Given a quiver with potential (Q,W) there are two associated n-dimensional complex manifolds: the space of stability conditions on the derived category of (Q,W), and the cluster variety of Q. These two spaces are related in an interesting way. I will explain this relationship in the examples arising from triangulations of surfaces, focusing on the simplest example - the A_2 quiver.

**Paolo Cascini** (Imperial): Birational geometry in positive characteristic

Many of the results in the Minimal Model Program over complex projective varieties depend on the Kodaira vanishing theorem and its generalisations. Because of the failure of these tools in positive characteristic, many of these results are still open in this case. I will survey some recent progress in the study of birational geometry of projective varieties defined over an algebraically closed field of positive characteristic.

**Mark Gross** (Cambridge): Degeneration formulas for logarithmic Gromov-Witten invariants

I will talk about joint work with Abramovich, Chen and Siebert aiming at a gluing formula for log Gromov-Witten invariants. Given a degeneration of varieties to a normal (or toroidal) crossings variety, the Gromov-Witten invariants of the general fibre are the same as the log Gromov-Witten invariants of the special fibre, the latter defiend by Gross-Siebert and by Abramovich-Chen. We show that in such a situation, the moduli space of stable log maps to the special fibre has a virtual decomposition into irreducible components indexed by rigid tropical curves.

**Anne-Sophie Kaloghiros** (Brunel): Volume preserving maps between uniruled log Calabi-Yau pairs

This talk will consider pairs (X, Delta_X) consisting of a Mori fibre space X and a reduced anticanonical section Delta_X, such that (X, Delta_X) is a log Calabi--Yau (it is an endproduct of the log-MMP). A birational map between two such pairs is called volume preserving, if the log discrepancies agree for every divisor E over X. I will discuss the decomposition of volume preserving birational maps into volume preserving Sarkisov links, and give some examples.

**Jonathan Pridham** (Edinburgh): Motives and derived Tannaka duality

Given a k-valued cohomology functor on the derived category of motives, Ayoub constructed a motivic Galois group as a Hopf algebra in the derived category of k, satisfying a weak universal property. I will explain how to strengthen this, recovering the derived category of motives from the Galois group. The key tool is a generalisation of Tannaka duality to dg coalgebras and dg categories, via derived Morita theory.

**Orsola Tommasi** (Hannover): Stable cohomology of toroidal compactifications of the moduli space of abelian varietiesIt is well known that the cohomology of the moduli space A_g of g-dimensional principally polarized abelian varieties stabilizes when the degree is smaller than g. This is a classical result of Borel on the stable cohomology of the symplectic group. By work of Charney and Lee, also the stable cohomology of the minimal compactification of A_g, the Satake compactification, is explicitly known.

In this talk, we consider the stable cohomology of toroidalcompactifications of A_g, concentrating on the perfect cone compactification and the matroidal partial compactification. We prove stability results for these compactifications and show that all stable cohomology is algebraic. This is joint work with Sam Grushevsky and Klaus Hulek.

**Claire Voisin** (Ecole Polytechnique): Decomposition of the diagonal and new stable birational invariants

We investigate new stable birational invariants which are possibly nontrivial for certain unirational varieties, thus allowing to detect further irrational or non-stably rational varieties which are unirational. These invariants are contained in the notion of Chow or cohomological decomposition of the diagonal.

**Geordie Williamson** (Bonn): Kazhdan-Lusztig conjectures and shadows of Hodge theory

Hodge theory provides deep structure on the cohomology ring of smooth projective varieties. Remarkably, shadows of this structure are present in certain representation theoretic objects ("Soergel bimodules") which (as far as we know) do not always have origins in geometry. I will discuss joint (ongoing) work with Ben Elias trying to make these structures explicit. A consequence of this work are new proofs of some conjectures of Kazhdan and Lusztig from 1979. I also hope to touch on the de Cataldo and Milgliorini's Hodge theoretic proof of the decomposition theorem, which was the major source of motivation for this work.