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Calabi-Yau manifolds have become a topic of study in both mathematics and physics, dissolving the boundaries between the two subjects.

A manifold is a type of geometrical space where each small region looks like normal Euclidean space. For example, an ant on the surface of the Earth sees its world as flat, rather than the curved surface of the sphere. Calabi-Yau manifolds are complex manifolds, that is, they can be disassembled into patches which look like flat complex space. What makes them so special is that these patches can only be joined together by the complex analogue of a rotation.

Proving a conjecture of Eugenio Calabi, Shing-Tung Yau has shown that Calabi-Yau manifolds have a property which is very interesting to physics. Einstein's equations show that spacetime curves according to the distribution of energy and momentum. But what if space is all empty? By Yau's theorem, not only is flat space a solution but so are Calabi-Yau manifolds. Furthermore, for this reason, Calabi-Yau spaces are possible candidates for the shape of extra spatial dimensions in String Theory.

Find out more from Oxford Mathematician Dr Andreas Braun in this latest instalment of our Oxford Mathematics Alphabet.

The Mathematical Finance Group in Oxford has long been a leader in research on financial mathematics. In recent years a number of research areas have become key focal points within the Group, notably behavioural finance and financial big data, robust pricing and machine learning. In particular, research has focused on financial stability, an area that became critical after the Financial Crisis in 2008. 

Prof. Doyne Farmer and some of his students are working on modelling complex financial systems and the ways that losses and financial distress can spread through such systems. In particular, the Group is looking at how to design a model that accurately captures the losses the banking system, shadow banking system and real economy could incur in another crisis. These models are called 'stress tests' and are nowadays conducted by most major banks. However, the current methodology of the stress tests does not capture the network of interbank connections, although this has been the major driver of losses in a crisis. Prof. Doyne Farmer's group is developing models that more accurately capture these interconnections. 

Today, a letter published in the Financial Times, discusses exactly this: the methodology of existing stress tests and their limitations.

When one wants to describe the symmetries of any object or system, in mathematics or everyday life, the right language to use is group theory. How might one go about understanding the universe of all groups and what kinds of novel geometry might emerge as we explore this universe?

The understanding of the possible geometries in dimension 3 is one of the triumphs of 20th century mathematics. In his Chairman's Inaugural Public Lecture, Professor Martin Bridson explains this triumph and why such an understanding is impossible in higher dimensions.

When you think about the founders of computing you may think Alan Turing, you may even think Charles Babbage. But you should definitely think about Ada Lovelace. Ada Lovelace is not only the link between Babbage and Turing, but a woman of fierce originality and intellectual interests whose ideas went beyond Babbage’s ideas of computers as manipulating numbers, and focused on their creative possibilities and their limits, the very issues with which we are wrestling today.

On 9 and 10 December the University of Oxford's celebrations of the 200th anniversary of Ada’s birth will culminate in a two-day Symposium exploring Ada's life and work, the scientific and cultural world around her and her continuing influence on science and the arts today. This Symposium will be streamed live.

Humans love to find an explanation that fits the facts, and fits them as closely as possible. But this often turns out to be a terrible way of learning about the world around us.

In the latest instalment of the Oxford Mathematics Alphabet we look at Bayes’ Theorem and how it is used in criminology, product recommendations, artificial intelligence, and recently in the search for the missing Malaysian Airliner MH370.

Tweedledum: "I know what you’re thinking about, but it isn’t so, nohow."
Tweedledee: "Contrariwise, if it was so, it might be; and if it were so, it would be: but as it isn’t, it ain’t. That’s logic."

If Charles Dodgson (Lewis Carroll) had not written Alice’s Adventures in Wonderland and Through the Looking-Glass, he’d probably be remembered as a pioneer photographer. But his Oxford ‘day job’ was as Lecturer in Mathematics at Christ Church. What mathematics did he do?  Find out in the latest in our poster series of Oxford Mathematicians.

 Lewis Carroll.pdf

A booklet about networks literacy developed by Mason Porter, Fellow of Somerville College and Professor of Nonlinear and Complex Systems in the University of Oxford's Mathematical Institute, in collaboration with colleagues from the USA, could help people understand all types of networks from social media to rabbit warrens. Mason was part of a team of over 30 network-science researchers, educators, teachers, and students who have written the booklet on networks literacy that schools can adapt to teach students the core concepts about networks.

"The concept of networks is truly interdisciplinary and knowing about general properties of networks allows students to see common patterns across disciplines, and thereby transcend disciplinary boundaries, said Hiroki Sayama, one of the partners on the project and Director of the Center for Collective Dynamics of Complex Systems and Associate Professor of Systems Science and Industrial Engineering at Binghamton University. It would be wonderful to see students studying various subjects - languages, history, social phenomena, biological organisms, engineered products, the Internet - all from a common lens of networks.

Porter, Sayama, and co-authors Catherine Cramer, Lori Sheetz, and Stephen Uzzo enumerated seven key concepts (with the input of numerous others) that characterise networks. The work was driven by one key question: what should every person living in the 21st century know about networks by the time they finish secondary education? The sooner future scientists know these core ideas, the sooner they can make networks around us more efficient, cost-effective, and safe.

The booklet, called 'Network Literacy: Essential Concepts and Core Ideas', breaks down the key ideas so that teachers can use it in the classroom or for lesson planning. The concepts have comparable importances, and they are ordered roughly according to difficulty level: the earlier concepts are easier to understand for everyone, whereas the latter ones may need more thinking and learning to grasp fully what they mean.

The project was done in collaboration with the New York Hall of Science and the U.S. Military Academy at West Point. The booklet has been translated into eight different languages so far, including Persian, Japanese, and German. The booklet (including all translations) is freely available online. 

A paper (with Sayama as the lead author), called 'What Are Essential Concepts About Networks?', about the procedure of creating the booklet appeared on 11 November as an advance-access article in the Journal of Complex Networks.