a rigorous „epsilon-expansion“ in this context. Our methods give access to a whole family of universality classes, and elucidate the behaviour of critical exponents etc. near the (lower-)critical dimension, which for this model is d=2. 

Based on joint work with Wen Zhang.

By work of Goodwillie-Weiss, given any manifold $M$ with boundary, there is a cosimplicial space whose totalization is a close approximation to the space of embedding of $[0,1]$ in $M$ with fixed behaviour at the boundary. The resulting homology spectral sequence is known to collapse rationally for $M=\mathbb{R}^n$ by work of Lambrechts-Turchin and Volic. I will explain a new proof of this result which can be generalized to a manifold of the form $M=X\times[0,1]$ with $X$ a smooth and proper complex algebraic variety. This involves constructing an action of some Galois group on the completion of the cosimplicial space. This is joint work with Pedro Boavida de Brito and Danica Kosanovic.

We consider a class of Lagrangian sections L contained in certain Calabi-Yau Lagrangian fibrations (mirrors of toric weak Fano manifolds). We prove that a form of the Thomas-Yau conjecture holds in this case: L is isomorphic to a special Lagrangian section in this class if and only if a stability condition holds, in the sense of a slope inequality on objects in a set of exact triangles in the Fukaya-Seidel category. This agrees with general proposals by Li. On
surfaces and threefolds, under more restrictive assumptions, this result can be used to show a precise relation with Bridgeland stability, as predicted by Joyce. Based on arXiv:2505.07228 and arXiv:2508.17709.

Dominik Lukeš from the AI Competency Centre will give an introductory survey of AI in relation to programming.

Zigzag persistence enables tracking topological changes in time-dependent data such as video streams. Nevertheless, traditional methods face severe computational and memory bottlenecks. In this talk, I show how the zigzag persistence of image sequences can be reduced to a graph problem, making it possible to leverage the near-linear time algorithm of Dey and Hou. By invoking Alexander duality, we obtain both H₀ and H₁ at the same computational cost, enabling fast computation of homological features. This speed-up brings us close to real-time analysis of dynamical systems, and, if time permits, I will outline how it opens the door to new applications such as the study of PDE dynamics using zigzag persistence, with the Gray-Scott diffusion equation as a motivating example.

theories on the lattice based on symmetry disentanglers: constant-depth
circuits of local unitaries that transform not-on-site symmetries into on-
site ones. When chiral symmetry can be realized not-on-site and such a
disentangler exists, the symmetry can be implemented in a strictly local
Hamiltonian and gauged by standard lattice methods. Using lattice ro-
tor models, we realize this idea in 1+1 and 3+1 spacetime dimensions
for U (1) symmetries with mixed ’t Hooft anomalies, and show that sym-
metry disentanglers can be constructed when anomalies cancel. As an
example, we present an exactly solvable Hamiltonian lattice model of the
(1+1)-dimensional “3450” chiral gauge theory, and we argue that a related
construction applies to the U (1) hypercharge symmetry of the Standard
Model fermions in 3+1 dimensions. Our results open a new route toward
fully local, nonperturbative formulations of chiral gauge theories.

In this talk, I'm going to give an introduction to my area of research, which concerns automorphic L-functions. We're going to start by introducing the ring of adeles and how it leads us to an integral representation of the Riemann zeta function. We'll see how this can be generalised for an arbitrary automorphic representation and pose general conjectures which resemble the Riemann Hypothesis. I'll finish by presenting the statement and an idea behind my recent result related to those conjectures.

Normal heart function relies of the fine-tuned synchronization of cellular components. In healthy hearts, calcium oscillations and physical contractions are coupled across a synchronised network of 3 billion heart cells. When the process of functional isolation of rogue cells isn’t successful, the network becomes maladapted, resulting in cardiovascular diseases, including heart failure and arrythmia. To advance knowledge on this normal-to-disease transition we must first address the lack of a mechanistic understanding of the plastic readaptation of these networks. In this talk I will explore coupling and loss of synchronisation using a mathematical model of calcium oscillations informed by experimental data. I will show some preliminary results pointing at the heterogeneity hidden behind seemingly uniform cell populations, as a causative mechanism behind disrupted dynamics in maladapted networks.

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.

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.

Flat space holography, if there really is such a thing, is intimately related to Carrollian geometry. I will give an introduction to Carrollian geometry, and discuss how many Carrollian spaces of interest arise as homogeneous spaces of the Poincaré group. Finally, I will discuss the construction of Cartan geometries modelled on these spaces.

Speaker Edward Tansley will talk about: 'Low-rank functions in machine learning'

Functions that vary along a low-dimensional subspace of their input space, often called multi-index or low-rank functions, frequently arise in machine learning. Understanding how such structure emerges can provide insight into the learning dynamics of neural networks. One line of work that explores how networks learn low-rank data representations is the Neural Feature Ansatz (NFA), which states that after training, the Gram matrix of the first-layer weights of a deep network is proportional to some power of the average gradient outer product (AGOP) of the network with respect to its inputs. Existing results prove this relationship for 2-layer linear networks under balanced initialization. In this work, we extend these results to general L-layer linear networks and remove the assumption of balanced initialization for networks trained with weight decay.

 Is it possible to define gauge theories on singular spaces? The answer to this question is emphatically yes​, and the prime example of such spaces are two-dimensional orbifolds known as spindles​. First, I will introduce spindles from a symplectic geometry perspective. Then I will discuss the notion of orbi-bundles, which allows one to consistently describe regular gauge fields/spinors on orbifolds.

In recent years, a more top-down approach to renormalisation for singular SPDEs has emerged within the theory of regularity structures, based on regularity structures of multi-indices. This approach adopts a geometric viewpoint, aiming to stably parametrise the solution manifold rather than the larger space of renormalised objects that typically arise in fixed-point formulations of the equation. While several works have established the construction of the renormalised data (the model) in this setting, less has been shown with regards to the corresponding solution theory since the intrinsic nature of the model leads to renormalised data that is too lean to apply Hairer’s fixed-point approach.

In this talk, I will discuss past and ongoing work with L. Broux and F. Otto addressing this issue for the Phi^4 equation in its full subcritical regime. We establish local and global well-posedness within the framework of regularity structures of multi-indices; first in a space-time periodic setting and subsequently in domains with Dirichlet boundary conditions.

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.

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. 

We adapt Joyce's theory of wall-crossing for enumerative invariants of $\mathbb C$-linear additive categories to Pandharipande-Thomas stable pairs on smooth projective Fano 3-folds of "type C or D", and investigate implications for Pandharipande-Thomas generating functions with descendent insertions.

By analyzing the wall-crossing behavior from a stability condition where pairs are unstable to the standard stability condition for PT stable pairs, we derive an explicit formula expressing the PT stable pair invariants $[P_n(X,\beta)]^{virt}$ in terms of sheaf-theoretic invariants $[\mathcal M^{ss}_{(0,0,\beta_i, n_i - \beta_i.c_1/2)}(\tau_-)]_{\rm inv}$ for the moduli space of Gieseker semistable coherent sheaves on $X$ with Chern character $(0,0,\beta_i, n_i - \beta_i.c_1/2)$.

These enumerative invariants are defined as elements in the Lie algebra on the rational Betti homology of the piecewise-linear rigidified higher moduli stack of objects in the bounded derived category of X. Under tensoring by a line bundle, we exhibit a control over the periodicity of sheaf-theoretic invariants with respect to the Euler characteristic $n_i$, which we use to show that the sheaf-theoretic invariants form a quasi-polynomial in $n_i$ of degree $2$ with period given by the divisibility of $\beta_i$ in the lattice $H_2(X,\mathbb Z)/\text{torsion}$.

We use this periodicity in the sheaf-theoretic invariants to show that the descendent generating series for Pandharipande-Thomas stable pairs is the Laurent expansion of a rational function over $\mathbb Q$ in this setting, thus confirming a conjecture due to Pandharipande-Thomas from 2007. Furthermore, we construct a counterexample to a conjecture due to Pandharipande from 2017 on the location of the poles of the descendent generating series, and give a direct proof of a slightly modified conjecture on the location of these poles using wall-crossing techniques.

 

We give examples to show that theorems of S Fisher and D Kielak 
relating the vanishing of L^2 cohomology to fibering over the 
circle for RFRS groups cannot be extended to some larger classes
of groups, and we introduce an L^2-torsion invariant that may 
prove useful. 

(Joint with Sam Hughes and Wolfgang Lueck) 

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.

A fundamental phenomenon in the representation theory of finite and compact groups is that irreducible characters tend to take smaller values on elements that are far from central. Character estimates of exponential type (that is, bounds of the form |chi(g)|<chi(1)^(1-epsilon)) are particularly useful for probabilistic applications, such as bounding the mixing time of random walks supported on conjugacy classes.

In 1981, Diaconis and Shahshahani established sharp estimates for irreducible characters of the symmetric group S_n, evaluated at a transposition t = (i j). As an application, they proved that roughly n*log(n) random transpositions are required to mix a deck of n playing cards. This was extended in 2007 by Muller--Schlage-Puchta to to arbitrary permutations in S_n. Exponential character bounds for finite simple groups were subsequently developed through a series of works by Bezrukavnikov, Liebeck, Shalev, Larsen, Guralnick, Tiep, and others. 

In this talk, Itay Glazer (Technion) will present recent progress on exponential character estimates for compact Lie groups.

This is based on joint work in progress with Nir Avni, Peter Keevash, and Noam Lifshitz.

Asymptotically Locally Flat (ALF) Ricci-flat metrics are expected to model certain long-time singularities in four-dimensional Ricci flow, so understanding their stability is essential. In this talk, I will discuss that conformally Kähler, non-hyperkähler Ricci-flat ALF metrics are dynamically unstable under Ricci flow. Our work establishes three key tools in this setting: a Fredholm theory for the Laplacian on ALF metrics, the preservation of the ALF structure along the Ricci flow, and an extension of Perelman’s λ-functional to ALF metrics. This is joint work with Tristan Ozuch.

In this seminar, I will explore how the sum over topologies in pure AdS_3​ quantum gravity furnishes a consistent statistical interpretation of the boundary CFT_2. By formulating a statistical version of the conformal bootstrap, which combines crossing symmetry with typicality at high energies, we will discover a large set of non-handlebody topologies in the bulk (of which I will give some examples) that are needed for consistency of the boundary description. Interestingly, this set contains only on-shell (i.e. hyperbolic) 3-manifolds, but not all of them. This suggests that the full sum over all on-shell saddles in 3d gravity may be a highly non-minimal solution of the statistical bootstrap. Based on the recent work 2601.07906 with Belin, Collier, Eberhardt and Liska.

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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In this talk I will give a hopefully not too technical introduction to one of the techniques that allowed Taylor and Wiles to prove the modularity theorem that was the final step for proving Fermat's Last Theorem.
After explaining how the patching works, I will present some generalisations of the method to different contexts. If time permits, I will also briefly explain how patching was used to produce a candidate for the p-adic local Langlands correspondence.

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. 

Quotients are a powerful tool used for constructing exotic embeddings in groups that act on negatively curved metric spaces.  Models for random quotients originate in work of Gromov, Arzhantseva and Ol’shanskii where relations are sampled from spheres in free groups to study genericity of properties like hyperbolicity.  I will introduce a new model for random quotients of groups that instead samples relations using random walks and describe how this model is well-adapted to studying quotients of groups with more flexible actions on hyperbolic spaces and discuss geometric tools used to establish when these more general forms of negative curvature are preserved in random quotients. These techniques also provide new examples of groups that are quasi-isometrically rigid and exotic common quotients.  This talk will be based on joint work with Abbott, Berlyne, Mangioni, and Rasmussen.

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.  

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.

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.

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.

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.

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.