There will be three mini lecture courses, by Charles Doran, Arno Kret and Colleen Robles. In addition we will have a few research talks, and question and answer sessions. In between, there will be time for discussions.

- Elliptic curves, K3 surfaces, and Calabi-Yau threefolds.
- Variations of Hodge structure and Picard-Fuchs ODE.
- Modular parametrization for lattice-polarized K3 surfaces.
- Calabi-Yau fibrations and degenerations.
- Mirror symmetry: old and new.

- Automorphic forms and Arthur's conjecture.
- The paper of Buzzard-Gee.
- Shimura varieties.
- Construction of Galois representions using the above; (part of) final lecture will be about my work with Shin on $GSp$ and $GSO$.

If you are thinking of preparing for this mini-course before the school, you could look at Taylor’s ICM paper on Galois representations or at the paper by Buzzard and Gee, The conjectural connections between automorphic representations and Galois representations.

- Introduction and motivation
- Polarized mixed Hodge structures and review of Schmid’s nilpotent orbit theorem.
- The Satake-Baily-Borel compactification of locally Hermitian symmetric spaces, and the Ash-Mumford-Rapoport-Tai toroidal desingularization.
- Generalizing SBB to arbitrary period maps.
- Generalizing AMRT to arbitrary period maps.

Generalizing finiteness theorems of Parshin, Arakelov, and Faltings, Deligne proved in 1987 that only finitely many $\mathbb Z$-local systems of a fixed rank underlie a polarized variation of Hodge structure, over a fixed quasiprojective variety. Deligne conjectured that an appropriate form of this finiteness also holds in families of quasiprojective varieties. In the 1990's, Simpson's refined this conjecture in the following form: the nonabelian Hodge locus is algebraic. I will discuss joint work with Salim Tayou that these conjectures are true when the algebraic monodromy group is cocompact.

I will report on a joint work with E. Colombo and G.P. Pirola, where we study asymptotic directions in the tangent bundle of the moduli space of curves of genus $g$, namely those tangent directions that are annihilated by the second fundamental form of the Torelli map. I will give examples of asymptotic directions for any $g$ at least $4$. I will show that if the rank $d$ of a tangent direction at $[C]$ (with respect to the infinitesimal deformation map) is less than the Clifford index of the curve $C$, then the tangent direction is not asymptotic. If the rank of a tangent direction $v$ is equal to the Clifford index of the curve, I will give sufficient conditions ensuring that the infinitesimal deformation $v$ is not asymptotic. Finally I will determine all asymptotic directions of rank $1$ and give an almost complete description of asymptotic directions of rank $2$.

On a Shimura variety, we have a collection of natural vector bundles, called automorphic bundles, which are constructed from group representations. The tautological ring of the Shimura variety is defined to be the ring generated by all Chern classes of these automorphic bundles; this may be considered on the level of cohomology, but also on more refined levels, such as the Chow ring. The construction extends to (sufficiently nice) toroidal compactifications of Shimura varieties. In my talk, I will explain that there is a very beautiful conjectural description of this tautological ring. I will present several examples and will explain what is the problem that remains open. If time permits, I will also explain why we know more in characteristic p>0; this is based on the theory of G-zips (about which I will not assume any knowledge).

I will sketch a modestly practical algorithm to compute all linear relations with algebraic coefficients between any given finite set of 1-periods. As a special case, we can algorithmically decide transcendence of 1-periods. This is based on the "qualitative description" of these relations by Huber and Wüstholz. We combine their result with the recent work on computing the endomorphism ring of abelian varieties. This is a work in progress with Jöel Ouaknine (MPI SWS) and James Worrell (Oxford).