Dynamical systems with multiple time scales appear in a range of problems from applications. Invariant manifolds theory forms the foundation of qualitative analysis for their dynamics. In this talk, we will show our recent results on invariant manifolds theory for two classes of fast-slow systems, i.e., normally hyperbolic invariant manifolds for fast-slow high-dimensional systems and invariant structures for neutral differential equations with small delays.

Given a family of complex (smooth projective) manifolds, one can measure its non-triviality by looking at how much the Hodge structures of the fibres change. This leads to the notion of maximal (infinitesimal) variation of Hodge structure (IVHS). In the case of families of curves, results of Lee-Pirola and of myself with Torelli imply that a general deformation of any curve has maximal IVHS. This is however not so clear if one wants the deformation to keep some further structure, such as the gonality of the curve or an embedding into a given surface. For example, it was only recently proved by Favale and Pirola that every smooth plane curve admits a deformation as a plane curve with maximal IVHS, and the question remains open for deformations of curves inside any other surface. In this talk I will present a joint work in progress with Sara Torelli extending this result to curves in P^1 x P^1, which turns out to be way more involved than the plane case.

The main focus of the talk will be a class of 2d singular mechanical systems on a surface S with a potential V having a finite number of singularities C := {c_1,..., c_n} of the form
V(q) ~ C_i d(c_i,q)^{-a_i}
where C_i>0, a_i >= 1 and q in O(c_i).
The first result I will present is an existence one: there are periodic solutions in (infinitely) many conjugacy classes of pi_1(S,C). Using this fact, I will construct an invariant set for the system which admits a semi-conjugation with a Bernoulli shift.
The second result I will discuss aims at identifying some situation in which the semi-conjugation is actually a conjugation and the invariant set constructed displays a chaotic behaviour. This happens, for instance, under some negativity condition on the curvature of S and for large values of the energy. Much emphasis will be put on the interplay between geometry, topology and variational methods.
This is a joint work with Gian Marco Canneori.

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

  This talk is based on joint work in progress with E. Feigin, M. Lanini and A. Pütz.
  We analyze a family of Quiver Grassmannians for the equioriented cycle, which are degenerations of the classical Grassmannians: for each one, we describe its irreducible components, find a cellular decomposition in terms of attracting sets, and give an overview of the underlying combinatorics. Then we introduce symplectic conditions and try to understand the associated subvarieties, which are degenerations of the classical isotropic Grassmannians.

  In this talk I give a description of the semistable Cohomological Hall algebra (CoHa) for extended Dynkin quivers with central slope in terms of generators and relations.
  This extends work of Franzen-Reineke who dealt with the case of the Kronecker quiver.

To an equivariant noncommutative principal bundle one associates an Atiyah sequence of braided derivations whose splittings give connections on the bundle. There is an explicit action of vertical braided derivations as infinitesimal gauge transformations on connections. From the sequence one derives a Chern—Weil homomorphism and braided Chern— Simons terms.
On the principal bundle of orthonormal frames over the quantum sphere S^{2n}_theta, the splitting of the sequence leads to a Levi-Civita connection on the corresponding module of braided derivations. The connection is torsion free and compatible with the 'round' metric. We work out the corresponding Riemannian geometry.
Note: This talk is part of the activity of the MUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)

Local entropy theory is a culmination of deep results in dynamics, ergodic theory and combinatorics. Given a dynamical system with positive entropy, it gives, in some sense, the location of where the entropy resides. It is a powerful tool that can be applied in a variety of settings. In this talk, we will show how the speaker (with his co-authors) has been able to apply local entropy theory to settle some problems in continuum theory, and in dynamics of maps on the space of finite measures.

Consider the moduli space of stable, n-marked curves M and the tautological subring R^*(M) of its Chow ring. The standard calculus for R^*(M) is based on the ''strata algebra" SA(M), which is constructed via the inductive structure of the boundary of M and the excess intersection formula, and in which calculations are expressed in terms of ''graph sums". In this talk I will discuss a new calculus for R^*(M), based on the introduction of a new ring L^*(M), built out of tropical geometry, and in which several standard calculations simplify significantly. I will explain how the comparison between SA and L is analogous to the comparison between equivariant cohomology and equivariant cohomology of the fixed locus in GKM theory. Finally, I will sketch how this idea can be used to give explicit formulas for the Brill-Noether cycles -- informally, the cycles on M parametrizing curves on which a line bundle of the form omega^k(sum a_ix_i) has at least r+1 linearly independent sections. This is a joint work with M. Abreu and N. Pagani.

The minimization of the action of a QFT with a constraint dictated by the block averaging procedure is an important part of the Balaban's approach to renormalization. It is particularly interesting for QFTs with non-trivial target spaces, such as gauge theories or non-linear sigma models on a lattice. We analyse this step for the O(4) non-linear sigma model in two dimensions and demonstrate in this case how various ingredients of the Balaban approach play together. First, using variational calculus on Lie groups, the equation for the minimum is derived. Then this non-linear equation is solved by the Banach fixed point theorem. This step requires a detailed control of lattice Green functions and their integral kernels via random walk expansions.

We discuss the embezzlement of entanglement and its relation to the classification of the latter, as well as its application to relativistic quantum field theory. Embezzlement (of entanglement), introduced by van Dam and Hayden, denotes the task of producing any entangled state to arbitrary precision from a shared entangled resource state, the embezzling state, using local operations without communication while perturbing the resource arbitrarily little. We show that Connes' classification of type III von Neumann algebras can be given a quantitative operational interpretation in terms of embezzlement. In particular, this quantification implies that all type III factors, apart from some type III_0 factors, host embezzling states. In contrast, semifinite factors (type I or II) cannot host embezzling states. Specifically, type III_1 factors are characterized as 'universal embezzlers', meaning every normal state is embezzling. The latter observation provides a simple explanation as to why relativistic quantum field theories maximally violate Bell inequalities. To understand the connection between embezzlement of entanglement and the classification of von Neumann algebras, we use a technique introduced by Haagerup and Størmer that associates to each normal state on a von Neumann algebra a state on the flow of weights. Our results then follow by quantifying the invariance of states on the flow of weights on the restriction of the dual modular flow. If time permits, we will also discuss the connection between embezzling states and embezzling families, as used by van Dam and Hayden.
This is joint work with Lauritz van Luijk, Reinhard F. Werner, and Henrik Wilming.