Our project involves tracking moving spots in a pattern as they evolve on a sphere. The spots are the result of numerical simulations of the Brusselator reaction-diffusion system. We use clustering algorithms from machine learning to extract the location of the spots from a large point cloud of time-dependent data. We then create traces showing the paths taken by the spots as they evolve.

Eventually, we hope to use our algorithm to help study the dynamics of splitting spots on a growing domain where spots are competing for space.

We aim to characterize the function space $\mathcal{D}(\Sigma)$, the set of all the holomorphic functions defined on $\Sigma$ of bounded Dirichlet norm, and the set of its boundary values on $\Gamma$.

In this poster, we will also characterize all the functions $\mathcal{D}_{harm}(\Sigma)$ that carry the jump decomposition. We then introduce an operator $I_f$, called Faber operator, between the space $\mathcal{D}(\Sigma)$ and a quotient space of $\overline{\mathcal{D}(\mathbf{D})}$ (the set of all anti-holomorphic functions on \textbf{D} of bounded Dirichlet norm) and show that under what circumstances $I_f$ is an isomorphism. The Faber operator generalizes the Faber operator of approximation theory of holomorphic functions on planar domains to Riemann surfaces. We will construct another operator $Gr_f$, called Grunsky operator, which can be interpreted as a generalization of a period matrix of a genus $g$ compact Riemann surface. The Dirichlet space is then characterized as the graph of this operator.

Finally, the connection between anti-holomorphic $1-$forms on $\Omega$ and holomorphic $1-$forms on $\Sigma$ and the important role of Schiffer operator will also be discussed.

Joint work with E. Schippers and W. Staubach

This work builds upon advances made by Baldoni, Vergne, and Walter (2016) who showed that, for fixed parameter sizes, the Kronecker coefficients are piecewise quasi-polynomials. They also provided theoretical computation methods, which are unfortunately too complex to be implemented in practice. More recently, Mishna, Rosas and Sundaram have found a way to reduce the complexity by linking the coefficients more directly to the number of integer points in specific polyhedra. This interpretation opens the door towards a complex analytic approach. We utilise techniques from the relatively new field of multivariate analytic combinatorics (described in 'Analytic Combinatorics in Several Variables' by Pemantle and Wilson, 2013) in order to determine asymptotic formulas for the Kronecker coefficients. Such formulas yield important information about the coefficients, notably, determining where the Kronecker coefficients are 0.

We present a new model for the edge motion of stable non-precipitating clouds. Our proposed model combines buoyancy driven fluid mechanics and the thermodynamic theory for phase change of water into a self-consistent Boussinesq theory. Further approximation including linearization of the thermodynamics about a cloud forming state and leveraging the low water content of the atmosphere leads to an analytically tractable free boundary model for cloud edge motion. Exact solutions to this model then serve as a theoretical benchmark against which full physics numerical models can be compared. In particular, we present results from the cm1 model (cloud model 1) developed by George Bryan out of the National Center for Atmospheric Research in Boulder Colorado. We finish by presenting an analytically calculated travelling wave solution on the clear/cloud interface for which the dispersion relation is derived and compared.