These seminars (unless otherwise stated) will take place on Tuesdays at 1pm via Zoom (https://ucl.zoom.us/j/97636994687) on an almost weekly basis.

**Zoom link: https://ucl.zoom.us/j/97636994687**

## 05 October 2021

**Speaker: Zonglun Li**

Supervisor: Prof A Zaikin

##### Title: Continuous models of molecular circuits for associative learning: a mathematical account of advanced behaviors

**Abstract:**

The development of synthetic biology has enabled us to make massive progress on biotechnology and to approach research questions from a brand new perspective. In particular, the design and study of gene regulatory networks have played an increasingly indispensable role in understanding and controlling biological phenomena. Among them, it is of great interest to understand how associative learning is formed at the molecular circuit level. Although the logic gate model is a commonly used approach in synthetic biology, increasing effort has been devoted to developing continuous models. In this work, we carry out an in-depth mathematical analysis of the canonical Fernando's model and demonstrate that the reinforcement effect can be achieved by properly selecting parameters. We also construct a novel circuit that can demonstrate forced dissociation, which is not observed in Fernando's model.

## 02 November 2021

**Speaker: Josh Smith**

Supervisor: Dr D Hewett

##### Title: Acoustic scattering by fractals: analysis and computation

**Abstract:**

Fractals are a natural model for objects which exhibit self-similarity and multi-scale roughness, such as coastlines, trees, and rivers. Furthermore, there are many physical and engineering problems which require us to compute the scattered field produced by an impinging wave incident upon objects which can be modelled as fractals, or have fractal geometry in their design. Examples include designing fractal noise barriers and antennas, as well as predicting how radiation is scattered by ice crystals in the atmosphere for climate models. A key underlying assumption in many of the tools required to analyse and compute such problems is that the geometry of the scatterer must be “sufficiently smooth”, that is (typically) the boundary of the scatterer must be Lipschitz or smoother. This presents many problems since most fractals do not have this property.

In this talk, we will consider the example of acoustic scattering by an impedance screen with fractal boundary, as a means to illustrate how pre-existing theory and methods can be modified to find a well-posed formulation of the problem and compute its solutions

## 16 November 2021

**Speaker: Alexandros Konstantinou**

Supervisor: Prof V Dokchitser

##### Title: ISOGENY RELATIONS BETWEEN JACOBIANS OF CURVES AND BSD

**Abstract:**

The main purpose of this talk is to present how one can use representation theoretic machinery to verify classical isogenies between Jacobians of curves, and apply it to recover known theorems related to the parity conjecture. At the heart of all of this lies the study of relations between permutation representations of finite groups. This presentation will heavily rely on examples, and we will see how to establish isogenies between Jacobians of curves using such relations (i.e. p-isogeny between 2 elliptic curves), and then use them to recover the parity of the rank of these Jacobians using local data.

## 23 November 2021

**Speaker: Maria Chivers**

Supervisor: Prof R Halburd

##### Title: Integrable systems as a reduction of ASDYM equations

**Abstract:**

Anti-self-dual Yang-Mills (ASDYM) equations is the “master of integrable systems”; a system of non-linear equations for Lie Algebra valued functions. They create the conditions that the curvature of a connection must be ASD with respect to the Hodge star operator. The ASDYM equations are integrable as they carry an associated linear problem and are a very rich source of Integrable Systems since most of the known Integrable Systems in dimensions three, two and one are embedded in the ASDYM equations, arising as a symmetry reduction of it. I will give a couple of such examples, including the Ernst equation (models the stationary axisymmetric gravitational fields in General Relativity) that we can obtain via dimensional reduction of ASDYM equations. I will also demonstrate the ways in which we can express the ASD condition; first as a system of first-order equations on the components of the gauge potential, and then as the commutativity condition on a Lax pair of operators.

## 30 November 2021

**Speaker: Yohance Osbourne**

Supervisor: Dr I Smears

##### Title: An Introduction to Forward-Backward PDE Systems from Mean Field Games

**Abstract:**

In large financial markets one can observe many agents following a variety of strategies while in competition over resources. In such markets, predicting trends, like the formation of price volatility for example, is generally a challenging endeavour due to the very large number of agents at play. To study situations involving many competing agents, a class of game-theoretic models called Mean Field Games (MFGs) has recently been introduced [1, 2]. Under simplifying assumptions, MFGs model Nash equilibria corresponding to stochastic differential games involving many players, and these equilibria can be described by a system of coupled nonlinear forward-backward differential equations which contain as special cases classical problems, such as the compressible Euler equations from Fluid Mechanics for instance.

In this talk I will focus on second order MFG systems, whereby I will illustrate their heuristic derivation via stochastic optimal control prior to discussing the issue of existence and uniqueness of solutions. Results concerning regularity theory for MFG systems will also be shared and I will conclude with a brief outline of existing numerical methods for Mean Field Games.

[1] Lasry, J.M. and Lions, P.L., 2006. Jeux à champ moyen. i–le cas stationnaire. *Comptes Rendus Mathématique*, *343*(9), pp.619-625.

[2] Lasry, J.M. and Lions, P.L., 2006. Jeux à champ moyen. II–Horizon fini et contrôle optimal. *Comptes Rendus Mathématique*, *343*(10), pp.679-684.

## 07 December 2021

**Speaker: Amalia Gjerloev**

Supervisor: Prof S Crowe/Prof C Pagel

##### Title: Simulations and Modelling Techniques in a Healthcare Context

**Abstract:**

The pandemic has put stress on all aspects of life, and unfortunately this has been particularly true for the NHS. The past (almost) 2 years has seen a huge influx of patients, a strain on hospital resources, and a demand to operate efficiently in order to save patient lives and distribute vaccinations. One method for addressing these problems is to use Operational Research (OR), which is a branch of mathematics that focuses on using analytical methods and mathematical modelling techniques to problem solve and improve operational decision making. Queuing theory is a commonly used OR technique used to simulate patient flow along a disease pathway. By modelling a patient pathway as a sequence of queues, queueing theory can be used to calculate waiting times, queue lengths, and steady state solutions. However, in more complex queuing networks that consider reneging customers and retrials, the simpler Jackson Network theory becomes inappropriate and fluid and diffusion approximations must be used. For these complex networks, discrete event simulation (DES) can be used and often serves as an ideal visual tool when talking with healthcare professionals. In this talk I will review how queuing theory and DES can be utilised to inform clinicians and healthcare staff, and I will discuss the implications of conducting effective OR research.

## 14 December 2021 (tbc)

**Speaker: Marialis Simoni**

Supervisor: Prof ER Johnson

##### Title: Applying the Kutta Condition on the Trailing Edges of Many Flat Plates in a Uniform Flow

**Abstract:**

Using series solution of the Laplace problems, the streamlines and equipotential lines for the uniform flow past many flat plates can be plotted. Laplace problems in multiply connected regions can be much easily solved using a series expansion to approximate for the streamfunction, rather than using conformal mapping, which makes the problem even more difficult than the original Laplace problem. Least-squares matching of boundary data with series expansion is used to solve Laplace’s problem. Also, the Kutta condition can be applied to the trailing edge of each individual plate. When applying the Kutta condition at the trailing edge of each plate, the flow needs to leave each plate with a finite velocity and parallel to each plate. In order to impose the Kutta condition, circulation of enough strength needs to be generated by the body about itself, when moving through the fluid. In this talk, I will show the analytical solution for the flow past one flat plate and then I will demonstrate the application of the Kutta condition at the trailing edges of many flat plates, using the series solution method.