♠
19.04.2016 // An overview of projective normal toric varieties
We start by giving the abstract definition of a toric variety, then explain how to construct affine toric varieties from cones and normal toric varieties from fans. We will then focus on projective normal toric varieties and their toric divisors and finally give an explicit description of their divisor class group.

♠
26.04.2016 // Toric geometric invariant theory
We will prove that every projective normal toric variety is constructed as a GIT quotient of a diagonalisable group acting on an affine space which is linearised by a character. In the first part of this talk, we explain how to construct GIT quotients of affine spaces with respect to a character and, in the second part of the talk, we give the proof of the above result.

♠
03.05.2016 // An introduction to symplectic geometry  Maik Pickl
We introduce symplectic manifolds and some examples (symplectic vector spaces, the cotangent bundle of a manifold and complex projective spaces). We introduce two types of morphisms in symplectic
geometry: symplectomorphisms (which are diffeomorphisms which respect the symplectic form) and Lagrangian correspondences (which are defined using Lagrangian submanifolds in the product).

♠
10.05.2016 // Group quotients in symplectic geometry
We consider the construction of symplectic quotients of actions of compact groups in symplectic geometry; this is known as symplectic reduction and is performed by taking a topological quotient of a level set of a moment map for the action. We prove the MarsdenWeinsteinMeyer Theorem, which states that if the group acts freely on this level set, then the reduction is a symplectic manifold.

♠
17.05.2016 // The KempfNess theorem
We prove the KempfNess theorem which states that the GIT quotient of a reductive group acting linearly on a smooth projective variety X over the complex numbers is homeomorphic to the symplectic reduction of the action of the maximal compact subgroup. The main step is to prove that an orbit meets the zero level set of the moment map if and only if the orbit is GIT polystable.

♠
24.05.2016 // The BialynickiBirula decomposition
Given an action of the multiplicative group on a smooth projective variety X, we prove that there are two decompositions of X into locally closed smooth subvarieties which are both indexed by the connected components of the fixed locus for the action. We explicitly describe the strata as locally trivial affine fibrations over the fixed loci.

♠
31.05.2016 // Variation of GIT for the multiplicative group
For a linear action of the multiplicative group on an affine variety, we describe the variation of GIT quotients as we twist by a character of the multiplicative group. There are three cases to consider: the trivial character, a positive character and a negative character. We use the BialynickiBirula decomposition to describe the morphisms from the positive (resp. negative) GIT quotient to the zero GIT quotient.

♠
07.06.2016 // Toric quotients and flips  Marta Pieropan
We study variation of GIT for the action of a subtorus of the torus for a toric variety associated to a polyhedron. The space of linearisations giving nonempty GIT quotients is closed related to the polyhedron defining this toric variety and on it we describe a polyhedral wall and chamber decomposition, and the birational transformations given by wallcrossings.

♠
14.06.2016 // No Seminar

♠
21.06.2016 // The space of linearisations

♠
28.06.2016 // Wallcrossings in variation of GIT

♠
5.07.2016 // The (Zariski local) description of the morphism to the wall

♠
12.07.2016 // The (étale local) description of the morphism to the wall

♠
19.07.2016 // Mori dream spaces and GIT  Marta Pieropan
