I shall discuss Zauner's conjecture about the existence of n^2 mutually equidistant points in complex projective space CP^{n-1} with its standard Fubini-Study metric. This is the so-called SIC-POVM problem, and is related to properties of the moment mapping that embeds CP^{n-1} into the Lie algebra su(n). In the case n=3, there is an obvious 1-parameter family of such sets of 9 points under the action of SU(3) and we shall sketch a proof that there are no others. This is joint work with Lane Hughston.

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.

The moduli space M_C of Higgs bundles over a complex curve X admits a hyperkaehler metric: a Riemannian metric which is Kaehler with respect to three different complex structures I, J, K, satisfying the quaternionic relations. If X admits an anti-holomorphic involution, then there is an induced involution on M_C which is anti-holomorphic with respect to I and J, and holomorphic with respect to K. The fixed point set of this involution, M_R, is therefore a real
Lagrangian submanifold with respect to I and J, and complex symplectic with respect to K, making it a so called AAB-brane. In this talk, I will explain how to compute the mod 2 Betti numbers of M_R using Morse theory. A key role in this calculation is played by the Abel-Jacobi map from symmetric products of X to the Jacobian of X.

Character varieties have been studied largely by means of their correspondence to the moduli space of Higgs bundles. In this talk we will report on a method to study their Hodge structure, in particular to compute their E- polynomials. Moreover, we will explain some applications of the given method such as, the study of the topology of the moduli space of doubly periodic instantons. This is joint work with A. González, V.Muñoz and P. Newstead.

Understanding the outcome of a collision between liquid drops (merge or bounce?) as well their impact and spreading over solid surfaces (splash or spread?) is key for a host of processes ranging from 3d printing to cloud formation. Accurate experimental observation of these phenomena is complex due to the small spatio-temporal scales or interest and, consequently, mathematical modelling and computational simulation become key tools with which to probe such flows.

Experiments show that the gas surrounding the drops can have a key role in the dynamics of impact and wetting, despite the small gas-to-liquid density and viscosity ratios. This is due to the formation of gas microfilms which exert their influence on drops through strong lubrication forces.  In this talk, I will describe how these microfilms cannot be described by the Navier-Stokes equations and instead require the development of a model based on the kinetic theory of gases.  Simulation results obtained using this model will then be discussed and compared to experimental data.

Some recent applications of supertwistors to superparticle mechanics will be reviewed.
First: Supertwistors allow a simple quantization of the  N-extended 4D massless superparticle, and peculiarities of massless 4D supermultiplets can then be explained by considering the quantum fate of a classical ``worldline CPT'' symmetry. For N=1 there is a global CPT anomaly, which explains why there is no CPT self-conjugate supermultiplet. For N=2 there is no anomaly but a Kramers degeneracy explains the doubling of states in the CPT self-conjugate hypermultiplet.
Second: the bi-supertwistor formulation of the N-extended massive superparticle in 3D, 4D and 6D makes manifest a ``hidden’’ 2N-extended supersymmetry. It also has a simple expression in terms of hermitian 2x2 matrices over the associative division algebras R,C,H.
Third: omission of the mass-shell constraint in this 3D,4D,6D bi-supertwistor action yields, as suggested  by holography, the action for a supergraviton in 4D,5D,7D AdS. Application to the near horizon AdSxS geometries of the M2,D3 and M5 brane confirms that the graviton supermultiplet has 128+128 polarisation states. 

 I will start with a motivation of what algebraic and model-theoretic properties an algebraically closed field of characteristic 1 is expected to have. Then I will explain how these properties forces one to follow the route of Hrushovski's construction leading to a a 'pseudo-analytic' structure which we identify as an algebraically closed field of characteristic 1 . Then I am able to formulate very precise axioms that such a field must satisfy.  The main theorem then states that under the axioms the structure has the desired algebraic and analytic properties. The axioms have a form of statements about existence of solutions to systems of equations in terms of a 'multi-dimensional' valuation theory and the validity of these statements is an open problem to be discussed. 
This is a joint work with Alex Cruz Morales.
 

1.Kremnitzer. I will explain an approach to constructing geometries relative to a symmetric monoidal 
category. I will then introduce the category of normed sets as a possible analytic geometry over 
the field with one element. I will show that the Fargues-Fontaine curve from p-adic Hodge theory and 
the Connes-Bost system are naturally interpreted in this geometry. This is joint work with Federico Bambozzi and 
Oren Ben-Bassat.