The goal of this talk is to algebraically model the S_r-equivariant homotopy type of the configuration space of r labeled and distinct points in d-dimensional Euclidean space. I will present and compare two models: the Barratt-Eccles simplicial set and the multisimplicial set of 'surjections'. I will introduce multisimplicial sets and discuss their connection to more well-known simplicial sets. Multisimplicial sets can model homotopy types using fewer cells, making them a highly useful tool. Following this, we will explore in detail how to recognize configuration spaces in the aforementioned models by playing with a graph poset. An explicit relationship between the models will also be presented. This is a joint work with Anibal M. Medina-Mardones and Paolo Salvatore.

  Vertex algebras of chiral differential operators on a complex reductive group G are "Kac-Moody" versions of the usual algebra of differential operators on G. Their categories of modules are especially interesting because they are related to the theory of D-modules on the loop group of G. That allows one to reformulate some conjectures of the (quantum) geometric Langlands program in the language of vertex algebras. For instance, in view of the geometric Satake equivalence, one may expect the appearance of the category of representations of the Langlands dual group of G.
  In this talk I will define this family of vertex algebras and we will see that they are classified by a certain parameter called level. Then, for generic levels, we will see that "to find" the Langlands dual group, it is necessary to perform a quantum Hamiltonian reduction. Finally I will build simple modules on the closely related equivariant W-algebra that match the combinatorics of the Langlands dual group.
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)

Inspired by the C*-algebra of observables for a conduction electron in an aperiodic material, we study dynamical systems associated to solvable Lie groups and their associated foliated spaces. We establish a relation between the homotopy type of the foliated space and the *-isomorphism class of the foliation C*-algebra which is naturally attached to it. This result can be viewed as a simple noncommutative analogue of the famous Borel conjecture in topology. We make use of the classification result for nuclear C*-algebras in terms of the Elliott invariant. In cases of C*-algebras of physical origin, the tracial part of such invariant can be interpreted as the integrated density of states of the system. This is joint work with H. Wang and H. Guo.

We present some recent results concerning elliptic and evolution problems driven by mixed operators, which are the sum of local and nonlocal ones under a peridynamical approach, as introduced by Silling few years ago.
NB:This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006

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