Topological defects in quantum field theory have received considerable attention in the last few years as generalizations of the concept of symmetry. In the context of two dimensional conformal field theory, the properties of topological defects have been studied since the 90s, in particular in a series of works by Froehlich, Fuchs, Runkel and Schweigert. In this talk, I will discuss some applications of these ideas from physics to the theory of vertex operator (super-)algebras. In particular, I will describe some recent results about topological defects in the Frenkel-Lepowsky-Meurman Monstrous module, as well as in the Conway module, i.e. the holomorphic vertex operator superalgebra at central charge 12 with no weight 1/2 states. Finally, I will speculate about possible generalizations of the Moonshine conjectures.

This is partially based on ongoing joint work with Roberta Angius, Stefano Giaccari, and Sarah Harrison.

A famous theorem by Mukai (1988) provides a classification of all possible finite groups admitting a faithful action by symplectic automorphisms on some K3 surface. In 2011, in collaboration with Gaberdiel and Hohenegger, we proposed that a 'physics version' of Mukai theorem should hold for certain two dimensional conformal quantum field theories, called non-linear sigma models (NLSM) on K3, that describe the dynamics of a superstring moving in a K3 surface. In particular, we provided a classification of all possible groups of symmetries of NLSM on K3, that commute with the N=(4,4) algebra of superconformal transformations. This result was later re-interpreted by Huybrechts as a classification of the finite groups of autoequivalences of the bounded derived category of coherent sheaves on a K3 surface. In the last few years, the concept of symmetry group in quantum field theory has been vastly generalized. In the context of two dimensional conformal field theories, these developments suggest that the idea of 'group of symmetries' should be replaced by 'fusion category of topological defects'. We discuss how our previous classification result could be extended to include fusion categories of topological defects in non-linear sigma models on K3. The geometric interpretation of these categories is still mysterious. This is based on joint work in collaboration with Roberta Angius and Stefano Giaccari.

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

The talk discusses a framework to analyze certain model-based reinforcement learning algorithm. Roughly speaking, this approach consists in designing a model to deal with situations in which the system dynamics is not known and encodes the available information about the state dynamics that an agent has as a measure on the space of functions. In this framework, a natural question is if whether the optimal policies and the value functions converge, respectively, to an optimal policy and to the value function of the real, underlying optimal control problem as soon as more information on the environment is gathered by the agent. We provide a positive answer in the linear-quadratic case and discuss some results also in the control-affine nonlinear case.
NB:This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006

Locally periodic structures show regions (grains) with different orientations and at the boundaries between grains there is the appearance of defects. This happens in physical systems (for instance at microscopic scales for metals or patterns in block copolymers) as well as in more geometric models (as local tassellations for partions and clusters, or optimal location problems). In all these cases the energy governing the systems concentrates at the grain boundaries. The understanding of this “surface tesion” is a key ingredient in order to reduce the complexity of the problem and work in a so to say sharp interface model. I will present some recent results in this direction focussing on a two dimensional model for grain boundaries in metals, which account for the elastic long range distorsion due to the presence of crystal defects (dislocations). The latter is inspired to a recent model proposed by Lauteri and Luckhaus. Its asymptotics as the lattice spacing tends to zero produces a sharp interface model for grain boundaries which confirms the Read-Shockley law for small angle grain boundaries.
Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027)

We will outline an efficient algorithmic approach to the desingularisation of plane algebraic curves. Applications include computing the genus, Riemann-Roch spaces, and testing whether the curve is hyperelliptic. Afterwards, we will see that the (apparently rustic-looking) problem of finding the antiderivative of an algebraic function is actually related to the (much cooler-sounding) ability to test whether certain divisors are torsion in the Picard group of a curve. We will show how to achieve this thanks to the algorithms outlined earlier, which will lead us to a complete integration algorithm for algebraic functions based on arithmetic geometry. The talk will feature many explicit examples.

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

  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)