According to physics literature, topologically ordered gapped ground states of 2-dimensional spin systems can be described by a topological quantum field theory. Many examples arise from microscopic models with local commuting projector Hamiltonians, namely Levin-Wen models.
In this talk, I will describe the general framework for classifying infinite volume gapped ground states (by Naaijkens, Ogata, et.al.) in the simple context of abelian Levin-Wen models. This framework is heavily inspired by the DHR analysis in relativistic quantum field theory. It applies to the doubled semion model whose anyon theory is a braided fusion category equivalent to the representation category of the twisted Drinfeld double of Z_2.
Based on joint work with Alex Bols and Alvin Moon, https://arxiv.org/abs/2306.13762.
Seminar schedule here: https://sites.google.com/view/oastorvergata/home-page.

In this presentation, we tackle the problem of reconstructing the optimal transport map $T$ between two absolutely continuous measures $mu, u in mathcal{P}(mathbb{R}^n)$, and for this approximation we employ flows generated by linear-control systems in $mathbb{R}^n$. We first show that, under suitable assumptions on the measures $mu, u$ and on the controlled vector fields, the optimal transport map is contained in the $C^0_c$-closure of the flows generable by the system. In the case that discrete approximations $mu_N, u_N$ of the measures $mu, u$ are available, we use a discrete optimal transport plan to set up an optimal control problem. With a $Gamma$-convergence argument, we prove that its solutions corresponds to flows that provide approximations of the optimal transport map $T$. Finally, in virtue of the Pontryagin Maximum Principle, we propose an iterative numerical scheme for the resolution of the optimal control problem, resulting in an algorithm for the practical computation of approximations of the optimal transport map. This approach can be interpreted as the construction of a ''Normalizing Flow'' by means of a Residual Neural Network (ResNet). Based on a joint work with Sara Farinelli. [1] A. Scagliotti, S. Farinelli. Normalizing flows as approximations of optimal transport maps via linear-control neural ODEs. arXiv preprint, 2023.
NB:This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C23000330006

Blow-ups are fundamental tools in algebraic geometry, and there are several results (e.g the famous Castelnuovo's theorem) that can be used to determine when a variety is obtained as a blow-up of a smooth variety along a smooth center. Weighted blow-ups play a similar role for stacks. In this talk I will present a criterion for finding out if a smooth DM stack is a weighted blow-up. I will apply this result for showing that certain alternative compactifications of moduli of marked elliptic curves are obtained via weighted blow-ups (and blow-downs). This in turn will prove to be useful in order to compute certain invariants, like Chow rings or Brauer groups. First part of this talk is a joint work with Arena, Inchiostro, Mathur, Obinna and Pernice; the second part of this talk is a joint work with L. Battistella.

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.