13:45 - 14:45 Tom Biesbrouck (KU Leuven)
15:15 - 16:15 Luka Bizjak (Université Libre de Bruxelles)
16:45 - 17:45 Aliaksandra Novik (Imperial College London)
Birational Zeta Functions - Tom Biesbrouck (KUL)
The motivic zeta function associated with a complex hypersurface is a rich invariant within singularity theory. It remains however quite mysterious, e.g. the cancellation of so-called 'candidate poles' is still not well-understood. In joint work with Nero Budur, Johannes Nicaise and Wim Veys, we define a birational analog of the motivic zeta function of a reduced polynomial in terms of minimal models, which one would hope is easier to handle. It admits an intrinsic meaning in terms of contact loci of arcs, an analog of a result of Denef and Loeser in the motivic case. For local plane curve singularities the poles of the birational zeta function essentially coincide with the poles of the motivic zeta function.
K-motivic sheaves and toric geometry - Luka Bizjak (ULB)
For an algebraic variety X over a field k equipped with an action of an algebraic group G over k, one can consider the category of motivic G-spectra over X, which is an algebraic analogue of the equivariant stable homotopy category from topology. This category contains the G-equivariant algebraic K-theory spectrum K_{G}(X) of X, which leads to a category of equivariant K-motives on X obtained by considering objects together with their K_{G}-module structure. However, these categories are defined using fairly abstract categorical constructions and are therefore difficult to describe explicitly. I will show that for toric varieties, under suitable assumptions, equivariant K-motives admit a combinatorial description in terms of the associated fans. If time permits, I will also discuss motivic Koszul duality for affine toric varieties over C.
Coherent systems on curves and stability conditions - Aliaksandra Novik (ICL)
Stability conditions on derived categories have become a powerful tool for extracting geometric information about an underlying variety via wall-crossing phenomena. However, for a smooth projective curve C of genus greater than 0, Macrì showed that the space of stability conditions on D^b(C) is trivial up to a certain group action. As a consequence, there is no room to deform stability conditions and apply wall-crossing techniques.