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Mathematics Colloquium 2016-17 Abstracts

14 October 2016 Tom Coates (Imperial) Mirror Symmetry and Classification
I will describe a program to understand and classify algebraic varieties -- the shapes defined by polynomial equations -- using a circle of ideas from theoretical physics known as Mirror Symmetry. This is work by and joint work with Alessio Corti, Mark Gross, Alexander Kasprzyk, Miles Reid, and others. The talks is aimed squarely at nonspecialists.

21 October 2016 Daniel Ueltschi (Warwick) From condensed matter physics to probability theory
The basic laws governing atoms and electrons are well understood, but it is impossible to make predictions about the behaviour of large systems of condensed matter physics. A popular approach is to introduce simple models and to use notions of statistical mechanics. I will review quantum spin systems and their stochastic representations in terms of random permutations and random loops. I will also describe the *universal* behaviour that is common to loop models in dimensions 3 and more.

28 October 2016 Raul Tempone (KAUST) On Monte Carlo, Multilevel Monte Carlo and Multi Index Monte Carlo
We will first recall, for a general audience, the use of Monte Carlo and Multi-level Monte Carlo (MLMC) methods. Then, we will discuss recent developments on MLMC. Later, we will describe and analyze the Multi-Index Monte Carlo (MIMC) and the Multi-Index Stochastic Collocation (MISC) methods for computing statistics of the solution of a PDE with random data. MIMC is both a stochastic version of the combination technique introduced by Zenger, Griebel and collaborators and an extension of the Multilevel Monte Carlo (MLMC) method first described by Heinrich and Giles. Instead of using first-order differences as in MLMC, MIMC uses mixed differences to reduce the variance of the hierarchical differences dramatically. These mixed differences yield new and improved complexity results, which are natural generalizations of Giles's MLMC analysis, and which increase the domain of problem parameters for which we achieve the optimal convergence. On the same vein, MISC is a deterministic combination technique based on mixed differences of spatial approximations and quadratures over the space of random data. Provided enough mixed regularity, MISC can achieve better complexity than MIMC. Moreover, we show that, in the optimal case, the convergence rate of MISC is only dictated by the convergence of the deterministic solver applied to a one-dimensional spatial problem.

04 November 2016 Vassili Gelfreich (Warwick) Arnold Diffusion and Multiple Time Scales in Hamiltonian Dynamics
This talk is an introduction to the modern Hamiltonian Dynamics with a special accent on the origin of multiple time scales and the problem of Arnold Diffusion. Examples of celestial mechanics will be used to illustrate theoretical concepts and to explain motivation behind recent developments of the theory. After quickly reviewing classical results, we will discuss models for Arnold Diffusion and try to visualise dynamics of a 4d symplectic map.

11 November 2016 Beatrice Pozzetti (Warwick) Higher Teichmüller Theory
The Teichmüller space of a surface S parametrizes marked hyperbolic structures on it. It can be realized as a connected component of the representation variety of the fundamental group of S in PSL(2,R). Higher Teichmüller spaces arises as connected components of representation varieties in more general Lie groups G, and share serveral algebraic and geometric properties with classical Teichmüller theory. I will give an introduction to this active field of research on the crossroad of low dimensional topology, algebraic geometry, ergodic theory and number theory.

18 November 2016 Alain Goriely (Oxford) The Mathematics and Mechanics of the Brain: from Axon to Organ, from Morphogenesis to Trauma.
The human brain is an organ of extreme complexity, the object of ultimate intellectual egocentrism, and a source of endless scientific challenges. Despite a clear evidence that mechanical factors play an important role in regulating brain activity, current research efforts focus mainly on the biochemical or electrophysiological activity of the brain. However, classical concepts from mechanics including deformations, stretch, strain, strain rate, pressure, and stress also play a crucial role in both shaping the brain and modulating its functions. In this talk, I will review our current understanding of the brain and present several important mechanical problems and mathematical models related to brain geometry, proper brain function, and brain pathology and trauma. In particular, I will present simple models for brain oedema formation and propagation, a dangerous consequence of traumatic brain injury.

25 November 2016 Ed Corrigan (York) Defects and Integrability
Defects, or discontinuities, of various kinds are ubiquitous in nature - a shock within a fluid flow being a common example where a velocity field changes abruptly from supersonic to subsonic. It has been noticed that many integrable systems (for example, sine-Gordon or KdV) can also support defects yet maintain their integrable nature. The purpose of this talk is to introduce the basic ideas, using straightforward examples, and then describe some of the curious properties of defects in this context, from both a classical and quantum field theory perspective, and one or two outstanding problems.

It should be suitable for undergraduates as well as more experienced colleagues.

02 December 2016 Sarah Zerbes (UCL) Elliptic curves and the conjecture of Birch and Swinnerton-Dyer
An important problem in number theory is to understand the rational solutions to algebraic equations. One of the first non-trivial examples, cubics in two variables, leads to the theory of so-called elliptic curves. The famous Birch—Swinnerton-Dyer conjecture, one of the Clay Millenium Problems, predicts a relation between the rational points on an elliptic curve and a certain complex-analytic function, the L-function on an elliptic curve. In my talk, I will give an overview of the conjecture and of some new results establishing the conjecture in special cases.

09 December 2016 Stephen Donkin (York) Representations of symmetric groups and general linear groups
The (discrete) representations of the symmetric groups and the (continuous) representations of the general linear groups are intimately connected in a way realised by Schur in his thesis in 1901, working over the complex numbers and building on work of Frobenius from the 1890s. This connection has been a rich source of inspiration and generalisation, particularly in the work of Brauer on the classical groups.

The problem solved by Schur, giving the irreducible characters in terms of Schur symmetric functions, is still wide open if one replaces the field of complex numbers by a field of characteristic p>0. But much of Schur's method is still applicable in this context, as explained in the enormously influential monograph by Sandy Green, published in 1980. This book reinterpreted fundamental work by Gordon James on the symmetric group by developing Schur's method in the modular context. It was an "instant hit", and has by now opened up the subject to several new generations. One of the strong points of the connection is that makes possible also the introduction of techniques from algebraic geometry and Lie theory.

This is still an area of intense investigation. I will survey some of this story and and bring some strands of it up to the present day.

13 January 2017 Julia Gog (Cambridge) Hunting for viral packaging signals
Influenza has a genome split into several segments, and this complicates virus particle assembly as each particle must have one of each of the segments. This means that each of the RNA segments must contain some signal, and that this signal ought to be fairly conserved. Is this enough to go and hunt them down using mathematics? The answer turns out to be yes. However, this required some creativity in algorithm design, drawing inspiration from a number of apparently unrelated problems. This hack seems to work, but leaves some interesting mathematical problems.

I’ll also briefly talk about some of the other problems in influenza and infectious disease that interest me, and general joys and challenges of being a mathematician trying to research biology.

19 January 2017 **Thursday!** Dusa McDuff (Columbia) Symplectic Topology Today
This talk will explain the basics of symplectic topology for a nonspecialist audience, outlining some of the classical results as well as some problems that are currently open.

27 January 2017 Alain Valette (Neuchatel) The Kadison-Singer problem
In 1959, R.V. Kadison and I.M. Singer asked whether each pure state of the algebra of bounded diagonal operators on ℓ2, admits a unique state extension to B(ℓ2). The positive answer was given in June 2013 by A. Marcus, D. Spielman and N. Srivastava, who took advantage of a series of translations of the original question, due to C. Akemann, J. Anderson, N. Weaver,... Ultimately, the problem boils down to an estimate of the largest zero of the expected characteristic polynomial of the sum of independent random variables taking values in rank 1 positive matrices in the algebra of n-by-n matrices.

03 February 2017 Gui-Qiang Chen (Oxford) Partial Differential Equations of Mixed Elliptic-Hyperbolic Type in Mechanics and Geometry
As is well-known, two of the basic types of linear partial differential equations (PDEs) are elliptic type and hyperbolic type, following the classification for linear PDEs first proposed by Jacques Hadamard in the 1920s; and linear theories of PDEs of these two types have considerably been established, respectively. On the other hand, many nonlinear PDEs arising in mechanics, geometry, and other areas naturally are of mixed elliptic-hyperbolic type. The solution of some longstanding fundamental problems in these areas greatly requires a deep understanding of such nonlinear PDEs of mixed type. Important examples include shock reflection-diffraction problems in fluid mechanics (the Euler equations) and isometric embedding problems in differential geometry (the Gauss-Codazzi-Ricci equations), among many others. In this talk we will present natural connections of nonlinear PDEs of mixed elliptic-hyperbolic type with these longstanding problems and will then discuss some of the most recent developments in the analysis of these nonlinear PDEs through the examples with emphasis on developing and identifying mathematical approaches, ideas, and techniques for dealing with the mixed-type problems. Further trends, perspectives, and open problems in this direction will also be addressed.

10 February 2017 Anne Taormina (Durham) The riches of Mathieu Moonshine
In 2009, three Japanese theoretical particle physicists observed that the elliptic genus of a K3 surface, when expressed in terms of mock modular forms, exposes numbers that can be linked to the dimensions of finite dimensional representations of the sporadic group Mathieu 24.

Since then, this intriguing connection has been studied from several points of view, other examples of the same type of phenomenon for other finite groups and mock modular forms have been discovered, and the research topic of 'New Moonshines’ has slowly caught the attention of researchers across fields.

In this talk, I will describe the 2009 observation, now referred to as 'Mathieu Moonshine’, and explain the challenges faced by the theoretical physics community in understanding the origin and role of the huge Mathieu 24 finite symmetry in the context of strings compactified on K3 surfaces. In particular, I will discuss how this phenomenon is related to the geometry of K3 surfaces and introduce the concept of symmetry surfing.

17 February 2017 Jean Bricmont (UCL, Belgium) What is the meaning of the wave function?
In quantum mechanics, the wave function or the quantum state has a perfectly well defined meaning as an instrument to predict results of measurements.
But what does it mean outside of measurements? To this question, no clear answer is given in quantum mechanics textbooks. We will first give a naive interpretation of what the wave function could mean outside of measurements and show that it is mathematically inconsistent. Then, we will briefly explain how the de Broglie-Bohm theory solves that problem.

24 February 2017 Emmanuel Breuillard (Münster) Hausdorff Banach Tarski paradox and dense free subgroups

I will start by recalling this celebrated paradox, which has for years served as a pretext to study and classify infinite groups and their geometric actions. In the 1930's it led von Neumann to define the notion of a (non) amenable group, which is intimately related to isoperimetry, to random walks on Cayley graphs, to Kazhdan's property T, and more generally the study of spectral gaps for group actions. The lecture will then focus on classical examples of discrete or dense subgroups of linear substitutions for which many of these questions can now be answered, using recently developed tools going from geometric group theory and random walks to number theory and combinatorics, and where many other problems remain open.

03 March 2017 Mihalis Dafermos (Cambridge) On falling into black holes
The celebrated “black hole” spacetimes of Schwarzschild and Kerr play a central role in our current understanding of Einstein’s general theory of relativity. Are these spacetimes stable, however, as solutions to the Einstein vacuum equations, in their exterior region? And what fate awaits physical observers who enter inside a “generic” black hole?
It turns out that these two questions are intimately related and the answer to the second may be more disturbing than previously thought. This talk will try to explain how so.

10 March 2017 Koji Fujiwara (University of Kyoto) Mapping class groups in Geometric group theory
Mapping class groups of closed surfaces play an important role in many areas of mathematics, including Geometric group theory. Our main tool to study
mapping class groups is "the curve complex" introduced by Masur-Minsky. One goal of the talk is to explain that the asymptotic dimension of a mapping class groups is finite. It is also meant to be an introduction to some main ideas in Geometric group theory.

17 March 2017 Mike Stillman (Cornell) Applications of computational algebraic geometry in physics


A number of problems in theoretical physics can be phrased in terms of computational algebra or computational algebraic geometry. In this talk, we focus on one such problem (describing the possible vacua of supersymmetric field theories. But we don't assume you know any of these words or notions!)

In one important case related to the standard model of particle physics, this problem boils down to a daunting computational algebra problem: computing the polynomial relations on 973 fairly complicated polynomials in 49 variables, seemingly a hopeless task!

It turns out that we can solve a large part of this problem. In this talk, we will assume no knowledge of physics, and we will keep the algebraic geometry to a minimum as well. We will very briefly describe the physics, and how this problem is translated into a computational one, and then we will describe the algebraic geometric and computational methods and results which allow the structure to become apparent.

This talk is based on joint work with the following authors, some of which appears in the paper, and some which has recently been obtained.

Joint work with:

Yang-Hui He (Oxford and City University London, UK) Vishnu Jejjala (University of the Witwatersrand, Johannesburg, South Africa) Cyril Matti (City University, London, UK) Brent Nelson (Northeastern University, USA)

28 April 2017 Rich Schwartz (Brown) 5 points on a sphere
This is an update of the colloquium talk I gave at Warwick 4 years ago about Thomson's 5 point problem. In the new talk, I will sketch my recent proof that there is some computable number S = 15.0477... such that the triangular bi-pyramid is the minimizer, amongst all 5-point configurations on the sphere, with respect to a power law potential with exponent s if and only if 0<s<=S. (The case s=1 is Thomson's problem, and the general result solves a conjecture of Melnyk-Knop-Smith from 1977.) I will illustrate the ideas with colorful computer demos.

12 May 2017 Tristan Riviere (ETH) How much does it turn the sphere inside out?
How much does it knot a closed simple curve ? To cover the sphere twice ? to realize such or such homotopy class ? ...etc. All these questions consisting of assigning a "canonical" number and possibly an optimal "shape" to a given topological operation are known to be mathematically very rich and to bring together notions and techniques from topology, geometry and analysis. In this talk we will concentrate on the operation consisting of everting the 2 sphere in the 3 dimensional space. Since Smale's proof in 1959 of the existence of such an operation the search for effective realizations of such eversions has triggered a lot of fascination and works in the math community. The absence in nature of matter that can interpenetrate and the quasi impossibility, up to the advent of virtual imaging, to experience this deformation is maybe the reason for the difficulty to develop an intuitive approach on the problem. We will present the optimization of Sophie Germain conformally invariant elastic energy for the eversion. Our efforts will finally bring us to consider more closely an integer number together with a mysterious minimal surface.

19 May 2017 Richard Thomas (Imperial) Counting Curves
For centuries mathematicians have generalised statements like “there is a unique line through any 2 points”, but with increasing technical difficulties. It was not until the late 1990s that new ideas from mathematics and string theory allowed rigorous definitions to be made of these “curve counting problems”.

I will outline two different definitions, assuming only a bit of undergraduate complex analysis.

The famous “MNOP conjecture” is that the two definitions give equivalent information. Its recent proof by Pandharipande and Pixton has enabled the solution of various counting problems, such as the “KKV conjecture” from string theory, expressing all curve counting problems on “K3 surfaces” in terms of modular forms.

26 May 2017 Bojan Mohar (Simon Fraser University & IMFM) Paths and trails of odd length and totally odd immersions of graphs
One of the milestones in graph theory is Menger's Theorem which states that the maximum number of edge-disjoint paths between two vertices u and v is equal to the minimum number of edges whose removal disconnects u from v. If we want paths from u to v having additional properties, for example being of odd length, this exact duality no longer holds. However, a kind of weak duality can be achieved. This problem and its resolution will be discussed and some of its applications will be presented. In particular, totally odd immersions of graphs are tightly related to this topic.

02 June 2017 Cameron Gordon (Austin, Texas) Left-orderability of 3-manifold groups
The fundamental group is a more or less complete invariant of a 3-dimensional manifold. We will discuss how the purely algebraic property of this group being left-orderable is related to two other aspects of 3-dimensional topology, one geometric and the other essentially analytic.

09 June 2017 Robert Kerr (Warwick) Scaling of Navier-Stokes trefoil reconnection
What is the'turbulence problem’? It depends upon your perspective. If you are an engineer or predict the weather, the question is 'Why do turbulent flows always dissipate finite energy in a finite time?’ Statistical physicists call this the 'dissipation anomaly’, for whom the real problem is explaining the underlying 'fluctuation-dissipation’ theorem. And for mathematicians, the real problem is the Clay Prize Navier-Stokes ‘regularity’ question. These communities often do not listen to one another. The preferred tool is usually analysis or simulations in unphysical periodic domains. That is Sobolev spaces and pseudo-spectral calculations. Worse, many of the simulations use highly symmetric initial conditions. But what if a real experiment tells us something that those methods suggest is impossible?

This possibility has been raised by the provocative results from experimental vortex knots shed by 3D-printed models, including trefoils and linked rings. These find that despite a change in their topology due to reconnection, a continuum measure of the linking number of those knots, the helicity, can be preserved. Based upon a series of high-resolution, huge domain three-dimensional simulations, not only is that experimental observation confirmed, but the scaling that underlies this regime could be showing us a path to answering each of the 'turbulence problems’ listed above.

23 June 2017 Carola-Bibiane Schönlieb (Cambridge) PDEs and variational approaches in inverse imaging and their adaption via learning
One of the most successful approaches to solve inverse problems in imaging is to cast the problem as a variational model. The key to the success of the variational approach is to define the variational energy such that its minimiser reflects the structural properties of the imaging problem in terms of regularisation and data consistency.

Variational models constitute mathematically rigorous inversion models with stability and approximation guarantees as well as a control on qualitative and physical properties of the solution. On the negative side, these methods are rigid in a sense that they can be adapted to data only to a certain extent.
Hence researchers started to apply machine learning techniques to “learn” more expressible variational models. The basic principle is to consider a bilevel optimization problem, where the variational model appears as the lower-level problem and the higher-level problem is the minimization over a loss function that measures the reconstruction error for the solution of the variational model. In this talk we discuss bilevel optimisation, its analysis and numerical treatment, and show applications to regularisation learning, learning of noise models and of sampling patterns in MRI.

This talk includes joint work with M. Benning, L. Calatroni, C. Chung, J. C. De Los Reyes, M. Ehrhardt, G. Maierhofer, F. Sherry, T. Valkonen, and V. Vladic.

27 June 2017 Christl Donnelly (Imperial College) What have numbers ever done for you?
There have been a lot of bad news stories about infectious disease lately. The largest Ebola epidemic the world had ever seen unfolded in West Africa. Zika virus, previously thought to be little - if any -threat, gave rise to a Public Health Emergency of International Concern. However, there are good news stories. In total, there have been six, yes 6, cases of poliovirus cases caused by wild poliovirus globally so far in 2017. That is the smallest number in a 6-month period that the world has ever seen.

Statistical and mathematical analyses are front and centre in the global flights against infectious diseases. I will highlight examples – explaining the challenges (computational, methodological, and data-related) faced and the most important achievements. I will look back as well as forward and speculate on what the future holds.