 Adrian Langer  On smooth projective Daffine varieties

Let D be the ring of differential operators on an algebraic variety. A variety is Daffine if Dmodules on this variety look like modules on an affine variety.
I will review some results concerning smooth projective Daffine varieties. Although the only known projective examples are some smooth homogeneous spaces, it seems rather difficult to check that a given variety is not Daffine.
One of the main aims of the talk is to give some criteria allowing to classify Daffine varieties in low dimensions.
 Kei Yuen Chan  Construction of simple quotients of BernsteinZelevinsky derivatives^{3}

The BernsteinZelevinsky derivative is originally introduced in classifying irreducible representations of general linear groups. A sequence of derivatives of generalized Steinberg representations can be used to construct simple quotients of BernsteinZelevinsky derivatives. In this talk, I shall begin with introducing BernsteinZelevinsky derivatives, and explain the analogous formulation on affine Hecke algebra of type A (joint work with Savin). Then I will explain how to use another derivatives introduced by C. Jantzen and independently by Minguez (which is related to crystal operators in quantum groups) to construct simple quotients of BernsteinZelevinsky derivatives.
 Lee Andrew Jenkins  On the geometry of nilpotent orbits for classical simple Lie superalgebras

Many aspects of the representation theory of a Lie algebra and its associated algebraic group are governed by the geometry of their nilpotent cone. In this talk, we will introduce an analogue of the nilpotent cone N for Lie superalgebras and show that for a simple classical Lie superalgebra the number of nilpotent orbits is finite. We will also show that the commuting variety X described by Duflo and Serganova, which has applications in the study of the finite dimensional representation theory of Lie superalgebras, is contained in N. Consequently, the finiteness result on N generalizes and extends the work on the commuting variety. For the general linear Lie superalgebra gl(mn), we will also discuss more detailed geometric results of N. In particular, we compute the dimensions of N and the centralizer of a nilpotent orbit, describe the irreducible components of N, and show that N is a complete intersection. This is joint work with Daniel Nakano from the University of Georgia.
 Thomas Gerber  Generalised symplectic Howe duality

The Howe duality is a classical result in Lie theory, which can be expressed as an identity between fundamental tensor multiplicities for g_{n}representations on the one hand and weight multiplicities for g_{m}representations on the other hand. Here g is a simple complex Lie algebra of fixed classical type (A, B, C or D) and n, m is the rank of the Lie algebra. In this talk, I will present a bijective proof of the above identity in the type A and C cases, relying on the combinatorics of tableaux and crystals. More precisely, weight multiplicities are counted by certain tableaux and fundamental tensor multiplicities are counted by sources in certain crystal graphs which we can explicitly put into onetoone correspondence.
If time allows, I will explain how to generalise the Howe duality in type C to multiplicities of tensor products of (restrictions of) simple modules (instead of fundamental tensor multiplicities) on the one hand and branching multiplicities (instead of weight multiplicities) on the other hand. From this, one can deduce that inducing to sp_{2n} from a Levi subalgebra is injective. This is joint work with J. Guilhot and C. Lecouvey.
 Ada Boralevi (Turin): A construction of equivariant bundles on the space of symmetric forms

I will report on a joint work with D. Faenzi and P. Lella, where we give a new construction of indecomposable vector bundles on the space of symmetric forms of degree d in n+1 variables, which are equivariant for the action of the complex Lie group SL_{n+1}, and moreover admit an equivariant free resolution of length 2. For n=1, we obtain new examples of indecomposable vector bundles of rank d1 on P^{d}, which are moreover equivariant for SL_{2}. The presentation matrix of these bundles attains the bound for matrices of linear forms of constant rank for that size and rank.
 Simone Marchesi  Generalized logarithmic sheaves

In this talk we will introduce a generalization of the definition of logarithmic tangent sheaf on a smooth surface associated to a pair (D,Z), being Z a set of fixed points of a divisor D. Furthermore, we will study when the pair (D,Z) can be recovered from the logarithmic sheaf, a property that is referred to as Torelli property. This is a joint work with S. Huh, J. PonsLlopis and J. Vallès.
 Michał Kapustka  Quaternary quartics and Gorenstein rings

By apolarity a quartic polynomial in four variables corresponds to an Artinian Gorenstein ring of regularity 4. I will present a classification of quartic polynomials in terms of rank and powersum decompositions according to the Betti table of the corresponding Artinian Gorenstein ring. This gives a stratification of the space of quartics which is incompatible with that given by NoetherLefschetz loci. We will then discuss powersum varieties for a general form in each stratum. Finally, we will provide various explicit constructions of general Artinian Gorenstein rings corresponding to each stratum and discuss their lifting to higher dimension. These provide constructions of codimension four varieties in various dimensions which include canonical surfaces, CalabiYau threefolds and Fano fourfolds.
 Leonardo Patimo  Precanonical Bases of Hecke Algebras

By apolarity a quartic polynomial in four variables corresponds to an Artinian Gorenstein ring of regularity 4. I will present a classification of quartic polynomials in terms of rank and powersum decompositions according to the Betti table of the corresponding Artinian Gorenstein ring. This gives a stratification of the space of quartics which is incompatible with that given by NoetherLefschetz loci. We will then discuss powersum varieties for a general form in each stratum. Finally, we will provide various explicit constructions of general Artinian Gorenstein rings corresponding to each stratum and discuss their lifting to higher dimension. These provide constructions of codimension four varieties in various dimensions which include canonical surfaces, CalabiYau threefolds and Fano fourfolds.
 Bea Schumann  Optimality of String Cone Inequalities and Potential Functions on Cluster Varieties

By apolarity a quartic polynomial in four variables corresponds to an Artinian Gorenstein ring of regularity 4. I will present a classification of quartic polynomials in terms of rank and powersum decompositions according to the Betti table of the corresponding Artinian Gorenstein ring. This gives a stratification of the space of quartics which is incompatible with that given by NoetherLefschetz loci. We will then discuss powersum varieties for a general form in each stratum. Finally, we will provide various explicit constructions of general Artinian Gorenstein rings corresponding to each stratum and discuss their lifting to higher dimension. These provide constructions of codimension four varieties in various dimensions which include canonical surfaces, CalabiYau threefolds and Fano fourfolds.