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 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).

  In this talk, based on joint work with Stephane Geudens and Marco Zambon, we develop the deformation theory of symplectic foliations, i.e. regular foliations equipped with a leafwise symplectic form. The main result is that each symplectic foliation is attached with a cubic L_∞-algebra controlling its deformation problem. Indeed, we establish a one-to-one correspondence between the small deformations of a given symplectic foliation and the Maurer-Cartan elements of the associated L_∞-algebra. Further, we prove that, under this one-to-one correspondence, the equivalence by isotopies of symplectic foliations agrees with the gauge equivalence of Maurer-Cartan elements. Finally, we show that the infinitesimal deformations of symplectic foliations can be obstructed.
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project Mat-Mod@TOV (CUP E83C23000330006)

  Bounded cohomology of groups is a variant of ordinary group cohomology introduced by Johnson in the 70s in the context of Banach algebras and then intensively studied by Gromov in his seminal paper "Volume and bounded cohomology" in relation to geometry and topology of manifolds. Since the 80s, bounded cohomology has then grown up as an independent and active research field. On the other hand, it is notoriously hard to compute bounded cohomology. For this reason it is natural to first investigate groups with trivial bounded cohomology groups. During this talk we survey recent advances around "acyclicity" in bounded cohomology and we will introduce a new algebraic criterion for the vanishing of bounded cohomology.
  This is part of a joint work with Caterina Campagnolo, Francesco Fournier-Facio and Yash Lodha.
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project Mat-Mod@TOV (CUP E83C23000330006)

We are concerned with a generalization to the singular case of a result of C.C. Chen e C.S. Lin [Comm. An. Geom. 1998] for Liouville-type equations with rough potentials. The singular problem is actually more delicate and results in a nontrivial variation of the regular case. Part of the arguments of Chen-Lin can be adapted to the singular case by means of an isoperimetric inequality for surfaces with conical singularities. The rest of the proof actually requires a different approach, due to the loss of translation invariance of the problem.