# Scientific

## Statistics of the Mulitiplicative Groups

For every positive integer n, the quotient ring Z/nZ is the natural ring whose additive group is cyclic. The "multiplicative group modulo n" is the group of invertible elements of this ring, with the multiplication operation. As it turns out, many quantities of interest to number theorists can be interpreted as "statistics" of these multiplicative groups. For example, the cardinality of the multiplicative group modulo n is simply the Euler phi function of n; also, the number of terms in the invariant factor composition of this group is closely related to the number of primes dividing n. Many of these statistics have known distributions when the integer n is chosen at random (the Euler phi function has a singular cumulative distribution, while the Erdös–Kac theorem tells us that the number of prime divisors follows an asymptotically normal distribution). Therefore this family of groups provides a convenient excuse for examining several famous number theory results and open problems. We shall describe how we know, given the factorization of n, the exact structure of the multiplicative group modulo n, and go on to outline the connections to these classical statistical problems in multiplicative number theory.

## Water Waves: Instabilities of Stokes Waves

The study of ocean waves, particularly surface waves, is crucial for predicting and preparing for natural disasters such as tsunamis. Although ocean waves naturally occur in three dimensions, there are instances when they can be analyzed within a two-dimensional framework. For example, waves that propagate from the epicenter of a storm can be treated as unidirectional. In this presentation, we will examine periodic traveling waves that occur at the free surface of an ideal (incompressible and inviscid) two-dimensional fluid of infinite depth. Specifically, we will introduce surface waves of permanent shape, also known as Stokes waves and discuss their stability.

## On Arnoux's coding of the geodesic flow on the modular surface.

I will present Pierre Arnoux's 1994 paper in which he applies Veech's notion of zippered rectangles in the genus one setting to coordinatize the unit tangent bundle of the modular surface and thereby win an explicit description of its geodesic flow. From this, Arnoux recovered a result of C. Series': the dynamical system defined by the Gauss map (underlying regular continued fractions) is a factor of a section to the geodesic flow on the aforementioned unit tangent bundle. Time permitting, I will sketch some further implications given in the paper.

## Easy detection of (Di)Graphical Regular Representations

Graphical and Digraphical Regular Representations (GRRs and DRRs) are a concrete way to visualise the regular action of a group, using graphs. More precisely, a GRR or DRR on the group G is a (di)graph whose automorphism group is isomorphic to the regular action of G on itself by right-multiplication.

For a (di)graph to be a DRR or GRR on G, it must be a Cayley (di)graph on G. Whenever the group G admits an automorphism that fixes the connection set of the Cayley (di)graph setwise, this induces a nontrivial graph automorphism that fixes the identity vertex, which means that the (di)graph is not a DRR or GRR. Checking whether or not there is any group automorphism that fixes a particular connection set can be done very quickly and easily compared with checking whether or not any nontrivial graph automorphism fixes some vertex, so it would be nice to know if there are circumstances under which the simpler test is enough to guarantee whether or not the Cayley graph is a GRR or DRR. I will present a number of results on this question.

This is based on joint work with Dave Morris and with Gabriel Verret.

## Counting Permutation Groups

What does a random permutation group look like? This talk will start with a brief survey of how we might go about counting subgroups of the symmetric group Sn, and talk about what is known about “most” subgroups.

To tackle the general problem, it would clearly be helpful to know how many subgroups there are. An elementary argument gives that there are at least 2n2/16 subgroups, and it was conjectured by Pyber in 1993 that up to lower order error terms this is also an upper bound. This talk will present an answer to Pyber's conjecture.

This is joint work with Gareth Tracey.

## Mathematical Biomedicine: Examples

Mathematical biomedicine is an area of research where questions that arise in medicine are addressed by mathematical methods. Each such question needs first to be represented by a network with nodes that includes the biological entities that will be used to address the medical question. This network is then converted into a dynamical system for these entities, with parameters that need to be computed, or estimated. Simulations of the model are first used to validate the model, and then to address the specific question. I will give some examples, mostly from my recent work, including cancer drug resistance, side effects and metastasis, autoimmune diseases, and chronic and diabetic wounds, where the dynamical systems are PDEs. In each example, I will write explicitly the biological network, but will not the details of the corresponding PDE system.

## Machine Learning for Functional Data

Functional data analysis (FDA) is a growing statistical field for analyzing curves, images, or any multidimensional functions, in which each random function is treated as a sample element. Functional data is found commonly in many applications such as longitudinal studies and brain imaging. In this talk, I will present a methodology for integrating functional data into deep neural networks. The model is defined for scalar responses with multiple functional and scalar covariates. A by-product of the method is a set of dynamic functional weights that can be visualized during the optimization process. This visualization leads to greater interpretability of the relationship between the covariates and the response relative to conventional neural networks. The model is shown to perform well in a number of contexts including prediction of new data and recovery of the true underlying relationship between the functional covariate and scalar response; these results were confirmed through real data applications and simulation studies.

## Topology and Azumaya algebras

An Azumaya algebra is something that is "locally" isomorphic to a matrix algebra. By varying the sense of "locally", we arrive at different incarnations of the concept. The motivating example is that of central simple algebras over a field. In this talk, I will concentrate on the topological aspects of the idea. I will give examples and show that the flexibility of topology allows one to produce counterexamples in algebra. At the end, I will mention some problems I do not know how to solve.

## On the Hardy Littlewood 3-tuple prime conjecture and convolutions of Ramanujan sums

The Hardy and Littlewood k-tuple prime conjecture is one of the most enduring unsolved problems in mathematics. In 1999, Gadiyar and Padma presented a heuristic derivation of the 2-tuples conjecture by employing the orthogonality principle of Ramanujan sums. Building upon their work, we explore triple convolution Ramanujan sums and use this approach to provide a heuristic derivation of the Hardy-Littlewood conjecture concerning prime 3-tuples. Furthermore, we estimate the triple convolution of the Jordan totient function using Ramanujan sums.

## On sums of coefficients of polynomials related to the Borwein conjectures

Peter Borewein empirically discovered quite a number of mysteries involving sign patterns of coefficients of polynomials of the form $f_{p,s,n}(q):=\prod_{j=0}^{n} \prod_{k=1}^{p-1} (1-q^{pj+k})^{s}$ ($p$ a prime and $s,n \in \mathbb{N}$). In the case $(p,s) \in \{(3,1), (3,2)\}$, he conjectured that the coefficients follow a repeating + - - pattern, and in the case $(p,s)=(5,1)$, it was conjectured that the coefficients follow a repeating + - - - - sign pattern. We consider a weaker problem of finding the signs of partial sums of coefficients along some arithmetic progressions. We use a combinatorial sieving principle by Li-Wan and elementary character theory to asymptotically estimate and find the signs of these partial sums. We find that the signs of these partial sums are compatible with the sign pattern in Borewein's conjectures. This is based on joint work with Ankush Goswami.