Past

A central challenge in applied mathematics is to extract predictive structure from data generated by complex dynamical systems. Koopman operator methods provide a principled framework for this task by embedding nonlinear dynamics into a linear operator acting on observables, reducing analysis and forecasting to questions about spectral approximation.

In this talk, I will present recent results on the analysis of data-driven Koopman methods, with an emphasis on when spectral quantities can be reliably approximated from finite data. I will describe a general framework that connects operator-theoretic properties of the Koopman operator with the behaviour of practical algorithms, clarifying phenomena such as spectral pollution and the role of continuous spectra. I will also discuss fundamental limitations: there exist classes of dynamical systems for which finite data cannot recover meaningful spectral information, placing intrinsic constraints on what Koopman-based approaches can achieve. Building on this, I will show how spectral approximation errors translate into quantitative bounds for forecasting, capturing how approximation and statistical errors propagate over time and ultimately limit long-term prediction. These results have implications for applications including fluid dynamics, molecular systems, and geophysical flows. I will conclude by highlighting open problems at the intersection of operator theory, numerical analysis, and scientific machine learning.

I am going to talk on a work aimed at formalising approximation procedures in physics.  The main new model-theoretic tool in this work is the notion of the ultraproduct in the classes of emerging metric structures which generalises the ultraproduct  of general structures developed by J.Kiesler. In particular, the structure of Minkowski spacetime with the action of the Lorentz group is an emerging metric ultraproduct of certain finite structures invariant under the action of appropriate finite groups. Also, it is shown that any compact simple Lie group is representable as emerging metric ultraproduct of finite groups.
 

Recent advances at the interface of stochastic analysis, rough path theory, stochastic filtering, stochastic control, and mean-field systems have led to a rapidly developing framework for analyzing stochastic dynamics conditioned on common/observation noise. This mini course  will survey how rough stochastic differential equations, introduced in 2021 by A. Hocquet, K. Lê and the speaker, lead to a unifying perspective across several areas of applied probability. (Additional coauthors include F. Bugini, J. Dause, W. Stannat, H. Zhang and P.Zorin-Kranich).

The boundedness of Fourier multipliers on non-commutative $L_p$-spaces ($1 < p < \infty$) is a fundamental problem in non-commutative analysis. Building on the non-commutative Cotlar identity introduced by Mei and Ricard (2017), which yields $L_p$-boundedness ($1 < p < \infty$) of Hilbert transforms on amalgamated free products of finite von Neumann algebras, their approach relies heavily on freeness in the underlying free product structure.

In this talk, Xiaoqi Lu introduces a new strategy that overcomes this limitation. Our approach combines a generalized Cotlar identity, which holds on suitable subspaces and captures non-freeness information, with an additional condition related to the property of Rapid Decay to control the remaining components. From this framework, we establish the $L_p$-boundedness ($1 < p < \infty$) of Rademacher-type Hilbert transforms on graph products of finite von Neumann algebras. This unified framework extends earlier results for free products of finite von Neumann algebras and for graph products of groups acting on right-angled buildings. This is a joint work with Runlian Xia.

Donaldson--Thomas invariants are virtual counts of coherent sheaves on complex Calabi–Yau 3-folds, where Brav–Bussi–Joyce's shifted Darboux theorem plays an important role. In this talk, I will present a Darboux-type theorem in characteristic $p>2$ for $(-1)$-shifted symplectic forms equipped with an "infinitesimal" structure suggested by Toën and Robalo. This result may be viewed as a first step towards exploring Donaldson–Thomas theory in positive characteristic.

Denote by p(k) the limit, as n tends to infinity, of the probability that a random permutation on n letters has some invariant set of size k. For example, p(1) = 1 - 1/e. I will discuss the asymptotic behaviour of p(k). Joint work with Mehtaab Sawhney.

Professor Ruoyu Sun will talk about: 'Understanding and Improving LLM Training via Hessian and Spectral Analysis' 

In the first part, we investigate the approximate block-diagonal Hessian structure of neural networks. We identify the conditions under which this structure emerges and give the first rigorous proofs based on random matrix theory. From this structural perspective, we explain why Adam works far better than SGD on Transformers. Following this structural guideline, we design the memory-efficient optimizer Adam-mini; Normuon is another optimizer developed under the same principle.

 In the second part, we adopt a spectral perspective to study and refine normalization layers for neural network training. We propose a preconditioning (PC) layer, an advanced weight-centric module built with low-degree polynomial preconditioning for scalable spectral control. Theoretically, for deep linear networks, we prove that bounding each layer's singular values ensures geometric convergence of gradient descent to global minima. Empirically, PC delivers consistent efficiency gains over a standard Transformer baseline in Llama2-1B pretraining.

Nilsequences are sequences coming from Lie groups which play the role of additive characters in higher order Fourier analysis. In this talk, I will define these and give some basic examples without assuming any prior knowledge. I'll use this to state an equidistribution result due to Green and Tao, and compare what happens in this setting to the familiar case of sequences in the torus.
 

Dehn Surgery is an operation on 3-manifolds that is ubiquitous in the field of low-dimensional topology. Concretely, given a link L in a 3-manifold N, Dehn surgery produces many new 3-manifolds. A classical result of Lickorish-Wallace states that from one fixed 3-manifold N one can obtain all other 3-manifolds by Dehn surgery using some link in N. However it remains unclear, in general, which manifolds can be obtained by Dehn surgery using a fixed manifold N and a fixed link L. I will discuss how one can algorithmically decide this question and then discuss applications of this algorithm. 

We consider a Kolmogorov-type PDE corresponding to a particle under white noise force. We are interested in stopping the process at a fixed position i.e. imposing Dirichlet conditions at a side boundary. We construct a simple Gaussian heat kernel inside the domain and investigate a boundary-layer kernel connected to some work by McKean. We show that this boundary layer heat kernel has a novel jump condition. We outline a polynomial expansion of for the heat kernels and then construct a Volterra equation for solving the original problem. The novel jump leads to a periodic structure of the Volterra equation.

Mr Oluwadamilola (Dami) Fasina will talk about; 'Learning with tensor paraproducts'

We discuss computational (Neural FIM) and analytical (tensor paraproducts) tools for learning structure of sets. In the first situation we focus on learning the metric amongst elements of a statistical manifold. To do so, we design a neural network which enables one to compute the Fisher information metric (FIM), so long the Jensen-Shannon divergences amongst probability distributions on the statistical manifold are preserved during training. In the second situation we focus on analyzing the structure of function compositions through separation of its low and high frequency components. This is accomplished by elaborating on J.M. Bony’s celebrated work on paraproducts by discretizing and allocating distinct scaling parameters along each dimension of the support of a function composition (with a prescribed regularity), permitting finer analytical control. A consequence of this extension is highlighted with a discussion of the regularity gains of kernels of integral operators. 

In 2008, Toms constructed a counterexample to the Elliott conjecture: a pair of simple, separable, nuclear and unital C*-algebras which are indistinguishable by the Elliott invariant, but are not isomorphic. The key to distinguishing this pair of carefully crafted C*-algebras lies with a rather refined invariant called the Cuntz semigroup. Consequently, Toms’s counterexample highlighted the importance of the Cuntz semigroup to the classification of C*-algebras.

In this talk, we will discuss the Cuntz semigroup in the context of graph C*-algebras, a highly diverse class of mostly non-simple C*-algebras. In particular, we will accentuate how the highly organised structure of a unital graph C*-algebra is reflected in its Cuntz semigroup and if enough time permits, mention properties of unital graph C*-algebras that are revealed by these Cuntz semigroups.

The celebrated additive kinematic formula expresses the mean volume of the Minkowski sum of two compact convex subsets of the Euclidean space placed at random. What about non convex subsets? What about other Lie groups than the Euclidean space? In a joint work with Andreas Bernig, we prove additive kinematic formulas for compact subanalytic sets of the Euclidean space and of the 3-sphere. The key is to generalize the Minkowski sum of convex bodies by a notion of convolution of subanalytic sets introduced by Schapira in the late 80s using Euler characteristic computations. The above will of course be an excuse to discuss integral geometric formulas and constructible functions.

In this talk, I will focus on a new construction of boundary correlators (or wavefunction coefficients in dS) that highlights simplicity at all spins and automatically imposes the conservation of boundary currents. This new construction is formulated in twistor space, a complex projective space that encodes solutions to equations of motion as holomorphic data. This is done via an isomorphism called the Penrose transform. First, I will discuss the case of AdS_3 and AdS_5, where bulk-to-boundary correlators naturally arise in minitwistor space. Then, I will show how in (A)dS₄ one can construct bulk correlation functions using only twistors, dual twistors, and the infinity twistor as building blocks. The relation to coordinate space arises now via nested Penrose transforms. The boundary limit of these correlators yields CFT correlators/wavefunction coefficients that satisfy the expected Ward identities. Finally, I will briefly discuss how this can be generalized to AdS_5 boundary correlators using ambitwistors.

Mechanistic ordinary differential equation (ODE) models are a powerful tool to study dynamic biological systems. However, their predictive power is constrained by gaps, biases, and inconsistencies in the literature. They typically also require quantitative time-lapse data for training, which is time-consuming to collect. At the same time, machine-learning approaches can capture complex patterns from data, but they are often harder to interpret and typically require large training datasets. Hybrid scientific machine learning (SciML) models offer a promising way to combine the strengths of both approaches by integrating mechanistic models with flexible data-driven modules. 
Despite this promise, the use of SciML in biology remains limited by insufficient infrastructure. Dedicated software is needed because coding end-to-end differentiable workflows for gradient-based training of hybrid models is technically challenging. In addition, model exchange is hindered by the lack of a standardized, reproducible format for specifying SciML training problems, analogous to the PEtab standard for ODE models. To address these challenges, we developed PEtab-SciML, an extension of the PEtab format, and implemented support for it in the state-of-the-art modeling toolboxes PEtab.jl and AMICI. In this seminar, I will introduce the PEtab-SciML format. Using real-data examples, I will show how PEtab-SciML enables the integration of diverse data modalities into dynamic model training; such as learning the kinetic parameters of an ODE model from omics and protein sequence data. I will also show how it supports machine-learning-based black-boxing of complex model components, such as quarantine strength in an SIR model. Finally, I will show how PEtab-SciML enables the use of efficient training strategies, such as curriculum learning, that make SciML models easier to train and apply in practice. 

Stability is the prototypical model theoretic dividing line. One interpretation is that a binary relation is stable if it is "close to unary": if the question $(x,y)\in E$ can be answered, at least most of the time, by knowing enough information about $x$, and separately enough information about $y$.

One natural question is asking how this can generalize to ternary (and higher-arity) relations. The connection to hypergraph regularity suggests an approach to identifying ternary stable-like properties, and also that there should be several versions, since a ternary relation could be almost unary, or almost binary, or a combination of these properties.

In this talk, I'll survey some of what we know about several of these "stable-like" ternary notions.

In many prediction and decision problems, the relevant inputs are path-valued covariates rather than static feature vectors. This paper studies asymptotic theory and empirical applications for path regression using signatures. We first establish \(L^2\) approximation rates for truncated signature representations. We prove a minimax-optimal approximation rate over a class of smooth coefficient functionals of observable It\^{o} diffusions. Building on this approximation theory, we then develop asymptotic results for three signature-based learning procedures: Signature-OLS, Signature-LASSO, and Signature-Logistic. These results establish asymptotic normality for least-squares path regression, sparse recovery for high-dimensional signature regression, and latent-score consistency for binary-response classification. Extensive empirical studies cover three real-data applications: foreign-exchange realized-volatility forecasting from intraday price paths, battery end-of-life prediction from early HPPC pulse paths, and epileptic seizure detection from short EEG windows. The empirical results show that signatures provide informative representations of path-valued covariates relative to handcrafted features.

Understanding the behaviour of L-functions of modular forms is a very classical and yet open problem. The Bloch-Kato conjecture predicts that the order of vanishing of the L-function of a modular form should be given by the rank of certain Bloch-Kato Selmer groups. In order to give a lower bound to these ranks in certain cases where the L-function vanishes, Bellaiche and Chenevier developed a clever strategy where they construct classes in the Selmer group via the geometry of points corresponding to certain lifts of modular forms on higher dimensional eigenvarieties. This strategy was successfully adapted for ordinary modular forms by Berger and Betina to give a lower bound in terms of the smoothness of Saito-Kurokawa points on a genus 2 Siegel eigenvariety. We generalise this work to finite slope and crucially infinite slope forms which are not seen on the Coleman-Mazur eigencurve - here we must develop the machinery of small parabolic eigenvarieties for the problem to be well defined. As a result we get new results towards the Bloch—Kato conjecture for infinite slope forms.

The ε-energy is a regularisation of the Dirichlet energy introduced by Tobias Lamm. Like the famous Sacks-Uhlenbeck regularisation this greatly improves the existence and regularity theory. When we take the limit of a sequence of ε-harmonic maps with the parameter ε decreasing to 0 these converge, in the standard bubbling sense, to harmonic maps, which we hope to extract information about. I will talk about some recent results for these sequences, being when we might hope to have no loss of energy and no neck forming and what sort of harmonic maps we can obtain in the limit.

Professor De Terran will talk about: 'New results on the inclusion of closure orbits and bundles of matrices and matrix pencils' 

Orbits of nxn matrices under similarity are sets of matrices with the same Jordan Canonical form (JCF). When computing the JCF (or just the eigenvalues) of a matrix, the knowledge of all possible JCFs of small perturbations of a given JCF can help to understand the output of the algorithm, which is affected by roundoff errors.

The JCFs that can be obtained after small perturbations of a given JCF, say J, correspond to orbits that ``dominate" the orbit of J. In other words, the orbit of J is in the closure of its dominant orbits. The hierarchy of orbit closures of general matrices is well-known, as well as that of the set of matrices with bounded rank.

For matrix pencils (namely, pairs of matrices with the same size) the inclusion relationship between orbit closures has been also considered since, at least the 1980's. In this case, the standard equivalence relation is the so-called strict equivalence, which preserves the eigenstructure of the pencil, and the canonical form for this relation is the Kronecker canonical form (KCF). The hierarchy of orbit closures of general pencils under strict equivalence is also well-known. However, when the pencil has some particular structure (e. g., symmetric or Hermitian) then we encounter a different problem if we want the perturbations to maintain this structure. Some effort has been devoted in recent years to the analysis of orbit closures of structured pencils.

In this talk, we will review some recent results on the inclusion relationship between orbit closures of general and bounded-rank structured matrix pencils. We will also consider the inclusion relation of bundle closures. Bundles are generalizations of orbits, allowing the eigenvalues to change, while keeping the KCF. 
 

Global symmetries have been generalized to non-invertible ones. For finite symmetries in $(1+1)$d, these are known as unitary fusion category symmetries. One natural question is: which fusion categories can arise as symmetries on a lattice? 
Progress has been made including the anyon chains, which realizes any fusion category symmetries. However, their Hilbert spaces do not admit the usual tensor product structure (tensor product of local Hilbert spaces over each site).
In [arxiv:2507.05185], Evans and Jones introduced an operator-algebraic framework and showed that a fusion category symmetry can be realized on a tensor product quasi-local algebra if and only if it is "integral". After reviewing this result, I will discuss a recent extension by Bunner and Jones [arxiv:2605.21327], who showed that this constraint disappears after stabilization with infinite-dimensional ancilla spaces on anyon chains. As a consequence, every unitary fusion category can be realized on tensor product Hilbert spaces.

Lorenzo Lazzarino will talk about: 'Error estimations for randomized low-rank approximations'

Randomized algorithms in numerical linear algebra have proven to be effective in ameliorating issues of scalability when working with large matrices, efficiently producing accurate low-rank approximations. A key remaining challenge, however, is to efficiently assess the approximation accuracy of randomized methods without additional expensive matrix accesses.

In this talk, we discuss a posteriori error estimation strategies for randomized low-rank approximations, with a focus on estimators that can be constructed from the same data used to compute the approximation or without matrix global accesses. These can serve both as certification tools and as algorithmic building blocks, enabling adaptive approximations and informed trade-offs between accuracy and computational cost. As a motivation and a case study, we include a discussion on spectromicroscopy experiments.

More than seventy years after its publication, Turing’s article “The Chemical Basis of Morphogenesis” is still able to surprise its reader, in particular for the power and the depth of its vision. If we know from his biographer, Andrew Hodges, that Turing became interested in embryology and morphogenesis because he wanted to build or, better, to grow a brain, many questions still arise for the reader of the original article: why did Turing – a mathematician, a logician, a cryptographer, one of the fathers of computer science – not use any informational metaphor associated with the notion of “genetic program” in his work on morphogenesis, preferring instead to develop a modelling approach based on a system of partial differential equations ? Where did he draw his modelling inspiration from, both from the point of view of the mathematics and from the point of view of references to biology ? In my presentation I will address these questions by highlighting the morphological connotations of Turing’s work in biology, that can be related to Turing’s interest, in D’Arcy Wentworth Thompson’s classic On Growth and Form (1917). The 1952 article is rather sparse in indications in this regard, which are, however, provided by Turing’s other writings, unpublished during his lifetime, in which he situates his work in continuity with Thompson’s morphological questions. I will also suggest that, as in a virtuous circle, Turing masterfully brings to life a synergy between a morphological look at the living (that implies that his work has a connotation in theoretical biology) and a mathematical exploration of the non-linear, helped by an appropriate and meaningful use of numerical calculus. 

In this talk, I will introduce the concept of Archimedean closedness - a concept from real non-commutative algebraic geometry which determines when "positivity" of an element (captured through *-representations) in a *-algebra can be completely certified algebraically. On the other hand, property FD is a representation theoretic property of groups depicting when any representation of a group can be approximated by finite representations in the unitary dual. I will try to connect these two seemingly very different concepts through some examples and speculations. This is a work in progress.

Given a quantum field theory, realising its global symmetries and anomalies on a lattice has been a fruitful approach to gain new insights of these symmetries. In this talk, we present an exact lattice model in 2+1D which hosts an exact microscopic avatar of its low-energy SU(2) valley symmetry and parity anomaly. We first show that our lattice model has a Lieb-Schultz-Mattis (LSM) anomaly of the “Onsager symmetries” in the UV, which indeed enforces that every Hamiltonian which is symmetric has to be gapless. We then show that the SU(2) Parity anomaly on the IR can be exactly matched by this LSM anomaly. Finally, we briefly discuss our results in relation to similar anomaly matching schemes in 1+1D and 3+1D. 

Recent advances at the interface of stochastic analysis, rough path theory, stochastic filtering, stochastic control, and mean-field systems have led to a rapidly developing framework for analyzing stochastic dynamics conditioned on common/observation noise. This mini course  will survey how rough stochastic differential equations, introduced in 2021 by A. Hocquet, K. Lê and the speaker, lead to a unifying perspective across several areas of applied probability. (Additional coauthors include F. Bugini, J. Dause, W. Stannat, H. Zhang and P.Zorin-Kranich).

The critical Fortuin–Kasteleyn random planar map with parameter q>0 is a model of random (discretised) surfaces decorated by loops, related to the q-state Potts model. For q<4, Sheffield established a scaling limit result for these discretised surfaces, where the limit is described by a so-called Liouville quantum gravity surface decorated by a conformal loop ensemble. At q=4 a phase transition occurs, and the correct rescaling needed to obtain a limit has so far remained unclear. I will talk about joint work with William Da Silva, XinJiang Hu, and Mo Dick Wong, where we identify the right rescaling at this critical value and prove a number of convergence results.

In the context of Fourier analysis on the real line, a \textit{one-sided problem} involves deducing properties of a function $f$ from some information about the restriction of its Fourier transform $\widehat{f}$ to a half-line, for instance to $\mathbb{R}_- := (-\infty, 0)$. A prototypical result, which is foundational to the theory of Hardy spaces on $\mathbb{R}$, asserts that if $f \in L^2(\mathbb{R})$ is non-zero and $\widehat{f}$ vanishes on a half-line, then $f$ satisfies the \textit{Szeg\H{o} condition} $\int_{-\infty}^\infty \frac{\log |f(x)|}{1+x^2} \, dx > -\infty$. 

Various problems in operator theory involve the study of functions $f$ satisfying a weaker condition of decay of $\widehat{f}$ on a half-line. In this setting, simple examples show that the Szeg\H{o} condition need not be satisfied. However, the following local Szeg\H{o}-type conditions hold: if the decay of $\widehat{f}$ is strong enough on a half-line, then the mass of the function $f \in L^2(\mathbb{R})$ must concentrate enough for the integral $\int_E \log |f(x)| dx$ to converge on a "massive" set $E$. 

In his talk, Bartosz Malman will describe this mass condensation phenomenon and its applications to operator-theoretic problems.

The `conifold gap' conjecture asserts that the polar part of the Gromov-Witten potential of a Calabi-Yau threefold near its conifold locus has a universal expression described by the logarithm of the Barnes G-function. In this talk I will describe a proof of the Conifold Gap Conjecture for the local projective plane, whereby the higher genus conifold Gromov-Witten generating series of local P2 are related to the thermodynamics of a certain statistical mechanical ensemble of repulsive particles on the positive half-line. As a corollary, this establishes the all-genus mirror principle for local P2 through the direct integration of the BCOV holomorphic anomaly equations.

Clean markings on surfaces were a key component in Masur and Minsky's hierarchy machinery, which proved to be a powerful tool in the study of mapping class groups. In this talk, I will briefly discuss the connection between clean markings and hierarchies, and I will explain how a natural analogue can be constructed for finite-type Artin groups.

We present a new dynamical way of establishing local laws for sparse random matrices, the Bernoulli flow method. It is based on a Markovian jump process, where the entries of the matrix jump independently from 0 to 1 at rate one. As an application, we show optimal (up to a constant) isotropic delocalisation for bulk eigenvectors of Erdös-Rényi graphs with edge probability p \geq (log N)^2/N. In the same regime, we obtain a local law with optimal (up to a constant) error bounds. Joint work with Antti Knowles.

Ken Brown reviews the history of the question in the title, and describes some recent progress towards answering it, including the identification of a "minimal criminal". The new material is joint work with Jason Bell (Waterloo) and Toby Stafford (Manchester).

Many systems in the natural sciences and beyond exhibit complex relational structure that changes over time. Social networks evolve as relationships change, traffic patterns vary throughout the day, and protein–protein interactions shift with cellular conditions. Learning these dynamics from data is a challenging problem. A recent approach in this area, Graph Neural Controlled Differential Equations, extends Neural CDEs from paths on Euclidean domains to paths on graph domains. In this talk, we discuss an extension of this framework that respects the geometry of the underlying set and is equivariant to permutations of the node ordering. We will discuss empirical advantages of this modification, as well as benefits of the formulation as a continuous-time model. 

Schwinger’s Closed Time Path formalism is the basis of modern treatments of cosmological field theories, hydrodynamics and open quantum systems. Its application to gauge theories at finite temperature is well studied, relying on KMS boundary conditions and complex-time contours. By contrast the discussion of gauge theories such as Yang-Mills out of equilibrium has been less well developed, in large part due to a lack of development of how to treat gauge issues and Faddeev-Popov-DeWitt ghosts on the CTP. I will show how to construct the CTP in the BRST formalism, where a single diagonal copy of BRST symmetry survives, and how to implement the boundary conditions for ghosts for arbitrary initial physical states. As an illustration I will discuss how Hard-thermal-loop EFTs can be viewed as open quantum systems, and how to construct an open EFT for a gauge theory in a Higgs phase. 

The cylindrical snap-fit is a ubiquitous fastening method that is both simple to manufacture and assemble, and yet secure. It consists of a partial cylindrical shell that ‘snaps’ onto a cylindrical object. We build on previous work to describe the mechanics of the cylindrical snap-fit as a naturally curved thin elastic shell placed atop a rigid cylinder; we investigate the shell's behaviour when subject to a point force pushing it onto or pulling it off the cylinder. We classify the possible contact regimes according to whether the shell has a nonzero lifting capacity. We term situations with lifting capacity ‘grip-fits’ and show that this includes both the snap-fit and a ‘stick-fit’ regime, which allows lifting despite not having the characteristic ‘snap’. We show that the different regimes may be characterized entirely by the shell/cylinder geometry and the coefficient of friction. We then consider different metrics for the lifting performance in the grip-fit regime. Our analysis reveals the trade-offs between assembly force, disassembly force, lifting force, and clamping force, providing design principles for secure lifting, easy detachment, and safe handling of fragile objects.

I will discuss some general PDE techniques, for example ``extended substitute kernel'' and ``linearized gluing'', and their application to various gluing constructions in Differential Geometry.

This paper advances the theory of \textit{Collective Finance}, as developed in \cite{BDFFM26}, \cite{DFM25} and \cite{F25}. Within a general semimartingale framework, we study the relationship between collective market efficiency and individual rationality. We derive necessary and sufficient condition for the existence of (possibly zero-sum) exchanges among agents that strictly increase their indirect utilities and characterize this condition in terms of the compatibility between agents’ preferences and collective pricing measures. The framework applies to both continuous and discrete-time models and clarifies when cooperation leads to a strict improvement in each participating agent’s indirect utility.

Fox’s trapezoidal conjecture, proposed in 1962, states that the absolute values of the coefficients of the Alexander polynomial of an alternating knot form a trapezoidal sequence: they strictly increase, possibly plateau, and then strictly decrease. In this talk, I will discuss recent progress on the conjecture, based on joint work with András Juhász and Tamás Kálmán. In particular, we prove the conjecture for diagrammatic plumbings of special alternating links and obtain partial results for alternating three-braid closures