I will report on a recent work in collaboration with F. Cavalletti and A. Lerario, where we study complex projective hypersurfaces seen as probability measures on the projective space.
  Our guiding question is: “What is the best way to deform a complex projective hypersurface into another one?"
  Here the word best means from the point of view of measure theory and mass optimal transportation. In particular, we construct an embedding of the space of complex homogeneous polynomials into the probability measures on the projective space and study its intrinsic Wasserstein metric.The Kähler structure of the projective space plays a fundamental role and we combine different techniques from symplectic geometry to the Benamou-Brenier dynamical approach to optimal transportation to prove several interesting facts. Among them we show that the space of hypersurfaces with the Wasserstein metric is complete and geodesic: any two hypersurfaces (possibly singular) are always joined by a minimizing geodesic. Moreover outside the discriminant locus, the metric is induced by a Kähler structure of Weil-Petersson type. In the last part I will give an application to the condition number of polynomial equations solving.
  This talk is part of the activity of the MIUR Excellence Department Project Mat-Mod@TOV (CUP E83C23000330006).

In this talk I will present a recent work in which the strong ill-posedness of the two-dimensional Boussinesq system is proven. I will show explicit examples of initial data with vorticity and density gradient in L^∞ (R²) for which the horizontal density gradient has a strong norm inflation in infinitesimal time. This is a joint work with Roberta Bianchini (CNR) and Lars Eric Hientzsch (Bielefeld University).
NB:This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C23000330006

Through the lens of Minimal Model Program, the classification of algebraic varieties can be summarised in 2 steps. First MMP allows to decompose a variety with mild singularities, birationally, into a tower of fibrations whose general fibres have ample, anti-ample or numerically trivial canonical divisor. The natural second step is to study these 3 classes, e.g. by their moduli theory or other properties that shed light on all elements in the classes. It turns out a very important property is boundedness: a collection of varieties is bounded when the elements of the given collection can be parametrised using a finite type geometric space. This is crucial in the construction of proper moduli spaces of finite type. Moreover if a given collection of algebraic varieties is bounded (in char 0) then the topological types of their underlying analytic spaces belong to finitely many homeomorphism classes and their topological invariants come in finitely many versions. While over the past 15 years several breakthroughs have completely settled the question of boundedness (and subsequent construction of moduli spaces) in the case of log canonical models (varieties/pairs with ample canonical divisor) and Fano varieties (anti-ample case), the situation is still quite unclear in the case of trivial numerical divisor. I will try to explain what is known or not, and which challenges make the situation quite more complicated than the other cases. I will explain how we can overcome most issues if we assume that a K-trivial variety is endowed with a fibration structure of relative dimenesion one. The seminar is based on various works I developed over the past 10 years with G. Di Cerbo, C. Birkar, S. Filipazzi and C. Hacon. Time permitting I will talk about work in progress with P. Engel, S. Filipazzi, F. Greer, M. Mauri showing various new boundedness results for K-trvial varieties fibered in K3 surfaces or abelian varieties

Operads are algebraic structures capturing multi-ary operations. The little disks operads, encoding operations on iterated loop spaces, are fundamental examples. Voronov introduced Swiss-Cheese operads, which generalize little disks to relative loop spaces of pairs of spaces. While the little disks operads are formal (their cohomology determines their rational homotopy type), the standard Swiss-Cheese operads are not. We will discuss why higher-codimensional Swiss-Cheese operads are formal, and why Voronov's original version is not. This non-formality result, stemming from joint work with R. V. Vieira, leads to questions about truncations and Massey products.
This talk is part of the activity of the MIUR Excellence Department Project Mat-Mod@TOV (CUP E83C23000330006).

We introduce energies E(V) of vector bundles V on Riemannian manifolds and more generally on Dirichlet spaces, commutative or not. We then derive a relationship between triviality of V and smallness of E(V).

Note: This talk is part of the activity of the MUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)

The Operator Algebra Seminar schedule is here: https://sites.google.com/view/oastorvergata/home-page

The moduli space of stable n-pointed rational curves is a fundamental object in algebraic geometry. Many aspects of the space, such as its intersection theory, have been completely understood. The two dimensional analogue, parameterizing KSBA stable configurations of n lines in the plane, is much more mysterious. It has been studied by a number of researchers, including Alexeev, Hacking-Keel-Tevelev, Lafforgue, and others. I will share a new perspective on this space, motivated by logarithmic geometry, and explain how this perspective can be used to endow the space with a virtual fundamental class, and puts it on essentially equal theoretical footing with its more well-studied sibling. I will then explain the combinatorial structure on the boundary, and discuss where we hope to take the story next. Based on ongoing joint work with Abramovich and Pandharipande.

  For a framed n-manifold M one can produce an explicit pairing between M and its one point compactification M⁺, taking values in Sⁿ, which on homology induces the Poincaré duality pairing. We show that this can be lifted to the level of operads to produce a stable equivalence between E_n , the little n-disks operad, and a shift of its Koszul dual. This gives a proof of the same celebrated result of Ching-Salvatore, but manages to avoid using technical results in geometry, homotopy theory, and even basic analysis that appear in their proof.
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006).

  The age of AI requires building capable and more efficient neural networks that are mainly achieved via:
    (I) developing and manufacturing more capable hardware;
    (II) designing smaller and more robust versions of neural networks that realize the same tasks;
    (III) reducing the computational complexities of learning algorithms without changing the structures of neural networks or hardware.
  The approach (I) is an industrial design and manufacturing challenge. The approach (II) is essentially the subject of network pruning. In this talk, we play on mathematicians' strengths and focus on a theoretical approach on (III) based on tropical arithmetics and geometry.
  I will first describe the setup of machine learning in simple mathematical terms and briefly introduce tropical geometry. After verifying that tropicalization will not affect the classification capacity of deep neural networks, I will discuss a tropical reformulation of backpropagation via tropical linear algebra.
  This talk assumes no preliminary knowledge of machine learning or tropical geometry: undergraduate-level math, and general curiosity will be sufficient for active participation.
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006).

The celebrated Osterwalder-Schrader (OS) reconstruction results provide conditions verified by Euclidean n-point correlation functions to produce a Wightman quantum field theory. This talk aims to show a conformal version of the OS axioms, including the linear growth condition, for n-point correlation functions defined for a reasonable class of unitary full Vertex Operator Algebras (VOAs). Such VOAs can be seen as extensions of commuting chiral and anti-chiral VOAs, introduced to describe compact 2D conformal field theories.

The Operator Algebra Seminar schedule is here: https://sites.google.com/view/oastorvergata/home-page?authuser=0

Isogeny theorems are a powerful number-theoretic tool to understand subgroup of elliptic curves over number fields and through those, properties of their rational points. As a prerequisite for such theorems, one needs to understand how the "height" h(E) of an elliptic curve E can change by an isogeny: if E and E' are elliptic curves over a number field K with an isogeny phi : E -> E' over K, the difference |h(E)-h(E')| is linearly bounded in terms of log(deg(phi)). In a recent paper, Griffon and Pazuki proved a similar result for elliptic curves over function fields of curves, which is surprisingly much more uniform on the degree of phi (for example, in characteristic 0 the height is invariant by isogeny !). In this talk, I will recall what are abelian varieties and isogenies between them and what are their heights, why Griffon-Pazuki's result is not as easily generalised as it seems (abelian varieties being the higher-dimensional avatars of elliptic curves) because of group schemes in characteristic p, and describe the optimal bounds we obtained with Griffon and Pazuki in the context of function fields