BMC2026

Monday 30 Mar

12:00
14:00

Registration & Welcome

13:30
14:00

Opening @ Abacws/0.01

14:00
15:00

Plenary TalkTalk @ Abacws/0.01

Cristiana de Filippis · University of Parma

Surfing regularity on nonlinear potentials

Abstract

The representation formula for the Poisson equation gives an explicit expression of solutions in terms of the data, yielding zero- and first-order pointwise bounds via convolution with appropriate Riesz potentials. The mapping properties of these potentials provide sharp regularity transfer from data to solutions, giving a complete description of the regularity features of solutions. I will outline key aspects of nonlinear potential theory that reproduce this behavior for nonlinear elliptic PDEs, where representation formulae are unavailable, and trace their regularity theory back to that of the Poisson equation up to the level. I will then present a novel potential-theoretic approach, altering a century old paradigm in nonlinear regularity theory, that resolves the longstanding problem of the validity of Schauder theory in nonuniformly elliptic PDEs.

15:05
15:30

Invited TalkTalk @ Abacws/5.05

Jessica Jay · Lancaster University

Connections between interacting particle systems and combinatorial objects

Abstract

In 2018 Balázs and Bowen gave a purely probabilistic proof to the Jacobi triple product identity, linking an infinite sum to an infinite triple product with interpretations across Mathematics and Physics. Probabilistically the identity is given as an equivalence of reversible stationary measures for two classical particle systems via the Exclusion–Zero-range correspondence. Recent research has found other examples of probabilistic proofs to identities of combinatorial significance. This connection can be very useful for proving new and sometimes surprising results both in combinatorics and probability. Based on joint works with Daniel Adams, Márton Balázs, Dan Fretwell and Benjamin Lees.

15:35
15:45

Contributed TalkTalk @ Abacws/5.05

Gilles Germain · University of Oxford

Normal approximation of the intrinsic volumes distribution of a convex body

Abstract

We introduce a new version of Stein's method of comparison of operators tailored to bounding the Wasserstein-1 distance between continuous and discrete distributions on the real line. Our approach rests on a new family of weighted discrete derivative operators called bespoke derivatives. The application concerns the convergence of the intrinsic volume distribution of a convex body to the normal distribution, a result that is currently conjectured. By means of our method, we prove the conjecture in the case of rectangular parallelotopes.

15:45
15:55

Contributed TalkTalk @ Abacws/5.05

Nero Z. Li · Imperial College London

Iterated Graph Systems: Brownian motion on fractals

Abstract

We introduce Iterated Graph Systems, a framework generating a broad class of fractal graphs via substitutions, building on classical studies of diffusions on fractals such as the Sierpiński gasket. We prove that, under appropriate rescaling, simple random walks on these graphs converge in the Gromov-Hausdorff-Prokhorov-Skorokhod topology to a diffusion on a compact metric measure space — the associated Brownian motion. We further study heat kernel behaviour and introduce the degree dimension as a quantitative tool unifying the locally finite and infinite-degree regimes within a single analytic framework.

16:00
16:30

Coffee Break

16:30
16:55

Invited TalkTalk @ Abacws/5.05

Tengyao Wang · London School of Economics

Coverage correlation: detecting singular dependencies between random variables

Abstract

We introduce the coverage correlation coefficient, a novel nonparametric measure of statistical association designed to quantify the extent to which two random variables have a joint distribution concentrated on a singular subset with respect to the product of the marginals. Our correlation statistic consistently estimates an f-divergence between the joint distribution and the product of the marginals, which is 0 if and only if the variables are independent and 1 if and only if the copula is singular. Using Monge–Kantorovich ranks, the coverage correlation naturally extends to measure association between random vectors. It is distribution-free, admits an analytically tractable asymptotic null distribution, and can be computed efficiently.

16:55
17:05

Contributed TalkTalk @ Abacws/5.05

Alberto Bordino · University of Warwick

Nonparametric inference for ratios of densities via uniformly valid and powerful permutation tests

Abstract

We propose the density ratio permutation test, a hypothesis test that assesses whether the ratio between two densities is proportional to a known function based on independent samples from each distribution. The test uses an efficient Markov Chain Monte Carlo scheme to draw weighted permutations of the pooled data, yielding exchangeable samples and finite sample validity. We introduce the shifted maximum mean discrepancy and prove minimax optimality of our test when a normalized difference between the densities lies in a Sobolev ball. We extend to the case of an unknown density ratio and derive type I error bounds as well as power results, allowing adaptation to conditional two sample testing — a versatile tool for assessing covariate-shift assumptions arising in transfer learning and causal inference.

17:05
17:15

Short Break

17:15
17:40

Invited TalkTalk @ Abacws/5.05

Marcel Ortgiese · University of Bath

The spatial Muller's ratchet

Abstract

The spatial Muller's ratchet is a spatial model of a population whose dynamics are shaped by the occurrence of deleterious mutations. The 'ratchet' refers to the effect that once a population has lost its fittest individuals it cannot recover these states. Mathematically, the model is described by a spatial birth-death process with rates depending on the local population density. We show that under appropriate re-scaling, the process converges weakly to an infinite system of PDEs. Under certain conditions, we analyse these PDEs and consider the behaviour of travelling waves exploring an empty habitat. Joint work with Joao de Oliveira Madeira and Sarah Penington.

17:40
17:50

Contributed TalkTalk @ Abacws/5.05

Anastasia Kovtun · Cardiff University

Multidimensional Dickman distribution and operator selfdecomposability

Abstract

The one-dimensional Dickman distribution arises in various stochastic models across number theory, combinatorics, physics, and biology. We propose to extend its definition to a class of vector-valued random elements, characterised as fixed points of a specific affine transformation involving a random matrix obtained from the matrix exponential of a uniformly distributed random variable. We prove that these distributions possess infinite divisibility and operator selfdecomposability, and identify several cases where this new distribution arises as a limit. A preprint is available at arXiv:2602.12988.

17:50
18:00

Contributed TalkTalk @ Abacws/5.05

Panqiu Xia · Cardiff University

Ergodicity and Gaussian fluctuations of stochastic heat equations

Abstract

I discuss the ergodicity of the stochastic heat equation driven by centred Gaussian noise, white in time and coloured in space, satisfying the Dalang condition. I provide a sufficient condition for ergodicity and classify the invariant measures based on their expectations. Assuming spatial correlation has a Riesz-type tail |x|^−γ, a Gaussian fluctuation result under diffusive scaling is established, and the diffusive scaling limit is shown to satisfy an Edwards–Wilkinson equation. Based on joint work with Le Chen, Alex Dunlap, Cheng Ouyang, and Samy Tindel.

18:00
21:00

LMS Social and Drinks Reception

Tuesday 31 Mar

09:00
10:00

Morning TalkTalk @ Abacws/0.01

Gui-Qiang Chen · University of Oxford

Partial differential equations of mixed type − analysis and connections

Abstract

Three of the fundamental types of partial differential equations (PDEs) — elliptic, hyperbolic, and parabolic — arise from the classical classification of linear PDEs. The linear theory for each of these types has been extensively developed. By contrast, many nonlinear PDEs arising in mathematics and the sciences are naturally of mixed type. A deep understanding of such nonlinear PDEs — particularly those of mixed elliptic-hyperbolic type — is essential for solving several longstanding fundamental problems. Notable examples include the multidimensional Riemann problem (formulated by Riemann in 1860 for the one-dimensional case) and related shock reflection/diffraction problems in fluid dynamics for the compressible Euler equations, as well as the isometric embedding problem in differential geometry governed by the Gauss-Codazzi-Ricci system. In this talk, we will present both classical and recent connections between nonlinear PDEs of mixed type and these fundamental problems, ranging from the Riemann problem to isometric embedding. We will then discuss recent developments in the analysis of nonlinear mixed-type PDEs, with an emphasis on unified ideas, approaches, and techniques for addressing such problems. Some further perspectives and open problems in this evolving direction will also be addressed.

Morning TalkTalk @ Abacws/2.26

Ivan Nourdin · University of Luxembourg

Quantitative CLT for deep neural networks

Abstract

I will discuss the asymptotic behavior at initialization of fully connected deep neural networks with Gaussian weights and biases when the widths of the hidden layers go to infinity. This talk is addressed to a broad mathematical audience. Apart from a basic knowledge of probability theory, no specific prerequisites are required, and all the necessary notions will be introduced progressively throughout the presentation. This is based on joint work with S. Favaro, B. Hanin, D. Marinucci and G. Peccati.

10:00
11:00

Morning TalkTalk @ Abacws/0.01

Enrico Le Donne · University of Freiburg

Metric Lie groups

Abstract

This talk explores the interplay between metric geometry and group theory through the framework of metric Lie groups. Starting from the notions of homogeneity and self-similarity, I will explain how natural geometric assumptions on a metric space lead to strong algebraic and differential structure, often forcing the space to arise from a Lie group endowed with an invariant distance. Emphasis will be placed on Carnot–Carathéodory spaces and sub-Finsler Lie groups, which provide a rich class of non-smooth yet highly structured geometries. I will also discuss how these spaces naturally appear in large-scale and infinitesimal geometry, for instance, through asymptotic cones, metric tangents, and their relation to volume growth.

Morning TalkTalk @ Abacws/2.26

Soheyla Feyzbakhsh · Imperial College London

Applications of Bridgeland stability conditions in algebraic geometry

Abstract

After a brief introduction to Bridgeland stability conditions on triangulated categories, I will explain several recent applications in algebraic geometry. These include their role in the Brill–Noether theory of vector bundles as well as in the enumerative geometry of Donaldson–Thomas theory.

11:00
11:30

Coffee Break

11:30
12:30

Plenary TalkTalk @ Abacws/0.01

Dennis Gaitsgory · MPIM, Bonn

Deligne-Lusztig theory as trace

Abstract

We will use the formalism of (higher) categorical trace to obtain a natural construction of Deligne-Lusztig representations. We will be able to recover a number of know results, and also obtain some new ones: independence of Deligne-Lusztig characters of the parabolic.

12:30
13:30

Lunch

13:30
13:45

BMC AGM Meeting @ Abacws/0.01

13:45
15:00

LMS Lecture and Society MeetingTalk @ Abacws/0.01

Péter Varjú · University of Cambridge

Self-similar sets and measures

Abstract

Abstract to be added.

15:05
15:30

Invited TalkTalk @ Abacws/5.05

Ellen Powell · Durham University

Scaling limits of critical FK-decorated maps at q=4.

Abstract

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.

15:35
15:45

Contributed TalkTalk @ Abacws/5.05

Sudeshna Bhattacharjee · University of Bristol

Law of fractional logarithm for the GUE minor process

Abstract

The Gaussian unitary ensemble (GUE) minor process is defined as the sequence of top n × n submatrices G_n of an infinite GUE matrix. It is well known that under appropriate centering and scaling the sequence of the largest eigenvalues λ_n of G_n converges weakly to the GUE Tracy-Widom distribution as n → ∞. The question of a law of fractional logarithm for this sequence was considered and partially answered by Paquette and Zeitouni (Ann. Probab., 2017). In particular, they showed that after a further scaling of (log n)^2/3 (resp. (log n)^1/3), the limsup (resp. liminf) of the centered and scaled sequence almost surely converges to some non-zero and finite constant. In this work we complete this picture by determining the explicit constant for the liminf, confirming their conjecture. A key ingredient for our work is understanding the correct decorrelation scale for the sequence. To achieve this, we crucially use a correspondence between the Brownian last passage percolation and the GUE minor process due to Baryshnikov (Probab. Theory and Related Fields, 2001).

15:45
15:55

Contributed TalkTalk @ Abacws/5.05

Ethan Baker · University of Birmingham

Polynomial Ergodicity of the Generalised Relativistic Langevin Equation

Abstract

We discuss the ergodicity of two relativistic Langevin equations, which describe the motion of a particle with relativistic kinetic energy subject to an external potential and noise. We introduce a Relativistic Langevin Equation with memory, the Generalised Relativistic Langevin Equation (GRLE), and re-formulate it as a Markov process by approximating the memory kernel as a sum of exponentials. We construct a Lyapunov function for the GRLE, both in the presence and absence of friction, to obtain polynomial convergence rates of the transition densities to the invariant Gibbs measure.

16:00
16:30

Coffee Break

16:30
16:55

Invited TalkTalk @ Abacws/5.05

María Dolores Ruiz Medina · University of Granada

Non-central limit theorems for sojourn measures of spatio-temporal random fields

Abstract

This paper derives noncentral limit results (NCLTs) for sojourn measures of spatially homogeneous and isotropic, and stationary in time, LRD Chi–Squared Spatiotemporal Random Fields (STRFs). The cases of connected and compact two point homogeneous spaces M_d⊂ℝ^d+1, and compact convex sets K⊂ℝ^d+1 whose interior has positive Lebesgue measure, are analyzed. These NCLTs are obtained in the first Laguerre Chaos by applying reduction theorems. The methodological approaches adopted in the derivation of these results are based on the pure point and continuous spectra of the Gaussian STRFs generating the chi–squared subordinators. Real-data applications where the results derived can be applied are discussed as well.

16:55
17:05

Contributed TalkTalk @ Abacws/5.05

Nenad Šuvak · University of Osijek

Spectral representation of the transition density of killed inverse-gamma diffusion on a bounded state space

Abstract

A one-dimensional inverse-gamma diffusion taking values in (0, K] with a space-dependent killing rate is investigated. By rewriting the infinitesimal generator in Sturm–Liouville form and imposing Dirichlet or Neumann boundary conditions at K, the associated operator is shown to possess a purely discrete spectrum of simple real eigenvalues and a complete orthogonal system of eigenfunctions. A closed-form eigenfunction expansion for the transition probability density is derived, providing an analytical framework for studying transition behaviour and related functionals of inverse-gamma diffusions.

17:05
17:15

Short Break

17:15
17:25

Contributed TalkTalk @ Abacws/5.05

Ivan Papić · University of Osijek

Stretched Non-Local Pearson Diffusions

Abstract

We introduce stretched non-local Pearson diffusions obtained by replacing the classical first-order time derivative in the Kolmogorov equation with a stretched non-local time operator. The resulting processes retain the quadratic diffusion coefficient and linear drift of the Pearson family, while exhibiting non-Markovian temporal behaviour. We derive explicit representations for transition densities via spectral expansions, with the stretched time operator yielding non-exponential relaxation with Kilbas–Saigo (Mittag–Leffler–type) asymptotics, offering analytically explicit examples of diffusion models with memory.

17:30
17:55

Invited TalkTalk @ Abacws/5.05

William Fitzgerald · University of Manchester

Ordered random walks and the Airy Line Ensemble

Abstract

The Airy line ensemble is a random collection of continuous ordered paths playing an important role within random matrix theory and the KPZ universality class. In the KPZ universality class, it describes the scaling limit of a large class of random interface growth models and interacting particle systems. In random matrix theory, it is the edge scaling limit of Dyson Brownian motion. I will discuss these connections and a universality property of the Airy line ensemble: a growing number of i.i.d. continuous-time random walks conditioned to stay in the same order converge in an edge scaling limit to the Airy line ensemble.

Evening

Conference Dinner

Wednesday 1 Apr

09:00
10:00

Morning TalkTalk @ Abacws/0.01

Giovanna Citti · University of Bologna

The Laplace operator on a plane in the Heisenberg group

Abstract

After recalling the main properties of sub-Riemannian manifolds, I'll discuss the geometry induced on sub-manifolds: the one induced by the immersion in the whole space, does not coincide in general with the geometry induced by the differential structure. This allows to give different notions of Laplacian. I'll consider the one whose fundamental solution is a power of the distance induced by the immersion. Using this operator, I'll prove a representation formula for functions belonging to the natural Sobolev spaces. This is a join work with Baldi, Cupini, and Galeotti.

Morning TalkTalk @ Abacws/2.26

Dustin Clausen · IHES, Paris

Some modern aspects of Weil's "Rosetta stone"

Abstract

In 1940, André Weil wrote a famous letter to his sister, in which he gushed about an analogy between number fields and Riemann surfaces, mediated by a third class of objects: the function fields with finite constant field. This threefold analogy remains as compelling now as it was in 1940. I will describe some of its modern aspects.

10:00
11:00

Morning TalkTalk @ Abacws/0.01

Daniel Huybrechts · University of Bonn

Brauer groups and geometry: The period-index conjecture

Abstract

Brauer-Severi varieties (so smooth fibrations with projective spaces as fibres) are not Zariski locally trivial. The failure is measured by the associated Brauer class to which two numerical invariants are attached: period and index. The precise relation between the two is unknown but the index is conjectured to be universally bounded by some power of the period. The talk will start with a gentle introduction into the general theory with a survey of things that are known. In the second half I will study the problem for hyperkähler varieties in which case a better bound is expected and in fact can be proved in interesting cases.

Morning TalkTalk @ Abacws/2.26

Luisa Beghin · Sapienza, Rome

Random processes through non-local operators: analytical versus stochastic approaches

Abstract

We describe two alternative approaches to introducing non-local operators in random models: the analytic approach versus the stochastic one. In the first case, fractional derivatives, in either the classical or generalized sense, appear in place of integer-order derivatives in the partial differential equations that govern the random processes. In the second case, on the other hand, non-local differential and integral operators are introduced directly into the definition of the generalized random process in infinite-dimensional spaces (with various types of measures). Both approaches can lead to models of anomalous diffusions and processes with memory.

11:00
11:30

Coffee Break

11:30
12:30

Plenary TalkTalk @ Abacws/0.01

Mikhail Kapranov · IPMU, Tokyo

Quantum symmetries in operator algebras and mathematical physics

Abstract

Abstract to be added.

12:30
14:00

Lunch

14:00
15:00

Plenary TalkTalk @ Abacws/0.01

Johannes Schmidt-Hieber · University of Twente

The mathematics behind spiking neural networks

Abstract

Artificial neural networks are inspired by the functioning of the brain but differ in several key aspects. In biological neural networks, information is encoded in the spiking times of neurons. In this survey talk, we first address the expressiveness of spiking neural networks and derive a universal representation theorem. Furthermore, it is implausible that biological learning is based on gradient descent. This has prompted researchers to propose various biologically inspired learning procedures. However, these methods lack a solid theoretical foundation. While statistical theory for artificial neural networks has been developed over the past years, the aim now is to extend this theory to biological neural networks, as the future of AI is likely to draw even more inspiration from biology. We will explore the challenges and present some recent theoretical results. Joint work with Niklas Dexheimer, Sascha Gaudlitz, Shayan Hundrieser, Insung Kong, and Philipp Tuchel.

15:05
15:30

Invited TalkTalk @ Abacws/5.05

Gesine Reinert · University of Oxford

Exponential random graph models analysed using Stein's method

Abstract

Exponential random graph models are popular models for the analysis of social networks, due to their flexibility and ability to capture complex dependence in networks. Using ideas from Stein's method we can characterize the model using a Stein operator and devise a kernelized Stein goodness-of-fit test based on this characterisation. We generate synthetic samples from the model underlying the observed data by mimicking the Stein operator dynamics. Finally, we show how a generalised approach can be used for anomaly detection. Based on joint works with Nathan Ross, Wenkai Xu, and Michal Kozyra.

15:35
15:45

Contributed TalkTalk @ Abacws/5.05

Ibrahim Kaddouri · University of Warwick

Late change-point in the preferential attachment random graph model

Abstract

We consider the problem of late change-point detection under the preferential attachment random graph model with time dependent attachment function, formulated as a hypothesis testing problem. It was conjectured that when observing only the unlabeled graph, detection of the change is not possible for Δ_n = o(n^1/2). We prove the impossibility of detecting the change when Δ_n = o(n^1/3). We also study change-point detection in the case where the labeled graph is observed and show that detection is possible if and only if Δ_n → ∞, exhibiting a strong difference between the two settings.

15:45
15:55

Contributed TalkTalk @ Abacws/5.05

Ollie Baker · University of Bristol

Entropy of Soft Random Geometric Graphs in General Geometries

Abstract

Knowledge of the entropy of a random network is important in information theory and probability. When considering the Soft Random Geometric Graph, there is currently a lack of understanding of how the geometry affects its information theoretic properties. We introduce the 'entropy graph' formalism, which allows us to study the amount of entropy that individual points contribute to the total entropy via their 'entropy mass'. We use this to prove a limit theorem in the small connection range limit, and quantify the effects of boundary components on the graph entropy in various geometries.

16:00
16:30

Coffee Break

16:30
16:55

Invited TalkTalk @ Abacws/5.05

Domenico Marinucci · University of Rome Tor Vergata

The Geometry of Random Neural Networks

Abstract

We study the geometric properties of random neural networks by investigating the boundary volumes of their excursion sets for different activation functions, as the depth increases. For activations which are not very regular (e.g., the Heaviside step function), the boundary volumes exhibit fractal behavior, with their Hausdorff dimension monotonically increasing with the depth. For more regular activations (e.g., ReLU, logistic and tanh), the expected boundary volumes can either converge to zero, remain constant or diverge exponentially, depending on a single spectral parameter. Based on joint works with Simmaco Di Lillo, Michele Salvi and Stefano Vigogna.

16:55
17:05

Contributed TalkTalk @ Abacws/5.05

Leoni Carla Wirth · University of Oxford

Private synthetic graph generation

Abstract

We focus on differential privacy in network generation. Starting with an input dataset viewed as vertex attributes, we construct a privatized network representation by first applying a privacy mechanism to vertex attributes and jointly constructing network representations based on both original and privatized data. We show the resulting procedure is differentially private and provide theoretical analysis of its utility, both in expectation and in distribution. Based on joint work with Gholamali Aminian (Alan Turing Institute) and Gesine Reinert (University of Oxford).

17:05
17:15

Short Break

17:15
17:25

Contributed TalkTalk @ Abacws/5.05

Alexander Kent · University of Warwick

Rate Optimality and Phase Transition for User-Level Local Differential Privacy

Abstract

We consider user-level local differential privacy, where each of n users holds T observations and wishes to maintain privacy of their entire collection. We obtain minimax optimal estimation rates for canonical statistical estimation problems including univariate and multidimensional mean estimation, sparse mean estimation, and non-parametric density estimation. We derive a general minimax lower bound showing that the risk cannot, in general, be made to vanish for a fixed number of users even when T is arbitrarily large. Matching upper and lower bounds are derived, and we observe phase transition phenomena in the minimax rates as T varies.

17:30
17:55

Invited TalkTalk @ Abacws/5.05

Tom Berrett · University of Warwick

Permutation testing under local differential privacy

Abstract

Personal and sensitive data is now collected at larger scales than ever before. Growing concern from data subjects and regulatory bodies has led to an increased demand for statistical procedures that do not compromise the privacy of individuals. In this talk I will discuss recent work on two-sample testing under a local differential privacy constraint where a permutation procedure is used to calibrate the tests. We design appropriate privacy mechanisms, both interactive and non-interactive, that allow for permutation tests. Our analysis shows that these lead to minimax optimal separation rates in both discrete and continuous settings, with interactive procedures being significantly more powerful. Joint work with Alexander Kent and Yi Yu.

18:00
19:30

Pizza Buffet

19:30
20:30

Public LectureTalk @ Centre for Student Life/2.06

Jens Marklof · University of Bristol

The Mathematics of Chaos: Predicting the Unpredictable

Abstract

We live in a chaotic world: from weather forecasts and natural disasters to political developments, it seems often impossible to even make the most basic predictions. In this lecture I will discuss one of the most fundamental mathematical principles behind such unpredictability: the inherent instability of systems with chaotic behaviour that leads to extreme amplification of the smallest errors in the data. In particular, I will explore with you the legendary question “Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?” (Lorenz 1972) and explain how chaos theory has established the best way of kneading bread dough! By the end of this lecture, I hope to have convinced you of the power of mathematics in the understanding of some of the world’s most complex systems. And although we will never be able to predict the future with complete certainty, we can forecast likelihoods. The more chaotic, the better!

Skein theory is an efficient tool for the graphical construction of topological field theories and modular functors, based on a given input category . We show that a Frobenius monoidal functor induces a relation between the associated skein theories. We present a situation where this relation is an equivalence. This turns out to encode information about correlators of two-dimensional conformal field theories.

Quantum many-body physics is a field of physics that investigates macroscopic properties of matter emerging from the collective behavior of many interacting quantum particles. The study of topological order tries to classify phases of matter by introducing equivalence or preorder relations between states of physical interest from a physically motivated point of view. This physical point of view introduces the concept of locality to the problem, a feature that has not been extensively investigated in purely mathematical research. In this talk, I would like to give an overview of this developing area of mathematical physics.