Abstract - This talk focuses on the recently introduced notion of bounded cohomology for quandles. We establish sufficient criteria ensuring that the second bounded cohomology of a quandle is infinite-dimensional. As a topological application, we prove that the second bounded cohomology of the fundamental quandles of most links is infinite-dimensional. If time permits, we will further explore this cohomology for families of quandles arising on surfaces.
Abstract - Symmetry and its anomaly constrain a system's dynamics and provide a universal characterization of its behaviors. It serves as a powerful tool to understand exotic phases of matter, especially symmetry-protected topological (SPT) orders. The notion of generalized symmetries requires extending our previous understanding of topological phases to those enriched by fusion category symmetries, yet new mathematical formalism is needed to implement unitary fusion category symmetries in a quantum many-body system. In this talk, I will introduce a general fixed-point lattice construction of (1+1)d SPTs with unitary fusion category symmetries, realized in a tensor-product Hilbert space with an “onsite” matrix-product-operator (MPO) version of the Hopf C*-algebra symmetry operators. Within this construction, I will address that the UV description of an anomaly-free fusion category symmetry must include the fiber functor, giving rise to a local symmetry action, a charge category and a trivial phase, and discuss an alternative characterization of SPT phases using the Q-system in the charge category. As an example, I will provide an explicit microscopic realization of all three 𝖱𝖾𝗉(D8) SPT phases, including a trivial phase, and further demonstrate the S3-duality among these three SPT phases.
Abstract. Adsorption is a ubiquitous chemical process where atoms, ions, or molecules from a fluid attach to the surface of a solid. Adsorption processes lie at the heart of many technologies, for example contaminants bind to porous media in water‑treatment columns, gases are selectively captured in purification and separation processes or for hydrogen storage, chromatography is based on the different adsorption-desorption rates of distinct materials, and organic molecules adsorb onto metal surfaces to form protective corrosion‑inhibiting films. Although these systems appear in very different technological settings, they share a common feature: their performance is governed by interfacial interactions whose kinetics, equilibria and transport are highly nonlinear. Understanding and predicting these behaviours requires mathematical models capable of linking microscopic adsorption mechanisms to macroscopic transport.
In this talk we will investigate the development and analysis of mathematical models for environmental contaminant capture in adsorption columns and also the application of corrosion inhibitors to pipes. The first couples an advection–diffusion equation, governing the transport of contaminants through the column, with a kinetic equation describing mass loss due to adsorption onto the solid phase. The second deals with the attachment of multiple layers of molecules to a substrate, described by a system of ODEs which, in certain cases, permit simple, closed form solutions.
The work demonstrates how classical and modern analytical techniques can yield simple solutions to complex physical problems, so permitting an understanding of the dominant physical processes and paving the way for process optimisation.
This is joint work with Michael Pinsker. A mixed identity for a group $G$ is a word $w(x_1, \dots, x_r, g_1,\dots, g_n)$ in the language of groups (with variables $x_1,\dots, x_r$ and constants $g_1, \dots, g_n\in G$) such that for any $h_1, \dots, h_r\in G$, $w(h_1, \dots, h_r, g_1,\dots, g_n)=1$. For example, in an Abelian group, $x y x^{-1} y^{-1}$ is a mixed identity (without constants). A mixed identity is singular if forgetting the constants and reducing the resulting resulting word, we get the identity. For example, $x g x^{-1}$ is singular, but $x g x$ is not. Recently, Bodirsky, Schneider, and Thom conjectured that if $G$ is the automorphism group of an $\omega$-categorical structure, then all of its mixed identities are singular. We prove that if $G$ has an action with no algebraicity, then all of its mixed identities are singular. Our result applies to the automorphism groups of a large class of $\omega$-categorical structures, including $(\mathbb{Q}, <)$, for which the aforementioned conjecture was open, but also to several other groups of interest to geometric group theory, such the Thompson group $F$, whose mixed identities were studied in works of Ivanov, Słanina, and Zarzycki, or the homeomorphism groups of manifolds. More generally, we prove that all mixed identities of a group $G$ are singular as long as $G$ admits an action for which algebraic closure forms a modular pregeometry and satisfies a certain higher dimensional variant of Neumann's lemma. This covers also infinite vector spaces over finite fields, whose mixed identities were studied by Bradford, Schneider, and Thom by different methods.
This talk is part of the Lancashire Yorkshire Model Theory Seminar.
Bounding Betti numbers of semialgebraic sets has a 70+ year history where questions are studied because they are interesting in their own right, and also because they have applications in many areas. Indeed, such bounds are crucial in incidence geometry and other closely related areas, especially in tools such as the seminal Polynomial Partitioning theorem proved by Guth and Katz. To push discrete geometry into settings where the sets are no longer semialgebraic, but are definable in any arbitrary o-minimal structure, one needs similar precise bounds on the Betti numbers of definable sets.
I will talk about some of my work along this line. In particular, I will talk about some of my work investigating if an analogue of the Polynomial Partitioning theorem can be established in the o-minimal world. I will then talk about our work in generalizing the polynomial partitioning theorem to settings that involve semi-Pfaffian sets. I will then discuss some avenues for future research that I think are particularly important.
This talk is part of the Lancashire Yorkshire Model Theory Seminar.
(Abelian) $ℓ$-groups are abelian groups equipped with a lattice order compatible with addition. A prototypical example is the additive group of continuous real-valued functions on a topological space.
As well as being an interesting variety of groups in their own right, $ℓ$-groups are also central objects in a wide array of modern mathematics, such as in functional analysis as reducts of rings of real-valued continuous functions on a topological space; and in valuation theory as the value groups of valued fields. In particular, a greater understanding of their model theory should hopefully advance our understanding of these areas also.
Work by Glass and Pierce in 1980 showed that the theory of $ℓ$-groups admits no model companion. In this talk, we will look at recent work of mine which seeks to rectify this problem. In particular, we define a multi-sorted extension for $ℓ$-groups, which behaves like the $ℓ$-group $C(X)$ equipped with the map $P : C(X) → Pow(X)$ sending $f$ to the set $\{x ∈ X | f(x) ⩾ 0\}$.
We will see that this theory admits well-behaved representations, generalising well-known results from the theory of $ℓ$-groups. Further, we will show that this theory is companionable, and this model companion is complete, with quantifier elimination in a small language extension. Time permitting, we will also briefly discuss other work in this area, including a similar result for the case of ordered abelian groups.
This talk is part of the Lancashire Yorkshire Model Theory Seminar.
Skew braces are a class of algebraic structures introduced by Guarnieri and Vendramin to study set-theoretic solutions of the Yang–Baxter equation. A skew brace consists of a set equipped with two group operations satisfying a compatibility condition known as skew-left distributivity. This framework bridges group theory, ring theory, and mathematical physics, and has attracted growing interest from an algebraic perspective in recent years. Notions such as solvability and nilpotency—particularly (strong) left nilpotency—have been developed and explored within this context.
In this talk, we investigate skew braces under the assumption of finite Morley rank, a model-theoretic concept that generalizes the idea of dimension from algebraic geometry. In particular, we present a complete classification of skew braces of Morley rank at most 3. Additionally, we prove that if both the additive and multiplicative groups of a skew brace are nilpotent, then the skew brace is strongly left nilpotent. Finally, under an assumption concerning the length of chains of left ideals, we show that if both the additive and multiplicative groups are solvable, then the skew brace is weakly solvable.
This talk is part of the Lancashire Yorkshire Model Theory Seminar
Urbanization shapes infectious disease dynamics through mobility, infrastructure, and spatial structure. Prior work shows how density and network connectivity influence epidemic spread [1,2], particularly for directly transmitted diseases such as COVID-19 and influenza [1,2,3]. However, many models assume pairwise transmission or use static mobility representations, which are less suitable for waterborne diseases like cholera, where risk depends on overlap between movement and shared infrastructure. While spatial cholera dynamics have been studied [4], few models combine mechanistic disease progression, behaviourally realistic mobility, and spatially explicit infrastructure. Common mobility models (e.g., gravity or radiation [2], commuting data [3]) also fail to capture repeated, preferential use of specific locations.
We develop a spatially explicit agent-based model combining a SEIRS process with a density-augmented Exploration and Preferential Return (d-EPR) mobility framework. Agents move between households and shared infrastructure, repeatedly visiting familiar locations while occasionally exploring new ones. Contamination accumulates at infrastructure sites and infection occurs through repeated exposure. The model is implemented on GIS-based urban layouts to examine how spatial structure, movement, and infrastructure jointly shape epidemic dynamics.
Results show that infection concentrates around shared infrastructure, particularly in main activity hubs. In baseline settings, mobility clusters agents into high-use areas, producing localised growth before outward spread. Mobility reduction limits spatial spread and lowers incidence but increases localised exposure. Decentralisation redistributes activity, reducing pressure on individual sites and lowering transmission by limiting repeated exposure. Across scenarios, concentrated movement leads to a small number of high-burden transmission points and faster outbreaks, while reduced or distributed movement spreads load, lowering peak exposure and flattening incidence. These findings highlight the central role of mobility, infrastructure interactions and suggest that targeted, structure-specific interventions are more effective than uniform approaches.
[1]Aguilar, J., Bassolas, A., Ghoshal, G., Hazarie, S., Kirkley, A., Mazzoli, M., ... & Sadilek, A. (2022). Impact of urban structure on infectious disease spreading. Scientific reports, 12(1), 3816.
[2] Wen, T. H., Hsu, C. S., & Hu, M. C. (2018). Evaluating neighborhood structures for modeling intercity diffusion of large-scale dengue epidemics. International journal of health geographics, 17, 1-15.
[3] Moss, R., Naghizade, E., Tomko, M., & Geard, N. (2019). What can urban mobility data reveal about the spatial distribution of infection in a single city?. BMC public health, 19, 1-16.
[4] Phelps, M. D., Azman, A. S., Lewnard, J. A., Antill´on, M., Simonsen, L., Andreasen, V., & Pitzer, V. E. (2017). The importance of thinking beyond the water-supply in cholera epidemics: A historical urban case-study. PLoS neglected tropical diseases, 11(11), e0006103.
Complex systems are often characterised by their intricate structures and behaviours, which can be challenging to analyse and understand. However, the principle of Occam’s Razor suggests that the simplest explanation is often the best. This idea, encapsulated in the concept of the toy model, is often perceived as a parsimonious or exploratory approach when dealing with problems involving many unknown mechanisms or restricted to aggregated data, and it is not always taken seriously. In this work, we connect the idea of model inference from data to rationalise our intuition about minimal models. Connecting ideas from information geometry and Bayesian inference, we show that toy models are often the optimal way to exploit the information in the data and, hence, are not only mathematically or computationally tractable but also optimal in the sense of information theory. We illustrate these ideas with examples from ecology, epidemiology, or astrochemistry. However, they can be applied to any field where data quality and quantity are limited, or the diversity is poor. We also discuss the implications of these ideas in the context of well-established ideas, such as the renormalisation group theory.
