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Mathematics Colloquium 2013-14 Abstracts

04 October 2013 Grégory Miermont (ENS, Lyon) Random maps and random 2-dimensional geometries
A map is a graph embedded in a 2-dimensional surface, considered up to homeomorphisms. In a way, it provides a discrete geometrization on this surface. Thus, a map chosen at random in some sense is a natural candidate for a notion of a geometry chosen at random on the surface. More precisely, one expects that, as the mesh of the map refines while the number of vertices blows up, random maps approximate a random surface, in the same way as random walks are the natural discrete approximations of Brownian motion. Like Brownian motion, the continuum surfaces that arise in this context are very wild, and far from being smooth Riemannian structures. It makes their study quite interesting, in the sense that one has to look for the geometric notions that still make sense in this context, such as distances and geodesics. By constrast with these "continuum" notions, we will see that the study of maps relies on tools that are of purely combinatorial nature, with an emphasis on the so-called Schaeffer bijection allowing to see maps as decorated trees. We will also discuss the conjectural connection of random maps with random conformally invariant objects.

01 November 2013 André Neves (Imperial college) Min-max theory in Geometry
I will survey various applications of min-max theory in Geometry since Poincare' asked 100 years ago whether every sphere admits a geodesic. Joint work with Fernando Marques.

08 November 2013 David Ruelle Hydrodynamic turbulence as a problem in nonequilibrium statistical mechanics

11 October 2013 Dan Kráľ (Warwick) Analytic methods in graph theory
The recently emerged theory of graph limits keeps attracting a substantial amount of attention. In the talk, we focus on limits of dense graphs with their applications in extremal combinatorics and theoretical computer science. We will present counterexamples to two conjectures of Lovasz and Szegedy on the structure of the topological space of typical vertices of the limit objects. At the end of the talk, we will discuss limits of sparse graphs initiated by Benjamini and Schramm and directions for unifying the two cases.

18 October 2013 Robert Calderbank (Duke) Golay, Heisenberg and Weyl
Sixty years ago, efforts by Marcel Golay to improve the sensitivity of far infrared spectrometry led to the discovery of pairs of complementary sequences. This talk will describe how these sequences arise in a universe that parallels classical Fourier analysis. Their construction involves binary counterparts of time and frequency shifts that are interchanged by a transform called the Walsh-Hadamard transform that parallels the Fourier transform in classical analysis.

25 October 2013 Ashkan Nikeghbali (Zurich) Random matrices and number theory: some probabilistic aspects
In the past decade, characteristic polynomials of random unitary matrices have played a major role in making spectacular predictions/conjectures for the Riemann zeta function. We shall revisit one of the most classical results in this direction but with a different point of view, based on classical probability theory. In addition we shall illustrate how this new point of view has provided us with new and deeper understanding both in random matrix theory and on the value distribution of the Riemann zeta function.

15 November 2013 Arieh Iserles (Cambridge) On the importance (and perils) of being skew-symmetric
In this talk we go back to the very basics of numerical analysis of PDEs, stability theory of finite difference schemes for linear evolution equations with variable coefficients. We prove that a universal "magic wand" renders numerical methods stable: the (first) space derivative should be discretised by a skew-symemtric matrix. The downside, however, is a barrier of 2 on the order of such methods on uniform grids. We derive a general theory coupling grid structure with the availability of skew-symmetric matrices corresponding to high-order methods.

22 November 2013 Sjoerd Verduyn Lunel (Utrecht) Wasserstein distances in the analysis of time series and dynamical systems
A new approach based on Wasserstein distances, which are numerical costs of an optimal transportation problem, allows to analyze nonlinear phenomena in a robust manner. The long-term behavior of a dynamical system represented by time series is reconstructed from time series, resulting in a probability distribution over phase space. Each pair of probability distributions is then assigned a numerical distance that quantifies the differences in their dynamical properties. From the totality of all these distances a low-dimensional representation in a Euclidean space is derived. This representation shows the functional relationships between the dynamical systems under study. It allows to assess synchronization properties and also offers a new way of numerical bifurcation analysis. Several examples are given to illustrate our results.

29 November 2013 Gitta Kutyniok (TU Berlin) Imaging Science meets Compressed Sensing
Modern imaging data are often composed of several geometrically distinct constituents. For instance, neurobiological images could consist of a superposition of spines (pointlike objects) and dendrites (curvelike objects) of a neuron. A neurobiologist might then seek to extract both components to analyze their structure
separately for the study of Alzheimer specific characteristics. However, this task seems impossible, since there are two unknowns for every datum. Compressed sensing is a novel research area, which was introduced in 2006, and since then has already become a key concept in various areas of applied mathematics, computer science, and electrical engineering. It surprisingly predicts that high-dimensional signals, which allow a sparse representation by a suitable basis or, more generally, a frame, can be recovered from what was previously considered highly incomplete linear measurements, by using efficient algorithms. Utilizing the methodology of Compressed Sensing, the geometric separation problem can indeed be solved both numerically and theoretically. For the separation of point- and curvelike objects, we choose a deliberately overcomplete representation system made of wavelets (suited to pointlike structures) and shearlets (suited to curvelike structures). The decomposition principle is to minimize the $\ell_1$ norm of the representation coefficients. Our theoretical results, which are based on microlocal analysis considerations, show that at all sufficiently fine scales, nearly-perfect separation is indeed achieved. This project was done in collaboration with David Donoho (Stanford University) and Wang-Q Lim (TU Berlin)."

6 December 2013 Andrea Malchiodi (Warwick) On the structure of spike-layers
Some structure formation in Reaction-Diffusions systems can be described in terms of Turing's bifurcation. Under a change of parameters in the system - for example a diffusivity coefficient - some homogeneous states tend to become unstable and generate non trivial patterns. These might in turn become unstable and
break-up to develop even more refined structures. We will describe the richness in the formation of spike-layers, which are highly concentrated solutions of semi-linear PDEs arising from systems such as the Gierer-Meinhardt's. The more complicated is the structure these solutions present, the more involved is the mathematical theory needed to describe them. In this respect, we will explore connections with Differential Geometry and Hamiltonian Systems.

10 January 2014 Andrew Blake (Microsoft Research) Machines that See, Powered by Probability
Machines with some kind of ability to see have become a reality in the last decade, and we see vision capabilities in cameras and photography, cars, graphics software and in the user interfaces to appliances. Such machines bring benefits to safety, consumer experiences, and healthcare, and their operation is based on mathematical ideas.

The visible world is inherently ambiguous and uncertain so estimation of physical properties by machine vision often relies on probabilistic methods. Prior distributions over shape can help significantly to make estimators for finding and tracking objects more robust. Learned distributions for colour and texture are used to make the estimators more discriminative. These ideas fit into a philosophy of vision as inference: exploring hypotheses for the contents of a scene that explain an image as fully as possible. More recently this explanatory approach has partly given way to powerful, direct estimation methods, whose operating parameters are learned from large data sets. Perhaps the most capable vision systems will come ultimately from some kind of fusion of the two approaches.

17 January 2014 Mark Pollicott (Warwick) Zeta functions in number theory, geometry and dynamics
Zeta functions are complex functions, the most famous of which is the Riemann zeta function in number theory. We will describe natural analogues of these in geometry and dynamical systems. In all of these settings the key idea is that the more we know about the domain of the zeta function the more information we have on natural objects in that context (eg prime numbers, closed geodesics, periodic orbits).

24 January 2014 Ian Melbourne (Warwick) A Huygens principle for anomalous diffusion in spatially extended systems
A consequence of the classical Huygens principle applied to the linear wave equation is that sound waves propagate in odd dimensions but not even dimensions. There also exist nonlinear formulations of this principle, based on the underlying Euclidean symmetry. Here (in joint work with Georg Gottwald), we show that for weakly chaotic dynamical systems with Euclidean symmetry, there is a related dichotomy where waves propagate diffusively (like Brownian motion) in even dimensions and superdiffusively (like a Levy process) in odd dimensions.

31 January 2014 Konstantin Khanin (Toronto) KPZ Universality and Random Hamilton-Jacobi Equation
Universal properties related to the KPZ equation is an extremely active research area right now. It is connected to many different areas in mathematics and mathematical physics: probability theory and statistical mechanics, PDEs, random matrices, integrable systems, and many others. While it was a big recent progress, mainly in deriving exact formulas for correlation functions, the problem of universality is still largely open. We shall discuss some recent results in this direction.

07 February 2014 David Loeffler (Warwick) Euler systems
One of the best-known open problems in number theory is the conjecture of Birch and Swinnerton-Dyer, which relates the existence of rational solutions to certain cubic equations ('elliptic curves') to the vanishing of a certain value of an associated complex-analytic function (an 'L-function'). In the late 1980s, Kolyvagin obtained significant results in the direction of this conjecture by using the existence of a certain algebraic widget -- an 'Euler system' -- that is essentially an algebraic counterpart of the L-function. Kolyvagin's method would also give powerful arithmetic applications in other settings if suitable Euler systems could be shown to exist; but only a few examples have been constructed so far. In this talk, I'll describe a recent joint project of Lei, Zerbes and myself which has led to the construction of several new classes of Euler systems, and some of the applications of these constructions to classical problems such as Birch--Swinnerton-Dyer and the Iwasawa main conjecture.

14 February 2014 Richard James (University of Minnesota/Oxford) Materials from Mathematics
We present some recent examples of new materials whose synthesis was guided by some essentially mathematical ideas. They are materials that undergo phase transformations from one crystal structure to another, with a change of shape but without diffusion. The underlying mathematical theory was designed to identify alloys that show low hysteresis and exceptional reversibility. The new alloys, of which Zn_45Au_30Cu_25 is the best example, do show unprecedented levels of these properties, but also raise fundamental questions for mathematical theory. In addition to the enhanced reversibility, some of these alloys have one phase that is a strong magnet, while the other is nonmagnetic. These can be used to convert heat to electricity (without the need of a separate electrical
generator), and provide interesting possible ways to recover the vast amounts of energy stored on earth at small temperature difference. (

21 February 2014 [**Room MS.04**] Tim Riley (Cornell) Hyperbolic groups, Cannon-Thurston maps, and hydra
Groups are Gromov-hyperbolic when all geodesic triangles in their Cayley graphs are close to being tripods. Despite being tree-like in this manner, they can harbour extreme wildness in their subgroups. I will describe examples stemming from a re-imagining of Hercules' battle with the hydra, where wildness is found in properties of "Cannon-Thurston maps" between boundaries. Also, I will give examples where this map between boundaries fails to be defined.

07 March 2014 Günter M. Ziegler (Freie Universität Berlin) The story of 3N points in a plane
Nearly 60 years ago, the Cambridge undergraduate Bryan Birch showed that any "3N points in a plane" can be split into N triples that span triangles with a non-empty intersection. He also conjectured a higher-dimensional version of this, which was proved by Helge Tverberg in 1964 (freezing, in a hotel room in Manchester). This is the beginning of a remarkable story that I will try to survey in this lecture. Highlights include the discovery that "3N-2 points in a plane" would have been enough (and perhaps even a better starting point); a topological version, proved by Bárány, Shlosman & Szücs for the case when N is a prime; a "colored version" of the problem proposed by Bárány & Larman in 1989, and finally proven in 2009 (joint with Blagojevic and Matschke); and the recent discovery, that from the topological version by Bárány--Shlosman--Szücs, one can get a lot of other results "nearly for free" (joint work with Pavle Blagojevic and Florian Frick).

14 March 2014 Robert Marsh (Leeds) Reflection group presentations arising from cluster algebras
A reflection group is a group of orthogonal transformations of Euclidean space generated by reflections; it is said to be crystallographic if it preserves an integral lattice. The finite crystallographic reflection groups play a key role in Lie theory, in particular in the study of simple Lie algebras and reductive algebraic groups, and have been classified by the Dynkin diagrams.

Such groups have good combinatorial presentations known as Coxeter presentations. I will explain joint work with Michael Barot in which we show that these presentations can be considered as part of a family of presentations corresponding to the quivers in the seeds of the Fomin-Zelevinsky cluster algebra of the same finite type, and also discuss generalizations.

21 March 2014 Colin McLarty (Case Western) Categorical Foundations Today
Saunders Mac Lane stressed foundations not as a priori philosophical justifications for mathematics but as "proposals for the organization of mathematics." He urged Lawvere's categorical foundations in this role. This talk will look at the current state of these categorical foundations in theory and in textbook practice. The talk will relate these foundations to the related sense of inquiry to find the minimal requirements for particular theorems or branches of mathematics, to some other styles of categorical foundations such as Homotopy Type Theory, and to objections to categorical foundations.

25 April 2014 Gunther Cornelissen (Utrecht) Differential geometric methods in number theory
A number theorist is not afraid to get her hands dirty on any kind of mathematics, as long as it proves an interesting number theoretical theorem. But differential geometry? This talk will feature some examples, such as: a lemma from the theory of soap bubbles that is applied to show the finiteness of the set of solutions to a diophantine equation; a Laplace operator that solves a conjecture about elliptic curves over function fields, and finally, a non-commutative space that illuminates a riddle in the theory of zeta functions.

02 May 2014 Peter Topping (Warwick) Ricci flow and uniformisation
Ricci flow is a natural way of deforming a Riemannian manifold under an essentially parabolic PDE. It was introduced by Hamilton in 1982, and has been extremely successful in applications to geometry and topology. We will take a look at the case of Ricci flow on surfaces, which is arguably the easiest situation in which to get a feel for the PDE, and yields the most general results. We will take a look at the beautiful results of Hamilton and Chow for compact surfaces from the 1980s, but focus mainly on the larger body of work in the noncompact case, which has been largely developed over the past few years. There will be few prerequisites for the talk. I will be talking about manifolds, Riemannian metrics and parabolic partial differential equations, but most concepts can be explained with pictures or by analogy.

16 May 2014 Peter Latham (UCL) Neuroscience from a physicist's perspective
The brain is arguably the most complicated object we have ever studied. So complicated, in fact, that it isn't even clear what it is we want to know about it. Here I'll try to settle on a specific set of problems faced by both the neuroscientist studying the brain and the brain itself, provide a sense of how hard they are, and point the way toward possible solutions. Along the way I'll provide background into what we know about the brain, and why standard physics approaches might not work so well. I'll end on an optimistic note, though: physicists do have much to contribute to neuroscience.

23 May 2014 Henry Abarbanel (UCSD) Data Assimilation as a Physics Inverse Problem
Data Assimilation, transferring information from observations of a complex system to a quantitative, predictive model of that complex system, has numerous applied mathematical challenges and substantial implications for many fields including meteorology and neuroscience, for example. This talk will summarize a general formulation of the data assimilation problem pointing out some applied mathematics opportunities and then focus on the practical ability to carry out the task of proving enough information to allow prediction using the nonlinear dynamical model built by the user. When the dynamics is chaotic, the jobs required in this are impeded by nonlinear instabilities, and we discuss how to regularize them, and how to extract additional information from observed data to permit accurate state and parameter estimation allowing accurate prediction using the model. Examples from nonlinear circuits, neurobiology, and weather prediction will be covered in some detail.

30 May 2014 Gabor Elek (Lancaster) Limits of finite structures
The basic motivation of this talk is the limit theory of bounded degree graphs. I would like to show, how the main arguments of the theory can be interpreted in other fields of mathematics.

06 June 2014 Benjamin Schlein (Bonn) Hartree-Fock dynamics for weakly interacting fermions
Fermions are quantum particles described by wave functions which are antisymmetric with respect to permutations. According to first principle quantum mechanics, the evolution of fermionic systems is described by the many body Schroedinger equation. We are interested in the so called mean field regime, which for fermions is naturally linked with a semiclassical limit. We will show that, in this regime, the Schroedinger dynamics can be approximated by the Hartree-Fock equation, providing precise bounds on the rate of the convergence.

20 June 2014 Yves Le Jan (Orsay) Markov loop ensembles
We introduce Markov loop ensembles and their relations to free fields, random spanning trees, percolation problems and Eulerian circuits.