Postgraduate Seminars Summer 2024
These seminars (unless otherwise stated) will take place on Tuesdays at 12pm-1pm on an almost weekly basis.
Talks are being given by 2nd and 3rd year Mathematics PhD students for PhD students.
25 April 2024 in 25 Gordon Street - Room 416 - 2pm-3pm
Speaker: Alexey Derkach
TITLE: some tools for spectral analysis of compact operators on hilbert space
Abstract:
Studying the asymptotics of eigenvalues is a classic question in Spectral theory. For a compact operator on a Hilbert space we usually estimate the decay of the s-numbers (singular values). In this talk I will discuss the connection between eigenvalues and s-values, the p-Schatten norms (and quasinorms), the Weidl operator classes and basic elements of Perturbation theory. We will see when it is easy to obtain an asymptotic formula and when we can give an asymptotic estimate only. These tools help to estimate the eigen/singular values of integral operators whose kernels satisfy certain conditions, (we will see how the rate of decay depends on the smoothness of the kernel). Mostly I am planning to discuss some general facts in this theory, but also want to mention the operators I study in my research, and the results obtained.
Zoom link: https://ucl.zoom.us/j/99473974347
4 June 2024 in 25 Gordon Street - Room 416 - 2pm-3pm
Speaker: Molly Brennan
TITLE: an asymptotic upscaling of transport through the bacterial membrane
Abstract:
Transport through the outer membrane of many microorganisms, such as gram-negative bacteria, is often restricted to specific channels and non-specific porins. These provide a size-restricted, hydrophilic passageway for small molecules through an otherwise impermeable membrane. Important examples include antibiotics, which must cross the outer membrane to effectively target gram-negative bacteria, and quorum sensing molecules which allow bacterial colonies to coordinate mass phenotypic changes such as the production of virulence factors. In mathematical models, this limiting transport mechanism is often represented via constitutive boundary conditions. In this work, we systematically derive the correct effective boundary conditions to impose across the membrane in terms of physical channel and porin properties. We use a hybrid mathematical approach, combining multiscale methodology such as asymptotic homogenisation and boundary layer theory with numerical simulations. We also analyse the implications of the upscaled equations we derive in the context of real biological systems.
Speaker: Alexis Farman
TITLE: in vitro modelling of car t-cell infiltration and interactions with tumour cells
Abstract:
CAR T-cell therapy, a form of immunotherapy, has proven highly effective against blood cancers, but it remains challenging for solid tumours. To explore the reasons behind this inefficacy, several in vitro experimental setups has been designed. One of these consists of two volumes—one containing the CAR T-cell product and the other housing cancer cells—connected by microchannels. The aim of this talk is to provide an overview of the mathematics required to model this experimental setup and the behaviour of the cells within it, focusing on three specific subproblems. First, we will determine the concentration of chemical particles produced by the tumour cells throughout the volume. Next, we will analyse CAR T-cell movement induced by the chemical concentration gradient, both within the volume and inside the microchannels. Finally, we will introduce a simple ODE model to describe the interactions between CAR T-cells and tumour cells. Throughout the talk, we will employ a combination of analytical and numerical techniques, comparing our results to existing data and motivating further theoretical work.