A list of talk abstracts and slides from the Mathematical Foundations of AI Conference can be found below.
Erik Bekkers
From Spinoza to Equivariance: Geometry as the Shared Structure of Mind and Matter
What is intelligence, and how does mind relate to matter? I want to start where Spinoza did, in 17th-century Amsterdam. He treated metaphysics with the rigor of geometry, and concluded that mind and matter are two attributes of one and the same substance, not two separate substances. I think that starting point is still the right one.
We never reach reality directly, only through representation, so we cannot leave the observer out of our account of the world, and matter as we know it is already a description shaped by mind. If mind and matter are two aspects of one reality, they are governed by the same laws, and since our best account of those laws is geometric, geometry is the structure they share. If our models are to represent reality faithfully, their internal organization should respect that geometry, which is why equivariance is not optional. In the talk I follow this line from Klein’s Erlangen Programme through geometric deep learning, and show that equivariance now comes at no extra cost, that scale does not replace it, and that the current frontier is to carry geometric structure into the latent space: world models, generation on manifolds, and novel view synthesis. I call this vision Ideal Machine Intelligence.
Rebekka Burkholz
Towards AI That Is Smart, Sparse and Social
Deep learning continues to achieve impressive breakthroughs across disciplines but relies on increasingly large neural network models that are trained on massive data sets. Their development inflicts costs that are only affordable by a few labs and prevent global participation in the creation of related technologies. But does it really have to be like this? We will identify some of the major challenges of deep learning at small scales and present solution strategies pertaining to the design of sparse training algorithms and problem specific neural network design. As examples, we will discuss NeuralODE models of gene regulatory dynamics and agentic networks, which hold the promise to overcome a fundamental trade-off between model specialization and trainability.
Kevin Buzzard
On autoformalization
Formalization is the art of translating mathematics from a human language such as English into a formal language such as Lean. Autoformalization is when an AI tool does it for you. I’ll speak about what’s been happening recently in the area.
Coralia Cartis
Towards understanding feature learning: low-rank functions and data properties
Low-rank or multi-index functions appear in both optimization and machine learning, displaying structure simplicity that allows dimensionality reduction and captures the important directions of variation of the landscape. This allows efficient optimization and learning or generalisation. We will discuss some relatively new occurrences of low rank objectives, such as when considering trained nets as functions of the training data, and the implications of this on robustness of training and the learning of important features. Time permitting, we will also discuss training data recovery algorithms, where we show both theoretically and numerically that given final and initial parameters of the trained network, as well as access to network gradients, and provided the network is sufficiently (but finitely) wide, we can recover training data samples to desired accuracy efficiently, by using only a subset of the parameters and low-rank properties of the data. Throughout our work, we will discover interesting connections between the data and the parameters of deep neural networks.
Oliver Clarke
Hierarchical embeddings and phylogenetic ranks of graphs
The phylogenetic rank of a finite metric space as introduced by Pachter and Sturmfels is the minimal number of metric trees needed to embed it isometrically. Here, the product of metric trees is endowed with the supremum norm. Intuitively the phylogenetic rank gives us a discretely measure of how far away a space is from being a tree. In this talk I will show how the phylogenetic rank is subadditive under certain constructions. I will also explain recent work where we develop both a greedy and an exact algorithm for computing phylogenetic ranks of metric spaces arising from graphs. Using our algorithms, we construct a database of phylogenetic ranks which includes all graphs on at most 7 vertices. In particular, we exhibit examples disproving: the phylogenetic rank is a hereditary property; bounded by half the number of points of the space; and a generalised 4-point conjecture by Pachter and Sturmfels. My work is joint with F. Ashworth, J. Giansiracusa, J. Jones, J. Quias-Acaves, Y. Ren.
Francesco Fabiano
Enhanced Decision-Making for AI Reasoning
This talk focuses on how AI decision-making can be enhanced through both existing logical or symbolic approaches and neural, experience-based methods. I present decision-making as the core process that connects reasoning, learning, memory, and action selection. The talk shows how decision-making can be improved in multiple directions: through robust and game-theoretic methods for uncertain environments, through cognitive architectures inspired by fast and slow thinking, and through neural approaches such as GNN-enhanced epistemic reasoning. Across these directions, the key idea is that better AI reasoning often comes from using the right heuristics and relaxations: mechanisms that reduce computational complexity, guide search, and allow agents to make effective decisions even when full reasoning is too expensive or uncertain.
Holly Francois
AI in Online Safety Regulation: Use Cases and Considerations
The rapid adoption of generative artificial intelligence is transforming the design and operation of online services, creating both new opportunities and emerging challenges for online safety regulation. This presentation explores the role of AI within the UK’s Online Safety Act framework and examines how regulators can address the risks and benefits associated with increasingly autonomous and generative systems. Drawing on experience from Ofcom’s Online Safety Technology team, the talk considers how AI is reshaping key safety-critical functions, with a focus on real world use cases and potential emerging risks. Particular attention is given to AI-driven recommender systems, the challenges of explainability and oversight in automated moderation, and the growing use of AI chatbots as companions, including potential impacts on children and vulnerable users. The discussion concludes by identifying priority research areas that can support evidence-based regulation and help ensure that advances in AI contribute to a safer online environment while preserving fundamental rights and freedoms.
Peter Grindrod
Next Generation and Generation After Next AI; mathematical underpinnings of creative disruption and fundamental understanding for responsible use
Mathematics will be both the innovator and the disruptor in the next phase of AI. It will move us beyond systems that merely correlate toward systems that reason, adapt, and justify their actions within known limits. It will help us replace blind trust with warranted confidence. And it will enable forms of creativity, not just in generating content, but in solving problems that are rigorous, accountable, and genuinely new. If we want AI that society can rely on, mathematics must be at its core. That is not a constraint on progress. It is the condition that makes progress sustainable.
Maz Hardey
The Liminal Place: AI, Math, and Human Variance in V Movements
Modern computational optimisation operates on a quiet, unsparing logic: treating friction, uncertainty, and human variance as structural defects to be engineered away. As machine learning models convert the warmth of human language, art, and reasoning into high-dimensional vector spaces, they succeed in minimising mathematical loss – yet in smoothing that terrain, they risk erasing the quiet interval where genuine understanding actually takes root. Drawing on her research across digital platforms, family systems, and computing cultures, Prof Mariann Hardey (Prof Maz) explores how algorithmic homeostasis penalises the atypical mind, swapping the spiky reality of real thinking for a safe, synthetic average. Moving through visual art, higher education’s ‘cringe deficit,’ William Thurston’s human-centred proofs, and neurodivergent variance, this talk challenges computer scientists, business leaders and mathematicians to reconsider the role of friction in intelligent systems. Rather than building high-speed off-ramps for cognitive and emotional discomfort, Hardey offers a constructive call for liminal-preserving architectures: computational systems designed to hold the pause, honour human variance, and safeguard the quiet space where original thought takes breath.
Kathryn Hess
Ring scores: quantifying circularity in data
I will give an overview of the theory of ring scores, which are numerical invariants of compact metric spaces that measure the prominence of circular structure, enabling a “circularity analysis” analogous to classical cluster analysis. Ring scores satisfy four axioms that together encode how closely a metric space resembles the standard circle equipped with its arc-length metric. I will explain how to characterize ring scores that factor through the degree-$1$ persistence diagram of the Vietoris-Rips filtration and sketch a stability theorem for the lifespan ring scores associated to it. To conclude I will outline an application of ring scores as a screening statistic for the detection of cyclic cell processes in single-cell data. (Joint work with Markus Youssef)
Stefanie Jegelka
Neural parameter symmetries: Learning on LoRAs, breaking symmetries and the effect of data
Parameter symmetries within neural networks govern important properties, including the optimization landscape and training behavior and model merging. They are also an important consideration when aiming to make predictions about neural networks from their weights, i.e., in the emerging area of weight space learning. In this talk, we will discuss aspects of neural parameter symmetries via illustrative examples. First, we build models to predict the performance of LoRA (low-rank adaptation) finetunes of generative models much faster than running the actual evaluation. Here, neural networks are inputs for our prediction, and due to the cost of learning on neural weights, taking into account symmetries can be particularly beneficial here. Second, we study the effect and behavior of symmetries within neural networks. To do so, we devise methods for removing such symmetries, as a tool for studying their effect. While doing so, we observe that the geometry of the data, although neglected by many works on symmetries and merging, plays an important role, too.
Roland Kwitt
Scaling-Up Persistent Homology Computation
This talk examines the computational challenges of scaling persistent homology for 3D point-cloud data. Scaling is considered along two complementary axes: increasing the number of points within each point cloud and processing large collections of point clouds in practical machine-learning pipelines. I will discuss applications in which the computational and memory requirements of current persistent-homology methods – and the subsequent vectorization of persistence diagrams – have so far limited their large-scale use. Finally, I will present some preliminary developments in learning directly from the underlying filtered simplicial complexes, using representations that capture topological and spectral information while bypassing explicit persistence reduction and diagram vectorization.
Richard Lane
Classifier Probability Calibration
Probabilities produced by AI models often do not reflect their true accuracy, being under- or over-confident in their predictions. If a model is 80% sure of an outcome, is it correct 80% of the time? Understanding calibration is important for assurance in safety or business-critical contexts and builds user trust in models. Probability calibration metrics measure the discrepancy between confidence and accuracy. We recently published a comprehensive review of such metrics, grouping them into four main families: point-based, bin-based, kernel or curve-based, and cumulative. This talk provides an overview of the metrics and how they can be used in practice.
Darrick Lee
The geometry of cochains on sampled point clouds
Manifold learning often begins by approximating an unknown continuum geometry with a graph, whose Laplacian can converge to the Laplace–Beltrami operator. However, a manifold also carries higher-order structure: differential forms, cohomology, and Hodge Laplacians. In this talk, we describe a framework for approximating this structure using simplicial complexes built from point-cloud data. We discuss the discretization of differential forms as simplicial cochains, which we equip with inner products based only on ambient distances. We provide probabilistic convergence results of these discrete constructions to their continuum analogues and consider their consequences for the discrete Hodge Laplacian. Based on joint work with Kelly Maggs.
Eng-Jon Ong
From Memorization to Simplification: A Three-Phase Explanation for Grokking
This talk concerns the deep learning phenomenon of “grokking,” where overparameterized neural networks exhibit delayed generalization, with accuracy on unseen data improving long after perfect training accuracy has been achieved. Existing work considers grokking through the lenses of the transition from lazy to rich training dynamics, late-stage neural collapse, and the information bottleneck principle. However, the precise mechanical evolution of internal hidden representations during this delay remains an open question. We show empirically, alongside a mathematical framework, how the learning process unfolds across three distinct phases: Phase 1) memorization, Phase 2) feature compression, and Phase 3) model simplification, with Grokking occuring in the last 2 phases. We demonstrate how Phase 2 is governed by an anisotropic spectral compression: the cross-entropy loss acts as a soft margin constraint that shields task-relevant weight matrix singular vectors, while weight decay compresses out the uninformative directions. Once this passive compression reaches its limit, Phase 3 begins. In this final phase, the weight vectors actively rotate, causing a collapse in the stable rank of each hidden layer. This alignment effectively produces a simplified model that exhibits improved generalization and robustness to both unseen and adversarial examples.
Yue Ren
Optimization in Polydisc Spaces: A Non-Archimedean Framework for Hierarchical Data
Hierarchical data is ubiquitous in many applications, whether it be inherent in the problem (phylogenetics, genomics, etc) or artificially introduced by humans (language processing, computer vision, etc). Prime example are international logistic networks, where locations are clustered first by city, then county, then country, and finally continent. An intrinsic characteristic of hierarchical data is the fact that distances do not add up. Regardless how far you move within a city, it will never get you out of the county. This property is called the non-Archimedean property. It is the reason why hierarchical structures are difficult to capture over the real numbers, and why their analysis is challenging.
Existing workarounds include working in hyperbolic space or working in very high dimensions (as seen in large language models). A natural approach is using non-Archimedean fields such as the p-adic numbers. While these fields are indispensable in number theory, where their hierarchical structure allows the study of polynomial equations prime by prime, optimisation in non-Archimedean spaces remains challenging: their totally disconnected topology prohibits the use of many standard optimisation techniques.
In this talk, we propose a new framework for analyzing hierarchical data in the form of so-called polydisc spaces over non-archimedean fields. Inspired by the theory of Berkovich geometry, we show these polydisc spaces retain the hierarchical structure of their non-archimedean field while acquiring many desirable geometric features absent from it. As such, they are capable of both serving as a representation space for hierarchical data as well as a domain amenable to standard optimisation algorithms.
Moreover, we present NonArchimedeanMachineLearning.jl, a new Julia library for optimisation in polydisc spaces. This is joint work with Paul Lezeau, Yiannis Fam, and Anthea Monod.
Tatiana Shavrina
AI Agents changing Scientific Discovery
AI research agents are rapidly transforming scientific discovery by automating complex research workflows. This talk surveys recent advances in frontier agentic systems, LLM-based research assistants, and evaluation benchmarks. We will examine agent performance across the scientific lifecycle—from hypothesis generation to experimentation and refinement—and discuss the remaining challenges toward truly autonomous scientific discovery.
Suvrit Sra
Tight generalization bounds in Inverse Optimization
Inverse optimization (IO) seeks to infer the parameters of a decision-maker’s objective from observed context–action data. We study noiseless IO, where demonstrations are generated by a ground-truth objective. We provide a high-probability O(d/T) generalization bound for the induced action set, where d is the number of unknown parameters and T is the size of the training dataset. We strengthen these guarantees under additional conditions that ensure uniqueness of the chosen action, bringing our IO guarantees in line with best-arm identification results in the bandit literature.
We further show that the O(d/T) rate is tight over all consistent estimators considered here, and extend the result to both instantaneous and cumulative regret. Notably, the resulting regret lower bound matches the corresponding upper bounds in the adversarial setting, indicating that the stochastic IO setting is effectively adversarial for the class of estimators studied here. Finally, we propose a parameter-free algorithm with lower per-iteration complexity than generic solvers. Experiments validate the predicted rates and illustrate the tightness of our bounds.
Marika Taylor
Physics inspired learning: geometry and symmetries
Physics-inspired neural networks (PINNs) can be thought of as models whose design and training incorporates physical principles such as symmetry and locality, as well as dynamical equations of motion. For example, rotation equivariant networks can be more efficient in classification of 3d images, while PINNs used for fluid dynamics problems penalise deviations from solving the fluid equations of motion. In this talk we will explore how underlying geometrical and symmetry structure can be built into networks, with a particular focus on networks used for analysis of quantum data.
Qiquan Wang
The Shape of Adversarial Influence: Characterising LLM Latent Spaces with Persistent Homology
Existing interpretability methods for large language models (LLMs) typically focus on linear directions or isolated features, which can miss the high-dimensional and nonlinear geometry of internal representations. In this talk, persistent homology is used to characterise the geometry of LLM latent spaces and to study how these representations are reshaped under adversarial inputs. Across multiple models (3.8B–70B parameters) and different attack settings, including indirect prompt injection and backdoor fine-tuning, a consistent phenomenon of topological compression is observed. Adversarial inputs induce a simplification of the latent space, where varied small-scale structure collapses into fewer dominant large-scale features. This signature is found to be architecture-agnostic, emerges early in the network, and remains highly discriminative across layers. By quantifying the shape of activation point clouds, this perspective reveals geometric structure in representational change that complements existing linear interpretability approaches.
George Williamson
From Possible to Proven: is the Erlangen Programme for AI a National Infrastructure programme?
Felix Klein’s Erlangen Programme didn’t just tidy up geometry: it gave mathematicians a shared language for understanding which properties were preserved across different geometrical transformations, and which were not. Modern AI still lacks an equivalent framework. We can build systems that work, often remarkably well, but we struggle to characterise in advance the conditions under which their behaviour will remain predictable, robust, and reliable. AI possesses many partial theories of generalisation and failure, but no unifying framework that plays a comparable role. That gap is no longer a purely academic concern. As AI becomes embedded in critical infrastructure, defence, and public services, uncertainty about its failure modes becomes a matter of national resilience rather than scientific tidiness. This talk argues, from inside the UK’s national institute for data science and AI, that the mathematical foundations agenda represented by this hub is not simply upstream of deployment, but must be intimately connected to it. I’ll reflect on what it would take to move from mathematically interesting to institutionally trusted, and ask whether AI needs its own analogue of the Erlangen Programme before it can be relied upon as national infrastructure.
Arne Wolf
Breaking Symmetry Bottlenecks in GNN Readouts
Graph neural networks (GNNs) are widely used for learning on structured data, yet their ability to distinguish non-isomorphic graphs is fundamentally limited. These limitations are typically attributed to message passing. This talk reveals an independent bottleneck at the readout stage. We introduce fundamental notions of finite-dimensional representation theory to prove that all linear permutation-invariant readouts, such as sum and mean pooling, inevitably project node embeddings onto a fixed subspace. This erases all non-trivial symmetry-aware components, regardless of how powerful the encoder is. To study and mitigate this bottleneck, we introduce projector-based invariant readouts that decompose node representations into symmetry-aware channels and summarize them with nonlinear invariant statistics. Finally, we demonstrate how modifying only the readout enables fixed encoders to separate WL-hard graph pairs and improves performance on symmetry-sensitive tasks.
Ka Man (Ambrose) Yim
Testing for Spatial Randomness with a Topological Stein Statistic
We propose a novel statistic for testing whether a given point cloud is completely spatially random. Usual spatial randomness tests rely on comparing test statistics (such as nearest neighbour distance distributions) of a given point cloud with those in the completely spatially random case. Our Stein Statistic enhances a test statistic by baking in properties of a completely spatially random model in the statistic itself. This is accomplished using a Stein operator for the Poisson point process (a model for completely spatially random point clouds). By augmenting topological test statistics with a Stein operator, we demonstrate its empirical effectiveness on synthetic point clouds and show theoretical asymptotic convergence results of this statistic. This talk features joint work with Gesine Reinert and Omer
