A heteroclinic network is a type of solution to a dynamical system consisting of a set of equilibrium solutions and connecting orbits between them. Heteroclinic networks can be thought of as an embedding of a directed graph into the phase space of the dynamical system, where vertices correspond to equilibria and directed edges to heteroclinic trajectories. The dynamics near a heteroclinic network is characterized by intermittent behaviour: solutions spend a long period of time close to one equilibrium before rapidly switching to another. The manner in which the transitions between equilibria occur can be incredibly rich: trajectories may visit all the equilibria in the network, or only a subset of them; the order in which equilibria are visited may be regular, or apparently chaotic. In spatially extended systems (modelled by partial differential equations), solutions near heteroclinic networks can arise as travelling or spiral waves. In this talk I will present some results demonstrating this exotic behaviour near heteroclinic networks, and discuss some of the ways in which we are able to analyse this behaviour.
West Nile virus (WNV) is a vector-borne pathogen causing major outbreaks of West Nile fever worldwide. Although the transmission is maintained via birds and mosquitoes, human infection is possible. Routine surveillance of WNV in the USA is performed by trapping mosquitos and testing for the presence of WNV during the transmission season by RT-qPCR testing. Apart from the general binary positive/negative outcome from these tests, they also generate cycle threshold (Ct) values. Motivated by findings from SARS-CoV-2 viral load dynamics in humans, we hypothesised that Ct values observed through routine pooled testing over time are sufficient to provide information on WNV epidemic dynamics. To investigate this, we introduce an agent-based model of mosquitoes and birds to model WNV epidemiological and viral load dynamics. In this model, we embed a within-host model of the viral load kinetics of mosquitoes. We simulate scenarios where mosquitoes are captured through routine surveillance and tested for WNV through pooled RT-qPCR testing, generating synthetic Ct values over time. We compare our model output to real Ct value data collected through WNV pooled testing in Nebraska in 2022 and 2023.
In this talk, we consider how much non-constructive principles are sufficient for Friedberg-Muchinik construction of degree $d$ such that $0<d<0'$. We will see that the only point we need a non-constructive principle is to show "if a recursive set $S$ of natural number has finite cardinality, then $S$ has an upper bound", which requires $\Sigma^0_1$ law of excluded middle.
This is based on the equally named paper which is on arxiv and to appear in Journal of Algebra. I will discuss the connection with the representations of reductive groups and Lie algebras: the hyper algebra, the restricted enveloping algebra and their centres. Then I will move to divided power algebras, truncated polynomial algebras and their invariants. I will discuss some results ($GL_n$ only) from my paper, a conjecture concerning the so-called “restriction property” and the connection with “special symmetrization map” from Okounkov-Olshanskii. If time allows, I will mention extensions of the above results to several matrices and vectors and covectors.
Let $\mathsf{M}$ be the weak set theory (with powersets) axiomatised by: $\textsf{Extensionality}$, $\textsf{Pair}$, $\textsf{Union}$, $\textsf{Infinity}$, $\textsf{Powerset}$, transitive containment ($\textsf{TCo}$), $\Delta_0\textsf{-Separation}$ and $\textsf{Set-Foundation}$. In this talk I will discuss the relationship between two alternative versions of the set-theoretic collection scheme: $\textsf{Collection}$ and $\textsf{Strong Collection}$. Both of these schemes yield $\mathsf{ZF}$ when added to $\mathsf{M}$, but when restricted the $\Pi_n$-formulae (denoted $\Pi_n\textsf{-Collection}$ and $\textsf{Strong } \Pi_n\textsf{-Collection}$) these alternative versions of set-theoretic collection differ. In particular, over the theory $\mathsf{M}$, $\textsf{Strong }\Pi_n\textsf{-Collecton}$ is equivalent to $\Pi_n\textsf{-Collection}+\Sigma_{n+1}\textsf{-Separation}$. And, $\mathsf{M}+\textsf{Strong }\Pi_n\textsf{-Collection}$ proves the consistency of $\mathsf{M}+\Pi_n\textsf{-Collection}$. In this talk I will show that, despite this difference in consistency strength, every countable well-founded model of $\mathsf{M}+\Pi_n\textsf{-Collection}$ satisfies $\textsf{Strong } \Pi_n\textsf{-Collection}$. If time permits I will outline how this argument can be refined to show that $\mathsf{M}+\Pi_n\textsf{-Collection}+\Pi_{n+1}\textsf{-Foundation}$ proves $\Sigma_{n+1}\textsf{-Separation}$.
Nichols algebras appear in several areas of mathematics, from Hopf algebras and quantum groups to Schubert calculus and conformal field theories. In this talk, I will review the main problems related to Nichols algebras and discuss some recent classification theorems.
We study the structure of the convolution semigroups of measures over definable groups. We isolate the property of generic transitivity and demonstrate that it is sufficient (and necessary) to develop stable group theory localizing on a generically stable type. We establish generic transitivity of generically stable idempotent types in important new cases, including abelian groups in arbitrary theories and arbitrary groups in rosy theories, and characterize them as generics of connected type-definable subgroups. Using tools from Keisler's randomization theory, we generalize some of these results from types to generically stable Keisler measures, and classify idempotent generically stable measures in abelian groups as (unique) translation-invariant measures on type-definable fsg subgroups. This provides a partial definable counterpart to the classical work of Rudin, Cohen and Pym for locally compact topological groups. Working over a countable NIP structure, we provide an explicit construction of a minimal left ideal in the convolution semigroup of measures from a minimal left ideal of types and the unique Haar measure on the ideal group. As a key ingredient, we prove the revised Ellis group conjecture of Newelski saying that under NIP, the so-called tau-topology on the ideal group is Hausdorff.
Joint work with Kyle Gannon and Krzysztof Krupiński.
There is an equivalence between the category of simplicial abelian groups and the category of differential graded abelian groups called the Dold-Kan equivalence. There is also a class of curious objects called Crossed-Simplicial Groups defined by a distributive law between a collection of groups indexed by the natural numbers and the simplicial category $\Delta$. There have been attempts to extend the Dold-Kan to crossed-simplicial setting by explicitly constructing extensions of the differential graded side. But these are few and far between. In this talk, I'll start by a new but a weakened analogue of the Dold-Kan by using certain induction and restriction functors, and by passing to the homotopy categories on both sides. Then show that this homotopical version of the equivalence extends to the crossed-simplicial setting.
In this talk, I will review some of the recent advances in developing mathematical and computational methods for 1D localised patterns (patterns that are embedded in a quiescent state) to 2D localised patterns. These patterns occur in a wide range of applications from buckling of cylinders, vegetation patches near deserts, to fluid mechanics. While the mathematical theory of these patterns in 1D is well-established in higher-dimensions, new tools are required. This work has appeared on the front cover of JFM 2015 and the January 2024 Nonlinearity journal, nominated for the IMA Lighthill-Thwaites prize 2021, and subject of the SIAM 2024 T. Brooke Benjamin prize.
