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# NoGAGS 2017

## Speakers

 Marian Aprodu (U Bucharest) Mateusz Michalek (MPI Leipzig) Sebastian Casalaina-Martin (U Colorado) Simon Pepin Lehalleur (FU Berlin) Thomas Krämer (HU Berlin) Michel van Garrel (U Hamburg) Victor Lozovanu (U Hannover)

## Registration

For registration please use the webform.

## Participants

full list of participants (last update: Thu, 16.11.2017 09:53:02)

## Programme

 Thursday, 16th November 01:00 pm Mateusz Michalek (MPI Leipzig) Real rank decompositions and their moduli 02:00 pm Coffee break 02:30 pm Michel van Garrel (U Hamburg) Comparing local and log GW invariants 03:30 pm Break 04:00 pm Marian Aprodu (U Bucharest) Green’s conjecture and vanishing of Koszul modules Friday, 17th November 09:30 am Thomas Krämer (HU Berlin) Holonomic D-modules on abelian varieties: From geometry to representation theory 10:30 am Coffee break 11:00 am Victor Lozovanu (U Hannover) Convex geometry and local positivity of an ample line bundle 12:00 pm Lunch 01:30 pm Sebastian Casalaina-Martin (U Colorado) The moduli space of cubic threefolds via intermediate Jacobians 02:30 pm Simon Pepin Lehalleur (FU Berlin) On the motive of the stack of vector bundles on a curve

## Abstracts

Marian Aprodu — Green’s conjecture and vanishing of Koszul modules

I report on a joint work in progress with G. Farkas, S. Papadima, C. Raicu and J. Weyman. Koszul modules are multi-linear algebra objects associated to an arbitrary subspace in a second exterior power. They are naturally presented as graded pieces of some Tor-s over the dual exterior algebra. Koszul modules appear in the Geometric Group Theory, in relations with Alexander invariants of groups. We prove an optimal vanishing result for the Koszul modules, and we describe explicitly the locus corresponding to Koszul modules that are not of finite length. To this end, we produce and verify a degenerate version of Green’s conjecture that holds for cuspidal curves and we use representation theory to connect the syzygies of rational cuspidal curves to some appropriate Koszul modules, called Weyman modules. We apply our vanishing result to Alexander invariants.

Sebastian Casalaina-Martin — The moduli space of cubic threefolds via intermediate Jacobians

Associating to a smooth cubic threefold its principally polarized intermediate Jacobian induces a rational period map from the GIT moduli space of cubic threefolds to the second Voronoi compactification of the moduli space of five dimensional principally polarized abelian varieties. In this talk I will describe a resolution of the period map, which allows for a geometric description of the boundary of the moduli space of intermediate Jacobians. This is joint work with Samuel Grushevsky, Klaus Hulek, and Radu Laza.

Thomas Krämer — Holonomic D-modules on abelian varieties: From geometry to representation theory

To any holonomic D-module on an abelian variety one may attach an algebraic group via the Tannakian formalism of convolution. Applying this to the intersection homology D-module of a closed subvariety, one obtains a bridge between geometry and representation theory: The highest weight theory of the arising groups is related to geometric topics such as Gauss maps, subvarieties dominated by a product of varieties, singularities of theta divisors, the Schottky problem, second order theta functions etc. In this talk, I will report on recent progress concerning these relations.

Victor Lozovanu — Convex geometry and local positivity of an ample line bundle

Seshadri constants measure the local positivity of an ample line bundle at a fixed point. Due to this, they show up in many areas of mathematics ranging from Kähler to algebraic or arithmetic geometry. In this talk I will try to explain how can one study these invariants using convex geometry. As a consequence of these methods, thus having local information, and singularity theory can lead to understanding of global properties on abelian varieties in terms of data on abelian submanifolds.

Mateusz Michalek — Real rank decompositions and their moduli

For a homogeneous polynomial $P$ it is a classical problem to understand its presentations as a sum of powers of linear forms $P=\sum_{i=1}^r l_i^r$. The smallest possible $r$ is known as the Waring rank of $P$. Over the reals, we may have a family of Euclidean open sets with consecutive Waring ranks. We will describe the geometry of these sets in special cases. Further, for a fixed $P$ we may have many Waring decompositions. Over the complex numbers their moduli - the Variety of Sums of Powers - has been studied by Mukai, Ranestad, Schreyer and others. The real decompositions correspond to a semialgebraic subset of the real part of the VSP. We will present several results concerning these subsets from a joint work with Moon, Sturmfels and Ventura.

Simon Pepin Lehalleur — On the motive of the stack of vector bundles on a curve

Following Grothendiecks vision that many cohomolgical invariants of of an algebraic variety should be captured by a common motive, Voevodsky introduced a triangulated category of mixed motives which partially realises this idea. After describing this category, I will explain how to define the motive of certain algebraic stacks in this context. I will then report on joint work in progress with Victoria Hoskins, in which we study the motive of the moduli stack of vector bundles on a smooth projective curve and show that this motive can be described in terms of the motive of this curve and its symmetric powers.

Michel van Garrel — Comparing local and log GW invariants

Let X be a smooth projective variety and let D be a smooth nef divisor on it. In this collaboration with Tom Graber and Helge Ruddat, we show that the genus 0 local Gromov-Witten (GW) invariants of the total space of O(-D) equal, up to a factor, the genus 0 log GW invariants of X with a single condition of maximal contact order along D.

## Venue

The lectures on the 16th of November will take place in Johann von Neumann building Rudower Chaussee 25 room 1.115 of the Humboldt University whereas those on the 17th of November will take place in the Erwin Schroedinger centre Rudower Chaussee 26 room 0.307.

## Dinner

There will be a dinner on Thursday evening at 19:00 at 12 Apostel Berlin Mitte.

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