We consider the question of whether almost-homomorphisms are close to honest homomorphisms. I’ll survey a few historical results, with different source/target collections of algebras, and also consider what to take as the definition of “almost-homomorphisms”. If we end up having time, I will sketch an elementary proof that almost-characters from commutative C*-algebras are close to honest characters.

Fix an Artin representation rho. Work in progress by Emmanuel Lecouturier and Loïc Merel claims that the special values L(f,rho,1) for certain modular forms f see some global data related to the L-function attached to rho. We first give a brief exposition on Mazur’s Eisenstein ideal, which lies at the heart of their work. We then describe this conjectural phenomenon in a few simple cases, the last being related to a conjecture of Harris and Venkatesh.

Recently a new inhabitant entered the zoo of convexity notions for vectorial variational problems: functional convexity. I would like to report of progress in understanding the corresponding integrands, but also new insight into fine properties of most general class of related integrands: It turns out that rank-one convex functions share surprisingly many pointwise differentiablity properties with ordinary convex functions.

The well-known "Dunkl operators" associated to a finite real reflection group constitute a commutative parameter family of deformations of the directional derivatives in Euclidean space. These operators are "differential-reflection" operators. Heckman and Cherednik have defined trigonometric versions of Dunkl's operators. The interest for these operators lies in their deep ties to Macdonald polynomials and hypergeometric functions, to the Calogero-Moser quantum integrable system, and to the representation theory of Hecke algebras. 

"Hypergeometric shift operators" are powerful tools to study Weyl group symmetric structures and functions in these contexts. In this talk, Eric Opdam presents a theorem of existence and uniqueness of ''nonsymmetric shift operators'' for the Dunkl operators. These are themselves differential reflection operators which "shift" the parameters of the Dunkl operators by integers by means of a "transmutation relation".

(Joint work with Valerio Toledano Laredo) 

Real complex systems exhibit rich collective behavior, yet identifying which components of an interaction network drive such dynamics remains a central challenge. Here, we show that complexity itself can resolve this problem. In large random and empirical networks, structural disorder and heterogeneity induce spectral localization, causing Laplacian modes to concentrate on small subsets of nodes. This converts global modes into identifiable dynamical units tied to specific structural components. Exploiting this principle, we develop a node-resolved stability framework that predicts instability onsets, identifies the nodes responsible for collective transitions, and restores interpretability in systems where classical modal theories fail. In heterogeneous reaction networks, the same mechanism enables collective states beyond those usually associated with homogeneous assumptions. More broadly, our results show that complexity can be revealed, rather than obscure, the microscopic drivers of macroscopic dynamics.

Tropicalization is a process by which we replace algebraic geometry with the geometry of piecewise linear (tropical) objects. One of the central questions in the field is when this process can be reversed: that is, when can we realize a tropical object with an honest algebraic one. In this talk, I'll discuss some recent work on the tropical to Lagrangian correspondence, and state under what conditions homological mirror symmetry allows us to transfer Lagrangian realizations into algebraic ones.

Large-scale linear least-squares problems arise in many areas of computational science and data analysis, where efficiency and scalability are crucial. In this talk, we introduce a randomized preconditioning framework for iterative solvers based on low-rank approximations of small sketches of the original problem. The key idea is to iteratively construct low-rank preconditioners that reshape the singular value distribution in a favorable way. By tightly coupling the preconditioning and Krylov solving phases within an iterative CUR decomposition -- a low-rank approximation built from selected of columns and rows of the original matrix -- the proposed algorithm achieves faster and earlier convergence than existing methods. The algorithm performs particularly well on problems that are large in both dimensions, as well as on sparse and ill-conditioned systems. This is a joint work with Coralia Cartis and Yuji Nakatsukasa.

The integration of AI into computational fluid dynamics (CFD) represents a transformative frontier for engineering, yet realizing this potential requires navigating the complexities inherent to fluid mechanics. Bridging the methodological gap between deep learning and traditional CFD simulation, this talk presents work (outlined in the recent preprint: Fluids Intelligence: A forward look on AI foundation models in computational fluid dynamics) to produce a novel scaling law tailored specifically for a fluids foundation model. We explore the theoretical and practical opportunities, analyzing the critical inflection points where model training compute begins to eclipse the high costs of traditional data generation. We conclude by discussing the technical challenges and opportunities the fluids and machine learning communities must collaboratively address to operationalize autonomous computational engineering.

Professor Po-Ling Loh will talk about; 'Private estimation in stochastic block models'

We study the problem of private estimation for stochastic block models, where the observation comes in the form of an undirected graph, and the goal is to partition the nodes into unknown, underlying communities. We consider a notion of differential privacy known as node differential privacy, meaning that two graphs are treated as neighbors if one can be transformed into the other by changing the edges connected to exactly one node. The goal is to develop algorithms with optimal misclassification error rates, subject to a certain level of differential privacy.

We present several algorithms based on private eigenvector extraction, private low-rank matrix estimation, and private SDP optimization. A key contribution of our work is a method for converting a procedure which is differentially private and has low statistical error on degree-bounded graphs to one that is differentially private on arbitrary graph inputs, while maintaining good accuracy (with high probability) on typical inputs. This is achieved by considering a certain smooth version of a map from the space of all undirected graphs to the space of bounded-degree graphs, which can be appropriately leveraged for privacy. We discuss the relative advantages of the algorithms we introduce and also provide some lower-bounds for the performance of any private community estimation algorithm.

This is joint work with Laurentiu Marchis, Ethan D'souza, and Tomas Flidr.

Prostate cancer progression and therapeutic resistance present significant clinical challenges, particularly in the transition to castration-resistant disease. Although androgen deprivation therapy and second-generation drugs have improved patient outcomes, resistance frequently develops, reflecting tumour heterogeneity and the influence of its microenvironment. This talk presents two interdisciplinary studies that address these issues through data-driven mathematical approaches. We show how integrating experimental data with mathematical and statistical modelling can improve our understanding of prostate cancer dynamics and inform more effective, context-specific therapeutic strategies. The first study examines drug resistance and tumour evolution under treatment. We develop a multi-scale hybrid modelling framework to capture processes occurring across different temporal scales. Partial differential equations describe the behaviour of drugs and other chemicals in the tumour microenvironment (over the ‘fast’ timescale), while a cellular automaton captures the dynamics of tumour cells (over the ‘slow’ timescale). Through computational analysis of the model solutions, we examine the spatial dynamics of tumour cells, assess the efficacy of different drug therapies in inhibiting prostate cancer growth, and identify effective drug combinations and treatment schedules to limit tumour progression and prevent metastasis. The second study focuses on the role of host–microbiome interactions in obesity-associated prostate cancer. Using data from experiments with the TRAMP mouse model, we apply statistical and machine learning methods, including generalised linear models, Granger causality, and support vector regression, to characterise microbial dynamics and their responses to treatment. These findings are incorporated into a dynamical systems framework that captures microbiome–tumour co-evolution under therapeutic and dietary perturbations, providing insight into how dietary fat and combination therapies involving enzalutamide and phytocannabinoids influence tumour progression and gut microbiota composition.

This week’s Fridays@2 session is intended to provide advice on exam preparation and how to approach the Part A, B, and C exams.  A panel consisting of past examiners and current students will answer any questions you might have as you approach exam season.

In a joint work with Yu Deng (University of Chicago) and Xiao Ma (University of Michigan), we extended Lanford’s theorem to long times—specifically, for as long as the solution of the Boltzmann equation exists. This allowed us to fully carry out Hilbert’s program and derive the fluid equations in the Boltzmann–Grad limit. The underlying strategy builds on earlier joint work with Yu Deng that resolved a parallel problem in which colliding particles are replaced by nonlinear waves, thereby establishing the mathematical foundations of wave turbulence theory. In this talk, we will review this progress and discuss some related problems and future directions. 

Associate Professor Nicolas Boumal will talk about: 'Smooth, globally Polyak-Łojasiewicz functions are nonlinear least-squares'

Polyak-Łojasiewicz (PŁ) functions abound in the literature, especially in nonconvex optimization. When they are also smooth, they become surprisingly simple---with an exotic twist. The plan is for us to discover the structure of those functions and of their sets of minimizers via gradient flow and fiber bundles.

Joint work with Christopher Criscitiello and Quentin Rebjock.

We study the clustering behavior of weakly interacting diffusions under the influence of sufficiently localized attractive interaction potentials on the one-dimensional torus. We describe how this clustering behavior is closely related to the presence of discontinuous phase transitions in the mean-field PDE. For local attractive interactions, we employ a new variant of the strict Riesz rearrangement inequality to prove that all global minimizers of the free energy are either uniform or single-cluster states, in the sense that they are symmetrically decreasing. We analyze different timescales for the particle system and the mean-field (McKean-Vlasov) PDE, arguing that while the particle system can exhibit coarsening by both coalescence and diffusive mass exchange between clusters, the clusters in the mean-field PDE are unable to move and coarsening occurs via the mass exchange of clusters. By introducing a new model for this mass exchange, we argue that the PDE exhibits dynamical metastability. We conclude by presenting careful numerical experiments that demonstrate the validity of our model.

One can learn a lot about a compact manifold if one can show that it fibres over the circle - in essence, this allows us to view a static n-dimensional manifold as a manifold of dimension n-1 that evolves in time.Being fibred (over the circle) is a relatively rare property. It is much more common to be virtually fibred, that is, to admit a finite cover that is fibred. For example, it was the content of a conjecture of William Thurston, now two theorems by Ian Agol and Dani Wise, that all finite-volume hyperbolic 3-manifolds are virtually fibred; in fact, this property is extremely common among irreducible 3-manifolds.The situation is less clear in higher dimensions. On the obstruction side, we know that virtually fibred manifolds must have vanishing Euler characteristic. This immediately shows that compact hyperbolic manifolds in even dimensions will not be virtually fibred. A more involved obstruction comes from L2-homology: virtually fibred manifolds must be L2-acyclic. The motivation behind the research I will present lies in trying to find situations in which the vanishing of L2-homology is is not only necessary, but also sufficient for virtual fibring. It turns our that a lot more can be said if we replace aspherical manifolds by their homological cousins: Poincare duality groups. Concretely, if G is an n-dimensional Poincare-duality group over the rationals, and if G satisfies the RFRS property, then G is L2-acyclic if and only if there is a finite-index subgroup G0 of G and an epimorphism from G0 onto the integers such that its kernel is a Poincare-duality group over the rationals of dimension n-1. (This last theorem is joint with Sam Fisher and Giovanni Italiano.)The RFRS property was introduced in Agol's work on Thurston's conjecture. A countable group is RFRS if and only if it is residually {virtually abelian and poly-Z}. All compact special groups in the sense of Haglund-Wise satisfy this property, so there is a ready supply of RFRS groups, also among fundamental groups of hyperbolic manifolds in high dimensions.

We provide a derivation of the fourth-order DLSS equation based on an interpretation as a chemical reaction network. We consider on the discretized circle the rate equation for the process where pairs of particles sitting on the same side jump simultaneously to the two neighboring sites, and the reverse jump where a pair of particles sitting on a common site jump simultaneously to the side in the middle. Depending on the rates, in the vanishing mesh size limit we obtain either the classical DLSS equation or a variant with nonlinear mobility of power type. We identify the limiting gradient structure to be driven by entropy with respect to a generalization of the diffusive transport type with nonlinear mobility via EDP convergence. Furthermore, the DLSS equation with nonlinear mobility of the power type shares qualitative similarities with the fast diffusion and porous medium equations, since we find traveling wave solutions with algebraic tails and polynomial compact support, respectively.    
       

Joint work with Alexander Mielke and Artur Stephan arXiv:2510.07149. The DLSS part is based on joints works with Daniel Matthes, Eva-Maria Rott and Giuseppe Savaré.

Everyone is invited to celebrate International Women in Mathematics Day with a pizza lunch! We will be watching ‘Journeys of Women in Mathematics’, a powerful 20-minute film by the International Mathematical Union showcasing the experiences of women mathematicians worldwide. It follows three mathematicians from India, Cameroon, and Brazil from their home institutions to the (WM)² international meeting, showing their research and what it’s like to be part of the global maths community.

Quantization is notoriously difficult and even when we have found quantizations of, say, coordinate rings of varieties, these often appear ad-hoc.  But, using ideas that were just emerging when the speaker was embarking on their research journey in the Institute some twenty years ago, a remarkable alternative approach has been developed. We will describe the journey, culminating in a recent result with Pressland showing that quantization and categorification are intimately linked. Braidings, grading and mirror symmetry also feature in our story.

The study of combinatorial designs includes some of the oldest questions at the heart of combinatorics. In a breakthrough result of 2014, Keevash proved the longstanding Existence Conjecture by showing the existence of (n,q,r)-Steiner systems (equivalently K_q^r-decompositions of K_n^r) for all large enough n satisfying the necessary divisibility conditions. Meanwhile, in recent decades, incremental progress has been made on the celebrated Nash-Williams' Conjecture of 1970, which posits that any large enough, triangle-divisible graph on n vertices with minimum degree at least 3n/4 admits a triangle decomposition. In 2021, Glock, Kühn, and Osthus proposed a generalization of these results by conjecturing a hypergraph version of the Nash-Williams' Conjecture, where their proposed minimum degree K_q^r-decomposition threshold is motivated by hypergraph Turán theory. By using the recently developed method of refined absorption and establishing a non-uniform Turán theory, we tie the K_q^r-decomposition threshold to its fractional relaxation. Combined with the best-known fractional decomposition threshold from Delcourt, Lesgourgues, and Postle, this dramatically closes the gap between what was known and the above conjecture. This talk is based on joint work with Luke Postle.

to follow

Cartan subalgebras in operator algebras are objects of dynamical nature that have a long history, both in von Neumann algebras and C*-algebras. A II_1 factor can behave in many different ways, from admitting no Cartan subalgebra, to having a unique one, to having unclassifiably many (up to suitable notions of equivalence).

Much less is known for C*-algebras; while many C*-algebras have canonical Cartan subalgebras, these are usually far from unique even if one prescribes certain topological features, as has been established by now mainly via applications of classification theory. In this talk from Grigoris Kopsacheilis, we will discuss some situations showcasing the variety of Cartans that a C*-algebra may exhibit, some relevant open questions, and we shall discuss some examples, namely extensions of C(S^1) by the compacts, where a form of rigidity occurs, in the sense that all their Cartan subalgebras can be described.

The talk is based on joint work with Wilhelm Winter, and joint work (in progress) with Philipp Sibbel.

In this talk, we give a construction of the Abelian Yang--Mills--Higgs measure on the two-dimensional torus via the variational approach initiated by Barashkov--Gubinelli. The construction is carried out through a disintegration of measures: we first construct the conditional Higgs measure given a rough gauge field, and then construct the gauge field marginal. This leads to iterated variational problems, one for the Higgs field and one for the gauge field. At the technical level, the starting point is the construction of the renormalised covariant Laplacian associated to a rough gauge field, together with the study of its resolvent. This allows us to define the covariant Gaussian free field, which serves as the reference Gaussian field for the conditional Higgs measure. Finally, we analyse the ratios of determinants that arise from the change-of-measure formula for Gaussian measures. This is joint work with Nikolay Barashkov, Ajay Chandra, Ilya Chevyrev, and Andreas Koller.
 

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Insects use flight as far more than a means of getting from A to B. Flight creates an aeiral theatre for interaction, whether between species or among members of the same species. For example, a male dragonfly must hunt for food, fend off rival males, and pursue evasive females in order to reproduce, tasks that all revolve around chasing fast-moving targets. Despite the remarkable diversity of insect species and their aerial behaviours, common patterns emerge in how they exploit speed and manoeuvrability to achieve these goals. Simple geometric guidance laws can describe these flight trajectories with surprising accuracy, revealing shared strategies that underpin insect aerial combat.

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