Harnack inequalities in sub-Riemannian settings

In the present thesis, we discuss the main notions of an axiomatic approach for an invariant Harnack inequality. This procedure, originated from techniques for fully nonlinear elliptic operators, has been developed by Di Fazio, Gutiérrez, and Lanconelli in the general settings of doubling Hölder quasi-metric spaces. The main tools of the approach are the so-called double ball property and critical density property: the validity of these properties implies an invariant Harnack inequality. We are mainly interested in the horizontally elliptic operators, i.e. some second order linear degenerate-elliptic operators which are elliptic with respect to the horizontal directions of a Carnot group. An invariant Harnack inequality of Krylov-Safonov type is still an open problem in this context. In the thesis we show how the double ball property is related to the solvability of a kind of exterior Dirichlet problem for these operators. More precisely, it is a consequence of the existence of some suitable interior barrier functions of Bouligand-type. By following these ideas, we prove the double ball property for a generic step two Carnot group. Regarding the critical density, we generalize to the setting of H-type groups some arguments by Gutiérrez and Tournier for the Heisenberg group. We recognize that the critical density holds true in these peculiar contexts by assuming a Cordes-Landis type condition for the coefficient matrix of the operator. By the axiomatic approach, we thus prove an invariant Harnack inequality in H-type groups which is uniform in the class of the coefficient matrices with prescribed bounds for the eigenvalues and satisfying such a Cordes-Landis condition.

Automorphisms of irreducible holomorphic symplectic manifolds and related problems

We study automorphisms and the mapping class group of irreducible holomorphic symplectic (IHS) manifolds. We produce two examples of manifolds of K3[2] type with a symplectic action of the alternating group A7. Our examples are realized as double EPW-sextics, the large cardinality of the group allows us to prove the irrationality of the associated families of Gushel-Mukai threefolds. We describe the group of automorphisms of double EPW-cubes. We give an answer to the Nielsen realization problem for IHS manifolds in analogy to the case of K3 surfaces, determining when a finite group of mapping classes fixes an Einstein (or Kähler-Einstein) metric. We describe, for some deformation classes, the mapping class group and its representation in second cohomology. We classify non-symplectic involutions of manifolds of OG10 type determining the possible invariant and coinvariant lattices. We study non-symplectic involutions on LSV manifolds that are geometrically induced from non-symplectic involutions on cubic fourfolds.

Estimating persistent Betti numbers for discrete shape analysis

Persistent Topology is an innovative way of matching topology and geometry, and it proves to be an effective mathematical tool in shape analysis. In order to express its full potential for applications, it has to interface with the typical environment of Computer Science: It must be possible to deal with a finite sampling of the object of interest, and with combinatorial representations of it. Following that idea, the main result claims that it is possible to construct a relation between the persistent Betti numbers (PBNs; also called rank invariant) of a compact, Riemannian submanifold X of R^m and the ones of an approximation U of X itself, where U is generated by a ball covering centered in the points of the sampling. Moreover we can state a further result in which, this time, we relate X with a finite simplicial complex S generated, thanks to a particular construction, by the sampling points. To be more precise, strict inequalities hold only in "blind strips'', i.e narrow areas around the discontinuity sets of the PBNs of U (or S). Out of the blind strips, the values of the PBNs of the original object, of the ball covering of it, and of the simplicial complex coincide, respectively.

Complex higher spins, Weyl invariance and tractors

In this thesis work I analyze higher spin field theories from a first quantized perspective, finding in particular new equations describing complex higher spin fields on Kaehler manifolds. They are studied by means of worldline path integrals and canonical quantization, in the framework of supersymmetric spinning particle theories, in order to investigate their quantum properties both in flat and curved backgrounds. For instance, by quantizing a spinning particle with one complex extended supersymmetry, I describe quantum massless (p,0)-forms and find a worldline representation for their effective action on a Kaehler background, as well as exact duality relations. Interesting results are found also in the definition of the functional integral for the so called O(N) spinning particles, that will allow to study real higher spins on curved spaces. In the second part, I study Weyl invariant field theories by using a particular mathematical framework known as tractor calculus, that enable to maintain at each step manifest Weyl covariance.

Automorphisms of O'Grady's sixfolds

We study automorphisms of irreducible holomorphic symplectic (IHS) manifolds deformation equivalent to the O’Grady’s sixfold. We classify non-symplectic and symplectic automorphisms using lattice theoretic criterions related to the lattice structure of the second integral cohomology. Moreover we introduce the concept of induced automorphisms. There are two birational models for O'Grady's sixfolds, the first one introduced by O'Grady, which is the resolution of singularities of the Albanese fiber of a moduli space of sheaves on an abelian surface, the second one which concerns in the quotient of an Hilbert cube by a symplectic involution. We find criterions to know when an automorphism is induced with respect to these two different models, i.e. it comes from an automorphism of the abelian surface or of the Hilbert cube.

Neurogeometry of stereo vision

This work aims to develop a neurogeometric model of stereo vision, based on cortical architectures involved in the problem of 3D perception and neural mechanisms generated by retinal disparities. First, we provide a sub-Riemannian geometry for stereo vision, inspired by the work on the stereo problem by Zucker (2006), and using sub-Riemannian tools introduced by Citti-Sarti (2006) for monocular vision. We present a mathematical interpretation of the neural mechanisms underlying the behavior of binocular cells, that integrate monocular inputs. The natural compatibility between stereo geometry and neurophysiological models shows that these binocular cells are sensitive to position and orientation. Therefore, we model their action in the space R3xS2 equipped with a sub-Riemannian metric. Integral curves of the sub-Riemannian structure model neural connectivity and can be related to the 3D analog of the psychophysical association fields for the 3D process of regular contour formation. Then, we identify 3D perceptual units in the visual scene: they emerge as a consequence of the random cortico-cortical connection of binocular cells. Considering an opportune stochastic version of the integral curves, we generate a family of kernels. These kernels represent the probability of interaction between binocular cells, and they are implemented as facilitation patterns to define the evolution in time of neural population activity at a point. This activity is usually modeled through a mean field equation: steady stable solutions lead to consider the associated eigenvalue problem. We show that three-dimensional perceptual units naturally arise from the discrete version of the eigenvalue problem associated to the integro-differential equation of the population activity.

New catalytic strategies for the manipulation of arenes

The aim of this Doctoral Thesis is the development of new catalytic synthetic methodologies in the context of the modern organic chemistry setting, with special focus on the use of cheap, sustainable catalytic materials. Specifically, during the course my PhD, I focused my research on two main distinct catalytic strategies, namely: the use of carbonaceous materials as catalysts (carbocatalysis) and nickel catalysis, also investigating a synergistic combination of the two. These methodologies were explored as means for the manipulation of (hetero)aromatic cores, representing ubiquitous, easily accessible and privileged scaffolds in medicinal or natural products chemistry. Both polar and radical reaction manifolds were engaged as complementary reactivities, capitalizing on metal- as well as organo-based activation modes. Particular attention has been devoted to addressing modern synthetic challenges or highly sought- after methodologies. Specifically, protocols for direct substitution of alcohols, dearomatization of arene nuclei, formation of C-S bonds, carbon dioxide fixation, C-C bond activation and fluoroalkylation were successfully achieved under carbo- or nickel catalyzed conditions.

Neurogeometry of reaching

The purpose of the thesis is to develop a model for the functional behaviour of neurons in the primary motor cortex (M1) responsible for arm reaching movements. From Georgopoulos neurophysiological data, we provide a first bundle structure compatible with the hypercolumnar organization and with the position-direction selectivity of motor cortical cells. We then extend this model to encode the direction of arm movement which varies in time, as experimentally measured by Hatsopoulos by introducing the notion of movement fragments. We provide a sub-Riemannian model which describes the time-dependent directional selectivity of cells though integral curves of the geometric structure we set up. The sub-Riemannian distance we define allows to implement a grouping algorithm able to detect a set of hand motor trajectories. These paths, identified by using a kernel defined in terms of kinematic variables, are compatible with the motor primitives obtained from neurophysiological results by spectral analysis applied directly on cortical variables. In a second part of the work, we propose geodesics in this space as an alternative model of models for arm movement trajectories. We define a special class of curves, called admissible, on which to study the geodesics problem: we provide a connectivity property in terms of admissible paths and the existence of normal length minimizers. Admissible geodesics are used as a model of reaching paths, finding a first validation through Flash and Hogan minimizing trajectories.