University of Oxford

In this talk I will introduce Hilbert's 10th Problem (H10) and the model-theoretic notions necessary to explore this problem from the perspective of mathematical logic. I will give a brief history of its proof, talk a little about its connection to decidability and definability, then close by speaking about generalisations of H10 - what has been proven and what has yet to be discovered.

An informal session for DPhil students, ECRs and undergraduates with an interest in probability. The aim is to gain exposure to areas outside of your own research interests in an informal and accessible way.

An informal session for DPhil students, ECRs and undergraduates with an interest in probability. The aim is to gain exposure to areas outside of your own research interests in an informal and accessible way.

An informal session for DPhil students, ECRs and undergraduates with an interest in probability. The aim is to gain exposure to areas outside of your own research interests in an informal and accessible way.

An informal session for DPhil students, ECRs and undergraduates with an interest in probability. The aim is to gain exposure to areas outside of your own research interests in an informal and accessible way.

There is a mismatch between levels of crime and its fear and often, cities might see an increase or a decrease in crime over time while the fear of crime remains unchanged. A model that considers fear of crime as an opinion shared by simulated individuals on a network will be presented, and the impact that different distributions of crime have on the fear experienced by the population will be explored. Results show that the dynamics of the fear is sensitive to the distribution of crime and that there is a phase transition for high levels of concentration of crime.

Many scientists, and in particular mathematicians, report difficulty in understanding thermodynamics. So why is thermodynamics so difficult? To attempt an answer, we begin by looking at the components in an exposition of a scientific theory. These include a mathematical core, a motivation for the choice of variables and equations, some historical remarks, some examples and a discussion of how variables, parameters, and functions (such as equations of state) can be inferred from experiments. There are other components too, such as an account of how a theory relates to other theories in the subject.

It will be suggested that theories of thermodynamics are hard to understand because (i) many expositions appear to argue from the particular to the general (ii) there are several different thermodynamic theories that have no obvious logical or mathematical equivalence (iii) each theory really is subtle and requires intense study (iv) in most expositions different theories are mixed up, and the different components of a scientific exposition are also mixed up. So, by presenting one theory at a time, and by making clear which component is being discussed, we might reduce the difficulty in understanding any individual thermodynamic theory. The key is perhaps separation of the mathematical core from the physical motivation. It is also useful to realise that a motivation is not generally the same as a proof, and that no theory is actually true.

By way of illustration we will attempt expositions of two of the simplest thermodynamic theories – reversible and then irreversible thermodynamics of homogeneous materials – where the mathematical core and the motivation are discussed separately. In conclusion we’ll relate these two simple theories to other, foundational and generalised, thermodynamic theories.

We will talk about the Cauchy problem of the three-dimensional isentropic compressible Navier-Stokes equations. When viscosity coefficients are given as a constant multiple of density's power, based on some analysis  of  the nonlinear structure of this system, by introducing some new variables and the initial layer compatibility conditions, we identify the class of initial data admitting a local regular solution with far field vacuum and  finite energy  in some inhomogeneous Sobolev spaces, which solves an open problem of degenerate viscous flow partially mentioned by Bresh-Desjardins-Metivier (2006, Anal. Simi. Fluid Dynam.),  Jiu-Wang-Xin (2014, JMFM) and so on. Moreover, in contrast to the classical well-posedness theory in the case of  the constant viscosity,   we show   that one can not obtain any global classical solution whose $L^\infty$  norm of $u$ decays to zero as time $t$ goes to infinity under the assumptions on the conservation laws of total mass and momentum.

Novel machine learning techniques based on deep learning, i.e., the data-driven manipulation of neural networks, have reported remarkable results in many areas such as image classification, game intelligence, or speech recognition. Driven by these successes, many scholars have started using them in areas which do not focus on traditional machine learning tasks. For instance, more and more researchers are employing neural networks to develop tools for the discretisation and solution of partial differential equations. Two reasons can be identified to be the driving forces behind the increased interest in neural networks in the area of the numerical analysis of PDEs. On the one hand, powerful approximation theoretical results have been established which demonstrate that neural networks can represent functions from the most relevant function classes with a minimal number of parameters. On the other hand, highly efficient machine learning techniques for the training of these networks are now available and can be used as a black box. In this talk, we will give an overview of some approaches towards the numerical treatment of PDEs with neural networks and study the two aspects above. We will recall some classical and some novel approximation theoretical results and tie these results to PDE discretisation. Afterwards, providing a counterpoint, we analyse the structure of network spaces and deduce considerable problems for the black box solver. In particular, we will identify a number of structural properties of the set of neural networks that render optimisation over this set especially challenging and sometimes impossible. The talk is based on joint work with Helmut Bölcskei, Philipp Grohs, Gitta Kutyniok, Felix Voigtlaender, and Mones Raslan

Matrix completion is an area of great mathematical interest and has numerous applications, including recommender systems for e-commerce. The recommender problem can be viewed as follows: given a database where rows are users and and columns are products, with entries indicating user preferences, fill in the entries so as to be able to recommend new products based on the preferences of other users. Viewing the interactions between user and product as links in a bipartite graph, the problem is equivalent to approximating a partially observed graph using clusters. We propose a divide and conquer algorithm inspired by the work of [1], who use recursive rank-1 approximation. We make the case for using an LP rank-1 approximation, similar to that of [2] by a showing that it guarantees a 2-approximation to the optimal, even in the case of missing data. We explore our algorithm's performance for different test cases.

[1]  Shen, B.H., Ji, S. and Ye, J., 2009, June. Mining discrete patterns via binary matrix factorization. In Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining (pp. 757-766). ACM.

[2] Koyutürk, M. and Grama, A., 2003, August. PROXIMUS: a framework for analyzing very high dimensional discrete-attributed datasets. In Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining (pp. 147-156). ACM.