Mini-workshop in Algebraic Geometry and Singularities

Mini-workshop in Algebraic Geometry and Singularities

Funded by ERC-NMST

BCAM-Basque Center for Applied Mathematic, Mazarredo 14, Bilbao, Basque Country, Spain


9:30-10:30 Alberto CASTAÑO (Chemnitz)

Hypergeometric D-modules, Hodge theory and statistics

D-modules have a big potential to be used as a powerful tool in other fields, such as representation theory, arithmetic geometry, mirror symmetry, Hodge theory or even statistics. In this talk I will explain two ongoing projects regarding the last two items. More concretely, on one hand, I will expound a joint work with Christian Sevenheck in which we have computed the irregular Hodge filtration of some irregular hypergeometric D-modules, arising in some geometrical contexts, by using recent and deep works of Mochizuki and Sabbah on mixed twistor modules. On the other, I will introduce the so-called Holonomic Gradient Method, a way to overcome some problems on statistics and optimization using D-modules, and if time permits, I will comment on a common project with Christian Sevenheck, Bernd Sturmfels, Carlos Améndola and Mateusz Michałek on some optimization problems that can be tackled using GKZ-hypergeometric systems.

10:45-11:45 Mathieu KLIMCZAK (Nantes)

Mixed Hodge structures and intersection spaces 

The construction of intersection spaces provide a way to restore Poincaré duality on singular spaces. Due to their method of construction, these spaces aren't algebraic varieties, even if the singular spaces they originate from are. Thus at first glance there should be no reason the rational cohomology of intersection spaces could be endowed with a mixed Hodge structure. However, Markus Banagl and Laurentiu Maxim showed that under some quite restrictive conditions this is the case. Using rational homotopy tools, we will show we can overcome these restrictions to endow with canonical mixed Hodge structures the rational cohomology of any intersection spaces coming from a complex projective algebraic variety with isolated singularities. We will then use these mixed Hodge structures to derive results about the rational homotopy type of intersection spaces and the formality of such spaces.


12:00- 13:00 Rory POTTER (Sheffield)

Equivariant Derived Categories and Godeaux Surfaces with Involutions

Motivated by examples of Godeaux surfaces with involutions studied by Calabri-Ciliberto-Mendes-Lopez and Mendes-Lopez-Pardini, we aim to study the equivariant derived category of a Godeaux surface with respect to an involution. We describe how we package the birational type of the quotient and the properties of the ramification divisor using the equivariant derived category and semi-orthogonal decompositions. We then example how they are constructed by studying the geometry of the associated quotient stack.

Lunch

15:00-16:00 Rene MBORO (Zurich)

On some botational invariants of projective varieties

We present some birational invariants relevant for rationality questions and present a
computation of these invariants in some cases; namely for some hypersurfaces and variety of lines of hypersurfaces. We will be especially interested in the Griffiths group Griff 1 , torsion Cycles annihilated by Deligne cycle map and the decomposition of the diagonal.

16:15-17:15 Marti LAHOZ (Paris)

Cubic fourfolds and non-commutative K3 surfaces


The derived category of coherent sheaves on a smooth cubic fourfold has a subcategory that can be thought as the derived category of a non-commutative K3 surface. This category has recently been studied by Kuznetsov and Addington-Thomas, among others. In this talk, I will present joint work with Bayer, Macro, and Stellari about the construction of Bridgeland stability conditions on this category.


21:00 Dinner

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