Gianluca Occhetta: The secret birational life of flags. Rational homogeneous varieties are remarkable objects from many points of view; however, within the framework of Mori theory, their geometry is rather simple: their Mori cone is simplicial and every extremal contraction is a smooth fibration. This behaviour is not shared by some of their quotients by torus actions. For instance the Grothendieck--Knudsen moduli space $\overline{M_{0,n}}$ is a quotient of the Grassmannian of lines $G(2,n)$ by its maximal torus, and the classical spaces of complete collineations, complete quadrics and complete skew forms can be realized as quotients of (classical, Lagrangian, orthogonal) Grassmannians with respect to a suitable $\mathbb C^*$-action.
In this talk, based on a joint work with Lorenzo Barban and Luis Solá Conde, I will present the construction of the quotient of the complete flag manifold of linear subspaces of the three dimensional complex projective space by its maximal torus. I will show that the quotient is a smooth threefold which is a Mori Dream Space, and I will describe in detail its rich birational geometry.
Alessio D'Alì: On strongly Koszul algebras and tidy Gröbner bases. Koszul algebras are quadratic algebras with many desirable homological properties. In this talk we investigate strong Koszulness, i.e., a strengthening of the Koszul property first introduced by Herzog, Hibi and Restuccia in 2000. We first show that strong Koszulness behaves well with respect to tensor and fiber products. We then prove that the existence of a quadratic revlex-universal Gröbner basis with a strong sparsity condition (that we name "tidiness") is a sufficient condition for strong Koszulness, and exhibit several concrete examples arising from determinantal objects and Macaulay’s inverse system, including an intriguing connection to (secant varieties of) Severi varieties. Finally, we exhibit examples of strongly Koszul algebras whose defining ideal does not admit a Gröbner basis of quadrics even after a linear change of coordinates, thus answering in the negative a question posed by Conca, De Negri and Rossi.
Leone Slavich: Convex real projective manifolds with Riemannian geometric structures. A properly convex real projective manifold is the quotient of a properly convex open subset of a real projective space by a discrete group of projective automorphisms.
We will characterise the geometries of compact locally symmetric spaces that support properly convex real projective structures and show that if a closed, indecomposable, convex projective four-
manifold admits a geometric decomposition in the sense of Thurston, then every piece is real hyperbolic. Joint work with Stefano Riolo and Andrea Seppi.
Emma Lepri: Unitally positive A-infinity algebras and the contraction algebra. By the work of Donovan and Wemyss, the functor of noncommutative deformations of a flopping irreducible rational curve C in a threefold X is representable by an algebra called the contraction algebra. This talk is based on a joint work with J. Karmazyn and M. Wemyss, where we construct a DG-algebra from the data of periodic projective resolution of the simple module on the contraction algebra, and prove that it reconstructs the A-infinity algebra $Ext^*_X(\mathcal{O}_C(-1), \mathcal{O}_C(-1))$, giving an alternative proof of the Donovan-Wemyss conjecture.
Carlo Sanna: Expected Number of Solutions to the Permuted Kernel Problem. The Permuted Kernel Problem (PKP) is a simple-to-state but hard-to-solve problem in combinatorial linear algebra: given an m x n matrix A and an n x 1 vector b over a finite field, find (if it exists) an n x n permutation matrix P such that APb = 0. This talk explains why the PKP is of interest to cryptographers and shows some results on the expected number of solutions to a "random" instance of the PKP. No prior knowledge of cryptography is required.
Benedetta Piroddi: A new symplectic orbifold. Symplectic orbifolds are singular analogues of hyperkaehler manifolds. One way to construct them is to take terminalizations of quotients of hyperkaehler manifolds by symplectic group actions. In this talk I'll present a new example of this construction, starting from an action of the group L_3(4) on a double-EPW sextic (a hyperkaehler fourfold). The resulting orbifold has some non-quotient terminal singularities.
This is a joint work in progress with M. Donten-Bury, G. Kapustka, T. Wawak.
Alessandro Oneto: Hadamard ranks of algebraic varieties. While additive decompositions and tensor rank are classically studied via secant varieties of Segre and Veronese varieties, this talk explores a different perspective: the decomposition of tensors as entry-wise products. I will present a geometric framework for this approach, defining Hadamard ranks for algebraic varieties through the Hadamard products of their secant varieties. The multiplicative structure of the problem naturally leads to the use of tropical geometry. Starting from motivations in algebraic statistics, I will provide a general overview of the theory and discuss recent results obtained in collaboration with Nick Vannieuwenhoven, Dario Antolini, and Guido Montúfar.
