We'll discuss our joint work with Ron Donagi and Tony Pantev on the construction of the Higgs bundles associated to Hecke eigensheaves for the geometric Langlands program in the case of rank 2 local systems on a curve of genus 2 . Recall that the moduli space of bundles in this case has two connected components: $\mathbb{P}^3$ and the intersection of two quadrics in $\mathbb{P}^5$. We look for Higgs bundles on these spaces with parabolic structure and logarithmic poles along the wobbly locus. This leads to the study of the geometry of the wobbly locus and its singularities, and the use of our Dolbeault higher direct image construction for the calculation of Hecke operators.[-]

The Bézier representation of homogenous polynomials has little and not the usual geometric meaning if we consider the graph of these polynomials over the sphere. However the graph can be seen as a rational surface and has an ordinary rational Bézier representation. As I will show, both Bézier representations are closely related. Further I consider rational spline constructions for spherical surfaces and other closed manifolds with a projective or hyperbolic structure.[-]

The Hopf algebra of Lie group integrators has been introduced by H. Munthe-Kaas and W. Wright as a tool to handle Runge-Kutta numerical methods on homogeneous spaces. It is spanned by planar rooted forests, possibly decorated. We will describe a canonical surjective Hopf algebra morphism onto the shuffle Hopf algebra which deserves to be called planar arborification. The space of primitive elements is a free post-Lie algebra, which in turn will permit us to describe the corresponding co-arborification process.
Joint work with Charles Curry (NTNU Trondheim), Kurusch Ebrahimi-Fard (NTNU) and Hans Z. Munthe-Kaas (Univ. Bergen).
The two triangles appearing at 24'04" and 25'19'' respectively should be understood as a #.[-]

We show that the spectrum of fundamental particles of matter and their symmetries can be encoded in a finite quantum geometry equipped with a supplementary structure connected with the quark-lepton symmetry. The occurrence of the exceptional quantum geometry for the description of the standard model with 3 generations is suggested. We discuss the field theoretical aspect of this approach taking into account the theory of connections on the corresponding Jordan modules.[-]

In this talk we shall discuss our recent results on the Adams' conjecture on theta correspondence. In more words, given a representation of a classical group (in our case, symplectic or even orthogonal) belonging to a local Arthur packet, Adams predicts that, under certain assumptions, its theta lift (i.e. a corresponding irreducible representation of the other group in a dual reductive pair), provided it is non-zero, is also in A-packet which can be easily described in terms of the original one. Mœglin gave some partial results, specifically, in case when the original representation is square-integrable. We are able to extend her results to the case of so called Arthur packets with the discrete diagonal restriction. Moreover, it seems that Arthur packet encapsulates lot of additional information even in relation to theta correspondence, e.g. we can easily read of from it the first occurrence index for the given representation in it. Adams conjecture takes an unexpectedly elegant form for the representations in discrete diagonal restriction packets. Also, we are able to pinpoint exactly how low in theta towers we can go with this description of the theta lifts which belong to Arthur packets, we can also address some other related conjectures due to Mœglin. This is joint work with Petar Baki.[-]

Matrix spherical functions associated to the symmetric pair $(G, K)=$ $\left(\mathrm{SU}(m+2), \mathrm{S}(\mathrm{U}(2) \times \mathrm{U}(m))\right.$, having reduced root system of type $\mathrm{BC}_{2}$ are studied. We consider a $K$-representation $\left(\pi, V_{\pi}\right)$ arising from the $\mathrm{U}(2)$-part of $K$, then the induced representation $\operatorname{Ind}_{K}^{G} \pi$ is multiplicity free. The corresponding spherical functions, i.e. $\Phi: G \rightarrow \operatorname{End}\left(V_{\pi}\right)$ satisfying $\Phi\left(k_{1} g k_{2}\right)=\pi\left(k_{1}\right) \Phi(g) \pi\left(k_{2}\right)$ for all $g \in G, k_{1}, k_{2} \in K$, are studied by studying certain leading coefficients. This is done explicitly using the action of the radial part of the Casimir operator on these functions and their leading coefficients. To suitably grouped matrix spherical functions we associate two-variable matrix orthogonal polynomials giving a matrix analogue of Koornwinder's 1970 s two-variable orthogonal polynomials, which are Heckman-Opdam polynomials for $\mathrm{BC}_{2}$. In particular, we find explicit orthogonality relations and the polynomials being eigenfunctions to a second order matrix partial differential operator. This is joint work with Jie Liu (Radboud $\mathrm{U}$ ).[-]