Non-recurrent dynamics - processes in which a system goes through a sequence of states while never returning to a previous state - encompasses a range of important problems, perhaps most notably the spreading of epidemics. We consider a system in which individuals go through a set of states as a consequence of interactions with their neighbours on an underlying network substrate. Predicting the evolution of such processes on a general network is an exponentially difficult problem. On certain classes of networks, however, very precise approximations can be given in the form of message-passing equations, which make use of the fact that non-recurrent dynamics can, in principle, be solved exactly on trees. Here we offer an intuitive interpretation of the message-passing approximation, discuss its connection to the non-backtracking matrix of the given network substrate and explore when (and how) this approximation is expected to fail.
Recall that a combinatorial solution of the Yang-Baxter equation is a tuple $(X,r)$, where $X$ is a non-empty set and $r: X \times X \rightarrow X \times X$ a (bijective) map such that on $X^3$ it holds that $$ (r \times \operatorname{id}_X) (\operatorname{id}_X \times r) (r \times \operatorname{id}_X) = (\operatorname{id}_X \times r)(r \times \operatorname{id}_X) (\operatorname{id}_X \times r).$$ One can then define the structure group $G(X,r)$/monoid $M(X,r)$ of a solution as the group/monoid generated by X with defining relations $xy = uv$ if $r(x,y)=(u,v)$. In this talk we talk we zoom in on the relation between $G(X,r)$ and a second group structure on this set, stemming from the behaviour of $r^2$. This led to the introduction of skew braces by Rump, and Guarnieri and Vendramin. Recall that a skew brace is a set $B$ with two group structures $(B,+)$ and $(B,\circ)$ that interact via a skew left distributivity condition, i.e. for any $a,b,c \in B$ one has that $a\circ (b+c) = (a \circ b) – a + (a\circ c)$. It turns out that these structures both generate and govern solutions. We will report in some recent advancements relating properties of skew braces to properties of its associated solution.
In the second part of the talk we will focus on the subclass of finite non-degenerate solutions. In the first part we discuss recent work on the structure of the monoid $M(X,r)$ and its monoid algebra $KM(X,r)$, where $K$ is an arbitrary field. In particular, we highlight the importance of the divisibility structure of $M(X,r)$ on the prime ideals of its algebra. Furthermore, we discuss how the homological properties of $KM(X,r)$ are akin to those of the polynomial algebra in several commuting variables. Concretely, a bound on the Gelfand-Kirillov and classical Krull dimension will be discussed. Moreover, some further homological properties can be shown to be equivalent to $r$ being an involution.
The source of angular momentum transport in stellar radiative zones remains an open, fundamental problem in stellar physics. One candidate mechanism is turbulence driven by the Tayler instability of toroidal magnetic fields. I will discuss a recent systematic revision of the linear stability analysis, followed by its implications for the efficiency of angular momentum transport out of evolved stellar cores. In particular, the Tayler instability is suppressed in the compositionally stratified regions of evolved low mass stars, suggesting that the Tayler instability cannot explain observations of core-envelope coupling on its own.
Preparing for the next pandemic: development of mathematical and computational tools for the assessment of the impact of novel antivirals at the individual and population level.
Some first modelling efforts towards the assessment of a new approach to combat pathogenic respiratory viruses using novel broadly neutralising antibodies will be presented.
I will present the Kinna-Wagner principles, which are weakenings of the Axiom of Choice and give a proof to what has so far been only a conjecture, which shows how these principles relate to the structure of intermediate models of ZF between a ground model and its forcing extension.
The intermediate model theorem states that whenever G is generic over V and V ⊆ M ⊆ V[G] are models of ZFC, then M is also a forcing extension of V . Unfortunately, this fails completely if we only assume ZF instead. Can more can be said? The goal of our talk is to present a generalization of the above theorem that works for ZF and talk about some of the recent progress made in the theory of symmetric extensions. This is joint with A. Karagila.
By a well-known procedure, usually referred to as "taking the classical limit", quantum systems become classical systems, equipped with a Hamiltonian stucture (symplectic or Poisson). From the deformation quantisation theory we know that a formal deformation of a commutative algebra $\mathcal{A}$ leads to a Poisson bracket on $\mathcal{A}$ and that the classical limit of a derivation on the deformation leads to a Hamiltonian derivation on $\mathcal{A}$ defined by the Poisson bracket. In this talk I present a generalisation of it for formal deformations of an arbitrary noncommutative associative algebra $\mathcal{A}$ [1]. I will show that a deformation leads to a commutative Poisson algebra structure on $\Pi(\mathcal{A}) := Z(\mathcal{A}) × (\mathcal{A}/Z(\mathcal{A}))$ and to the structure of a $\Pi(\mathcal{A})$-Poisson module on $\mathcal{A}$, where $Z(\mathcal{A})$ denotes the centre of $\mathcal{A}$. The limiting derivations are then still derivations of $\mathcal{A}$, but with the Hamiltonians belong to $\Pi(A)$, rather than to $A$. We illustrate our construction with several cases of formal deformations, coming from known quantum algebras, such as the ones associated with the Kontsevich integrable map, the quantum plane, the quantised Grassmann algebra and quantisations of the Volterra hierarchy [2, 3, 4].
This talk is based on a joint work with Pol Vanhaecke [1].
References
[1] Alexander V. Mikhailov and Pol Vanhaecke. Commutative Poisson algebras from deformations of noncommutative algebras. Lett. Math. Phys., 114(5), 1-51, 2024, arXiv:2402.16191v2.
[2] Alexander V. Mikhailov Quantisation ideals of nonabelian integrable systems. Russ. Math. Surv., 75(5):199, 2020, (arXiv:2009.01838), 2020).
[3] Sylvain Carpentier, Alexander V. Mikhailov and Jing Ping Wang. Quantisation of the Volterra hierarchy. Lett. Math. Phys., 112:94, 2022, (arXiv:2204.03095).
[4] Sylvain Carpentier, Alexander V. Mikhailov and Jing Ping Wang. Hamiltonians for the quantised Volterra hierarchy. Nonlinearity, 37(9), 095033 2024, arXiv:2312.12077
Effective collective decision-making in human and animal groups requires robust mechanisms to form consensus, typically via feedback loops in which individuals adapt their behaviour based on their perception of others. Such behaviour has been observed and theorised across scales from nucleosomes to entire societies. Of equal importance, but far less well studied, is the question of how consensus is overturned. In many contexts it is vital that group decisions do not remain fixed in the face of new evidence; echo-chamber effects must be suppressed so that the collective preferences which are expressed are not too strongly entrenched. In this talk I will discuss a new mathematical theory for how consensus can be overturned in symmetric binary choice problems, and compare the theoretical predictions to experiments with human and animal groups.
Changes in the Earth's rotation originate mostly from torques by the atmosphere and oceans, with a relatively small contribution from the Earth's core. The angular momentum exchange between core and mantle is however a long-standing problem. In this talk I will describe some of the mechanisms potentially at play and how they can be used to infer key properties of the core-mantle boundary. I will focus in particular on torsional Alfvén eigenmodes. The energy in these modes is equal parts magnetic and kinetic, and their motion is mostly columnar. The latter property has been used in the past to build approximate inviscid 1-D models. We have built a 3-D numerical model including viscosity, an electrically conductive inner core, and a thin, electrically conductive layer at the bottom of the mantle allowing the outer core to exchange angular momentum with the solid inner core and the mantle. I will present a systematic study the most relevant properties of these modes, particularly their columnarity, their torques, and their lifetimes as functions of the magnetic diffusivity and viscosity of the outer core, as well as functions of the electrical conductance of the bottom of the mantle.
CW complexes are topological spaces built up dimension by dimension from Euclidean cells, with a subset declared to be open if its intersection with each of these cells is open. Unfortunately when you take the product of two CW complexes, the product topology does not in general satisfy this requirement. I will explain when exactly it does; it turns out that it depends on the cardinal $\mathfrak{b}$. For the old hands who have seen this talk multiple times before, there will also be something new, with details that I realised last week I ought to draw out more.
