University of Oxford

The first part of Galois' Second Mémoire, less than three pages of manuscript written in 1830, is devoted to an amazing insight, far ahead of its time. Translated into modern mathematical language (and out of French), it is the theorem that a primitive soluble finite permutation group has prime-power degree. This, and Galois' ideas, and counterexamples to some of

them, will be my theme.

The visceral endoderm (VE) is an epithelium of approximately 200 cells

encompassing the early post-implantation mouse embryo. At embryonic day

5.5, a subset of around 20 cells differentiate into morphologically

distinct tissue, known as the anterior visceral endoderm (AVE), and

migrate away from the distal tip, stopping abruptly at the future

anterior. This process is essential for ensuring the correct orientation

of the anterior-posterior axis, and patterning of the adjacent embryonic

tissue. However, the mechanisms driving this migration are not clearly

understood. Indeed it is unknown whether the position of the future

anterior is pre-determined, or defined by the movement of the migrating

cells. Recent experiments on the mouse embryo, carried out by Dr.

Shankar Srinivas (Department of Physiology, Anatomy and Genetics) have

revealed the presence of multicellular ‘rosettes’ during AVE migration.

We are developing a comprehensive vertex-based model of AVE migration.

In this formulation cells are treated as polygons, with forces applied

to their vertices. Starting with a simple 2D model, we are able to mimic

rosette formation by allowing close vertices to join together. We then

transfer to a more realistic geometry, and incorporate more features,

including cell growth, proliferation, and T1 transitions. The model is

currently being used to test various hypotheses in relation to AVE

migration, such as how the direction of migration is determined, what

causes migration to stop, and what role rosettes play in the process.

Abstract: Nonlinear models have been widely employed to characterize the

underlying structure in a time series. It has been shown that the

in-sample fit of nonlinear models is better than linear models, however,

the superiority of nonlinear models over linear models, from the

perspective of out-of-sample forecasting accuracy remains doubtful. We

compare forecast accuracy of nonlinear regime switching models against

classical linear models using different performance scores, such as root

mean square error (RMSE), mean absolute error (MAE), and the continuous

ranked probability score (CRPS). We propose and investigate the efficacy

of a class of simple nonparametric, nonlinear models that are based on

estimation of a few parameters, and can generate more accurate forecasts

when compared with the classical models. Also, given the importance of

gauging uncertainty in forecasts for proper risk assessment and well

informed decision making, we focus on generating and evaluating both point

and density forecasts.

Keywords: Nonlinear, Forecasting, Performance scores.

This talk will introduce various aspects of modern cryptography. After introducing RSA and some factoring algorithms, I will move on to how elliptic curves can be used to produce a more complex form of Diffie--Hellman key exchange.

We investigate a class of weakly interactive particle systems with absorption. We assume that the coefficients in our model depend on an "absorbing" factor and prove the existence and uniqueness of the proposed model. Then we investigate the convergence of the empirical measure of the particle system and derive the Stochastic PDE satisfied by the density of the limit empirical measure. This result can be applied to credit modelling. This is a joint work with Dr. Ben Hambly.

TBA