# Mathematics Colloquium 2014-15 Abstracts

**03 October 2014 Miles Reid (Warwick) The structure of higher dimensional algebraic varieties**

Abstract:

I give a brief overview of the development of the classification of n-dimensional algebraic varieties, that start with Mori's first papers and my work on 3-folds from my first years at Warwick in the late 1970s, and that has reached a certain maturity over the intervening 40 years. For a related view on the same topic, see János Kollár's talk at the recent ICM on YouTube

http://www.youtube.com/watch?v=7-qPC_6PAp0

and for more details of my involvement, my biographical notes

http://www.maths.warwick.ac.uk/~miles/Unpub/CPR_book/OPV.ps

http://www.warwick.ac.uk/staff/Miles.Reid/Personal/obituary

Longer abstract:

The LMS Pólya Prize is thought of as a "life-time achievement" award. Stripped of the euphemism, it means I get it for being a worthy old codger. My main claim to fame is the development of the 3-fold minimal model program, which dates back to my first years at Warwick in the late 1970s.

The theory of algebraic surfaces was first developed around 1900 by the Italian school, and updated several times during the 20th century. Setting up algebraic surfaces involves some tricky foundational points, but once past these, the subject is harmonious and beautiful, and it acquired widespread popularity in different areas of math a long time before I came onto the scene.

Higher dimensions always loomed as attractive but mysterious distant peaks, but the various attempts to study 3-folds made little headway up to my work. With hindsight, these failed because they worked only with nonsingular varieties -- at first sight this is a natural assumption, but it causes insuperable difficulties if one wishes to control the sign of the canonical class. In place of this, my insight was to set up a framework of 3-folds allowed certain singularities, but with a positivity condition on the canonical class.

Zariski and Hironaka had established the resolution of singularities of complex algebraic varieties in all dimensions (Hironaka's 1970 Fields medal). This means that given a singular variety X, one can make a nonsingular model Y by a sequence of blowups, thus unpacking the geometry around a singularity into a standard neighbourhood of a normal crossing divisor.

Meanwhile, there were simple examples kicking around of naturally occurring singular varieties with the paradoxical property that resolving their singularities actually makes them much more complicated. In the years before I came to Warwick in 1978, I was lucky enough to stumble on a number of these examples, and developed this thread into my theory of canonical singularities (1979) and terminal singularities (1981). In parallel with this, Mori introduced his key notion of extremal rays, and the road to 3-fold minimal models was clear, albeit still strewn with formidable difficulties for my more technically minded colleagues. See my "25 years of 3-folds" for a detailed description of these developments.

**10 October 2014 John Smillie (Warwick) Closed orbits on polygonal billiard tables**

Abstract:

I will describe the problem of estimating the asymptotics of the number of closed billiard trajectories on polygonal billiard tables with angles which are rational multiples of pi. In particular I will explain why there is one such orbit and I will connect the question with recent work in the field of Teichmuller dynamics.

**17 October 2014 Keith Ball (Warwick) Noise sensitivity and Gaussian surface area**

Abstract:

The sensitivity to noise of Boolean functions is a problem that has attracted a fair amount of attention from theoretical computer scientists. This talk will describe the problems and their Gaussian analogues which serve as a model for what one expects in the Boolean case.

**24 October 2014 Alex Veslov (Loughborough) From hyperplane arrangements to Deligne-Mumford moduli spaces: Kohno-Drinfeld way**

Abstract:

Gaudin subalgebras are abelian Lie subalgebras of maximal dimension spanned by generators of the Kohno-Drinfeld Lie algebra t_n, associated to A-type hyperplane arrangement. It turns out that Gaudin subalgebras form a smooth algebraic variety isomorphic to the Deligne-Mumford moduli space M_{0,n+1} of

stable genus zero curves with n+1 marked points. A real version of this result allows to describe the moduli space of integrable n-dimensional tops and

separation coordinates on the unit sphere in terms of the geometry of Stasheff polytope. The talk is based on joint works with L. Aguirre and G. Felder and with K. Schoebel.

**31 October 2014 Michael Berry (H H Wills Physics Laboratory) Curl forces and beyond**

Abstract:

Forces depending on position but which are not derivable from a potential, that is, forces with non-zero curl, give rise to dynamics that is not Hamiltonian or Lagrangian, while also being non-dissipative. Noether’s theorem does not apply, so the link between symmetries and conservation laws is broken. The physical existence of curl forces has been controversial and the subject of intense debate among engineers. But an example is familiar in optics: force on a dielectric particle in an optical field. Motion under curl forces near optical vortices can be understood in detail, and the full series of ‘superadiabatic’ correction forces derived, leading to an exact slow manifold in which fast (internal) and slow (external) motion of the particle is separated.

**07 November 2014 Nathanael Berestycki (Cambridge) Geometry of Random Surfaces**

Abstract:

What is a random surface? And what does it look like? In the last 10-15 years probabilists have been trying to answer these questions. I will explain two approaches to these questions. In the discrete approach, one studies the combinatorics of large random planar maps. In the continuous approach, one studies a universal

object known as the Gaussian Free Field. I will outline some of the major achievements in this field and discuss some outstandng conjectures.

**14 November 2014 Wolfgang Lueck (Bonn) Introduction to $L^2$-cohomology**

Abstract:

I will give an introduction to $L^2$-cohomology and present some of its striking applications to topology, geometry, algebra and analysis. Since this is a colloquium talk, I will start with a basic survey on finite simplicial complexes and ordinary homology in the classical sense and its applications before I will explain its extension to the $L^2$- setting for universal coverings using von Neumann algebras.

**21 November 2014 Trevor Wooley (Bristol) A translation-invariant perspective on arithmetic (and) harmonic analysis**

Abstract:

We learn about translation-invariance at mathematical birth. Thus, given a three term arithmetic progression, such as 10, 20, 30, we learn that one can obtain another such progression by shifting every entry by the same integer, as in the triple 17, 27, 37. Very recently, a method has emerged that applies non-linear translation-invariant equations to analyse Fourier series having polynomial arguments (such as Weyl sums). There are many circumstances in which this method achieves best possible estimates, with consequent applications of Diophantine type such as Waring’s problem, equidistribution of polynomial sequences modulo one, and Fourier restriction problems. In this talk, while we outline the underlying ideas, we emphasise applications. If time permits we will speculate concerning the potential for future applications to such topics as randomness extractors, work on decoupling conjectures of Bourgain and Demeter, and approximately translation-invariant systems.

**28 November 2014 Reidun Twarok (York) Viruses and geometry –new insights into virus structure and the mechanisms underpinning infection**

Abstract:

Viruses are remarkable examples of order at the nanoscale. Many viruses, including the common cold, have protein containers that are organized with icosahedral symmetry, and group theory can therefore be used to characterize their architectures. We present here group theoretical tools that reveal a molecular scaling principle between different viral components, and we demonstrate that it can be used to gain a better understanding of structural transitions in the viral protein containers that are important for infection. We also demonstrate applications of these mathematical tools in other areas of Science, such as quasicrystals and fullerenes.

References:

[1] P. Dechant, J. Wardman, T. Keef & R. Twarock (2014) Viruses and fullerenes - symmetry as a common thread? Acta Cryst A 70:162-7– see also: Nature Physics(“Know your onion”, Vol 10, p. 244, April 2014)

[2] T. Keef, J.P. Wardman, N.A. Ranson, P.G. Stockley & R. Twarock (2013) Structural constraints on the three-dimensional geometry of simple viruses: case studies of a new predictive tool. Acta Crystallogr A. 69, 140-50

[3] P.-P. Dechant, C. Bœhm & R. Twarock (2013) Affine extensions of non-crystallographic Coxeter groups induced by projection, J. Math. Phys. 54, 093508

[4] P.-P. Dechant, C. Bœhm & R. Twarock (2012) Novel Kac-Moody-type affine extensions of non-crystallographic Coxeter groups. J. Phys. A 45(28), 285202

[5] P. Cermelli, G. Indelicato & R. Twarock (2013) Nonicosahedral pathways for capsid expansion, Phys. Rev. E 88, 032710

[6] G. Indelicato, P. Cermelli, D.G. Salthouse, S. Racca, G. Zanzotto & R. Twarock (2012) A crystallographic approach to structural transitions in icosahedral viruses. J. Math. Biol. 64,745-73

[7] G. Indelicato, T. Keef, P. Cermelli, D.G. Salthouse, R. Twarock & G. Zanzotto (2012) Structural transitions in quasicrystals induced by higher dimensional lattice transitions, Proc R Soc A 468: 1452-1471

**5 December 2014 Impact Case Studies
**An opportunity to learn about some of the impact case studies Warwick Maths & Stats submitted to the REF, what was involved in putting them together, and what might be good cases for next time. Approximate timetable:

16.05 Caroline Series, View from the REF panel and Mathematics in Digital Art

16.15 Saul Schleimer, More on Mathematics in DIgital Art

16.20 Matt Keeling, Modelling influenza and Livestock Infection

16.35 David Firth, Overview of Statistics' Case Studies

16.50 Discussion

**09 January 2015 David Damanik (Rice) Cantor sets and Cantor measures**

Abstract:

A subset of the real line is called a Cantor set if it is compact, perfect, and nowhere dense. Cantor sets arise in many areas; in this talk we will discuss their relevance in the spectral theory of Schrodinger operators. We discuss several results showing that the spectrum of such an operator is a Cantor set, from the discovery of the first example by Moser to a genericity result by Avila, Bochi and Damanik. A Cantor measure is a probability measure whose topological support is a Cantor set. A primary example in the spectral theory context is given by the density of states measure in situations where the spectrum is a Cantor set. A conjecture of Simon claims a strict inequality between the dimensions of the set and the measure for the Fibonacci potential. If time permits, we will discuss a recent result of Damanik, Gorodetski and Yessen, which establishes this conjecture in full generality.

**16 January 2015 Jürgen Berndt (KCL) Symmetries in Riemannian Geometry**

Abstract:

Symmetry is one of the fundamental concepts in geometry. In the talk I plan to review a few major applications of symmetry in Riemannian geometry and present some recent developments and open questions.

**23 January 2015 Robert MacKay (Warwick) A kinematic explanation for gamma-ray bursts**

Abstract:

Gamma-ray bursts are flashes of light observed from all directions in space, lasting from milliseconds to a few minutes, which start as gamma rays then soften progressively to X-rays and ultimately to radio waves. They have been attributed to cataclysmic events. Colin Rourke and I propose, however, that many of them may be optical illusions, simply the result of our entry into the region illuminated by a continuously emitting object. At such an entry, the emitter appears infinitely blue-shifted and infinitely bright. We demonstrate the phenomenon in de Sitter space, where much can be calculated explicitly, and then extend the idea to more general space-times.

**30 January 2015 Felix Fischer (Cambridge) Optimal Impartial Selection**

Abstract:

I will talk about joint work with Max Klimm on impartial mechanisms for selecting a member of a set of agents based on nominations by agents from that set. Here, impartiality means that nominations submitted by an agent do not affect its own chance of selection. Our main result concerns a randomized mechanism that in expectation selects an agent with at least half the maximum number of nominations. Subject to impartiality, this is optimal. If time permits I will also discuss some special cases, strong impossibility results for deterministic mechanisms, and other mechanisms for impartial agents.

**06 February 2015 Jean-Pierre Eckmann (Geneve) Statics and dynamics of 2D and 3D topological glasses**

Abstract:

I will discuss work with Pierre Collet and Maher Younan about the dynamics of 2 and 3-dimensional triangulations of the sphere. A stochastic dynamic is introduced by defining a local energy of the triangulation (e.g. the deviation of the degree of a node from the expected average degree). One then shows how a Metropolis algorithm produces slowing down of return to equilibrium which is of glass-like nature. While the 2d problem is satisfactorily solved, many open problems remain in the 3d case.

**13 February 2015 Iain Gordon (Edinburgh) Wronskians, Galois Theory and Schubert Calculus**

Abstract:

Beginning with a conjecture about the critical points of a rational function in one variable (which has been confirmed), I will discuss an elementary problem that arose in enumerative geometry. The enumerative geometry here is to do with counting intersections of lines, planes, and so forth in projective spaces, or in other words Schubert Calculus. Normally such intersections don’t behave well over the real numbers; it is necessary to work over the complex numbers because one needs to find roots to polynomial equations. But amazingly, there is an explicit recipe that demonstrates the ‘Reality of Schubert Calculus’, meaning that all the intersections work over the real numbers. This was proved by Mukhin-Tarasov-Varchenko using methods from integrable systems and Lie theory. Happily, their work leads to many intriguing new questions connecting combinatorics, geometry and representation theory.

**20 February 2015 Mark Gross (Cambridge) Mirror symmetry and tropical geometry
**Abstract:

Mirror symmetry is a subject which originated around 1990, coming out of some rather astonishing calculations by string theorists which made predictions about solutions to algebro-geometric counting problems. In particular, early calculations by Candelas, de la Ossa, Green and Parkes calculated number of rational curves of all degrees in a quintic hypersurface in complex projective four-space. Since then the subject has blossomed in many different directions. In this talk I will outline the history of the subject, and explain how over the last ten years a type of combinatorial geometry known as "tropical geometry" has come to be seen as providing a conceptual explanation for mirror symmetry.

**27 February 2015 Jürg Fröhlich (ETH) ABC in Quantum Mechanics**

Abstract:

I discuss the foundations of the quantum theory of observations and measurements. After a general introduction, I first sketch what I think is the correct theory of projective (von Neumann) measurements. In particular, I discuss the difference between potential and empirical/objective properties of a quantum-mechanical system. I then outline the theory of indirect- and, in particular, non-demolition measurements. Along the way, I clarify what is meant by an event in quantum theory. To conclude, I will explain a tantalizing connection between indirect measurements and classical statistical mechanics and apply it to describe "quantum jumps" and particle tracks.

**06 March 2015 Konstantin Ardakov (Oxford) Localisation of p-adic representations of p-adic Lie groups**

Abstract:

p-adic Lie groups, and their representations, are of interest in several branches of modern algebraic number theory. I will try to explain how to study these representations geometrically using modules over the sheaf of infinite order differential operators on p-adic rigid analytic manifolds.

**13 March 2015 Mark Haskins (Imperial) A brief history of G_2**

Abstract:

This colloquium will tell the tale of the exceptional simple Lie group G_2: from its unexpected discovery in 1887 to the present, where it has come to play an important role both in Differential Geometry and in Theoretical Physics (M-theory). This is a tale suitable for the whole family (of mathematicians); several reversals of fortune and periods of exile will befall our hero en route to present day influence.

**20 March 2015 Frances Kirwan (Oxfrord) Non-reductive geometric invariant theory and applications in algebraic and hyperkahler geometry**

Abstract:

Mumford's geometric invariant theory (GIT) provides a method for constructing (projective completions of) quotient varieties for linear actions of complex reductive groups on affine and projective varieties, and has many applications (for example in the construction of moduli spaces in algebraic geometry). The aim of this talk is to discuss an extension of Mumford's GIT to actions of linear algebraic groups which are not necessarily reductive, and some of its applications. Non-reductive GIT is linked to the construction known as symplectic implosion (due to Guillemin, Jeffrey and Sjamaar) and more recently an analogous construction in hyperkahler geometry. If time permits I will finish with some speculations about possible links to derived categories of sheaves.

**01 May 2015 Gregory Seregin (Oxford) Certain Classes of Ancient Solutions to Navier-Stokes equations**

Abstract:

The relationship between the so-called ancient (backwards) solutions to the Navier-Stokes equations in the space or in a half space and the global well-posedness of initial boundary value problems for these equations will be explained. It will be shown how necessary conditions for the nite time blowup can be reduced to a Liouville type problem for smooth bounded ancient solutions to the Navier-Stokes equations.

**08 May 2015 Anton Bovier (Bonn) Extremes of Gaussian Processes on Trees**

Abstract:

Gaussian processes indexed by trees form an interesting class of correlated random fields where the structure of extremal processes can be studied. One popular example is Branching Brownian motion, which has received a lot of attention over the last decades, non the least because of its connection to the KPP equation.

In this talk I review the construction of the extremal process of standard and variable speed BBM (with Arguin, Hartung, and Kistler).

**15 May 2015 Wendelin Werner (ETH) An elementary approach to the renormalization flow**

Abstract:

We will present a formalism in which one can interpret the renormalization flow in some models from statistical physics. The critical continuous (sometimes

proved) scaling limits should then appear in this formalism as stationary distributions for some rather simple Markov processes on the state of discrete graphs with edge-weights. We will illustrate this with the concrete case of the two-dimensional uniform spanning tree (which is treated in joint work with Stéphane Benoit and Laure Dumaz). This talk does not require any particular background.

**22 May 2015 David Brydges (UBC) The Coulomb gas in two dimensions**

Abstract:

This will be an introduction for a general audience to the statistical mechanics of an infinite assembly of positively and negatively charged particles on a lattice $Z^2$. There are some surprising theorems: for example, at large values of the parameter that represents temperature, empirical charge densities in disjoint regions are almost independent, even though the Coulomb interaction is very long range. This is called Debye screening. At small values of the temperature there is

the famous Kosterlitz-Thouless phase where all charges are bound into neutral configurations; the system has the same long distance correlations as a random field known as the massless free field. Recent progress by Pierluigi Falco has made it possible to understand the transition into this phase in detail, but this subject will never cease to generate interesting new questions, and I will point out some of those along the way.

**29 May 2015 Thierry Levy (Paris 6) Eigenvalues of Hermitian matrices**

Abstract:

Taking as a pretext a leisurely discussion of a celebrated theorem of Wigner, we will introduce some ideas and techniques which are fundamental in the theory of large random matrices.

**19 June 2015 Ben Green (Oxford) On the probability that a random permutation fixes a set of size k**

Abstract:

Let pi be a random permutation of {1,...,n}. It is a well-known fact that the probability of pi fixing some point (that is, pi(x) = x for some x) tends to 1 - 1/e as n -> infty. What is the probability that pi fixes some set of size k? We will discuss this question, placing particular emphasis on connections with number theory. The talk should be accessible to a general audience.

**05 June 2015 Yujiro Kawamata (Tokyo) Derived Mckay correspondence for finite Abelian group quotients.**

Abstract: