After recalling the definition of Grassmann tensors, we will describe the varieties parametrizing them. In particular, we will give some contributions in the case of bifocal tensors and discuss some generalizations for trifocal tensors. The results presented in this talk, and those in progress, are obtained in collaboration with Marina Bertolini and Cristina Turrini.

Strassen's conjecture is a famous statement first uttered by the German mathematician Volker Strassen in 1973. He conjectured in his article Vermeidung von Divisionen that the multiplicative complexity of the union of two linear systems in two disjoint sets of variables is equal to the sum of the complexities of the two systems. Translated: he conjectured that the rank of the direct sum of two tensors is in general equal to the sum of the ranks of the two tensors. The problem of Strassen's conjecture remained unsolved until 2019, when Yaroslav Shitov proved in his article Counterexamples to Strassen's direct sum conjecture that this conjecture is in fact false, constructing a procedure to find a counterexample for tensors in a large number of variables. We propose to get to the core of this proof and go over the procedure constructed by Yaroslav Shitov, analyzing it in detail and providing graphical representations.

If Z is a set of four general points in projective 3-space then a general projection of Z to a general plane is clearly the complete intersection of two conics. If |Z|=ab and Z lies on a smooth quadric surface Q and is the intersection of a lines in one ruling and b lines in the other ruling then again it is clear that a general projection of Z is a complete intersection, now of type (a,b). Beyond this, examples are very far from being obvious. But they exist, and we say that Z is (a,b)-geproci if a GEneral PROjection is the Complete Intersection of a curve of degree a and one of degree b. A great deal of work has, at this point, gone into the study of geproci sets but they are far from being completely understood. One can relax this condition. For example, consider sets of six points in projective 3-space in linear general position. A general projection does not lie on a conic, but the locus of points of projection for which the image does lie on a conic (hence has a chance to be a complete intersection of type (2,3)) turns out to be a surface of degree 4 called the Weddle surface. This leads to the more general notion of the d-Weddle locus for a finite set. To study such a locus, one approach that has borne fruit is via Macaulay duality. This allows us to translate the problem into one involving the cokernel of a certain multiplication map with a linear form, leading to a strong connection between the d-Weddle locus and the non-Lefschetz locus of a certain algebra. We will describe these connections. All the results described here are joint with Luca Chiantini, Łucja Farnik, Giuseppe Favacchio, Brian Harbourne, Tomasz Szemberg and Justyna Spond.

The classical Waring problem for homogeneous polynomials can be translated into geometric terms, using the notion of defectivity and identifiability for secant varieties. The defectivity problem was completely solved by Alexander-Hirschowitz using classical degeneration techniques. On the other hand identifiability has recently been addressed by Mella and Galuppi. In this talk I will briefly explain the relationship between defectivity and identifiability in a more general setting and give bounds for a generalized Waring problem, introduced by Fröberg, Ottaviani, and Shapiro.  In particular, we will see how the union of classical degeneration techniques combine with techniques borrowed from toric geometry, allowing us to give very sharp bounds on identifiability and defectivity in a much more general context. In the last part of the talk I will show how to generalize the previous approach to singular toric varieties. This is a joint work (in progress) with Elisa Postinghel.

Introduction to the Bhattacharya-Mesner product for tensors of dimension n. This product is particularly suitable for the study of neural networks described by acyclic directed graphs in which each node decides its own activation through tensors of choice. The final state of the network can be determined from the product of node tensors.

I will discuss how some problems of elementary projective geometry can be studied and solved by using the Veronese process.