Let us consider a complex abelian scheme endowed with a section. On some suitable open subsets of the base it is possible to define the period map, i.e. a holomorphic map which marks a basis of the period lattice for each fiber. Since the abelian exponential map of the associated Lie algebra bundle is locally invertible, one can define a notion of abelian logarithm attached to the section. In general, the period map and the abelian logarithm cannot be globally defined on the base, in fact after analytic continuation they turn out to be multivalued functions: the obstruction to the global existence of such functions is measured by some monodromy groups. In the case when the abelian scheme has no fixed part and has maximal variation in moduli, we show that the relative monodromy group of ramified sections is non-trivial and, under some additional hypotheses, it is of full rank. As a consequence we deduce a new proof of Manin's kernel theorem and of the algebraic independence of the coordinates of abelian logarithms with respect to the coordinates of periods. (Joint work with Paolo Dolce, Westlake University.)

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

After an introduction to (formal) deformation quantization of Poisson manifold, focusing on the deformation theory of associative algebras following Gerstenhaber, I will introduce constraint algebras as a way to study (symmetry) reduction in this setting. Constraint algebras encode the additional structure on the algebra of functions on a manifold needed for reduction and their deformations yield (formal) quantizations compatible with reduction. I will discuss the deformation theory of these constraint algebras and present some results about an adapted Hochschild-Kostant-Rosenberg Theorem computing the cohomology controlling this deformation theory.
This talk is part of the activity of the MIUR Excellence Department Project Mat-Mod@TOV (CUP E83C23000330006)

Biexactness and the (AO)-property can be considered as analytic counterparts of hyperbolicity for discrete groups. Motivated by the problem of determining whether they are equivalent, I will discuss an approach to the study of regularity properties of boundary actions/representations based on measurable dynamics. This approach will be used to study SL(3,Z) and to answer a question posed by C. Anantharaman-Delaroche.

Some bibliography:
https://arxiv.org/abs/2111.13885
https://arxiv.org/abs/2305.16277
https://arxiv.org/abs/2410.01447

abstract

A basic problem in symplectic topology is the classification of Lagrangian submanifolds up to Hamiltonian isotopy. There is growing evidence that this is impossible to solve, but one can hope to have a coarser classification by proving that finitely many Lagrangians generate the Fukaya category. I will illustrate some concrete examples where we know how to do this, some in which we do not, and a new technique called decoupling that could partially bridge the gap.

In this talk I will present the recent paper by Fewster, Janssen, Loveridge, Waldron and myself: "Quantum reference frames, measurement schemes and the type of local algebras in quantum field theory." In this work we show how mathematically rigorous notion of quantum reference frames allows to generalize the results of Chandrasekaran, Longo, Penington and Witten on observables in de Sitter space. The main idea is to study the joint algebra associated to the system together with the reference frame in the presence of symmetries. If both the system and the reference frame are covariant under some symmetry group, the construction of the invariant joint algebra very naturally involves crossed products.

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

We will discuss modular compactifications of M_{g,n} (the moduli space of smooth curves) and their birational geometry within the framework of the Hassett-Keel program. We classify the open substacks of canonically polarized curves with nodes, cusps, and tacnodes having a proper good moduli space. Using the S- and Theta-completeness criteria, we transform the problem into a combinatorial one where compactifications and flips can be described using cluster algebra theory. This approach yields a complete description of the Q-factorialization fan of (overline{M}_{g,n}(7/10)) as a cluster fan.

  The induced character formula in classical representation theory can be used, among other things, to describe the dimension of coinvariants of a representation in terms of its character. In this talk, I will explain how this formula is related to the multiplicativity of Euler characteristics in algebraic topology, and, in a more homotopy-coherent context, to the composability of so-called Becker-Gottlieb transfers, which are "wrong-way maps" in singular (co)homology; by describing a general formula to compute "dimensions of homotopy colimits". If time permits, I will discuss the most general case in which composability of Becker-Gottlieb transfers is now known. This is based on joint works with Carmeli-Cnossen-Yanovski, Klein-Malkiewich (and the last part with Volpe-Wolf).
  This talk is part of the activity of the MIUR Excellence Department Project Mat-Mod@TOV (CUP E83C23000330006).