Fridays@4

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

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

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Anna Railton, Technical Staff, Smith Institute

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Michele Taroni, Senior Project Manager, Roxar

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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.

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. 

Some relish the idea of working with users of research and having an impact on the outside world - some view it as a ridiculous government agenda which interferes with academic freedom.  We’ll give an overview of  the political and practical aspects of impact and identify things you might want to consider when deciding whether, and how, to get involved.