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

We will prove that Palais-Smale sequences for Liouville type functionals on closed surfaces are precompact whenever they satisfy a bound on their Morse index. As a byproduct, we will obtain a new proof of existence of solutions for Liouville type mean-field equations in a supercritical regime. Moreover, we will also discuss an extension of this result to the case of singular Liouville equations.
NB:This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C23000330006

In the recent work arXiv:2303.17190v2, G. Höhn and S. Möller propose a classification of vertex operator superalgebras (VOSAs) with central charge at most 24 and with trivial representation theory. VOSAs with the latter property are usually called self-dual or holomorphic. From the point of view of the operator algebraic approach to chiral Conformal Field Theory, it is a natural question whether they give rise to graded-local conformal nets of von Neumann algebras. Indeed, this happens if one proves that those VOSAs are unitary and they satisfy the strong graded locality condition, according to the correspondence given by Carpi, Gaudio and Hillier, which extends to the Fermi case the one by Carpi, Kawahigashi, Longo and Weiner of 2018. In this seminar, based on the work arXiv:2410.07099v2, we discuss the unitarity of holomorphic VOSAs with central charge at most 24. To do that we need to recall the notion of complete unitarity for vertex operator algebras and the recent developments regarding their extensions. Then we move to their strong graded locality, which is established for most of them. Indeed, this property remains an open problem for only 59 out of 969 VOSAs in the central charge 24. Nevertheless, we obtain many new examples of holomorphic graded-local conformal nets. They can be considered as the Fermi counterparts of the models recently built from the so-called Schellekens list.

We will survey developments in the minimal model program for Mgbar over the last 15 years. After summarizing various strategies to determine moduli interpretations of the log canonical models of Mgbar, we will highlight open questions in the field.

The classical Courant's nodal domain theorem states that the n-th eigenfunction of the Laplacian on a compact manifold has at most n nodal domains. The same holds for Steklov eigenfunctions on a compact manifold with boundary. The classical argument of the proof, however, does not apply to Dirichlet-to-Neumann eigenfunctions, which are the traces of Steklov eigenfunctions on the boundary. We disprove the conjectured validity of Courant's theorem for D-t-N eigenfunctions. Namely, given a smooth manifold M, and integers K,N, we built a Riemannian metric on M for which the n-th D-t-N eigenfunction has at least K nodal domains for all n=1,...,N. Based on a joint work with Angela Pistoia (Sapienza Università di Roma) and Alberto Enciso (ICMAT Madrid).