&nbsp; The relation between the geometric and the algebraic structure in algebraic quantum field theory is an intriguing topic that has been studied through several mathematical areas. A fundamental concept in Algebraic Quantum Field Theory (AQFT) is the relation between the localization property and the geometry of models. In the recent work with K.-H. Neeb, we rephrased and generalized some aspects of this relation by using the language of Lie theory. <br> &nbsp; We will start the talk introducing fundamental algebraic features of AQFT, in particular the Haag-Kastler axioms and the one particle formalism, and the presenting algebraic construction of the free field due to R.Brunetti, D. Guido and R. Longo. We will explain how this picture can be generalized. Firstly, how to determine some fundamental localization region, called wedge regions, at the Lie theory level and how a general Lie group can support a generalized AQFT. Then we show a classification of the simple Lie algebras supporting abstract wedges in relation with some special wedge configurations. The construction is possible for a large family of Lie groups and provides several new models in a generalized framework. Such a description of AQFT model generalization does not need a supporting manifold even if it is a desirable object. Time permitting, we will comment on recent developments about symmetric manifolds such models. <br> &nbsp; Based on V. Morinelli and K.-H. Neeb, <em>Covariant homogeneous nets of standard subspaces</em>, Commun. in Math. Phys <strong>386</strong> (1), 305-358 (2021). <br> &nbsp; <strong>N.B.:</strong> please <a href="https://teams.microsoft.com/l/meetup-join/19%3a342c23eced2540e4bcbb6f938e999db6%40thread.tacv2/1614163430980?context=%7b%22Tid%22%3a%2224c5be2a-d764-40c5-9975-82d08ae47d0e%22%2c%22Oid%22%3a%22cf31dbee-7758-4272-af72-503d7694f2ea%22%7d" target="Teams">click <strong>HERE</strong> to attend the talk</a> in streaming.

I will report on a joint work with Yuchen Liu and Taro Sano in which we construct infinitely many families of Sasaki-Einstein metrics on odd-dimensional spheres that bound parallelizable manifolds, proving in this way conjectures of Boyer-Galicki-Kollar and Collins-Szekelyhidi. The construction is based on showing the K-stability of certain Fano weighted orbifold hypersurfaces.

We consider the problem of prescribing Gaussian and geodesic curvatures for a conformal metric on the unit disk, which is equivalent to a Liouville-type PDE with nonlinear Neumann boundary conditions. We build a family of solutions which blow up on the boundary at a critical point of a functional which is a combination of the curvatures we are prescribing. The talk is based on joint works with M. Medina and A. Pistoia. <br> <b>NB</b>:<i>This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006</i>

We propose nonparametric estimators for the second-order central moments of spherical random fields within a functional data context. We consider a measurement framework where each field among an identically distributed collection of spherical random fields is sampled at a few random directions, possibly subject to measurement error. The collection of fields could be i.i.d. or serially dependent. Though similar setups have already been explored for random functions defined on the unit interval, the nonparametric estimators proposed in the literature often rely on local polynomials, which do not readily extend to the (product) spherical setting. We therefore formulate our estimation procedure as a variational problem involving a generalized Tikhonov regularization term. The latter favours smooth covariance/autocovariance functions, where the smoothness is specified by means of suitable Sobolev-like pseudo-differential operators. Using the machinery of reproducing kernel Hilbert spaces, we establish representer theorems that fully characterize the form of our estimators. We determine their uniform rates of convergence as the number of fields diverges, both for the dense (increasing number of spatial samples) and sparse (bounded number of spatial samples) regimes. We moreover validate and demonstrate the practical feasibility of our estimation procedure in a simulation setting. Based on a joint work with Julien Fageot, Matthieu Simeoni and Victor M. Panaretos. This talk is part of the activity of the MIUR Excellence Department Project CUP E83C18000100006.

We will present recent results for a class of Choquard type equations in the limiting Sobolev dimension in which one has the Riesz logarithmic kernel in the nonlocal part and the nonlinearity exhibits the highest possible growth, which is of exponential type. The competition between the logarithmic kernel and the exponential nonlinearity demands for new tools. A proper function space setting is provided by a new weighted version of the Pohozaev--Trudinger inequality which enables us to prove the existence of variational, in particular finite energy solutions. Equivalence issues with connected higher order fractional Scroedinger-Poisson systems will be also discussed, as well as related open problems. <br> <b>NB</b>:<i>This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006</i>

  I will report on the realization of the simple Lie superalgebra <em>G</em>(3) as symmetry superalgebra of various geometric structures - most importantly super-versions of the Hilbert-Cartan equation and Cartan's involutive system that exhibit <em>G</em>(2) symmetry - and compute, via Spencer cohomology groups, the Tanaka-Weisfeiler prolongation of the negatively graded Lie superalgebras associated with two particular choices of parabolics. I will then discuss non-holonomic superdistributions with growth vector (2|4 , 1|2 , 2|0) obtained as super-deformations of rank 2 distributions in a 5-dimensional space, and show that the second Spencer cohomology group gives a binary quadric, thereby providing a "square-root" of Cartan's classical binary quartic invariant for (2,3,5)-distributions. <br>   This is a joint work with B. Kruglikov and D. The. <br> &nbsp; <strong>N.B.:</strong> please <a href="https://teams.microsoft.com/l/meetup-join/19%3a342c23eced2540e4bcbb6f938e999db6%40thread.tacv2/1614163430980?context=%7b%22Tid%22%3a%2224c5be2a-d764-40c5-9975-82d08ae47d0e%22%2c%22Oid%22%3a%22cf31dbee-7758-4272-af72-503d7694f2ea%22%7d" target="Teams">click <strong>HERE</strong> to attend the talk</a> in streaming.

We discuss some results about the existence and multiplicity problem of different kind of entire solutions for some systems of semilinear elliptic equations, including the Allen Cahn and the NLS type models, under weak global non degeneracy conditions. <br> <b>NB</b>:<i>This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006</i>

<a href="~ricerca/analis/seminario_Candela.pdf">Link to the abstract</a> <br> <b>NB</b>:<i>This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006</i>

I report on two joint works: with my current student Siqing Zhang, and with Davesh Maulik (MIT), Junliang Shen (Yale) and Siqing Zhang. The Dolbeault moduli space of Higgs bundles over a complex algebraic curve is one of the ingredients in the Nonabelian Hodge Theory of the curve. Much is known and much is not known about this theory. From my current point of view, I consider some of the structures on the cohomology ring of these moduli spaces. I will start by introducing the P=W conjecture in Nonabelian Hodge Theory, mostly as motivation for the two joint works. The first work provides a cohomological shadow of a (strictly speaking non-existing) Nonabelian Hodge Theory for curves over fields of positive characteristic, and it unearths a new pattern for moduli of Higgs bundles in positive characteristic, which we call p-multiplicativity. The second work applies the first over a finite field to provide indirect evidence for the P=W conjecture over the complex numbers.