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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.