L1

Categorification of knot polynomials -- Daniele Celoria

Classically, the most powerful and versatile knot invariants take the form of polynomials. These can usually be defined by simple recursive equations, known as skein relations; after giving the main examples of polynomial knot invariants (Alexander and Jones polynomials), we are going to informally introduce categorifications. Finally we are going to present the Knot Floer and the Khovanov homologies, and show that they provide a categorification of the aforementioned polynomial knot invariants.

Network science for online social media: an x-ray or a stethoscope for society -- Mariano Beguerisse

The abundance of data from social media outlets such as Twitter provides the opportunity to perform research at a societal level at a scale unforeseen. This has spurred the development of mathematical and computational methods such as network science, which uses the formalism and language of graph theory to study large systems of interacting agents. In this talk, I will provide a sketch of network science and its application to study online social media. A number of different networks can be constructed from Twitter data, which can be used to ask questions about users, ranging from the structural (an 'x-ray' to see how societies are connected online) to the topical ('stethoscope' to feel how users interact in the context of specific event). I will provide concrete examples from the UK riots of 2011, applications to medical anthropology, and political referenda, and will also highlight distinct challenges such as the directionality of connections, the size of the network, the use of temporal information and text, all of which are active areas of research.

Do you find yourself agreeing to things when actually you want more – or less? In this session we will look at how to be clear about what you want, and how to use assertiveness and negotiation skills and strategies to achieve win-win outcomes when working with others. 

This session will look at the range of courses available to early career researchers and graduate students from MPLS. It will also discuss the benefits of training and development for researchers and how it can help you in enhancing your career inside and outside academia.
 

Featuring
Peter Grindrod, Director of the Oxford-Emirates Data Science Lab, Oxford Mathematical Institute

Geraint Lloyd, Senior Software Engineer, Schlumberger

[[{"fid":"45910","view_mode":"media_square","fields":{"format":"media_square","field_file_image_alt_text[und][0][value]":"Geraint Lloyd","field_file_image_title_text[und][0][value]":"Geraint Lloyd"},"type":"media","attributes":{"alt":"Geraint Lloyd","title":"Geraint Lloyd","height":"258","width":"258","class":"media-element file-media-square"}}]]

Mick Pont, VP Research and Development, Numerical Algorithms Group (NAG)

[[{"fid":"45911","view_mode":"media_square","fields":{"format":"media_square","field_file_image_alt_text[und][0][value]":"Mick Pont","field_file_image_title_text[und][0][value]":"Mick Pont"},"type":"media","attributes":{"alt":"Mick Pont","title":"Mick Pont","height":"258","width":"258","class":"media-element file-media-square"}}]]

Anna Railton, Technical Staff, Smith Institute

[[{"fid":"45912","view_mode":"media_square","fields":{"format":"media_square","field_file_image_alt_text[und][0][value]":"Anna Railton","field_file_image_title_text[und][0][value]":"Anna Railton"},"type":"media","attributes":{"alt":"Anna Railton","title":"Anna Railton","height":"258","width":"258","class":"media-element file-media-square"}}]]
Michele Taroni, Senior Project Manager, Roxar

[[{"fid":"45913","view_mode":"media_square","fields":{"format":"media_square","field_file_image_alt_text[und][0][value]":"Michele Taroni","field_file_image_title_text[und][0][value]":"Michele Taroni"},"type":"media","attributes":{"alt":"Michele Taroni","title":"Michele Taroni","height":"258","width":"258","class":"media-element file-media-square"}}]]

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.