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Mathematics Colloquium 2017-18 Abstracts

MONDAY 2 October 2017 (4pm, B3:03): Hinke Osinga (Auckland) Shaken but not stirred: using mathematics in earthquakes

06 October 2017: Matthias Schacht (Hamburg) Extremal problems for uniformly dense hypergraphs
Extremal problems for hypergraphs concern the maximum density of large k-uniform hypergraphs H that do not contain a copy of a given k-uniform hypergraph F. Determining or estimating this maximum density is a classical and central problem in extremal combinatorics. While for k=2 this problem is well understood, due to the work of Turán and of Erdös and Stone, only very little is known for k-uniform hypergraphs for k>2. We consider a variation of the problem, where the large hypergraphs H satisfy additional hereditary density conditions. Questions of this type were suggested by Erdös and Sós about 30 years ago. We present recent results in that direction, which were obtained in joint with Reiher and Rödl.

13 October 2017: Beth Wingate (Exeter) The story of mathematics and weather/climate prediction: Triumphs of the past & challenges for the oncoming era of exascale computing

One of the first major breakthroughs in scientific computing occurred just after World War II when a group of mathematicians and scientists came together to create the world’s first numerical weather prediction on one of the world's earliest computers. Perhaps the most important mathematical lessons learned from this endeavour was that there is an intimate relationship between the underlying mathematical structure of the governing equations and their numerical approximation.

A new grand challenge is on our doorstep, the challenge of next generation computers, which have been designed in new ways to address physical limitations in the manufacture of transistors and energy consumption. To run well on these new computer architectures weather and climate modeling algorithms will be required to exploit on the order of hundred-million-way parallelism. This degree of parallelism far exceeds anything possible even in today's most sophisticated models.

In this talk I will discuss one of the mathematical issues that leads to computational limitations for climate and weather prediction models – oscillatory stiffness in the PDEs that leads to time scale separation. I will discuss the historical context of the first mathematical discoveries of how nonlinear phenomenon give rise to low-frequency solutions and its relationship to fast singular limits studied in PDE’s analysis and numerical analysis. I will discuss some of the latest research directions aimed at quantifying and discretising the low-frequency part of the solutions, strategies that are aimed at addressing the limitations inherent in fast singular limits. Finally, I will close by describing potential research directions where mathematics and statistics could provide solutions.

20 October 2017: Anibal Rodriguez-Bernal (Madrid Complutense) Reaction diffusion equations in R^N, function spaces and asymptotic behavior
Starting from basic arguments with ODEs, we will review the importance of considering reaction diffusion equations PDEs in the Euclidean space R^N. We will also discuss possible functional settings to search for solutions and some of their advantages and inconveniences. We will finally discuss questions related to asymptotic behavior of solutions for large times and present results on how linear diffusion and nonlinear reaction must collaborate for solutions to have well defined asymptotic states in terms of a global attractor.

03 November 2017: Weiyi Zhang (Warwick) From smooth to almost complex
An almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex or symplectic manifold is an almost complex manifold, but not vice versa.

Transversality is the notion of general position in manifold topology. If two submanifolds intersect transversely in some ambient manifold, then their intersection is a manifold. We will discuss differential topology of almost complex manifolds, explain how to use transversality statements for smooth manifolds to formulate and prove corresponding results for an arbitrary almost complex manifold. The examples include intersection of almost complex manifolds, pseudoholomorphic maps and zero locus of certain harmonic forms.

10 November 2017: Bertrand Rémy (Ecole Polytechnique) On some non-linear (and simple) groups acting on exotic buildings
We will propose an introduction to geometric group theory through one basic question, namely producing simple infinite groups. The idea is to use suitable metric spaces, with sufficiently many symmetries to admit nice group actions. The reason why geometric arguments are requested is due to the fact that finitely generated matrix groups are, in some sense, too nice to be simple.

17 November 2017: Kate Smith-Miles (University of Melbourne) Optimization in the Darkness of Uncertainty: when you don't know what you don't know, and what you do know isn't much!
Many industrial optimisation problems involve the challenging task of efficiently searching for optimal decisions from a huge set of possible combinations. The optimal solution is the one that best optimises a set of objectives or goals, such as maximising productivity while minimising costs. If we have a nice mathematical equation for how each objective depends on the decisions we make, then we can usually employ standard mathematical approaches, such as calculus, to find the optimal solution. But what do we do when we have no idea how our decisions affect the objectives, and thus no equations? What if all we have is a small set of experiments, where we have tried to measure the effect of some decisions? How do we make use of this limited information to try to find the best decisions?

This talk will present a common industrial optimisation problem, known as expensive black box optimisation, through a case study from the manufacturing sector. For problems like this, calculus can’t help, and trial and error is not an option! We will introduce some methods and tools for tackling expensive black-box optimisation. Finally, we will discuss new methodologies for assessing the strengths and weaknesses of optimisation methods, to ensure the right method is selected for the right problem.

24 November 2017:
Ulrike Tillmann (Oxford) Topological field theories in homotopy theory
The axiomatic definition of conformal and topological field theories by Segal and Atiyah in the late 1980s has inspired many mathematicians working in geometry and topology. In this lecture I will explain the homotopy theoretic approach to topological field theory, survey some of the major results, and provide new evidence that the stable homotopy category is a natural target category.

1 Decembre 2017:
Igor Krasovsky (Imperial College) Toeplitz determinants in the theory of the Ising model and random matrices.
We will review the history of asymptotic formulas for Toeplitz determinants, especially the Strong Szego Limit Theorem. We will discuss the importance of these results for the central property of the 2-dimensional Ising model, the phase transition. We will also discuss more recent applications of Toeplitz determinants and their analogues in the theory of random matrices.

8 December 2017: Sergey Nazarenko (Warwick) Gravitational Waves and Turbulence in the Early Universe
Gravitational waves are spacetime ripples predicted by Einstein in 1916 and experimentally observed last year by the the Laser Interferometer Gravitational-Wave Observatory (LIGO) team. This experimental discovery was recognised by the 2017 Nobel Prize in Physics. The gravitational waves may interact with each other because they are described by Einstein's equations which are nonlinear. The nonlinearity may play role when the metric disturbances are large, which could be expected during collisions of massive astrophysical objects, like black holes and neutron stars, or in the Early Universe. The nonlinear interaction transfers energy among different length scales and, as such, may be important for understanding the evolution of the Early Universe, in particular during the inflation era. In my talk I will describe a statistical theory of interacting gravitational waves derived from the vacuum Einstein equations, and I will discuss some interesting scaling solutions of this theory and their possible significance for the Early Universe.

12 January 2018: Xavier Buff (Toulouse III) Holomorphic dynamics, transversality and algebraic curves
We consider rational maps $f:{\mathbb S}\to {\mathbb S}$ of the Rieman sphere ${\mathbb S}$ as dynamical systems, i.e., we study sequences
defined recursively by $z_n = f(z_{n-1}) = f^{\circ n}(z_0)$.
Periodic points, i.e., points $z$ such that $f^{\circ n}(z) = z$ for some $n\geq 1$ play a special role. The multiplier
at such a point is the eigenvalue $\lambda$ of the automorphism $D_zf : T_z {\mathbb S}\to T_z{\mathbb S}$.
We are particularly interested in the set ${\rm Per}_n(\lambda)$ of quadratic rational maps which have a periodic
point of period $n$ and multiplier $\lambda$. What do those sets look like: Are they smooth? Are they connected? How do they intersect?

19 January 2018: Dmitri Vassiliev (UCL) Spectral theory of differential operators: what's it all about and what is its use
I will give a popular overview of the spectral theory of partial differential operators, charting its development from the non-rigorous work of physicists to modern rigorous mathematical results.

26 January 2018: James Robinson (Warwick) Rigorous numerics for the Navier-Stokes equations without full existence and uniqueness results
Why bother proving existence and uniqueness results for mathematical models of physical systems? One possible answer is that it is dangerous to try to solve an equation numerically without being sure that there is actually a solution there to be approximated.

However, in this talk I will show that it in the case of the Navier-Stokes equations, for which there is no full existence and uniqueness theory currently available, it is possible to show that a "sufficiently good" numerically computed solution is close to a unique solution of the exact equations.

With a little more work one can also show that if there is a regular solution arising from some given choice of initial condition then this can be verified in finite time by numerical calculation.

Many of the arguments can be illustrated using the simple toy model ODE, bypassing the technicalities that arise in the full proof.

The key idea here is due to Sergey Chernysehnko (Imperial), and the results are based on joint work with Sergey, Peter Constantin, and Edriss Titi, after conversations we had at a meeting in Warwick in 2005.

2 February 2018: Andrew Ranicki (Edinburgh) Simplicial complexes
A simplicial complex is a combinatorial prescription K for building a topological space
|K| from points, lines, triangles etc. Simplicial complexes were introduced by Poincare some 120 years ago, in his proof of the duality between the homology and cohomology of a manifold. Simplicial complexes are the foundational objects of piecewise linear topology, a Warwick specialty 50 years ago. They then disappeared from view somewhat, being subsumed in the theory of simplicial sets. However, simplicial complexes have now made a triumphant comeback in the age of big data, being the key ingredient of persistent homology. The talk will be largely concerned with the use of simplicial complexes in surgery theory, specifically the result that for n>4 the polyhedron |K| of a finite simplicial complex K is homotopy equivalent to a compact n-dimensional topological manifold if and only if |K| has the appropriate amount of n-dimensional Poincare duality. This is a kind of converse of the original duality of Poincare.

9 February 2018: Andrea Mondino (Warwick) Smooth and non-smooth aspects of Ricci curvature lower bounds
After recalling the basic notions coming from differential geometry, the colloquium will be focused on spaces satisfying Ricci curvature lower bounds.
The idea of compactifying the space of Riemannian manifolds satisfying Ricci curvature lower bounds goes back to Gromov in the ‘80s and was pushed by Cheeger and Colding in the ‘90s who investigated the fine structure of possibly non-smooth limit spaces.
A completely new approach via optimal transportation was proposed by Lott-Villani and Sturm almost ten years ago. Via such an approach one can give a precise definition of what means for a non-smooth space to have Ricci curvature bounded below. Such an approach has been refined in the last years giving new insights to the theory and yielding applications which seems to be new even for smooth Riemannian manifolds.

16 February 2018: Louise Dyson (Warwick) Stochastic problems in biology and epidemiology
Stochasticity can be a blessing or a curse in biological systems. I will talk about some systems in which the addition of random effects can lead to significant changes to the outcome. The talk is split into three sections: the emergence of noise-induced bistable states when an ant colony is dividing its workers; the effect of systematic non-attendence during mass drug administration for a disease eradication campaign; and some current work looking for the existence of critical slowing down as diseases approach eradication.

9 March 2018: Mohamed Saidi (Exeter) Recent Advances in Galois Theory

I will review the foundational results of Neukirch, Uchida, and Pop on the Galois characterisation of finitely generated fields, or the so-called Grothendieck birational anabelian conjectures. I will then present a new result of Saidi and Tamagawa on the Galois characterisation of number fields, which is a substantial sharpening of the Neukirch-Uchida theorem, and which reads as follows. Two number fields are isomorphic if their 3-step solvable Galois groups are isomorphic. Further, the isomorphy type of the maximal abelian extension of a number field is functorially encoded in the isomorphy type of its 6-step solvable Galois group.

16 March 2018: Hugo Parlier (Luxembourg) The asymptotic geometries of puzzles and moduli spaces
Certain puzzles have natural configuration spaces with interesting geometries, often encoded by graphs. You get a family of spaces by modifying the size of the underlying puzzle, for instance by varying the side length of Rubik’s cubes. This talk is about the geometry of puzzle spaces, and how they seem to emulate phenomena that you can observe in types of combinatorial moduli spaces, such as flip-graphs of polygons where one looks at distances between triangulations.

27 April 2018: David Sauzin (CNRS, Paris and Pisa) Introduction to Resurgence theory and Alien Calculus
Resurgence theory was developed by J.Ecalle in the 1980s to deal with divergent series originating with dynamical system problems. Based on Borel-Laplace summation, it defines interesting subalgebras of the space of formal series, in which the so-called alien derivations act. This is an infinite family of operators satisfying the Leibniz rule, which happen to generate a free Lie algebra. Using these operators, one is naturally led to go from series to transseries to measure resummation ambiguities. Resurgence was studied in mathematical problems involving differential or difference equations (giving rise to moduli spaces for analytic dynamical systems), but also in mathematical physics, in relation with WKB expansions and more recently string theory and quantum field theory. I'll illustrate the basic definitions and facts of resurgence theory on some examples.

4 May 2018: Andrew Brooke-Taylor (Leeds) Products of CW complexes: the full story
CW complexes are used extensively in algebraic topology, but the product of two CW complexes need not be a CW complex. Whilst Whitehead and Milnor gave sufficient conditions for the product to be a CW complex, all previous characterisations of those pairs of CW complexes which have product a CW complex rely on extra set-theoretic axioms, like the Continuum Hypothesis. In this talk I will provide a complete characterisation of when the product of two CW complexes is a CW complex, without assuming any extra set-theoretic axioms.

11 May 2018: Geoffrey Robinson (Aberdeen) Some open questions in representation theory
We will survey some open questions in the representation theory of finite groups and discuss their relationship with other areas of Mathematics in some cases. One motivation will be to discuss how modular representation theory can assist in the understanding of the structure of finite linear groups ( ie groups of matrices) even over the complex field.

18 May 2018: Paul Milewski (Bath) Understanding the Complex Dynamics of Faraday Pilot Waves
Faraday pilot waves are a newly discovered hydrodynamic structure that consists a bouncing droplet which creates, and is propelled by, a Faraday wave. These pilot waves can behave in extremely complex ways and result in dynamics which mimic behaviour usually thought to be unique to quantum mechanics. I will show some of this fascinating behaviour and will present a surface wave-droplet fluid model that captures many of the features observed observed in experiments, focussing on the statistical emergence of complex states.

25 May 2018:  Hendrik Weber (Warwick) The stochastic quantisation equation - scaling limits, meta-stability and the role of infinity
This talk is concerned with stochastic partial differential equations (SPDEs) driven by a singular noise term namely space-time white noise. These equtions naturally appear in diverse contexts, but their rigorous mathematical treatment is challenging, because solutions are in general very irregular. They often have to be interpreted as Schwartz distributions rather than as functions, and equations often has to be interpreted in a “renormalized” sense, i.e. some formally infinite terms have to be removed.

I will focus on a particular SPDE, namely the stochastic quantisation equation. This equation was first proposed by Parisi and Wu in the early 80s in the context of constructive field theory, but I will discuss mainly two very different situations where the equation arises naturally: as a scaling limit for an interacting particle system and as a model for meta-stability. I will explain in particular, which role the “infinite terms” play in each of these situations.

1 June 2018: John Greenlees (Warwick) Invariants of spaces with a group action.
When studying topological spaces X with an action of a compact Lie group G, algebraic topology suggests one should apply cohomological invariants. Indeed, there are many options, some of which will be recalled, and the point of the talk is to explain that if the invariants take values in rational vector spaces, one can hope to understand all possible invariants: not just a classification, but also a means of construction. Thus one can construct new invariants from formally analgous structures, such as those from algebraic geometry.

The talk will be largely restricted to the circle group G=T. In that case the classical Smith theorem says that if X is a rational homology sphere then its fixed point set X^T is also a homology sphere. The talk will recall the proof through cohomology of the Borel construction, and in fact the classification of rational T-equivariant cohomology theories is essentially based on this argument (together with the usual formal apparatus relating different subgroups).

In effect it says that an arbitratary cohomology theory is built from one piece of data from each closed subgroup (T itself or the finite cyclic groups of order 1,2, 3, 4, ….). This is precisely analgous to the fact that abelian groups can be built adelically using one piece of data for each prime ideal ( (0) or the maximal primes (2), (3), (5), (7), ….).

(The talk will allude to joint work with S.Balchin, D. Barnes, M.Kedziorek and B.Shipley)

8 June 2018:  Alessio Figalli (ETH, Zürich) Regularity of interfaces in phase transitions via obstacle problems
The so-called Stefan problem describes the temperature distribution in a homogeneous medium undergoing a phase change, for example ice melting to water. An important goal is to describe the structure of the interface separating the two phases. In its stationary version, the Stefan problem can be reduced to the classical obstacle problem, which consists in finding the equilibrium position of an elastic membrane whose boundary is held fixed and which is constrained to lie above a given obstacle. The aim of this talk is to give a general overview of the classical theory of the obstacle problem, and then discuss recent developments on the structure of interfaces, both in the static and the parabolic settings.

15 June 2018: Marcelo Viana (IMPA) The Mañé-Bochi theorem is false for non-invertible maps
At his 1983 ICM address, Ricardo Mañé announced that the Lyapunov exponents of area-preserving C^1 diffeomorphisms on surfaces can always be anihilated by C^1-small perturbations, unless the diffeomorphism is Anosov.
A complete proof of this claim was provided in 2001 by Jairo Bochi, we also formulated a corresponding statement for C^0 linear cocycles over aperiodic homeomorphisms.

From the beginning it seemed that this was a typically C^0 phenomenon, and Lyapunov exponents should be better behaved with respect to C^r topology when r > 0. A number of developments over the last 15 years or so partially confirmed this expectation.

On the other hand, Jiagang Yang and I have recently realized that the Lyapunov exponents of linear cocycles may vary continuously even with respect to the C^0 topology, when the base map is non-invertible.