The gluing maps on the moduli space of curves are integral to much of the enumerative geometry of curves. For example, Gromov-Witten invariants satisfy recursive relations with respect to the gluing maps. For log Gromov-Witten invariants, counting curves with tangency conditions, this fails at the very first step as logarithmic curves cannot be glued, by a simple tropical obstruction. I will describe a certain logarithmic enhancement of (M_{g,n}) from joint work with David Holmes that does admit gluing maps. With this enhancement, we can geometrically see a recursive structure appearing in log Gromov-Witten invariants. I will present how this leads to a pullback formula for the log double ramification cycle (roughly a log Gromov-Witten invariant of P^1). Time permitting, I will sketch how this extends to general log Gromov-Witten invariants (joint work with Leo Herr and David Holmes). This story tropicalises by replacing log curves with tropical curves (metrised dual graphs) and algebraic geometry by polyhedral geometry. In this language both the logarithmic enhancement and the recursive structure admit a simpler formulation. I will keep this tropical story central throughout.

Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027)

For a quasilinear equation involving the n-Laplacian and an exponential nonlinearity, I will discuss quantization issues for blow-up masses in the non-compact situation, where the exponential nonlinearity concentrates as a sum of Dirac measures. A fundamental tool is provided here by some Harnack inequality of sup + inf type, a question of independent interest that we prove in the quasilinear context through a new and simple blow-up approach. If time permits, we will also discuss some recent progress concerning sharp sup+inf inequalities. Joint work with Marcello Lucia.
NB:This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006

We introduce local invariants of algebraic spaces and stacks which measure how far they are from being a scheme. Using these invariants, we develop mostly topological criteria to determine when the moduli space of a stack is a scheme. In the setting of a stack admitting a separated (good) moduli space this also yields a criterion for when the moduli space is a scheme. As an application we identify all separated good moduli spaces of vector bundles over a smooth projective curve which are schemes. This is joint work with Andres Fernandez Herrero and Xucheng Zhang.

Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027)

  The noncommutative differential geometry of quantum flag manifolds has seen rapid growth in recent years, following the remarkable finding of a complex structure for flag manifolds of irreducible type by Heckenberger and Kolb. With a large part of the theory for the irreducible cases already figured out, it is now time to tackle the question of how to obtain the same structure for other types of flag manifolds. In this work in collaboration with R. Ó Buachalla and J. Razzaq, we give a complex structure for the full flag manifold of quantum SU(3), that includes the differential calculus discovered by Ó Buachalla and Somberg as its holomorphic sub-complex.
  I shall review this construction that makes use of Lusztig quantum root vectors, while at the same time giving a general overview of the theory of noncommutative differential calculi for quantum homogeneous spaces.
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006).

  Persistent homology, a popular method in Topological Data Analysis, encodes geometric information of data into algebraic objects called persistence modules. Invoking a decomposition theorem, these algebraic objects are usually represented as multisets of points in the plane, called persistence diagrams, which can be fruitfully used in data analysis in combination with statistical or machine learning methods.
  Wasserstein distances between persistence diagrams are a common way to compare the outputs of the persistent homology pipeline. In this talk, I will explain how a notion of p-norm for persistence modules leads to an algebraic version of Wasserstein distances which fit into a general framework for producing distances between persistence modules. I will then present stable invariants of persistence modules which depend on Wasserstein distances and can be computed efficiently. The use of these invariants in a supervised learning context will be illustrated with some examples.
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project Mat-Mod@TOV (CUP E83C23000330006)

Schrödinger-type equations model a lot of natural phenomena and their solutions have interesting and important properties. This gives rise to the search for normalised solutions, i.e., when the mass is prescribed. In this talk, I will exploit a novel variational approach, introduced in the context of autonomous Schrödinger equations, to find a least-energy solution to a problem involving the m-Laplacian and a Hardy-type potential. The growth of the non-linearity is mass-supercritical at infinity and at least mass-critical at the origin. An important step in this approach is to show that all the solutions satisfy the Pohozaev identity, which in the presence of a Hardy-type potential was previously known only in the spherical case with m = 1. This talk is based on a joint article with Bartosz Bieganowski and Jaroslaw Mederski, about energy-subcritical non-linearities, and a joint preprint with Bartosz Bieganowski and Olímpio H. Miyagaki, concerning exponential critical non-linear terms in dimension N = 2m.
NB:This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006

Il contesto naturale della congettura di Greenberg è la teoria di Iwasawa. In questo colloquio introdurrò la teoria di Iwasawa e la congettura di Greenberg, per poi presentare risultati recenti ottenuti in collaborazione con Pietro Mercuri.

Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027)

The solution of systems of non-autonomous linear ordinary differential equations is crucial in various applications, such as nuclear magnetic resonance spectroscopy. We introduced a new solution expression in terms of a generalization of the Volterra composition. Such an expression is linear in a particular algebraic structure of distributions, which can be mapped onto a subalgebra of infinite matrices. It is possible to exploit the new expression to devise fast numerical methods for linear non-autonomous ODEs. As a first example, we present a new method for the operator solution of the generalized Rosen-Zener model, a system of linear non-autonomous ODEs from quantum mechanics. The new method’s computing time scales linearly with the model’s size in the numerical experiments. A second example is the analysis of temporal network, where the new expression might lead to novel extension of subgraph centrality indexes.

Enumerating genus g curves passing through g points in an abelian surface is a natural problem, whose difficulty highly depends on the degree of the curves. For "primitive" degrees, we have an easy explicit answer. For "divisible" classes, such a resolution is quite demanding and often out of reach. Yet, the invariants for divisible classes easily express in terms of the invariants for primitive classes through the multiple cover formula, conjectured by G. Oberdieck a few years ago. In this talk, we'll show how tropical geometry enables to prove the formula without any kind of concrete enumeration.

Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027), Prin 2022 Moduli Spaces and Birational Geometry and Prin PNRR 2022 Mathematical Primitives for Post Quantum Digital Signatures

We discuss the unique continuation property for linear differential operators of the form sum of squares of vector fields satisfying Hörmander's bracket generating condition. We provide some negative and some positive results.
NB:This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006