School Colloquium
The general purpose of the School Colloquia is to provide accessible and engaging introductions to a broad range of topics of current research interest in any branch of Mathematics or its applications.
The University website provides detailed directions for reaching the Mile End campus. The Mathematics Building is number 4 on the campus map.

DateRoomSpeakerTitle

03/10/2016 5:00 PMMathematics Lecture TheatreFrank den Hollander (Leiden)Breaking of ensemble equivalence for complex networksIn statistical physics the mathematical description of interacting particle systems is based on what are called “Gibbs ensembles”. These represent several a priori choices for the probability distribution on the configuration space of the system capturing physically relevant situations. One of the corner stones of statistical physics is the assumption that, in the thermodynamic limit when the system becomes very large, the microcanonical ensemble as a function of “energy” coincides with the canonical ensemble as a function of “temperature”. However, various examples of systems have been found for which these two ensembles are not equivalent. A complete theory of this intriguing phenomenon is still missing. We show that ensemble nonequivalence can manifest itself also in random graphs with topological constraints. In particular, we show that if we consider a random graph in which the degrees of the nodes are fixed (= hard constraint), respectively, the degrees of the nodes are fixed on average (= soft constraint), then there is no equivalence in the limit as the graph becomes very large. This fact has important consequences for how realworld networks must be modelled. The talk is aimed at a general mathematical audience and requires no prior knowledge of statistical physics or random graphs.

31/10/2016 5:00 PMMathematics Lecture TheatreAndrew Thomason (Cambridge)Containers in combinatorics
A hypergraph with vertex set, say, {1,2,...,n} is a collection of subsets of the vertex set of some fixed size – these subsets are called edges. For example, the subsets might be all triples that form an arithmetic progression. An independent set in the hypergraph is a subset of the vertices that contain no edge – in the example, it would be a set of integers containing no 3term arithmetic progression. It has recently been discovered that the independent sets in any hypergraph must be structured in some way: they are all contained within one of a small collection of "independentlike" subsets. We shall discuss this discovery and its applications.

05/12/2016 5:00 PMMathematics Lecture TheatreJens Marklof (Bristol)Kinetic transport, renormalization and measure rigidity
The theory of homogeneous flows provides a powerful mathematical toolkit that has recently contributed to the solution of a number of longstanding problems. I will survey some of the recent progress, including a study of distances in multiloop networks (circulant graphs), the coin exchange problem and the derivation of a new kinetic equation which captures surprising transport phenomena in crystals and quasicrystals. This lecture is aimed at a broad audience.

16/01/2017 4:30 PMLecture Theatre PP2, People's PalaceMihalis Dafermos (Cambridge)Mathematical problems in general relativity
I'll give an overview of some of the major open problems in general relativity and the progress that mathematicians have made in recent years towards their solution.

20/03/2017 4:30 PMLecture Theatre PP2, People's PalaceVaughan Jones (UC Berkeley)Unitary representations of R. Thompson’s groups inspired by quantum spin chains.
Critical phenomena in physics are supposed to involve some kind of scale invariance. The Thompson groups are composed of local scaling transformations so one might expect them to be relevant to critical phenomena on lattices. Motivated by this thought we will construct a family of unitary representations of the Thompson groups whose coefficients are of some interest.

08/05/2017 4:30 PMLecture Theatre PP2, People's PalaceAntoine Joux (Paris 6)Technical history of discrete logarithms in small characteristic finite fields
Due to its use in cryptographic protocols such as the DiffieHellman key exchange, the discrete logarithm problem attracted a considerable amount of attention in the past 40 years. In this talk, we summarise the key technical ideas and their evolution for the case of discrete logarithms in small characteristic finite fields. This road leads from the original belief that this problem was hard enough for cryptographic purpose to the current state of the art where the algorithms are so efficient and practical that the problem can no longer be considered for cryptographic use.

02/10/2017 4:30 PMLecture Theatre PP2, People's PalaceStefaan Vaes (KU Leuven)Classification of von Neumann algebras
The theme of this talk is the dichotomy between amenability and nonamenability. Because the group of motions of the threedimensional Euclidean space is nonamenable (as a group with the discrete topology), we have the BanachTarski paradox. In dimension two, the group of motions is amenable and there is therefore no paradoxical decomposition of the disk. This dichotomy is most apparent in the theory of von Neumann algebras: the amenable ones are completely classified by the work of Connes and Haagerup, while the nonamenable ones give rise to amazing rigidity theorems, especially within Sorin Popa's deformation/rigidity theory. I will illustrate the gap between amenability and nonamenability for von Neumann algebras associated with countable groups, with locally compact groups, and with group actions on probability spaces.

06/11/2017 4:30 PMLecture Theatre PP2, People's PalaceRobert MacKay (Warwick)Three topics in electricity system management
With the move to more renewable sources of electricity, three things are becoming necessary:
 storage systems
 demand response
 monitoring and control of oscillations in power flow.
I will give an overview of some mathematical work on these. Firstly, with Lisa Flatley and Mike Waterson, we have designed an algorithm for optimal use of an energy store; this can be used to maximise profit from variations in the price or to minimise variations in the mismatch between supply and demand. Secondly, with Ellen Webborn, we have analysed the effect of making thermostatically controlled loads frequencysensitive; this can reduce variations in the mismatch between supply and demand but there are dangers that large uptake of frequencysensitive technology could lead to instabilities; in our model the uniform distribution of phases appears to be stable and the fully synchronised one is unstable. Thirdly, I am developing a Gaussian process method to detect oscillations in power flow from phasor measurement unit data, and estimate their frequency, damping rate and mode shape; it is essential to detect these modes early enough to design control for them.

20/11/2017 4:30 PMLecture Theatre PP2, People's PalaceDan Romik (UC Davis)The moving sofa problem
The moving sofa problem is a wellknown open problem in geometry. It asks for the planar shape of largest area that can be moved around a rightangled corner in a twodimensional hallway of width 1. Although deceptively easy to state, it turns out to be highly nontrivial to analyze, and has a rich structure that is intriguing to amateurs and experts alike. In this talk I will survey the known results about the problem, including a new moving sofa shape with an interesting algebraic structure, and new bounds on the area of a moving sofa I derived recently in collaboration with Yoav Kallus.

12/12/2017 5:00 PMLecture Theatre PP2, People's Palace (TBC) Seminar series: School ColloquiumLeonid Pastur/Леонід Пастур (Verkin Institute, Kharkov)Disordered systems and related probabilistic structures
We discuss several problems of modern physics requiring probabilistic ideas and techniques, mostly in the frameworks of spectral analysis of differential and finitedifference selfadjoint operators with random coefficients and hermitian random matrices of large size. We give an outline of certain basic results (selfaveraging, ergodic opertors, dense point spectrum, spectral rigidity and universality, etc.) and discuss their spectral, probabilistic and physical content, including results that appeared recently.

15/01/2018 4:30 PMLecture Theatre 2.10, LawsImre Leader (Cambridge)Tiling with arbitrary tiles
It is easy to tile the plane with dominoes (2x1 rectangles). It is also easy to tile the plane with trominoes (three 1x1 squares forming an Lshape). Of course, not every shape tiles the plane. What happens when we increase the number of dimensions from two to three, or beyond?

12/02/2018 4:30 PMLecture Theatre PP2, People's PalaceShiri ArtsteinAvidan (Tel Aviv)Around Godbersen's conjecture for mixed volumes
We shall discuss some recent progress regarding a conjecture of Godbersen on mixed volumes, and other measures of symmetry, for which the simplex is conjectured, or proved, to be extremal.

12/03/2018 4:30 PMLecture Theatre PP2, People's PalaceTim Roughgarden (Stanford)Distributionfree models of social and information networksThe mathematical study of social and information networks has historically centered around generative models for such networks (preferential attachment, the ChungLu random graph model, Kronecker graphs, etc.). This talk proposes distributionfree models of social and information networks – novel classes of graphs that capture all plausible such networks. Our models are motivated by triadic closure, the property that vertices with one or more mutual neighbors tend to also be neighbors; this is one of the most universal signatures of social networks. We prove structural results on the clustering properties of such graphs, and give algorithmic applications to clustering and cliquefinding problems.Includes joint work with Jacob Fox, Rishi Gupta, C. Seshadhri, Fan Wei, and Nicole Wein.

23/04/2018 4:30 PMLecture Theatre PP2, People's PalaceAndrew Granville (UCL)The latest on pretentiousnessAbout 8 years ago, Soundararajan and I proposed an alternative approach to the whole subject of analytic number theory. This "pretentious" approach has now permeated the subject and has been the basis of several spectacular recent advances. In this talk we will give some idea of what is different about this approach and explain the interest in some of these advances.

10/12/2018 4:00 PMBernd Sturmfels (UC Berkeley and Max Plank Institute, Leipzig)This Colloquium is cancelled

25/01/2016 4:30 PMProf. Alexander Gamburd (CUNY, New York)Markoff Triples and Strong Approximation
Markoff triples are integer solutions of the equation $x^2+y^2+z^2=3xyz$ which arose in Markoff's spectacular and fundamental work (1879) on diophantine approximation and has been henceforth ubiquitous in a tremendous variety of different fields in mathematics and beyond. After reviewing some of these (Product Replacement Algorithm; algebraic solutions of Painleve VI equations) we will present recent joint work with Bourgain and Sarnak on the connectedness of the set of solutions of the Markoff equation modulo a prime $p$ under the action of the group generated by Vieta involutions. Similar results for composite moduli enable us to establish certain new arithmetic properties of Markoff numbers, for instance, the fact that almost all of them are composite.

15/02/2016 4:30 PMProf. Martin Hairer FRS (Warwick)Taming infinities
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 now.

14/03/2016 4:30 PMProf. Keith Ball FRS (Warwick)Where is a convex set?
Convex domains play a central role in many areas of analysis and mathematical computer science. During the last 15 years considerable progress has been made in understanding the distribution of mass within such domains in high dimensions. The most telling point is that such domains exhibit characteristics that we normally associate with product probability measures: the joint laws of independent random variables. The talk will survey some of the progress, explain the main open problems and show how our intuition can be badly misleading when we try to think about highdimensional objects.

05/10/2015 4:30 PMProf. Misha Sodin (Tel Aviv)Topology of zero sets of smooth random functions

19/10/2015 2:57 PMProf. Ivan Smith (Cambridge)Flat surfaces and symplectic topology
Surfaces with a (perhaps singular) flat Euclidean metric play a prominent role in Teichmueller theory and dynamics, and are connected to classical questions in billiards. More recently, this field has been connected to sixdimensional symplectic topology and to the homological algebra study of algebras defined by quivers with potential. We will review this circle of ideas, largely reporting on joint work with Tom Bridgeland.

07/12/2015 2:59 PMProf. Günter M. ZieglerGeometry vs. Topology: On 4polytopes and 3spheres and their parameters
There has been massive efforts to understand the parameter spaces of convex polytopes  and great results such as the “gTheorem” on the way. On the other hand, key questions are still open, such as the “fatness problem” for 4dimensional polytopes, and the related question whether we expect the same answers for convex polytopes (discretegeometric objects) and for cellular spheres (a topological model). I will argue that we should not, and give some evidence for this.

09/02/2015 4:30 PMProf. Nick Trefethen FRS (Oxford)Mathematics of the Faraday Cage
Everybody has heard of the Faraday cage effect, in which a wire mesh does a good job of blocking electric fields. Surely the mathematics of such a famous and useful phenomenon has been long ago worked out and written up in the textbooks? It seems to be not so. One reason may be that that the effect is not as simple as one might expect: it depends on the wires having finite radius. Nor is it as strong as one might imagine: the shielding improves only linearly as the wire spacing decreases. This talk will present results by Jon Chapman, Dave Hewett and myself including (a) numerical simulations, (b) a theorem proved by conformal mapping, (c) a continuous model derived by multiple scales analysis, (d) a discrete model derived by energy minimization, (e) a connection with the periodic trapezoidal rule for analytic integrands, and (f) a physical explanation.

02/03/2015 4:30 PMProf. Mark Pollicott (Warwick)Counting circles in the Apollonian Circle Packing
The classical Apollonian circle packing is constructed by taking three mutually tangent circles in the unit plane and repeatedly inscribing circles in the (curved) triangles between them. This results in infinitely many circles, whose radii tend to zero. Furthermore, there is a remarkably simple asymptotic formula due to Kontorovich and Oh for the number of these circles with radii > r, as r tends to infinity. This talk will describe this and related results.

30/03/2015 4:30 PMProf. Christian Krattenthaler (Vienna)FussCatalan Combinatorics
The past few years have seen the emergence of a new research branch at the interface of combinatorics, algebra and geometry: "FussCatalan combinatorics," as some people call it. It has been observed that there are several sets of objects associated to reflection groups, such as the (generalised) noncrossing partitions of Armstrong, the generalised cluster complex of Fomin and Reading, certain regions in the extended Shi arrangement, the geometric chains of Athanasiadis, etc., which are all enumerated by the (Fu\ss)Catalan numbers associated with the underlying reflection group. Why this is so is only partially understood. I shall give an introduction into this fascinating subject of algebraic combinatorics, not assuming any prerequisites.

03/11/2014 4:30 PMProf. Jon Keating FRS (Bristol)Primes and Polynomials in Short Intervals
Understanding the statistical distribution of the primes is one of the outstanding problems in mathematics. It is, for example, the origin of the Riemann Hypothesis. The question of how many primes typically lie in an interval of a given size goes back to Gauss. It is the subject of an important conjecture due to Goldston and Montgomery. I will review the history of this problem and then explain a remarkable analogy between the primes and certain polynomials. In the case of polynomials, the analogue of GoldstonMontgomery conjecture can be proved using ideas from mathematical physics relating to random matrices.

17/11/2014 4:30 PMProf. Toby Gee (Imperial)The modularity of elliptic curves 20 years on
Twenty years ago, Wiles and TaylorWiles proved the modularity of semistable elliptic curves over the rational numbers. Subsequently this was extended by BreuilConradDiamondTaylor to prove the modularity of all elliptic curves over the rational numbers. I will review what this means, and explain some of the progress since then on extending these results to more general number fields.

01/12/2014 4:30 PMProf. Helen Byrne (Oxford)Seeing the wood for the trees with mathematical modelling
The eye is a complex organ and, as such, represents a rich source of fascinating problems for applied mathematicians, interested in understanding its anatomy and physiology and how these change during ageing and in disease. In this talk attention will focus on photoreceptors, lightsensing retinal cells whose length fluctuates on a daily basis. I will start by presenting a simple mathematical model, formulated as a free boundary problem, which can be used to determine whether the observed fluctuations in healthy photoreceptors may be attributed to changes in oxygen demand during periods of light and dark. I will then focus on retinitis pigmentosa, a degenerative disease that targets the photoreceptors and causes progressive loss of visual function. I will present a second mathematical model developed in order to determine whether hyperoxia, exposure to elevated oxygen levels, may be responsible for the patterns of photoreceptor degeneration associated with retinitis pigmentosa. If time permits, I will also explain how simple continuum models may be combined with experimental data on corneal angiogenesis in order to compare the mechanisms by which different angiogenic factors act.

14/01/2019 4:00 PMGE Fogg Lecture TheatreSergei Kuksin (Universite ParisDiderot)Ergodicity, mixing and KAMTo prove the ergodicity for Hamiltonian systems of big or infinite dimension is a highly important problem which is notoriously complicated. But what we often have in physics are not Hamiltonian systems, but damped/driven Hamiltonian systems, i.e., systems of the form<Hamiltonian system> + <small dissipation> + <small random forcing>,where the random forcing may be very degenerate (i.e., affect only a few modes). For such systems an analogy of the ergodicity is called mixing.In my talk I will recall the definition of mixing and explain how the KAMtheory provides a powerful tool to prove the mixing for the systems above.The talk is based on a joint work with Armen Shirikyan and Vahagn Nersesyan, arXiv:1802.03250v2

12/11/2018 4:00 PMLecture Theatre PP1, People's PalaceSean Meyn (University of Florida)Fastest Convergence for Reinforcement Learning
Stochastic approximation algorithms are used to approximate solutions to fixed point equations that involve expectations of functions with respect to possibly unknown distributions. Among many algorithms in machine learning, reinforcement learning algorithms such as TD and Qlearning are two of its most famous applications. This talk will provide an overview of stochastic approximation, with focus on optimizing the rate of convergence. Based on this general theory, the well known slow convergence of Qlearning is explained: the CLT variance of the algorithm is typically infinite. Three new Qlearning algorithms are introduced to dramatically improve performance:
(i) The Zap Qlearning algorithm that has provably optimal asymptotic variance, and resembles the NewtonRaphson method
(ii) The PolSA algorithm that is based on Polyak's momentum technique, but with a specialized matrix momentum, and
(iii) The NeSA algorithm based on Nesterov's acceleration technique
Analysis of (ii) and (iii) require entirely new analytic techniques. One approach is via coupling: conditions are established under which the parameter estimates obtained using the PolSA algorithm couple with those obtained using the NewtonRaphson based algorithm. Numerical examples confirm this behavior, and the remarkable performance of these algorithms. 
01/10/2018 4:00 PMLecture Theatre PP1, People's PalacePeter Keevash (Oxford)Designs and decompositions
In this talk I will give an introduction to Design Theory from the combinatorial perspective of (hyper)graph decompositions. I will survey some recent progress on the existence of decompositions, with particular attention to triangle decompositions of graphs, which provide a simple (yet still interesting) illustration of more general results.

13/02/2019 3:00 PMScape, Room: 0.14Gwyneth Stallard (Open University)Complex dynamics: the intriguing case of wandering domains
Complex dynamics concerns the iteration of analytic functions of the complex plane. For each function, the plane is split into two sets: the Fatou set (where the behaviour of the iterates is stable under local variation) and the Julia set (where the behaviour is chaotic). The dynamical behaviour of the iterates inside the periodic components of the Fatou set was classified into four different types by the founders of the subject and this classification has played a key role in the development of the subject. One of the most dramatic breakthroughs was given by Sullivan in the 1980s when he proved that, for rational functions, all components of the Fatou set are eventually periodic and there are no socalled wandering domains. For transcendental functions, however, wandering domains can exist and the rich variety of possible behaviours that can occur is only just becoming apparent.

18/09/2019 3:00 PMMathematical Sciences Building, Room MB503Mihai Putinar (UC, Santa Barbara)ChristoffelDarboux AnalysisAn inner product structure on the polynomial algebra provides vectors representing point evaluation functionals. Putting together these vectors one forms a reproducing kernel. This basic object of modern mathematics, known today as the ChristoffelDarboux kernel, appeared as early as mid XIXth Century in various impersonations. We will review some history and basic observations, with the aim at showing in the second part of the lecture how the ChristoffelDarboux kernel continues to surprise scientists and impact some contemporary explorations, mostly revolving around inverse problems.

30/10/2019 3:00 PMMathematical Sciences Building, Room: MB 503Johannes Carmesin (Birmingham)Embedding simply connected 2complexes in 3space
I will start my talk with an introduction about embedding graphs in the plane. A classical theorem of Kuratowski characterises graphs embeddable in the plane by two obstructions. More precisely, a graph is planar if and only if it does not contain the complete graph \(K_5\) or the complete bipartite graph \(K_{3,3}\) as a minor.
Can you characterise embeddability of 2dimensional simplicial complexes in 3space in a way analogous to Kuratowski’s characterisation of graph planarity? 
12/02/2020 2:30 PMMB 503Clément Mouhot (University of Cambridge)From Hörmander to De Giorgi through Boltzmann
The theory of hypoellipticity of Hörmander is concerned with recovering the regularity of solutions to partial differential equations when a diffusion is present but degenerate in some directions while other transport terms provide mixing. The theory of De GiorgiNashMoser is concerned with the Hölder regularity of solutions to parabolic or elliptic equations with rough coefficients (merely measurable). This talk will present some general introduction to these theories for nonspecialists, and sketch recent developments about their interplay. These recent developments are motivated by some conjectures about the behaviour of solutions to the Boltzmann equation in kinetic theory.

19/02/2020 2:30 PMMB503. (We will transfer to MLT if the capacity of MB503 is exceeded.)Kevin Buzzard (ICL)Computers and mathematics
Abstract: Mathematics is done by humans, and this causes problems. Our system is imperfect, but also vastly successful and highly (but not 100%) accurate. Can we use computer proof checkers to make the system 100% accurate? This seems extremely hard to do, both in theory and in practice. If more young people start using these systems however, the difficulties will start disappearing. I will give an overview of the area. No background in computer proof checkers will be assumed and the talk will be accessible to undergraduates.