Past Colloquia

The theory of metric measure spaces with Ricci curvature from below is growing very quickly, both in the "Riemannian" class RCD and the general  CD one. I will review some of the most recent results, by illustrating the key identification results and technical tools (at the level of calculus in metric measure spaces) underlying these results.
 

The fourth QBIOX Colloquium will take place in the Mathematical Institute on Friday 10th November (5th week) and feature talks from Professor Paul Riley (Department of Pathology, Anatomy and Genetics / BHF Oxbridge Centre for Regenerative Medicine, https://www.dpag.ox.ac.uk/research/riley-group) and Professor Eleanor Stride (Institute of Biomedical Engineering, http://www.ibme.ox.ac.uk/research/non-invasive-therapy-drug-delivery/pe…).

1600-1645 - Paul Riley, "Enroute to mending broken hearts".
1645-1730 - Eleanor Stride, "Reducing tissue hypoxia for cancer therapy".
1730-1800 - Networking and refreshments.

We very much hope to see you there. As ever, tickets are not necessary, but registering to attend will help us with numbers for catering.
Please see the following link for further details and a link to register.
https://www.eventbrite.co.uk/e/qbiox-colloquium-michelmas-term-2017-tic…

Abstracts
Paul Riley - "En route to mending broken hearts".
We adopt the paradigm of understanding how the heart develops during pregnancy as a first principal to inform on adult heart repair and regeneration. Our target for cell-based repair is the epicardium and epicardium-derived cells (EPDCs) which line the outside of the forming heart and contribute vascular endothelial and smooth muscle cells to the coronary vasculature, interstitial fibroblasts and cardiomyocytes. The epicardium can also act as a source of signals to condition the growth of the underlying embryonic heart muscle. In the adult heart, whilst the epicardium is retained, it is effectively quiescent. We have sought to extrapolate the developmental potential of the epicardium to the adult heart following injury by stimulating dormant epicardial cells to give rise to new muscle and vasculature. In parallel, we seek to modulate the local environment into which the new cells emerge: a cytotoxic mixture of inflammation and fibrosis which prevents cell engraftment and integration with survived heart tissue. To this end we manipulate the lymphatic vessels in the heart given that, elsewhere in the body, the lymphatics survey the immune system and modulate inflammation at peripheral injury sites. We recently described the development of the cardiac lymphatic vasculature and revealed in the adult heart that they undergo increased vessel sprouting (lymphangiogenesis) in response to injury, to improve function, remodelling and fibrosis. We are currently investigating whether increased lymphangiogenesis functions to clear immune cells and constrain the reparative response for optimal healing. 

Eleanor Stride - "Reducing tissue hypoxia for cancer therapy"
Hypoxia, i.e. a reduction in dissolved oxygen concentration below physiologically normal levels, has been identified as playing a critical role in the progression of many types of disease and as a key determinant of the success of cancer treatment. It poses a particular challenge for treatments such as radiotherapy, photodynamic and sonodynamic therapy which rely on the production of reactive oxygen species. Strategies for treating hypoxia have included the development of hypoxia-selective drugs as well as methods for directly increasing blood oxygenation, e.g. hyperbaric oxygen therapy, pure oxygen or carbogen breathing, ozone therapy, hydrogen peroxide injections and administration of suspensions of oxygen carrier liquids. To date, however, these approaches have delivered limited success either due to lack of proven efficacy and/or unwanted side effects. Gas microbubbles, stabilised by a biocompatible shell have been used as ultrasound contrast agents for several decades and have also been widely investigated as a means of promoting drug delivery. This talk will present our recent research on the use of micro and nanobubbles to deliver both drug molecules and oxygen simultaneously to a tumour to facilitate treatment.

Coding theory revolves around the question of what can be accomplished with only memory and redundancy. When we ask what enables the things that transmit and store information, we discover codes at work, connecting the world of geometry to the world of algorithms.

This talk will focus on those connections that link the real world of Euclidean geometry to the world of binary geometry that we associate with Hamming.

A cubic polynomial equation in four or more variables tends to have many integer solutions, while one in two variables has a limited number of such solutions. There is a body of work establishing results along these lines. On the other hand very little is known in the critical case of three variables. For special such cubics, which we call Markoff surfaces, a theory can be developed. We will review some of the tools used to deal with these and related problems.

Joint works with Bourgain/Gamburd and with Ghosh
 

The cover of the December 2016 AMS Notices shows an eye-like region picked out by blue and red dots and surrounded by green rays. The picture, drawn by Yasushi Yamashita, illustrates Gaven Martin’s search for the smallest volume 3-dimensional hyperbolic orbifold. It represents a family of two generator groups of isometries of hyperbolic 3-space which was recently studied, for quite different reasons, by myself, Yamashita and Ser Peow Tan.

After explaining the coloured dots and their role in Martin’s search, we concentrate on the green rays. These are Keen-Series pleating rays which are used to locate spaces of discrete groups. The theory also suggests why groups represented by the red dots on the rays in the inner part of the eye display some interesting geometry.
 

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. 

Permutations of finitely many elements are often drawn as permutation diagrams. We take this point of view as a motivation to construct and describe more complicated algebras arising for instance from differential operators, from operators acting on (co)homologies, from invariant theory, or from Hecke algebras. The surprising fact is that these diagrams are elementary and simple to describe, but at the same time describe relations between cobordisms as well as categories of represenetations of p-adic groups. The goal of the talk is to give some glimpses of these phenomena and indicate which role categorification plays here.
 

The law of quadratic reciprocity and the celebrated connection between modular forms and elliptic curves over Q are both examples of reciprocity laws. Constructing new reciprocity laws is one of the goals of the Langlands program, which is meant to connect number theory with harmonic analysis and representation theory.

In this talk, I will survey some recent progress in establishing new reciprocity laws, relying on the Galois representations attached to torsion classes which occur in the cohomology of arithmetic hyperbolic 3-manifolds. I will outline joint work in progress on better understanding these Galois representations, proving modularity lifting theorems in new settings, and applying this to elliptic curves over imaginary quadratic fields.

Self-organization is observed in systems driven by the “social engagement” of agents with their local neighbors. Prototypical models are found in opinion dynamics, flocking, self-organization of biological organisms, and rendezvous in mobile networks.

We discuss the emergent behavior of such systems. Two natural questions arise in this context. The underlying issue of the first question is how different rules of engagement influence the formation of clusters, and in particular, the emergence of 'consensus'. Different paradigms of emergence yield different patterns, depending on the propagation of connectivity of the underlying graphs of communication.  The second question involves different descriptions of self-organized dynamics when the number of agents tends to infinity. It lends itself to “social hydrodynamics”, driven by the corresponding tendency to move towards the local means. 

We discuss the global regularity of social hydrodynamics for sub-critical initial configurations.

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.

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.

Let X be a complex algebraic variety containing a point x. One of the central ideas of deformation theory is that the local structure of X near the point x can be encoded by a differential graded Lie algebra. In this talk, Jacob Lurie will explain this idea and discuss some generalizations to more exotic contexts.

One of the big ideas in linear algebra is {\em eigenvalues}. Most matrices become in some basis {\em diagonal} matrices; so a lot of information about the matrix (which is specified by $n^2$ matrix entries) is encoded by by just $n$ eigenvalues. The fact that lots of different matrices can have the same eigenvalues reflects the fact that matrix multiplication is not commutative.

I'll look at how to make these vague statements (``lots of different matrices...") more precise; how to extend them from matrices to abstract symmetry groups; and how to relate abstract symmetry groups to matrices.

Eigenvectors of square matrices are central to linear algebra. Eigenvectors of tensors are a natural generalization. The spectral theory of tensors was pioneered by Lim and Qi around 2005. It has numerous applications, and ties in closely with optimization and dynamical systems.  We present an introduction that emphasizes algebraic and geometric aspects

The question of deriving Fluid Mechanics equations from deterministic
systems of interacting particles obeying Newton's laws, in the limit
when the number of particles goes to infinity, is a longstanding open
problem suggested by Hilbert in his 6th problem. In this talk we shall
present a few attempts in this program, by explaining how to derive some
linear models such as the Heat, acoustic and Stokes-Fourier equations.
This corresponds to joint works with Thierry Bodineau and Laure Saint
Raymond.

Most networks and graphs encountered in empirical studies, including internet and web graphs, social networks, and biological and ecological networks, are very sparse.  Standard spectral and linear algebra methods can fail badly when applied to such networks and a fundamentally different approach is needed.  Message passing methods, such as belief propagation, offer a promising solution for these problems.  In this talk I will introduce some simple models of sparse networks and illustrate how message passing can form the basis for a wide range of calculations of their structure.  I will also show how message passing can be applied to real-world data to calculate fundamental properties such as percolation thresholds, graph spectra, and community structure, and how the fixed-point structure of the message passing equations has a deep connection with structural phase transitions in networks.

The talk will consider three well-defined problems which can be interpreted as mathematical tests of the physical reality of black holes: Rigidity, stability and formation of black holes.

There has been a great deal of attention paid to "Big Data" over the last few years.  However, often as not, the problem with the analysis of data is not as much the size as the complexity of the data.  Even very small data sets can exhibit substantial complexity.  There is therefore a need for methods for representing complex data sets, beyond the usual linear or even polynomial models.  The mathematical notion of shape, encoded in a metric, provides a very useful way to represent complex data sets.  On the other hand, Topology is the mathematical sub discipline which concerns itself with studying shape, in all dimensions.  In recent years, methods from topology have been adapted to the study of data sets, i.e. finite metric spaces.  In this talk, we will discuss what has been
done in this direction and what the future might hold, with numerous examples.

I will review Bott's classical periodicity result about topological K-theory (with period 2 in the case of complex K-theory, and period 8 in the case of real K-theory), and provide an easy sketch of proof, based on the algebraic periodicity of Clifford algebras. I will then introduce the `higher real K-theory' of Hopkins and Miller, also known as TMF. I'll discuss its periodicity (with period 576), and present a conjecture about a corresponding algebraic periodicity of `higher Clifford algebras'. Finally, applications to physics will be discussed.

Some physical and mathematical theories have the unfortunate feature that if one takes them at face value, many quantities of interest appear to be infinite! Various techniques, usually going under the common name of “renormalisation” have been developed over the years to address this, allowing mathematicians and physicists to tame these infinities. We will tip our toes into some of the mathematical aspects of these techniques and we will see how they have recently been used to make precise analytical statements about the solutions of some equations whose meaning was not even clear until recently.

Optimization methods for large-scale machine learning must confront a number of challenges that are unique to this discipline. In addition to being scalable, parallelizable and capable of handling nonlinearity (even non-convexity), they must also be good learning algorithms. These challenges have spurred a great amount of research that I will review, paying particular attention to variance reduction methods. I will propose a new algorithm of this kind and illustrate its performance on text and image classification problems.

Based upon our joint work with M. Marcolli, I will introduce some algebraic geometric models in cosmology related to the "boundaries" of space-time: Big Bang, Mixmaster Universe, and Roger Penrose's crossovers between aeons. We suggest to model the kinematics of Big Bang using the algebraic geometric (or analytic) blow up of a point $x$. This creates a boundary  which consists of the projective space of tangent directions to $x$ and possibly of the light cone of $x$. We argue that time on the boundary undergoes the Wick rotation and becomes purely imaginary. The Mixmaster (Bianchi IX) model of the early history of the universe is neatly explained in this picture by postulating that the reverse Wick rotation follows a hyperbolic geodesic connecting imaginary time axis to the real one. Roger Penrose's idea to see the Big Bang as a sign of crossover from "the end of the previous aeon" of the expanding and cooling Universe to the "beginning of the next aeon" is interpreted as an identification of a natural boundary of Minkowski space at infinity with the Bing Bang boundary.

Quantum Mechanics presents a radically different perspective on physical reality compared with the world of classical physics. In particular, results such as the Bell and Kochen-Specker theorems highlight the essentially non-local and contextual nature of quantum mechanics. The rapidly developing field of quantum information seeks to exploit these non-classical features of quantum physics to transcend classical bounds on information processing tasks.

In this talk, we shall explore the rich mathematical structures underlying these results. The study of non-locality and contextuality can be expressed in a unified and generalised form in the language of sheaves or bundles, in terms of obstructions to global sections. These obstructions can, in many cases, be witnessed by cohomology invariants. There are also strong connections with logic. For example, Bell inequalities, one of the major tools of quantum information and foundations, arise systematically from logical consistency conditions.

These general mathematical characterisations of non-locality and contextuality also allow precise connections to be made with a number of seemingly unrelated topics, in classical computation, logic, and natural language semantics. By varying the semiring in which distributions are valued, the same structures and results can be recognised in databases and constraint satisfaction as in probability models arising from quantum mechanics. A rich field of contextual semantics, applicable to many of the situations where the pervasive phenomenon of contextuality arises, promises to emerge.

Universal fluctuations are shown to exist when well-known and widely used numerical algorithms are applied with random data. Similar universal behavior is shown in stochastic algorithms and algorithms that model neural computation. The question of whether universality is present in all, or nearly all, computation is raised. (Joint work with G.Menon, S.Olver and T. Trogdon.)