Past

Heckman introduced N operators on the space of polynomials in N variables, such that these operators form a covariant set relative to permutations of the operators and variables, and such that Jack symmetric polynomials are eigenfunctions of the power sums of these operators. We introduce the analogues of these N operators for Macdonald symmetric polynomials, by using Cherednik operators. The latter operators pairwise commute, and Macdonald polynomials are eigenfunctions of their power sums. We compute the limits of our operators at N → ∞ . These limits yield a Lax operator for Macdonald symmetric functions. This is a joint work with Evgeny Sklyanin.

What's the continuous analog of randn?  In recent months I've been exploring such questions with Abdul-Lateef Haji-Ali and other members of the Chebfun team, and Chebfun now has commands randnfun, randnfun2, randnfunsphere, and randnfundisk.  These are based on finite Fourier series with random coefficients, and interesting questions arise in the "white noise" limit as the lengths of the series approaches infinity and as random ODEs become stochastic DEs.    This work is at an early stage and we are grateful for input from stochastic experts at Oxford and elsewhere.

Multi-agent systems in nature oftentimes exhibit emergent behaviour, i.e. the formation of patterns in the absence of a leader or external stimuli such as light or food sources. We present a non-local two species crossinteraction model with cross-diffusion and explore its long-time behaviour. We observe a rich zoology of behaviours exhibiting phenomena such as mixing and/or segregation of both species and the formation of travelling pulses.

 Locality is not expected to be a fundamental aspect of a full theory of quantum gravity; it should be emergent in an appropriate semiclassical limit.  In the context of general holography, I'll define a new construct - the causal state - which provides a necessary and sufficient condition for a boundary state to have a holographic semiclassical dual causal geometry (and thus be "local").  This definition illuminates some general features of holographic quantum gravity: for instance, I'll show that the emergence of locality is "all or nothing" in the sense that it exhibits features of quantum error correction and quantum secret sharing.  In the special case of AdS/CFT, I'll also argue that the causal state is the natural boundary dual to the so-called causal wedge of a region. 

Part of the series "What do historians of mathematics do?"  
In the last year of 14th century, a French mathematician/geometer Jean Mignot, was called from Paris to help with the construction of the Cathedral of Milan. Thus was created one of the most famous stories about how mathematics literally supports great works of art, helping them stand the test of time. This talk will look at some patterns that begin to become apparent in the investigations of the relationship between architecture and mathematics and the creativity that is common to the pursuit of both. I will present the case on how this may matter to someone who is interested in the history of mathematics. To make this more intelligible, I will partly talk also of my personal journey in investigating this relationship and the issues I have researched and written about, and how these in turn changed my view of the nature of mathematics education. 

Various concepts of weak solution have been suggested for the fundamental equations of fluid dynamics over the last few decades. However, such weak solutions may be non-unique, or at least their uniqueness is unknown. Nevertheless, a conditional notion of uniqueness, the so-called weak-strong uniqueness, can be established in various situations. We present some recent results, both positive and negative, on weak-strong uniqueness in the realm of incompressible and compressible fluid dynamics. Applications to the convergence of numerical schemes will be indicated.

After giving an introduction to functorial field theories I will explain a natural generalization thereof, called "twisted" field theories by Stolz-Teichner. The definition uses the notion of lax or oplax natural transformations of strong functors of higher categories for which I will sketch a framework. I will discuss the fully extended case, which gives a comparison to Freed-Teleman's "relative" boundary field theories. Finally, I will explain some examples, one of which explicitly arises from factorization homology and whose target is the higher Morita category of E_n-algebras, bimodules, bimodules of bimodules etc.

In certain cases of (linear) partial differential equations random perturbations have been observed to cause regularizing effects, in some cases even producing the uniqueness of solutions. In view of the long-standing open problems of uniqueness of solutions for certain PDE arising in fluid dynamics such results are of particular interest. In this talk we will extend some known results concerning the well-posedness by noise for linear transport equations to the nonlinear case.

Recent work in regularity structures has provided a robust solution theory for a wide class of singular SPDEs. While much progress has been made on understanding the analytic and algebraic aspects of renormalisation of the driving signal, the action of the renormalisation group on the equation still needed to be performed by hand. In this talk, we aim to give a systematic description of the renormalisation procedure directly on the level of the PDE, which allows for explicit computation of the form of the renormalised equation. Joint work with Yvain Bruned, Ajay Chandra, and Martin Hairer.

The moduli space M(G) of Higgs bundles for a complex reductive group G on a compact Riemann surface carries a natural hyperkahler structure and it comes equipped with an algebraically completely integrable system through a flat projective morphism called the Hitchin map. Motivated by mirror symmetry, I will discuss certain complex Lagrangians (BAA-branes) in M(G) coming from real forms of G and give a proposal for the mirror (BBB-brane) in the moduli space of Higgs bundles for the Langlands dual group of G.  In this talk, I will focus on the real groups SU^*(2m), SO^*(4m) and Sp(m,m). The image under the Hitchin map of Higgs bundles for these groups is completely contained in the discriminant locus of the base and our analysis is carried out by describing the whole
(singular) fibres they intersect. These turn out to be certain subvarieties of the moduli space of rank 1 torsion-free sheaves on a non-reduced curve. If time permits we will also discuss another class of complex Lagrangians in M(G) which can be constructed from symplectic representations of G.

A heterotic $G_2$ system is a quadruple $([Y,\varphi], [V, A], [TY,\theta], H)$ where $Y$ is a seven dimensional manifold with an integrable <br /> $G_2$ structure $\varphi$, $V$ is a bundle on $Y$ with an instanton connection $A$, $TY$ is the tangent bundle with an instanton connection $\theta$ and $H$ is a three form on $Y$ determined uniquely by the $G_2$ structure on $Y$. Further, H  is constrained so that it satisfies a condition that involves the Chern-Simons forms of $A$ and $\theta$, thus mixing the geometry of $Y$ with that of the bundles (this is the so called anomaly cancelation condition).  In this talk I will describe the tangent space of the moduli space of these systems. We first prove that a heterotic system is equivalent to an exterior covariant derivative $\cal D$ on the bundle ${\cal Q} = T^*Y\oplus {\rm End}(V)\oplus {\rm End}(TY)$ which satisfies $\check{\cal D}^2 = 0$ for some appropriately defined projection of the operator $\cal D$.  Remarkably, this equivalence implies the (Bianchi identity of) the anomaly cancelation condition. We show that the infinitesimal moduli space is given by the cohomology group $H^1_{\check{\cal D}}(Y, {\cal Q})$ and therefore it is finite dimensional.   Our analysis leads to results that are of relevance to all orders in $\alpha’$.  Time permitting, I will comment on work in progress about the finite deformations of heterotic $G_2$ systems and the relation to differential graded Lie algebras.

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.

Please email @email to register

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