University of Oxford

We consider the classic problem of establishing a statistical ranking of a set of n items given a set of inconsistent and incomplete pairwise comparisons between such items. Instantiations of this problem occur in numerous applications in data analysis (e.g., ranking teams in sports data), computer vision, and machine learning. We formulate the above problem of ranking with incomplete noisy information as an instance of the group synchronization problem over the group SO(2) of planar rotations, whose usefulness has been demonstrated in numerous applications in recent years. Its least squares solution can be approximated by either a spectral or a semidefinite programming (SDP) relaxation, followed by a rounding procedure. We perform extensive numerical simulations on both synthetic and real-world data sets (Premier League soccer games, a Halo 2 game tournament and NCAA College Basketball games) showing that our proposed method compares favorably to other algorithms from the recent literature.

We propose a similar synchronization-based algorithm for the rank-aggregation problem, which integrates in a globally consistent ranking pairwise comparisons given by different rating systems on the same set of items. We also discuss the problem of semi-supervised ranking when there is available information on the ground truth rank of a subset of players, and propose an algorithm based on SDP which recovers the ranks of the remaining players. Finally, synchronization-based ranking, combined with a spectral technique for the densest subgraph problem, allows one to extract locally-consistent partial rankings, in other words, to identify the rank of a small subset of players whose pairwise comparisons are less noisy than the rest of the data, which other methods are not able to identify. 
 

Bubble Dynamics

We shall discuss certain generalisations of the Rayleigh Plesset equation for bubble dynamics

Self-assembly of a filament by curvature-inducing proteins

We explore a simplified macroscopic model of membrane shaping by means of curvature-sensing proteins. Equations describing the interplay between the shape of a freely floating filament in a fluid and the adhesion kinetics of proteins are derived from mechanical principles. The constant curvature solutions that arise from this system are studied using weakly nonlinear analysis. We show that the stability of the filament’s shape is completely characterized by the parameters associated with protein recruitment and establish that in the bistable regime, proteins aggregate on the filament forming regions of high and low curvatures. This pattern formation is then followed by phase-coarsening that resolves on a time-scale dependent on protein diffusion and drift across the filament, which contend to smooth and maintain the pattern respectively. The model is generalized for multiple species of proteins and we show that the stability of the assembled shape is determined by a competition between proteins attaching on opposing sides.

I will compare features of (classical) cohomology theory of groups to the rather exotic features of bounded (or continuous bounded) cohomology of groups.
Besides giving concrete examples I will state classical cohomological tools/features and see how (if) they survive in the case of bounded cohomology. Such will include the Mayer-Vietoris sequence, the transfer map, resolutions, classifying spaces, the universal coefficient theorem, the cup product, vanishing results, cohomological dimension and relation to extensions. 
Finally I will discuss their connection to each other via the comparison map.

In this meeting we will talk about the first two chapters of Robert Ghrist's book "Elementary Applied Topology". The book is freely available at the following link: https://www.math.upenn.edu/~ghrist/notes.html

A successful programme of personalised discounts and recommendations relies on identifying products that customers want, based both on items bought in the past and on relevant products that the customers have not yet purchased. Using basket-level grocery shopping data, we aim to use clustering ("community detection") techniques to identify groups of shoppers with similar preferences, along with the corresponding products that they purchase, in order to design better recommendation systems.

Stochastic block models (SBMs) are an increasingly popular class of methods for community detection. In this talk, I will expand on some work done by Newman and Clauset [1] that uses a modified SBM for community detection in annotated networks. In these networks, additional information in the form of node metadata is used to improve the quality of the inferred community structure. The method can be extended to bipartite networks, which contain two types of nodes and edges only between nodes of different types. I will show some results obtained from applying this method to a bipartite network of customers and products. Finally, I will discuss some desirable extensions to this method such as incorporating edge weights and assessing the relationship between metadata and network structure in a statistically robust way.

[1] Structure and inference in annotated networks, MEJ Newman and A Clauset, Nature Communications 7, 11863 (2016).

Note: This talk will cover similar topics to my presentation in the InFoMM group meeting on Friday, November 25 but it won't be exactly the same. I will focus more on the mathematical details for my JAMS talk.
 

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.

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.