# Mathematics

## Multiple fission cycles in Chlamydomonas

In this talk I will present a "dynamical paradigm" for modeling networks of interacting genes and proteins that regulate every aspect of cell physiology. The paradigm is based on dynamical systems theory of nonlinear ODEs, especially one- and two-parameter bifurcation diagrams. I will show how we have used this paradigm to unravel the mechanisms controlling "multiple fission" cycles in the photosynthetic green alga Chlamydomonas. While most eukaryotic cells maintain a characteristic size by executing binary division after each mass doubling, Chlamydomonas cells can grow more than eight-fold during daytime before undergoing rapid cycles of DNA replication, mitosis and cell division at night, which produce up to 16 daughter cells. We propose that this unusual strategy of growth and division (which is clearly advantageous for a photosynthetic organism) can be governed by a size-dependent bistable switch that turns on and off a limit cycle oscillator that drives cells through rapid cycles of DNA synthesis and mitosis. We show that this simple ‘sizer-oscillator’ arrangement reproduces the experimentally observed features of multiple-fission cycles and the response of Chlamydomonas cells to different light-dark regimes. Our model makes unexpected predictions about the size-dependence of the time of onset of cell-cycle oscillations after cells are transferred from light to dark conditions, and these predictions are confirmed by single-cell experiments.

Collaborators: Stefan Heldt and Bela Novak (Oxford Univ) on the modeling; Fred Cross (Rockefeller Univ) on the experiments.

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

Let $L/K$ be a Galois extension of number fields with Galois group $G$, and let $C⊂G$ be a conjugacy class. Attached to each unramified prime ideal p in OK is the Artin symbol $\sigma p$, a conjugacy class in $G$. In 1922 Chebotarev established what is referred to his density theorem (CDT). It asserts that the number $\pi C(x)$ of such primes with $\sigma p=C$ and norm $Np≤x$ is asymptotically $\left|C\right|\left|G\right|\mathrm{Li} (x)$ as $x\rightarrow\infty$ where $\mathrm{Li} (x)$ is the usual logarithmic integral. As such, CDT is a generalisation of both the prime number theorem and Dirichlet's theorem on primes in arithmetic progressions. In light of Linnik's result on the least prime in an arithmetic progression, one may ask for a bound for the least prime ideal whose Artin symbol equals C. In 1977 Lagarias and Odlyzko proved explicit versions of CDT and in 1979 Lagarias, Montgomery and Odlyzko gave bounds for the least prime ideal in the CDT. Since 2012 several unconditional explicit results of these theorems have appeared with contributions by Zaman, Zaman and Thorner, Ahn and Kwon, and Winckler. I will present several recent results we have proven with Das, Ng, and Wong.

## 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 $\left{t_g : g \in G\right}, 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

A variety of strategies and approaches have been proposed, and implemented by governments, for COVID mitigation. In this presentation, I introduce some of these, briefly discuss some of the resulting difficulties - in particular in the context of the northern Netherlands, where I have been working most recently. We then take a preliminary look at the possibility of `targeted quarantine' . Many questions, both mathematical, clinical, logistical and ethical remain to be answered, and as such, this presentation will be closer to a discussion session than the usual Mathbio Works in progress seminars. All feedback appreciated and welcome

## 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

A large part of stochastic portfolio theory, as initiated by Robert Fernholz in the 1990s, is concerned with construction of practical equity portfolios that can beat the stock market index by active rule-based trading. The truly remarkable part of the theory is that it requires no probabilistic modeling on the future behavior of stock prices. There is a Monge-Kantorovich optimal transport problem that naturally arises in the construction of such portfolios. This transport problem is a multiplicative analog of the well-studied quadratic Kantorovich- Wasserstein transport with equally striking properties. We will see aspects of this transport problem from theoretical uses such as defining gradient flows in a non-metric setting to practical uses such as in determining the right frequency of trading. Interesting probability theory comes in as we consider entropic relaxation of this problem giving rise to multiplicative Schrodinger bridges.

## 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

A random nxm matrix gives a random linear transformation from $\mathbb{Z}^m$ to $\mathbb{Z}^n$ (between vectors with integral coordinates). Asking for the probability that such a map is injective is a question of the non-vanishing of determinants. In this talk, we discuss the probability that such a map is surjective, which is a more subtle integral question. We show that when $m=n+u$, for $u$ at least 1, as n goes to infinity, the surjectivity probability is a non-zero product of inverse values of the Riemann zeta function. This probability is universal, i.e. we prove that it does not depend on the distribution from which you choose independent entries of the matrix, and this probability also arises in the Cohen-Lenstra heuristics predicting the distribution of class groups of real quadratic fields. This talk is on joint work with Hoi Nguyen.

## 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.