University of Oxford

What does abstract nonsense (category theory) have to do with the apparently opposite proverbial concreteness of physics? In this talk I will try to convey the importance of understanding physical theories from a compositional and structural perspective, where the fundamental logic of interaction between systems becomes the real protagonist. Firstly, we will see how different classes of symmetric monoidal categories can be used to model physical processes in a very natural and intuitive way. We will then explore the claim that category theory is not only useful in providing a unified framework, but it can be also used to perfect and modify preexistent models. In this direction, I will show how the introduction of a trace in the symmetric monoidal category describing QIT can be used to talk about quantum interactions induced by cyclic causal relationships.

Nets of lines are line arrangements satisfying very strict intersection conditions. We will see that nets can be defined in a very natural way in algebraic geometry, and, thanks to the strict intersection properties they satisfy, we will see that a lot can be said about classifying them over the complex numbers. Despite this, there are still basic unanswered questions about nets, which we will discuss. 
 

Fermat’s two-squares theorem is an elementary theorem in number theory that readily lends itself to a classification of the positive integers representable as the sum of two squares. Given this, a natural question is: what is the minimal number of squares needed to represent any given (positive) integer? One proof of Fermat’s result depends on essentially a buffed pigeonhole principle in the form of Minkowski’s Convex Body Theorem, and this idea can be used in a nearly identical fashion to provide 4 as an upper bound to the aforementioned question (this is Lagrange’s four-square theorem). The question of identifying the integers representable as the sum of three squares turns out to be substantially harder, however leaning on a powerful theorem of Dirichlet and a handful of tricks we can use Minkowski’s CBT to settle this final piece as well (this is Legendre’s three-square theorem).

Medical data often comes in multi-modal, streamed data. The challenge is to extract useful information from this data in an environment where gathering data is expensive. In this talk, I show how signatures can be used to predict the progression of the ALS disease.

The core idea behind metric spaces is the triangular inequality. Metrics have been generalized in many ways, but the most tempting way to alter them would be to "flip" the triangular inequality, obtaining an "anti-metric". This, however, only allows for trivial spaces where the distance between any two points is 0. However, if we intertwine the concept of antimetrics with the structures of partial linear--and cyclic--orders, we can define a structure where the anti-triangular inequality holds conditionally. We define this structure, give examples, and show an interesting result involving metrics and antimetrics.

There have been two main attempts so far to forecast the level of development of artificial intelligence (or ‘computerisation’) over time, Frey and Osborne (2013, 2017) and Manyika et al (2017). Unfortunately, their methodology seems to be flawed. Their results depend upon expert predictions of which occupations will be automatable in 2050, but these predictions are notoriously unreliable. Therefore, we develop an alternative which does not depend upon these expert predictions. We build a dataset of all the start-ups, firms, and university research laboratories working on automating different types of tasks, and use this to build a dynamic network model of them and how they interact. How automatable each type of task is ‘emerges’ from the model. We validate it, predicting the level of development of supervised learning in 2017 using data from the year 2000, and use it to forecast of the automatability of each of these task types from 2018 to 2050. Finally, we discuss extensions for our model; how it could be used to test the impact of public policy decisions or forecast developments in other high-technology industries.

In this talk we will introduce quantifier elimination and give various examples of theories with this property. We will see some very useful applications of quantifier elimination to algebra and geometry that will hopefully convince you how practical this property is to other areas of mathematics.

We continue the study by Melo and Winter [arXiv:1712.01763, 2017] on the possible intersection sizes of a $k$-dimensional subspace with the vertices of the $n$-dimensional hypercube in Euclidean space. Melo and Winter conjectured that all intersection sizes larger than $2^{k-1}$ (the “large” sizes) are of the form $2^{k-1} + 2^i$. We show that this is almost true: the large intersection sizes are either of this form or of the form $35\cdot2^{k-6}$ . We also disprove a second conjecture of Melo and Winter by proving that a positive fraction of the “small” values is missing.

In this talk, we shall introduce various identities among partitions of integers, and how these can be expressed via formal power series. In particular, we shall look at the Rogers Ramanujan identities of power series, and discuss possible combinatorial proofs using partitions and Durfree squares.

Graphs are a central object of study in various scientific fields, such as discrete mathematics, theoretical computer science and network science. These graphs are typically studied using combinatorial, algebraic or probabilistic methods, each of which highlights the properties of graphs in a unique way. I will discuss a novel approach to study graphs: the simplex geometry (a simplex is a generalized triangle). This perspective, proposed by Miroslav Fiedler, introduces techniques from (simplex) geometry into the field of graph theory and conversely, via an exact correspondence. We introduce the graph-simplex correspondence, identify a number of basic connections between graph characteristics and simplex properties, and suggest some applications as example.

Reference: https://arxiv.org/abs/1807.06475