University of Oxford

KW states that every finite abelian extension of the rationals is contained in a cyclotomic extension. In a previous talk, this was reduced to considering cyclic extensions of the local fields Q_p of prime power order l^r. When l\neq p, general theory is sufficient, however for l=p, more specific (although not necessarily more abstruse) descriptions become necessary.
I will focus on the simple structure of Q_p's extensions to obstruct the remaining obstructions to KW (and hopefully provoke some interest in local fields in those less familiar). Time-permitting, I will talk about this theorem in the context of class field theory and/or Hilbert's 12th problem.

The Kronecker-Weber theorem states that every finite abelian extension of the rationals is contained in some cyclotomic field. I will present a proof that emphasizes the standard local-global philosophy by first proving it for the p-adics and then deducing the global case.

Many singular stochastic PDEs are expected to be universal objects that govern a wide range of microscopic models in different universality classes. Two notable examples are KPZ and \Phi^4_3. In these cases, one usually finds a parameter in the system, and tunes according to the space-time scale in such a way that the system rescales to the SPDE in the large-scale limit. We justify this belief for a large class of continuous microscopic growth models (for KPZ) and phase co-existence models (for Phi^4_3), allowing microscopic nonlinear mechanisms far beyond polynomials. Aside from the framework of regularity structures, the main new ingredient is a moment bound for general nonlinear functionals of Gaussians. This essentially allows one to reduce the problem of a general function to that of a polynomial. Based on a joint work with Martin Hairer, and another joint work in progress with Chenjie Fan and Jiawei Li. 

In recent years, much attention has been given to single-cell RNA sequencing techniques as they allow researchers to examine the functions and relationships of single cells inside a tissue. In this study, we combine single-cell RNA sequencing data with protein–protein interaction networks (PPINs) to detect active modules in cells of different transcriptional states. We achieve this by clustering single-cell RNA sequencing data, constructing node-weighted PPINs, and identifying the maximum-weight connected subgraphs with an exact Steiner-Tree approach. As a case study, we investigate RNA sequencing data from human liver spheroids but the techniques described here are applicable to other organisms and tissues. The benefits of our novel method are two-fold: First, it allows us to identify important proteins (e.g., receptors) which are not detected from a differential gene-expression analysis as they only interact with proteins that are transcribed in higher levels. Second, we find that different transcriptional states have different subnetworks of the PPIN significantly overexpressed. These subnetworks often reflect known biological pathways (e.g., lipid metabolism and stress response) and we obtain a nuanced picture of cellular function as we can associate them with a subset of all analysed cells.

Aura Raulo (Ecological and Evolutionary Dynamics) and Marie-Claire Koschowitz (Vertebrate Palaeobiology) discuss their work and its mathematical challenges.

Aura Raulo

" Aura Raulo is a graduate student in Zoology Department working on transmission of symbiotic bacteria in the social networks of their animal hosts"
Title: Heaps in networks - How we share our microbiota through kisses
Abstract: Humans, like all vertebrates have a microbiome, a diverse community of symbiotic bacteria that live in and on us and are crucial for our functioning. These bacteria help us digest food, regulate our mood and function as a key part of our immune system. Intriguingly, while they are part of us, they are, unlike our other cells, in constant flux between us, challenging the traditional definition of a biological individual. Many of these bacteria need intimate social contact to be transmitted from human to human, making social network analysis tools handy in explaining their community dynamics.What then is a recipe for a ``good microbiome”? Theories and evidence implies that the most healthy and immunologically robust microbiome composition is both diverse, semi-stable and somewhat synchronized among closely interacting individuals, but little is known about what kind of transmission landscapes determine these bacterial cocktails. In my talk, I will present humanmicrobiome as a network trait: a metacommunity of cells shaped by an equilibrium of isolation and contact among their hosts. I propose that we do notnecessarily need to think of levels of life (e.g. cells, individuals, populations) as being neatly nested inside of each other. Rather, aggregations of cooperating cells (both bacteria and human cells) can be considered as mere tighter clusters in their interaction network, dynamically creating de novo defined units of life. I will present a few game theoretical evolutionary dilemmas following from this perspective and highlight outstanding questions in mapping how network position of the host translates into community composition of bacteria in flux.

Marie Koschowitz
“Marie Koschowitz is a PhD student in the Department of Zoology and the Department of Earth Sciences, working on comparative physiology and large scale evolutionary patterns in reptiles such as crocodiles, birds and dinosaurs."
Title: Putting the maths into dinosaurs – A zoologist's perspective
Abstract: Contemporary palaeontology is a subject area that often deals with sparse data.Therefore, palaeontologists became rather inventive in pursuit of getting the most out of what is available. If we find a dinosaur’s skull that shows prominent, but puzzling, bony ridges without any apparent function, how can we make meaningful interpretations of its purpose in the living animal that was? If we are confronted with a variety of partially preserved bones from animals looking anatomically similar, but not quite alike, how can we infer relationships in the absence of genetic data?Some methods that resolve these questions, such as finite element analysis, were borrowed from engineering. Others, like comparative phylogenetics or MCMC generalised mixed effects models, are even more directly based on mathematical computations. All of these approaches help us to calculate things like a raptors bite-force and understand the ins and outsof their skulls anatomy, or why pterosaurs and plesiosaurs aren’t exactly dinosaurs. This talk aims to presents a selection of current approaches to applied mathematics which have been inspired by interdisciplinary research – and to foster awareness of all the ways how mathematicians can get involved in “dinosaur research”, if they feel inclined to do so.


 

What actually happens when you submit an article to a journal? How does refereeing work in practice? How can you keep editors happy as an author or referee? How does one become a referee or editor? What does 'publication' mean with the internet and arXiv?

In this panel we'll discuss what happens between finishing writing a mathematical paper and its final (?) publication, looking at the various roles that people play and how they work best.

Featuring Helen Byrne, Rama Cont and Jonathan Pila.

 

Simple mathematical models have had remarkable successes in biology, framing how we understand a host of mechanisms and processes. However, with the advent of a host of new experimental technologies, the last ten years has seen an explosion in the amount and types of data now being generated. Increasingly larger and more complicated processes are now being explored, including large signalling or gene regulatory networks, and the development, dynamics and disease of entire cells and tissues. As such, the mechanistic, mathematical models developed to interrogate these processes are also necessarily growing in size and complexity. These detailed models have the potential to provide vital insights where data alone cannot, but to achieve this goal requires meeting significant mathematical challenges. In this talk, I will outline some of these challenges, and recent steps we have taken in addressing them.