L1

A continuum of expanders -- David Hume

Expanders are a holy grail of networking; robustly connected networks of arbitrary size which require minimal resources. Like the grail, they are also notoriously difficult to construct. In this talk I will introduce expanders, give a brief overview of just a few aspects of their diverse history, and outline a very recent result of mine, which states that there are a continuum of expanders with fundamentally different large-scale geometry.

What makes cities successful? A complex systems approach to modelling urban economies -- Neave O'Clery

Urban centres draw a diverse range of people, attracted by opportunity, amenities, and the energy of crowds. Yet, while benefiting from density and proximity of people, cities also suffer from issues surrounding crime, congestion and density. Seeking to uncover the mechanisms behind the success of cities using novel tools from the mathematical and data sciences, this work uses network techniques to model the opportunity landscape of cities. Under the theory that cities move into new economic activities that share inputs with existing capabilities, path dependent industrial diversification can be described using a network of industries. Edges represent shared necessary capabilities, and are empirically estimated via flows of workers moving between industries. The position of a city in this network (i.e., the subnetwork of its current industries) will determine its future diversification potential. A city located in a central well-connected region has many options, but one with only few peripheral industries has limited opportunities.

We develop this framework to explain the large variation in labour formality rates across cities in the developing world, using data from Colombia. We show that, as cities become larger, they move into increasingly complex industries as firms combine complementary capabilities derived from a more diverse pool of workers. We further show that a level of agglomeration equivalent to between 45 and 75 minutes of commuting time maximizes the ability of cities to generate formal employment using the variety of skills available. Our results suggest that rather than discouraging the expansion of metropolitan areas, cities should invest in transportation to enable firms to take advantage of urban diversity.

This talk will be based on joint work with Eduardo Lora and Andres Gomez at Harvard University.

Tim Harford, Financial Times columnist and presenter of Radio 4's "More or Less", argues that politicians, businesses and even charities have been poisoning the value of statistics and data. Tim will argue that we need to defend the value of good data in public discourse, and will suggest how to lead the defence of statistical truth-telling.

Please email @email to register 

Mathematical methods are increasingly being used to design auctions. Paul Klemperer will talk about some of his own experience which includes designing the U.K.'s mobile phone licence auction that raised £22.5 billion, and a new auction that helped the Bank of England in the financial crisis. (The then-Governor, Mervyn King, described it as "a marvellous application of theoretical economics to a practical problem of vital importance".) He will also discuss further development of the latter auction using convex and "tropical" geometric methods.

Hinke Osinga, University of Auckland
joint work with: Bernd Krauskopf and Stefanie Hittmeyer (University of Auckland)

Dynamical systems of Lorenz type are similar to the famous Lorenz system of just three ordinary differential equations in a well-defined geometric sense. The behaviour of the Lorenz system is organised by a chaotic attractor, known as the butterfly attractor. Under certain conditions, the dynamics is such that a dimension reduction can be applied, which relates the behaviour to that of a one-dimensional non-invertible map. A lot of research has focussed on understanding the dynamics of this one-dimensional map. The study of what this means for the full three-dimensional system has only recently become possible through the use of advanced numerical methods based on the continuation of two-point boundary value problems. Did you know that the chaotic dynamics is organised by a space-filling pancake? We show how similar techniques can help to understand the dynamics of higher-dimensional Lorenz-type systems. Using a similar dimension-reduction technique, a two-dimensional non-invertible map describes the behaviour of five or more ordinary differential equations. Here, a new type of chaotic dynamics is possible, called wild chaos. 

Meteorologist Ed Lorenz was one of the founding fathers of chaos theory. In 1963, he showed with just three simple equations that the world around us could be both completely deterministic and yet practically unpredictable. More than this, Lorenz discovered that this behaviour arose from a beautiful fractal geometric structure residing in the so-called state space of these equations. In the 1990s, Lorenz’s work was popularised by science writer James Gleick. In his book Gleick used the phrase “The Butterfly Effect” to describe the unpredictability of Lorenz’s equations. The notion that the flap of a butterfly’s wings could change the course of future weather was an idea that Lorenz himself used in his outreach talks.

However, Lorenz used it to describe something much more radical than can be found in his three simple equations. Lorenz didn’t know whether the Butterfly Effect, as he understood it, was true or not. In fact, it lies at the heart of one of the Clay Mathematics Millennium Prize problems, and is still an open problem today. In this talk I will discuss Lorenz the man, his background and his work in the 1950s and 1960s, and will compare and contrast the meaning of the “Butterfly Effect" as most people understand it today, and as Lorenz himself intended it to mean. The implications of the “Real Butterfly Effect" for understanding the predictability of nonlinear multi-scale systems (such as weather and climate) will be discussed. No technical knowledge of the field is assumed. 

Please email @email to register

Further reading:
T.N.Palmer, A. Döring and G. Seregin (2014): The Real Butterfly Effect. Nonlinearity, 27, R123-R141.

In two dimensional topological phases of matter, processes depend on gross topology rather than detailed geometry. Thinking in 2+1 dimensions, the space-time histories of particles can be interpreted as knots or links, and the amplitude for certain processes becomes a topological invariant of that link. While sounding rather exotic, we believe that such phases of matter not only exist, but have actually been observed (or could be soon observed) in experiments. These phases of matter could provide a uniquely practical route to building a quantum computer. Experimental systems of relevance include Fractional Quantum Hall Effects, Exotic superconductors such as Strontium Ruthenate, Superfluid Helium, Semiconductor-Superconductor-Spin-Orbit systems including Quantum Wires. The physics of these systems, and how they might be used for quantum computation will be discussed.

Featuring

Professor Alison Etheridge, Professor of Probability in the Mathematical Institute and Department of Statistics, Oxford

Professor Ben Green, Waynflete Professor of Pure Mathematics, Oxford

Dr Heather Harrington, Royal Society University Research Fellow in the Mathematical Institute, Oxford

Professor Jon Keating, Henry Overton Wills Professor of Mathematics, Bristol and Chair of the Heilbronn Institute for Mathematical Research

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Dr Christopher Voyce, Head of Research Facilitation in the Mathematical Institute, Oxford

Approximate prime numbers -- James Maynard

I will talk about the idea of an 'almost prime' number, and how this can be used to make progress on some famous problems about the primes themselves.

Mathematical biology: An early career retrospective -- Thomas Woolley

Since 2008 Thomas has focused his attention to the application of mathematical techniques to biological problems. Through numerous fruitful collaborations he has been extremely fortunate to work alongside some amazing researchers. But what has he done in the last 8 years? What lessons has he learnt? What knowledge has he produced?

This talk will encompass a brief overview of a range of applications, from animal skin patterns to cellular mechanics, via zombies and Godzilla.

Wondering about how to organise your DPhil? How to make the most of your supervision meetings? How to guarantee success in your studies? Look no further!

In this session we will explore the fundamentals of a successful DPhil with help from faculty members, postdocs and DPhil students.

In the first half of the session Andreas Münch, the Director of Graduate Studies, will give a brief overview of the stages of the DPhil programme in Oxford; after this Marc Lackenby will talk about his experience as a PhD student and supervisor.

The second part of the session will be a panel discussion, with panel members Lucy Hutchinson, Mark Penney, Michal Przykucki, and Thomas Woolley. Senior faculty members will be kindly asked to leave the lecture theatre to ensure that students feel comfortable about discussing their experiences with later year students and postdocs/research fellows.

At 5pm senior and junior faculty members, postdocs and students will reunite in the Common Room for Happy Hour.

About the speakers and panel members:

Andreas Münch received his PhD from the Technical University of Munich under the supervision of Karl-Heinz Hoffmann. He moved to Oxford in 2009, where he is an Associate Professor in Applied Mathematics. As the Director of Graduate Studies he deals with matters related to training and education of graduate students. 

Marc Lackenby received his PhD from Cambridge under the supervision of W. B. Raymond Lickorish. He moved to Oxford in 1999, where he has been a Professor of Mathematics since 2006. 

Lucy Hutchinson is a DPhil student in the Mathematical Biology group studying her final year.

Mark Penney is a fourth-year DPhil student in the Topology group.

Michal Przykucki received his PhD from Cambridge in 2013 under the supervision of Béla Bollobás; he is a member of the Combinatorics research group, and has been a Drapers Junior Research Fellow at St Anne's College since 2014. 

Thomas Woolley received his DPhil from Oxford in 2012 under the supervision of Ruth Baker, Eamonn Gaffney, and Philip Maini. He is a member of the Mathematical Biology Group and has been a St John’s College Junior Research Fellow in Mathematics since 2013.

What is the minimal size of a separating set? -- Emilie Dufresne

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Abstract: The problem of classifying objects up to certain allowed transformations figures prominently in almost all branches of Mathematics, and Invariants are used to decide if two objects are equivalent. A separating set is a set of invariants which achieve the desired classification. In this talk we take the point of view of Invariant Theory, where the objects correspond to points on an affine variety (often a vector space) and equivalence is given by the action of an algebraic group on this affine variety. We explain how the geometry and combinatorics of the group action govern the minimal size of separating sets.

Predator-Prey-Subsidy Dynamics and the Paradox of Enrichment on Networks -- Robert Van Gorder

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Abstract: The phrase "paradox of enrichment" was coined by Rosenzweig (1971) to describe the observation that increasing the food available to prey participating in predator-prey interactions can destabilize the predator's population. Subsequent work demonstrated that food-web connectance on networks can stabilize the predator-prey dynamics, thereby dampening the paradox of enrichment in networked domains (such as those used in stepping-stone models). However, when a resource subsidy is available to predators which migrate between nodes on such a network (as is actually observed in some real systems), we may show that predator-prey systems can exhibit a paradox of enrichment - induced by the motion of predators between nodes - provided that such networks are sufficiently densely connected.