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)

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.

In the last years there has been a renewed interest around a long-standing conjecture by Griffiths characterizing which should be the positive characteristic forms for any Griffiths positive vector bundle. This conjecture can be interpreted as the differential geometric counterpart of the celebrated Fulton–Lazarsfeld theorem on positive polynomials for ample vector bundles. In this talk, we present some results that confirm the above conjecture for several characteristic forms. The positivity of these forms is due to a theorem which provides the version at the level of representatives of the universal push-forward formula for flag bundles valid in cohomology.

The class of Generalized Locally Toeplitz (GLT) sequences has been introduced as a generalization both of classical Toeplitz sequences and of variable coefficient differential operators and, for every sequence of the class, under mild assumptions it has been demonstrated that it is possible to give a rigorous description of the asymptotic spectrum in terms of a function (the symbol) that can be easily identified. The GLT class has nice algebraic properties and indeed it has been proven that it is stable under linear combinations, products, and inversion when the sequence which is inverted shows a sparsely vanishing symbol (sparsely vanishing symbol = a symbol which vanishes at most in a set of zero Lebesgue measure). Furthermore, the GLT class virtually includes any approximation of partial differential equations (PDEs), fractional differential equations (FDEs), integro-differential equations (IDEs) by local methods (Finite Difference, Finite Element, Isogeometric Analysis etc). In the present talk, we discuss the foundations of the theory and its impact with special attention to new tools and to new directions as those based on symmetrization tricks, on the extra-dimensional approach, and on blocking operations, blocking structures.

The Torelli locus - the image of the moduli space of curves (M_g) in the moduli space of abelian varieties (A_g) - is much-studied but still mysterious. In characteristic p, A_g has a beautiful stratification by the isomorphism type of A[p], and examples show that Mg is far from transverse to this stratification. In an ongoing project, we develop tools to understand (and perhaps make principled conjectures about) which strata of A_g meet M_g. In this talk, we explain some of the structures involved and give new results about them. Parts of this are joint work with Bryden Cais and Rachel Pries.

In this talk we consider inverse problems for the partial differential equations describing the behavior of certain fluids. Our focus will be on the fluid-structure interaction problem and the objective is to determine the moving domain where the equations are satisfied, based on external measurements. We concentrate on a one-dimensional fluid-solid interaction problem for the Burgers equation, and we will prove uniqueness and conditional stability results. This work is in collaboration with J. Apraiz, E. Fernandez-Cara and M. Yamamoto [1].

[1] J. Apraiz, A. Doubova, E. Fernandez-Cara, M. Yamamoto, "Inverse problems for one-dimensional fluid-solid interaction models", Communications on Applied Mathematics and Computation, https://doi.org/10.1007/s42967-024-00437-3

NB:This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C23000330006

We consider radial solutions of fully nonlinear, uniformly elliptic equations posed in punctured balls, in presence of radial singular quadratic potentials. We discuss both the principal eigenvalues problem and the case of equations having also absorbing superlinear zero order terms: for the former problem, we explicitly compute the principal eigenvalues, thus obtaining an extension in the fully nonlinear framework of the Hardy-Sobolev constant; for the latter case, we provide a complete classification of solutions based on their asymptotic behavior near the singularity. The results are based on joint papers with I. Birindelli and F. Demengel.

The tautological ring is a certain subring of the Chow ring of the moduli space of curves. It is generated by the algebraic cycles that arise from the modular nature of the moduli space, and is one of the most studied objects in enumerative geometry. In this talk, I will explain that any semi stable family of algebraic varieties -- in particular the compactified Jacobians over the moduli space of curves --gives rise to a tautological ring, and discuss the relationship between the tautological rings of compactified Jacobians and the usual tautological ring of the moduli space of curves.

  Let G be a finite group. It is not hard to see that for any representation ρ : G ⟶ GL(V) for V a real vector space, there exists a G-invariant bilinear form β on V, i.e., a non-degenerate bilinear form such that β(ρ(gv,ρ(g)w) = β(v,w) for all g ∈ G, v, w ∈ V. If ρ is "orthogonally stable" (so it is a sum of even-dimensional irreducible real representations) then the square class of the determinant of the Gram matrix for any basis (the "orthogonal determinant") does not depend on the choice of β, giving us interesting invariants of our group G. Richard Parker conjectured that these orthogonal determinants are always "odd", for any finite group. We will see that the conjecture holds for the symmetric groups, as well as the general linear groups GL(q) for q a power of an odd prime. In the discussion, important concepts like (standard) Young tableaux and Iwahori-Hecke algebras will come up. This talk has the additional purpose of giving a small introduction (with many examples) into the representation theory of finite groups. As such, no previous knowledge in that area will be assumed.
N.B.: this talk is part of the activity of the MIUR Excellence Department Project Mat-Mod@TOV (CUP E83C23000330006)

  It is known by the works of Adamović and Perše that the affine simple vertex algebras associated with G₂ and B₃ at level -2 can be conformally embedded into L_-2(D₄).
In this talk, I will present a join work with Tomoyuki Arakawa, Xuanzhong Dai, Justine Fasquel, Bohan Li on the classification to the irreducible highest weight modules of these vertex algebras.
I will also describe their associated varieties: the associated variety of that corresponding to G₂ is the orbifold of the associated variety of that corresponding to D₄ by the symmetric group of degree 3 which is the Dynkin diagram automorphism group of D₄. This provides new interesting examples in the context of orbifold vertex algebras. These vertex algebra also appear as the vertex operator algebras corresponding to rank one Argyres-Douglas theories in four dimension with flavour symmetry G₂ and B₃.
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project Mat-Mod@TOV (CUP E83C23000330006)