Past

Hinke Osinga, University of Auckland
joint work with: Bernd Krauskopf and Stefanie Hittmeyer (University of Auckland)

Dynamical systems of Lorenz type are similar to the famous Lorenz system of just three ordinary differential equations in a well-defined geometric sense. The behaviour of the Lorenz system is organised by a chaotic attractor, known as the butterfly attractor. Under certain conditions, the dynamics is such that a dimension reduction can be applied, which relates the behaviour to that of a one-dimensional non-invertible map. A lot of research has focussed on understanding the dynamics of this one-dimensional map. The study of what this means for the full three-dimensional system has only recently become possible through the use of advanced numerical methods based on the continuation of two-point boundary value problems. Did you know that the chaotic dynamics is organised by a space-filling pancake? We show how similar techniques can help to understand the dynamics of higher-dimensional Lorenz-type systems. Using a similar dimension-reduction technique, a two-dimensional non-invertible map describes the behaviour of five or more ordinary differential equations. Here, a new type of chaotic dynamics is possible, called wild chaos. 

For a prime number p, we will consider completed group algebras, or Iwasawa algebras, of the form kG, for G a complete p-valued group of finite rank, k a field of characteristic p. Classifying the ideal structure of Iwasawa algebras has been an ongoing project within non-commutative algebra and representation theory, and we will discuss ideas related to this topic based on previous work and try to extend it. An important concept in studying ideals of group algebras is the notion of controlling ideals, where we say a closed subgroup H of G controls a right ideal I of kG if I is generated by a subset of kH. It was proved by Konstantin Ardakov in 2012 that for G nilpotent, every faithful prime ideal of kG is controlled by the centre of G, and it follows that the prime spectrum of kG can be realised as the disjoint union of commutative strata. I am hoping to extend this to a more general case, perhaps to when G is solvable. A key step in the proof is to consider a faithful prime ideal P in kG, and an automorphism of G, trivial mod centre, that fixes P. By considering the Mahler expansion of the automorphism, and approximating the coefficients, we can examine sequences of bounded k-linear functions of kG, and study their convergence. If we find that they converge to an appropriate quantized divided power, we can find proper open subgroups of G that control P. I have extended this notion to larger classes of automorphisms, not necessarily trivial mod centre, using which this proof can be replicated, and in some cases extended to when G is abelian-by-procyclic. I will give some examples, for G with small rank, for which these ideas yield results.

I will explain the model theoretic notion of ampleness
and present the geometric context of recent constructions.

Symmetry has played a critical role both for composers and in the creation of musical instruments. From Bach’s Goldberg Variations to Schoenberg’s Twelve-tone rows, composers have exploited symmetry to create variations on a theme. But symmetry is also embedded in the very way instruments make sound. The lecture will culminate in a reconstruction of nineteenth-century scientist Ernst Chladni's exhibition that famously toured the courts of Europe to reveal extraordinary symmetrical shapes in the vibrations of a metal plate.

The lecture will be preceded by a demonstration of the Chladni plates with the audience encouraged to participate. Each of the 16 plates will have their own dials to explore the changing input and can accommodate 16 players at a time. Participants will be able to explore how these shapes might fit together into interesting tessellations of the plane. The ultimate idea is to create an aural dynamic version of the walls in the Alhambra.

The lecture will start at 5pm, but the demonstration will be available from 2.30pm.

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The aim of this talk is to explain how to axiomatize Hilbert's Theorem 90, in the setting of (the cohomology with finite coefficients of) profinite groups. I shall first explain the general framework.  It includes, in particular, the use of divided power modules over Witt vectors; a process which appears to be of independent interest in the theory of modular representations. I shall then give several applications to Galois cohomology, notably to the problem of lifting mod p Galois representations (or more accurately: torsors under these) modulo higher powers of p. I'll also explain the connection with the Bloch-Kato conjecture in Galois cohomology, proved by Rost, Suslin and Voevodsky. This is joint work in progress with Charles De Clercq.

What can maths tell us about this topic? Do mathematicians even have a seat at the table, and should we? What do we know about directed networks and dynamical systems that can contribute to this?

We consider the implications of the mathematical modelling and analysis of neurone-to-neurone dynamical complex networks. We explain how the dynamical behaviour of relatively small scale strongly connected networks lead naturally to non-binary information processing and thus to multiple hypothesis decision making, even at the very lowest level of the brain’s architecture. This all looks a like a a loose  coupled array of  k-dimensional clocks. There are lots of challenges for maths here. We build on these ideas to address the "hard problem" of consciousness - which other disciplines say is beyond any mathematical explanation for ever! 

We discuss how a proposed “dual hierarchy model”, made up from both externally perceived, physical, elements of increasing complexity, and internally experienced, mental elements (which we argue are equivalent to feelings), may support a leaning and evolving consciousness. We introduce the idea that a human brain ought to be able to re-conjure subjective mental feelings at will. An immediate consequence of this model  is that finite human brains must always be learning and forgetting and that any possible subjective internal feeling that might be fully idealised only with a countable infinity of facets, could never be learned completely a priori by zombies or automata: it may be experienced more and more fully by an evolving human brain (yet never in totality, not even in a lifetime). 

We study risk-sharing equilibria with trading subject to small proportional transaction costs. We show that the frictionless equilibrium prices also form an "asymptotic equilibrium" in the small-cost limit. To wit, there exist asymptotically optimal policies for all agents and a split of the trading cost according to their risk aversions for which the frictionless equilibrium prices still clear the market. Starting from a frictionless equilibrium, this allows to study the interplay of volatility, liquidity, and trading volume.
(This is joint work with Johannes Muhle-Karbe, University of Michigan.)
 

We describe a convergence acceleration technique for generic optimization problems. Our scheme computes estimates of the optimum from a nonlinear average of the iterates produced by any optimization method. The weights in this average are computed via a simple linear system, whose solution can be updated online. This acceleration scheme runs in parallel to the base algorithm, providing improved estimates of the solution on the fly, while the original optimization method is running. Numerical experiments are detailed on classical classification problems.
 

TBA

The Robin-Schensted-Knuth insertion algorithm provides a bijection between non-negative integer matrices and pairs of semistandard Young tableau. However, by relaxing the conditions on the correspondence, it allows us to define the Poirer-Reutenauer bialgebra, which exactly describes the algebra of symmetric functions viewed as generated by the Schur polynomials. This gives an interesting combinatorial decomposition of symmetric products of Schur polynomials, called a Littlewood Richardson rule, which we will discuss. We will then power through as many generalisations as I have time for: Hecke insertion and stable Grothendieck polynomials, shifted insertion and Schur P-functions, and shifted Hecke insertion and weak shifted stable Grothendieck polynomials

Meteorologist Ed Lorenz was one of the founding fathers of chaos theory. In 1963, he showed with just three simple equations that the world around us could be both completely deterministic and yet practically unpredictable. More than this, Lorenz discovered that this behaviour arose from a beautiful fractal geometric structure residing in the so-called state space of these equations. In the 1990s, Lorenz’s work was popularised by science writer James Gleick. In his book Gleick used the phrase “The Butterfly Effect” to describe the unpredictability of Lorenz’s equations. The notion that the flap of a butterfly’s wings could change the course of future weather was an idea that Lorenz himself used in his outreach talks.

However, Lorenz used it to describe something much more radical than can be found in his three simple equations. Lorenz didn’t know whether the Butterfly Effect, as he understood it, was true or not. In fact, it lies at the heart of one of the Clay Mathematics Millennium Prize problems, and is still an open problem today. In this talk I will discuss Lorenz the man, his background and his work in the 1950s and 1960s, and will compare and contrast the meaning of the “Butterfly Effect" as most people understand it today, and as Lorenz himself intended it to mean. The implications of the “Real Butterfly Effect" for understanding the predictability of nonlinear multi-scale systems (such as weather and climate) will be discussed. No technical knowledge of the field is assumed. 

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Further reading:
T.N.Palmer, A. Döring and G. Seregin (2014): The Real Butterfly Effect. Nonlinearity, 27, R123-R141.

We analyse the numerical approximation of functions using radial basis functions in the context of frames. Frames generalize the notion of a basis by allowing redundancy, while being restricted by a so-called frame condition. The theory of numerical frame approximations allows the study of ill-conditioning, inherently due to their redundancy, and suggests discretization techniques that still offer numerical stability to machine precision. We apply the theory to radial basis functions.

The joint spectral radius (JSR) of a set of matrices characterizes the maximum growth rate that can be achieved by multiplying them in arbitrary order. This concept, which essentially generalizes the notion of the "largest eigenvalue" from one matrix to many, was introduced by Rota and Strang in the early 60s and has since emerged in many areas of application such as stability of switched linear systems, computation of the capacity of codes, convergence of consensus algorithms, tracability of graphs, and many others. The JSR is a very difficult quantity to compute even for a pair of matrices. In this talk, we present optimization-based algorithms (e.g., via semidefinite programming or dynamic programming) that can either compute the JSR exactly in special cases or approximate it with arbitrary prescribed accuracy in the general case.

Based on joint work (in different subsets) with Raphael Jungers, Pablo Parrilo, and Mardavij Roozbehani.
 

We find a new duality  for form factors of lightlike Wilson loops
in planar N=4 super-Yang-Mills theory. The duality maps a form factor
involving a lightlike polygonal super-Wilson loop together with external
on-shell states, to the same type of object  but with the edges of the
Wilson loop and the external states swapping roles.  This relation can
essentially be seen graphically in Lorentz harmonic chiral (LHC) superspace
where it is equivalent to planar graph duality.

Part of the series "What do historians of mathematics do?"  

I will address this question by turning to another: "What is algebra?"  In answering this second question, and surveying the way that the answer changes as we move through the centuries, I will highlight some of the problems that face historians of mathematics when it comes to interpreting historical mathematics, and give a flavour of what it means to study the history of mathematics.

Results on the existence and stability of clustered spike patterns for biological reaction‐diffusion systems with two small diffusivities will be presented. In particular we consider a consumer chain model and the Gierer‐Meinhardt activator-inhibitor system with a precursor gradient. A clustered spike pattern consists of multiple spikes which all approach the same limiting point as the diffusivities tend to zero. We will present results on the asymptotic behaviour of the spikes including their shapes, positions and amplitudes. We will also compute the asymptotic behaviour of the eigenvalues of the system linearised around a clustered spike pattern. These systems and their solutions play an important role in biological modelling to account for the bridging of lengthscales, e.g. between genetic, nuclear, intra‐cellular, cellular and tissue levels, or for the time-hierarchy of biological processes, e.g. a large‐scale structure, which appears first, induces patterns on smaller scales. This is joint work with Juncheng Wei.
 

An efficient way to descibe binary operations which are associative only up to coherent homotopy is via simplicial spaces. 2-Segal spaces were introduced independently by Dyckerhoff--Kapranov and G\'alvez-Carrillo--Kock--Tonks to encode spaces carrying multivalued, coherently associative products. For example, the Waldhausen S-construction of an abelian category is a 2-Segal space. It describes a multivalued product on the space of objects given in terms of short exact sequences. 
The main motivation to study spaces carrying multivalued products is that they can be linearised, producing algebras in the usual sense of the word. For the preceding example, the linearisation yields the Hall algebra of the abelian category. One can also extract tensor categories using a categorical linearisation procedure.
In this talk I will discuss double 2-Segal spaces, that is, bisimplicial spaces which satisfy the 2-Segal condition in each variable. Such bisimplicial spaces give rise to multivalued bialgebras. The second iteration of the Waldhausen S-construction is a double 2-Segal space whose linearisation is the bialgebra structure given by Green's Theorem. The categorial linearisation produces categorifications of Zelevinsky's positive, self-adjoint Hopf algebras.
 

In this work we study a stochastic three-dimensional Landau-Lifschitz-Gilbert equation perturbed by pure jump noise in the Marcus canonical form. We show existence of weak martingale solutions taking values in a two-dimensional sphere $\mathbb{S}^3$ and discuss certain regularity results. The construction of the solution is based on the classical Faedo-Galerkin approximation, the compactness method and the Jakubowski version of the Skorokhod Theorem for nonmetric spaces. This is a joint work with Utpal Manna (Triva

This talk will address a new link from stochastic differential equations (SDEs) to nonlinear parabolic PDEs. Starting from the necessary and sufficient condition of the path-independence of the density of Girsanov transform for SDEs, we derive characterisation by nonlinear parabolic equations of Burgers-KPZ type. Extensions to the case of SDEs on differential manifolds and the case od SDEs with jumps as well as to that of (infinite dimensional) SDEs on separable Hilbert spaces will be discussed. A perspective to stochastically deformed dynamical systems will be briefly considered.