Invited speaker abstracts are available to download here.

Abstracts arranged alphabetically by surname.

Kevin Buzzard (Imperial College London)

When will computers prove theorems?

Computers can now beat us at chess and at go. When will they start to beat us at proving theorems? What kind of theorems might they prove, and what might their proofs look like? What is a proof anyway? Do humans prove theorems, or do they just sketch proofs? If the proof of a theorem is "known to the experts" and then one day there are no experts left, is this still a proof? Will humans and computers be able to collaborate in the future? And when will all this happen -- in 5 years or in 50 years? I will survey the state of the art.

Bute 1-2, Wednesday

Jose Antonio Carrillo (Oxford)

Nonlocal Aggregation-Diffusion Equations: entropies, gradient flows, phase transitions and applications

This talk will be devoted to an overview of recent results understanding the bifurcation analysis of nonlinear Fokker-Planck equations arising in a myriad of applications such as consensus formation, optimization, granular media, swarming behavior, opinion dynamics and financial mathematics to name a few. We will present several results related to localized Cucker-Smale orientation dynamics, McKean-Vlasov equations, and nonlinear diffusion Keller-Segel type models in several settings. We will show the existence of continuous or discontinuous phase transitions on the torus under suitable assumptions on the Fourier modes of the interaction potential. The analysis is based on linear stability in the right functional space associated to the regularity of the problem at hand. While in the case of linear diffusion, one can work in the L2 framework, nonlinear diffusion needs the stronger Linfty topology to proceed with the analysis based on Crandall-Rabinowitz bifurcation analysis applied to the variation of the entropy functional. Explicit examples show that the global bifurcation branches can be very complicated. Stability of the solutions will be discussed based on numerical simulations with fully explicit energy decaying finite volume schemes specifically tailored to the gradient flow structure of these problems. The theoretical analysis of the asymptotic stability of the different branches of solutions is a challenging open problem. This overview talk is based on several works in collaboration with R. Bailo, A. Barbaro, J. A. Canizo, X. Chen, P. Degond, R. Gvalani, J. Hu, G. Pavliotis, A. Schlichting, Q. Wang, Z. Wang, and L. Zhang. This research has been funded by EPSRC EP/P031587/1 and ERC Advanced Grant Nonlocal-CPD 883363.

Bute 3-5, Clyde 3-5, Hillhead 3-5, Thursday

Mark Gross (Cambridge)

Intrinsic Mirror Symmetry

Mirror symmetry was a phenomenon discovered by physicists around 1989: they observed that certain kinds of six-dimensional geometric objects known as Calabi-Yau manifolds seemed to come in pairs, with a strange relationship between different kinds of geometric objects on the pairs. Since then, the subject has blossomed into a vast field, with many different approaches and philosophies. I will give a brief introduction to the subject, and explain how one of these approaches, developed with Bernd Siebert, has led to a general construction of mirror pairs.

Bute 1-2, Thursday

Heather A. Harrington (Oxford)

Algebraic Systems Biology

Signalling pathways can be modelled as a biochemical reaction network.

When the kinetics follow mass-action kinetics, the resulting mathematical model is a polynomial dynamical system. I will overview approaches to analyse these models with steady-state data using computational algebraic geometry and statistics. Then I will present how to analyse such models with time-course data using differential algebra and geometry for model identifiability. Finally, I will present how topological data analysis can provide additional information to distinguish these models and experimental data from wild-type and mutant molecules. These case studies showcase how computational geometry, topology and dynamics can provide new insights in the biological systems, specifically how changes at the molecular scale (e.g. MEK WT and mutants) result in phenotypic changes (e.g.fruit fly mutations).

Bute 1-2, Friday

Sascha Hilgenfeldt (Illinois)

Foams: A Prototype for the Structure and Dynamics of Interfaces

Liquid foams are fascinating materials that provide study cases to mathematicians, physicists, engineers, and biologists. The increasing importance of mechanical and dynamical processes involving interfaces on small scales in both hard and soft condensed matter has prompted questions that can often be most rigorously studied in foams, where interfaces are mechanically simple. We will show how modeling tools applied to foams allow for quantitative insight into two very different systems: (i) The mechanical energy landscape of metastable states in cellular matter such as biological tissues is complicated and poorly understood. A new approach infers mechanical energy from statistical information of the disordered structure. Results from foams can be quantitatively translated to other systems with much more complicated energy functionals, to elucidate the mechanics of tissue layers purely from visual snapshot information. (ii) Liquid foams have long been used as model systems for rearrangements in atomic lattices. It is shown here that they can be used to model fracture events as well, providing the first detailed experimental and theoretical insight into the existence of a velocity gap, i.e., a minimum speed required for steady propagation of a crack. Modeling based on fluid dynamics phenomena such as film instability and non-linear dissipation results in highly dynamical, robust fracture behavior that predicts the stress distribution and speed of propagating cracks.

Spanning such a wide range of physical phenomena both static and dynamic, material properties both solid and liquid, and structures both discrete and continuum, illustrates that foams are an enormously versatile and vital tool for interfacial modeling, capable of yielding new modeling approaches as well as insight of practical use.

All Sococo rooms, Wednesday

Jon Keating (Oxford)

Random Matrix Theory, Integrable Systems, and Discrete Probability

I will review some connections between Random Matrix Theory, aspects of Discrete Probability, and certain integrable systems. I will then outline how these connections have led to recent progress in computing various moments of characteristic polynomials of random matrices that had previously been inaccessible.

Bute 1-2, Thursday

Ailsa Keating (Cambridge)

An invitation to symplectic mapping class groups

Given a symplectic manifold, one can define its symplectic mapping class group: diffeomorphisms preserving the symplectic form, up to symplectic isotopy. In dimension two, this agrees with the classical mapping class group of the space, which has been extensively studied. This talk will aim to give an overview of what is known in the higher-dimensional case. No prior knowledge of symplectic topology will be assumed.

Bute 1-5, Tuesday

Minhyong Kim (Oxford)

Recent Progress on Diophantine Equations in Two Variables

The study of rational or integral solutions to polynomial equations f(x_1, x_2,.., x_n)=0 is among the oldest subjects in mathematics. After a brief description of its modern history, we will review a few of the breakthroughs of the last few decades and some recent geometric approaches to describing sets of solutions when the number of variables is 2.

Clyde 1-2, Wednesday

Daniela Kuhn (Birmingham)

Proof of the Erdos-Faber-Lovasz conjecture

The Erdos-Faber-Lovasz conjecture (posed in 1972) states that the chromatic index of any linear hypergraph on n vertices is at most n. (Here the chromatic index of a hypergraph H is the smallest number of colours needed to colour the edges of H so that any two edges that share a vertex have different colours.) Erdos considered this to be one of his three most favorite combinatorial problems and offered $500 for the solution of the problem.

In joint work with Dong-yeap Kang, Tom Kelly, Abhishek Methuku and Deryk Osthus, we prove this conjecture for every large n. We also provide `stability versions' of this result, which confirm a prediction of Kahn.

In my talk, I will discuss some background, some of the ideas behind the proof as well as some related open problems.

Clyde 1-2, Thursday

Xin Li (Glasgow)

Interactions between C*-algebras, topological dynamics and group theory

C*-algebras are algebras of bounded linear operators on Hilbert spaces. Originally introduced as a mathematical foundation for quantum physics, these structures turn out to be interesting on their own right and exhibit a rich interplay with several other mathematical disciplines. Indeed, as I will explain, several key ideas which were initially developed to classify C*-algebras also lead to classification results for topological dynamical systems. At the same time, it was discovered recently that all C*-algebras which have been classified arise from dynamics in a precise sense. Surprisingly, this circle of ideas also leads to new constructions of groups which answered several open questions in group theory.

Clyde 1-2, Friday

Antony Maciocia (Edinburgh)

An Update on Moduli of Sheaves and Bridgeland Stability Conditions

To a variety or space we can associate moduli spaces of sheaves. These moduli spaces have their own intrinsic interest as well as helping us to understand the geometry of the original space. They also arise in a number of other ways and playing constructions off against each other allows us to understand their structure better. One such way is through the very general notion of Bridgeland stability. In dimension two the situation is fairly well understood but very little is known in dimension three. In this talk I will describe some of the background ideas and demonstrate a few of the tools we are using currently to analyse moduli spaces and especially in dimension three.

Clyde 1-2, Thursday

Ciprian Manolescu (Stanford)

Khovanov homology and four-manifolds

Over the last forty years, most progress in four-dimensional topology came from gauge theory and related invariants. Khovanov homology is an invariant of knots in of a different kind: its construction is combinatorial, and connected to ideas from representation theory. There is hope that it can tell us more about smooth 4-manifolds; for example, Freedman, Gompf, Morrison and Walker suggested a strategy to disprove the smooth 4D Poincare conjecture using Rasmussen's invariant from Khovanov homology. It is yet unclear whether their strategy can work, and I will explain some of its challenges. I will also review other topological applications of Khovanov homology, with regard to smoothly embedded surfaces in 4-manifolds.

All Sococo rooms, Thursday

David Rand (Warwick)

Geometry, information and genetics

Can we persuade biologists to take seriously the ideas expressed in Rene Thom's famous book Stabilité structurelle et morphogénèse. This was meant to lay out a new approach to morphogenesis and developmental biology and contained much more than what is now understood as Catastrophe Theory. Although Thom was in discussion with C H Waddington, a revered biologist, its deeply philosophical mathematical approach is very far from how biologists think and to take it seriously they need to see how to profit from such an approach. In my opinion Thom also made one serious philosophical mistake: in his Field’s medal autobiography Thom states “… catastrophe theory is dead … it died of its own success … as soon as it became clear that the theory did not permit quantitative prediction”. One of my aims is to show that, on the contrary, combined with other geometrical and dynamical systems theory and advanced statistics, it does permit quantitative prediction in precisely the area where Thom was most interested and this is what is needed to sell it to biologists. Moreover, today we have a huge advantage in increased understanding of genetics and hugely powerful biological data technologies. I will try and convince the audience that new approaches based on geometry, dynamics and information which combines qualitative and quantitative approaches and are very much in tune with Thom's viewpoint can provide new insights into biology, particularly developmental biology and genetics. This is based on joint work with many people, notably Eric Siggia (Rockefeller), James Briscoe (Crick), Merixell Saez (Warwick & Crick), Elena Camacho Aguilar (Rice), Robert Blassberg (Crick), Michael White (Manchester) and Andrew Millar (Edinburgh).

Hillhead 1-5, Tuesday

Colva Roney-Dougal (St Andrews)

Finite simple groups and computational complexity

This talk will describe connections between structural results about the finite simple groups and the complexity of computational algorithms for permutation groups.

The first part of the talk will survey both old and very new results on the minimal number of generators and the base size of a permutation group, two invariants which influence the complexity of the vast majority of permutation group algorithms.

After this, we shall introduce the complexity class NP, and discuss group-theoretic questions for which there is no known polynomial time solution. In particular, we shall present a new approach to computing the normaliser of a primitive group G in an arbitrary subgroup H of S_n. One key tool is Babai’s recent breakthrough on the graph isomorphism problem, and Helfgott's improvement of Babai's bound.

Clyde 1-5, Tuesday

Bernd Schroers (Heriot-Watt)

The hidden geometry of magnetic skyrmions

This talk is about a model for topological defects in planar ferromagnetic materials, and the geometrical techniques which can be used to obtain infinitely many exact static solutions for certain choices of coupling constants. The defects, called magnetic skyrmions in this context, are widely studied in physics because of their potential role in future magnetic information storage. The applicability of techniques from complex geometry and gauge theory is surprising, and leads to interesting links with the theory of gravitational lensing. Finally, the dynamics of magnetic skyrmions in response to an applied current, which is of practical interest, also contains unexpected geometry and can be understood in terms of quaternionic Moebius transformations.

Clyde 1-2, Friday

Rebecca Shipley (UCL)

Multiscale Models of Tumour Fluid and Drug Distribution: Integrating Imaging and Computation

Understanding how drugs and other therapeutics are delivered to a diseased tissue, and their subsequent spatial and temporal distribution, is a key factor in the development of effective, targeted therapies. Capturing the physiological variation in complete, intact tissue specimens is particularly useful in tumours, which can be highly heterogeneous, both between tumour types and even within individual tumours. However, it is virtually impossible to quantify drug delivery across whole tumour samples through experimental imaging alone.

Here we propose the development of multiscale models of tumour fluid and drug distribution, integrated with high resolution microstructure imaging, to gain insight into this problem. We present a suite of imaging modalities and data sets which provide both high-resolution microstructure data for whole tumours, as well as tissue-scale perfusion information. We present a series of computational models that capture transport phenomena across these length scales, and are calibrated and validated against the modelling data. Finally, we present modelling predictions which provide insight into treatment strategies for a range of cancer therapeutics.

All Sococo rooms, Friday

Michael Singer (UCL)

Brilliant Corners

When we first meet functions of two variables, examples such as xy/(x^2+y^2) are introduced, usually to show that a function can be quite bad even though its partial derivatives exist everywhere. This example is bad because if (x,y) goes to zero, then the limit depends on the direction of approach to the origin. On the other hand the function is completely smooth in polar coordinates around the origin.

While this example may appear to be pathological, constructed specifically to scare students, such direction-dependent limits are very frequently encountered in the study of PDE problems in non-compact settings or when there is small parameter giving a singular limiting problem.

The title of the talk refers to a circle of ideas, pioneered by Richard Melrose, in which manifolds with corners are used systematically to study such problems. I aim to survey these ideas through geometric examples which illustrate how brilliant corners can be.

Bute 1-2, Friday

Catharina Stroppel (Bonn)

Tensor products and branching - changes of perspective

Understanding restrictions of group actions to subgroups or tensor products of representations is a classical problem with a long history. Depending on the viewpoint and context one might call it well-understood or vastly unclear. In this talk I like to illustrate how the perspective of attacking such problems changed over the years and indicate how it lead to the development of important new concepts and surprising connections. A new way of drawing classical partitions and standard tableaux for instance allows already to make direct connections with 2-dimensional TQFTs, certain Fukaya categories, supergroups etc. which might have been impossible without a change of perspective.

All Sococo rooms, Tuesday

Bernd Sturmfels (Berkeley)

Linear PDE with Constant Coefficients

We discuss algebraic methods for solving systems of homogeneous linear partial differential equations with constant coefficients. The setting is the Fundamental Principle established by Ehrenpreis and Palamodov in the 1960’s. Our approach rests on recent advances in commutative algebra, and it offers new vistas on schemes and coherent sheaves in computational algebraic geometry.

All Sococo rooms, Tuesday

Nathalie Wahl (Copenhagen)

Strings in manifolds

Chas and Sullivan defined 20 years ago a “topological string interaction” on manifolds and asked what these interactions could tell us about the manifold they live in. I’ll give an introduction to this rich world, now known as string topology, that has since spread its tentacles to algebra, geometry or even geometric group theory.

Bute 1-2, Wednesday

Neshan Wickramasekera (Cambridge)

Allen–Cahn equation and the existence of prescribed-mean-curvature hypersurfaces in Riemannian manifolds

An n-dimensional hypersurface of a Riemannian manifold has n principal curvatures (relative to a chosen unit normal vector field) at each of its points. The sum of the principal curvatures is the scalar mean curvature, which is a real valued function on the hypersurface. A basic question in Riemannian geometry is whether, given a real valued function g on the ambient manifold, there exists a boundaryless hypersurface, together with a choice of unit normal, such that the scalar mean curvature of the hypersurface (relative to the unit normal) is given by g at every point. The case g = 0 corresponds to extensively-studied minimal hypersurfaces. The lecture will discuss progress on this question for closed ambient manifolds, focusing on a recently developed PDE theoretic approach (joint work with Costante Bellettini at UCL). This method utilises the elliptic and parabolic Allen–Cahn equations on the ambient space. It brings to bear on the question certain elementary, and yet very effective, variational and gradient flow principles in semi-linear elliptic and parabolic PDE theory—principles that serve as a conceptually and technically simpler replacement for the Geometric Measure Theory machinery developed four decades ago in the pioneering work of Almgren and Pitts to prove existence of a minimal hypersurface. A key outcome ofthe PDE method is an affirmative answer to the above question, in all dimensions n ≥ 2, when g is any non-negative (or non-positive) Lipschitz function: for such g, any closed Riemannian manifold contains a C 2 quasi-embedded, boundaryless, mean-curvature-g hypersurface (which, if n ≥ 7, may contain a closed singular set of Hausdorff dimension ≤ n − 7).

Clyde 1-2, Wednesday