# Another Sandbox

Feel free to use this page for any experiments that you would like to save.

LaTeX test: If $$a > 0$$, then $$a^2 > 0$$.

experiment with tables:

 September 9 (Monday!), 4pm Hendrik Süß (Edinburgh) By Klyachko's work there is an equivalence of categories between equivariant vector bundles on toric varieties and families of vector space filtrations. In this talk I will discuss an generalization of this equivalence to bundles on varieties $$X_3$$ with smaller torus actions. Now, vector space filtrations are replaced by filtrations of vector bundles on some quotient space. This description comes with a nice splitting criterion and allows to prove that vector bundles of low rank on projective space, which are equivariant with respect to special subtori of the maximal acting torus must split. September 9 (Monday!), 5pm Francois Petit (Edinburgh) Fourier-Mukai transform in the quantized settingAfter reviewing some elements of the theory of Deformation Quantization modules (DQ-modules), I will show that a coherent DQ-kernel induces an equivalence between the derived categories of coherent DQ-modules if and only if the graded commutative kernel associated to it induces an equivalence between the derived categories of coherent O-modules. September 19, 4 pm Evgeny Shinder (Edinburgh) Exceptional collections on fake projective planesFake projective planes are surfaces of general type with the same rational cohomology as a projective plane CP^2. These surfaces appear as quotients of a complex 2-ball by an arithmetic subgroup of PU(2,1) and are completely classified into finitely many isomorphism classes. We study the derived category of coherent sheaves on fake projective planes. For fake projective planes with an automorphism group of order 21 we show that O, O(-1), O(-2) form an exceptional collection in the derived category. The orthogonal complement to this collection provides an example of a quasi-phantom (a category with vanishing Hochschild homology). September 19, 5pm Ciaran Meachan (Edinburgh) TBA October 3 GLEN in Liverpool October 4, 9:30 am - 6:00 pm Maxwell colloquium on Combinatorial Algebraic Geometry November 7 Gregory Sankaran (Bath) TBA

In my talk I will construct a functor from some class of skew fields to some class of commutative fields with $$h_3$$ map
$$Q C x, x \to Q C x, y$$
$$Q C[x, \partial_x] \to Q C[x, y]$$ as the main example. I will also provide some link between this construction and some known results and conjectures on skew fields.

By Klyachko's work there is an equivalence of categories between equivariant vector bundles on toric varieties and families of vector space filtrations. In this talk I will discuss an generalization of this equivalence to bundles on varieties $$X_3$$ with smaller torus actions. Now, vector space filtrations are replaced by filtrations of vector bundles on some quotient space. This description comes with a nice splitting criterion and allows to prove that vector bundles of low rank on projective space, which are equivariant with respect to special subtori of the maximal acting torus must split.