# Mathematics

## Explicit results about primes in Chebotarev's density theorem

## Regular Representations of Groups

A natural way to understand groups visually is by examining objects on which the group has a natural permutation action. In fact, this is often the way we first show groups to undergraduate students: introducing the cyclic and dihedral groups as the groups of symmetries of polygons, logos, or designs. For example, the dihedral group D_8 of order 8 is the group of symmetries of a square. However, there are some challenges with this particular example of visualisation, as many people struggle to understand how reflections and rotations interact as symmetries of a square.

Every group G admits a natural permutation action on the set of elements of G (in fact, two): acting by right- (or left-) multiplication. (The action by right-multiplication is given by {t_g : g in G}, where t_g(h) = hg for every h in G.) This action is called the "right- (or left-) regular representation" of G. It is straightforward to observe that this action is regular (that is, for any two elements of the underlying set, there is precisely one group element that maps one to the other). If it is possible to find an object that can be labelled with the elements of G in such a way that the symmetries of the object are precisely the right-regular representation of G, then we call this object a "regular representation" of G.

A Cayley (di)graph Cay(G,S) on the group G (with connection set S, a subset of G) is defined to have the set G as its vertices, with an arc from g to sg for every s in S. It is straightforward to see that the right-regular representation of G is a subset of the automorphism group of this (di)graph. However, it is often not at all obvious whether or not Cay(G,S) admits additional automorphisms. For example, Cay(Z_4, {1,3}) is a square, and therefore has D_8 rather than Z_4 as its full automorphism group, so is not a regular representation of Z_4. Nonetheless, since a regular representation that is a (di)graph must always be a Cayley (di)graph, we study these to determine when regular representations of groups are possible.

I will present results about which groups admit graphs, digraphs, and oriented graphs as regular representations, and how common it is for an arbitrary Cayley digraph to be a regular representation.

## In Progress COVID-19 modelling

## Computational properties of network dynamical systems

Dynamical systems concepts have mostly been developed to understand the behaviour of autonomous, i.e. input-free, nonlinear systems. Even in this case, it is well recognized that such systems can display a wide range of dynamical behaviours. Understanding how non-autonomous systems behave is an additional mathematical challenge that gives insight into how complex systems can perform computational activities in response to inputs. In this talk I will discuss ways that the dynamics of network attractors can be used to describe and predict not only the how the systems perform computations, but also how they may make errors during the computations.

### Speaker Biography

Peter Ashwin is Professor of Mathematics at the University of Exeter (UK) since 2007. His main interests are in nonlinear dynamical systems and applications: bifurcation theory and dynamical systems, especially synchronization problems, symmetric chaotic dynamics, spatially extended systems and nonautonomous systems. Applications of dynamical systems include climate (bifurcations, tipping points), fluids (bifurcations and mixing), laser systems (synchronization), neural systems (computational properties), materials and electronic systems (digital signal processing) and biophysical modelling (cell biology).

## From portfolio theory to optimal transport and Schrodinger bridge in-between

## Variation in the descent of genome: modeling and inference

In meiosis, DNA is copied from parents to offspring, so that individuals who share common ancestors may have identical DNA copies from those ancestors through repeated meiosis. This identical-by-descent (IBD) DNA underlies the similarities between relatives, at both the family level and at the population level. However, the process of meiosis is quite variable, and DNA is inherited generation-to-generation in large segments. The patterns of IBD genome among relatives are complex, and in remote relatives segments of IBD DNA are rare but not short. Modern genetic data on millions of markers across the genome allows estimation of shared DNA, but accurate estimation requires modelling the processes that give rise to these complex IBD patterns. IBD must be estimated jointly among individuals and across the genome. Pedigree information, if available, provides prior probabilities of IBD patterns. Where inferred IBD is discordant with pedigree information, there is potential to detect selection or other processes distorting the outcomes of the meiotic process.

**Speaker Biography:** Elizabeth Thompson received her B.A. and Ph.D. in mathematics from Cambridge University, UK. After postdoctoral work in genetics at Stanford University, she joined the mathematics faculty of the University of Cambridge in 1976. She was a Professor of Statistics at the University of Washington from 1985 until her (semi-) retirement in 2018. Her research is in the development of methods for model-based likelihood inference from genetic data on both humans and other species, including inference of relationships among individuals and among populations. Dr. Thompson has received an Sc.D degree from the University of Cambridge, the Jerome Sacks award for cross-disciplinary statistical research, the Weldon Prize for contributions to Biometric Science, and a Guggenheim fellowship. She is an honorary fellow of Newnham College, Cambridge, and an elected member of the International Statistical Institute, the American Academy of Arts and Sciences, and the US National Academy of Sciences.

## Surjectivity of random integral matrices on integral vectors

## Scalable approximation of integrals using non-reversible methods: from Riemann to Lebesgue, and why you should care

How to approximate intractable integrals? This is an old problem which is still a pain point in many disciplines (including mine, Bayesian inference, but also statistical mechanics, computational chemistry, combinatorics, etc).

The vast majority of current work on this problem (HMC, SGLD, variational) is based on mimicking the field of optimization, in particular gradient based methods, and as a consequence focusses on Riemann integrals. This severely limits the applicability of these methods, making them inadequate to the wide range of problems requiring the full expressivity of Lebesgue integrals, for example integrals over phylogenetic tree spaces or other mixed combinatorial-continuous problems arising in networks models, record linkage and feature allocation.

I will describe novel perspectives on the problem of approximating Lebesgue integrals, coming from the nascent field of non-reversible Monte Carlo methods. In particular, I will present an adaptive, non-reversible Parallel Tempering (PT) allowing MCMC exploration of challenging problems such as single cell phylogenetic trees.

By analyzing the behaviour of PT algorithms using a novel asymptotic regime, a sharp divide emerges in the behaviour and performance of reversible versus non-reversible PT schemes: the performance of the former eventually collapses as the number of parallel cores used increases whereas non-reversible benefits from arbitrarily many available parallel cores. These theoretical results are exploited to develop an adaptive scheme approximating the optimal annealing schedule.

My group is also interested in making these advanced non-reversible Monte Carlo methods easily available to data scientists. To do so, we have designed a Bayesian modelling language to perform inference over arbitrary data types using non-reversible, highly parallel algorithms.

## If Space Turned out to be Time: Resonances and Patterns in the Visual Cortex

When subjects are exposed to full field flicker in certain frequencies, they perceive a variety of complex geometric patterns that are often called flicker hallucinations. On the other had, when looking at high contrast geometric patterns like op art, shimmering and flickering is observed. In some people, flicker or such op art can induce seizures. In this talk, I describe a simple network model of excitatory and inhibitory neurons that comprise the visual area of the brain. I show that these phenomena are reproduced and then give an explanation based on symmetry breaking bifurcations and Floquet theory. Symmetric bifurcation theory also shows why one expects a different class of patterns at high frequencies from those at low frequencies.

On the other hand, the visual system is also very sensitive to specific spatial frequencies and this sensitivity can be pathological in the case of so-called pattern-senstive epilepsy. It has been shown that certain types of "op art"can cause visual discomfort. We show that the network that we used in flicker is also sensitive to spatially periodic inputs and suggest that a Hopf bifurcation instability is responsible for the discomfort and seizures.

#### Speaker Biography

Bard Ermentrout received his PhD in Theoretical Biology at the University of Chicago and was a postdoctoral fellow at the NIH from 1979-1982. He has been at the University of Pittsburgh since then. He is the author of over 200 papers and two books as well as the simulation package, XPPAUT. He is a Sloan Fellow and a SIAM Fellow and received the Mathematical Neuroscience Prize in 2015.