In this thesis we study the relation between algebraic torus actions on complex projective varieties and the birational geometry of their geometric quotients. Given a C*-action on a normal projective variety X, there exist two unique connected components of the fixed point locus, called the sink Y− and the source Y+, containing the limit at ∞ and 0 of the general orbit. Let GX− (resp. GX+) be the variety parametrizing the orbits converging to the sink (resp. the source). Since there exists an open subset of points converging to Y±, we obtain a birational map ψ: GX->GX+. By choosing different linearizations of ample line bundles on X, we obtain a factorization of the birational map ψ among inner geometric quotient, parametrizing different open subsets of stable points. In this setting, we investigate the local analytic geometry of the birational map ψ. On one hand we link certain birational transformations, called rooftop flips, with varieties with two projective bundles structures. On the other we study when the birational map ψ can be locally described by a toric flip of Atiyah type. If on one side a C*-action naturally induces a birational map among geometric quotients, it is meaningful to study the opposite direction: more precisely, given a birational map φ: Z+->Z− among normal projective varieties, how can we construct a normal projective variety X, endowed with a C*-action, such that Z− is the sink, Z+ is the source, and the natural birational map ψ constructed above coincide with φ? Such an X is called a geometric realization of the birational map φ. We propose a construction of a geometric realization of φ, whose geometry reflects the factorization of the map as a composition of flips, blow-ups and blow-downs. We describe in particular the case in which φ is a small modification of dream type, namely a birational map which is an isomorphism in codimension 1 associated to a finitely generated multisection ring. Moreover, we show that the cone of divisors associated to such multisection rings admits a chamber decomposition where the models are the geometric quotients of the C*-action. If in addition Z± are assumed to be toric varieties, we construct a function in SageMath to compute the polytope of the associated toric geometric realization.

The influence of Chow groups on singular cohomology is motivated by classical results by Mumford and Roitman and has been investigated extensively. On the other hand, the converse influence is rather conjectural and it takes place in the framework of the ``philosophy of mixed motives'', which is mainly due to Grothendieck, Bloch and Beilinson. In the spirit of exploring this influence, Voisin formulated in 1996 a conjecture on 0--cycles on the self--product of surfaces of geometric genus one. There are few examples in which Voisin's conjecture has been verified, but it is still open for a general $K3$ surface. Our aim is to present a new example in which Voisin's conjecture is true, a family of Todorov surfaces. We give an explicit description of the family as quotient of complete intersection of four quadrics in $mathbb{P}^{6}$. We verify Voisin's conjecture for the family of Todorov surfaces of type $(2,12)$. Our main tool is Voisin's ``spreading of cycles'', we use it to establish a relation between 0--cycles on the Todorov surface and on the associated K3 surface. We give a motivic version of this result and some interesting motivic applications.

Le mappe comomento omotopiche sono una generalizzazione della nozione di mappa momento introdotta al fine di estendere il concetto di azione hamiltoniana al contesto della geometria multisimplettica. L'obiettivo di questa tesi è fornire nuove costruzioni esplicite ed esempi concreti di azioni di gruppi di Lie su varietà multisimplettiche che ammettono delle mappe comomento. Il primo risultato è una classificazione completa delle azioni di gruppi compatti su sfere multisimplettiche. In questo caso, l'esistenza di mappe comomento omotopiche dipende dalla dimensione della sfera e dalla transitività dell'azione di gruppo. Il secondo risultato è la costruzione esplicita di un analogo multisimplettico dell’inclusione dell'algebra di Poisson di una varietà simplettica dentro il corrispondente algebroide di Lie twistato. E’ possibile dimostrare che questa inclusione soddisfa una relazione di compatibilità nel caso di varietà multisimplettiche gauge-correlate in presenza di un'azione di gruppo Hamiltoniana. Tale costruzione potrebbe giocare un ruolo nella formulazione di un analogo multisimplettico della procedura di quantizzazione geometrica. L’ultimo risultato è una costruzione concreta di una mappa comomento omotopica relativa all'azione multisimplettica del gruppo di diffeomorfismi che preservano la forma volume dello spazio Euclideo. Questa mappa ammette naturalmente un’interpretazione idrodinamica, nello specifico trasgredisce alla mappa comomento idrodinamica introdotta da Arnol'd, Marsden, Weinstein e altri. La mappa comomento così costruita può essere inoltre messa in relazione alla teoria dei nodi avvalendosi dell’approccio ai link nel formalismo dei vortici. Questo punto di apre la strada a un'interpretazione semiclassica del polinomio HOMFLYPT nel linguaggio della quantizzazione geometrica. / Homotopy comomentum maps are a higher generalization of the notion of moment map introduced to extend the concept of Hamiltonian actions to the framework of multisymplectic geometry. Loosely speaking, higher means passing from considering symplectic $2$-form to consider differential forms in higher degrees. The goal of this thesis is to provide new explicit constructions and concrete examples related to group actions on multisymplectic manifolds admitting homotopy comomentum maps. The first result is a complete classification of compact group actions on multisymplectic spheres. The existence of a homotopy comomentum maps pertaining to the latter depends on the dimension of the sphere and the transitivity of the group action. Several concrete examples of such actions are also provided. The second novel result is the explicit construction of the higher analogue of the embedding of the Poisson algebra of a given symplectic manifold into the corresponding twisted Lie algebroid. It is also proved a compatibility condition for such embedding for gauge-related multisymplectic manifolds in presence of a compatible Hamiltonian group action. The latter construction could play a role in determining the multisymplectic analogue of the geometric quantization procedure. Finally a concrete construction of a homotopy comomentum map for the action of the group of volume-preserving diffeomorphisms on the multisymplectic 3-dimensional Euclidean space is proposed. This map can be naturally related to hydrodynamics. For instance, it transgresses to the standard hydrodynamical co-momentum map of Arnol'd, Marsden and Weinstein and others. A slight generalization of this construction to a special class of Riemannian manifolds is also provided. The explicitly constructed homotopy comomentum map can be also related to knot theory by virtue of the aforementioned hydrodynamical interpretation. Namely, it allows for a reinterpretation of (higher-order) linking numbers in terms of multisymplectic conserved quantities. As an aside, it also paves the road for a semiclassical interpretation of the HOMFLYPT polynomial in the language of geometric quantization.

In this Thesis, we analyze three variational and geometric problems, that extend classical Euclidean issues of the calculus of variations to more general classes of spaces. The results we outline are based on the articles [Ved21; MV21] and on a forthcoming joint work with Nicolussi Golo and Serra Cassano. In the first place, in Chapter 1 we provide a general introduction to metric measure spaces and some of their properties. In Chapter 2 we extend the classical Talenti’s comparison theorem for elliptic equations to the setting of RCD(K,N) spaces: in addition the the generalization of Talenti’s inequality, we will prove that the result is rigid, in the sense that equality forces the space to have a symmetric structure, and stable. Chapter 3 is devoted to the study of the Bernstein problem for intrinsic graphs in the first Heisenberg group H^1: we will show that under mild assumptions on the regularity any stationary and stable solution to the minimal surface equation needs to be intrinsically affine. Finally, in Chapter 4 we study the dimension and structure of the singular set for p-harmonic maps taking values in a Riemannian manifold.

The work is situated within the theory of slice analysis, a generalization of complex analysis for hypercomplex numbers, considering function of both quaternionic and Clifford variables, in both one and several variables. %We first characterize some partial slice sets of The primary focus of the thesis is on the harmonic and polyharmonic properties of slice regular functions. We derive explicit formulas for the iteration of the Laplacian on slice regular functions, proving that their degree of harmonicity increases with the dimension of the algebra. Consequently, we present Almansi-type decompositions for slice functions in several variables. Additionally, using the harmonic properties of the partial spherical derivatives and their connection with the Dirac operator in Clifford analysis, we achieve a generalization of the Fueter and Fueter-Sce theorems in the several variables context. Finally, we establish that regular polynomials of sufficiently low degree are the unique slice regular functions in the kernel of the iteration of the Laplacian, whose power is less than Sce index.

In 2020 Parusiński and Rond proved that every algebraic set X ⊂ R^n is homeomorphic to an algebraic set X’ ⊂ R^n which is described globally (and also locally) by polynomial equations whose coefficients are real algebraic numbers. In general, the following problem was widely open: Open Problem. Is every real algebraic set homeomorphic to a real algebraic set defined by polynomial equations with rational coefficients? The aim of my PhD thesis is to provide classes of real algebraic sets that positively answer to above Open Problem. In Chapter 1 I introduce a new theory of real and complex algebraic geometry over subfields recently developed by Fernando and Ghiloni. In particular, the main notion to outline is the so called R|Q-regularity of points of a Q-algebraic set X ⊂ R^n. This definition suggests a natural notion of a Q-nonsingular Q-algebraic set X ⊂ R^n. The study of Q-nonsingular Q-algebraic sets is the main topic of Chapter 2. Then, in Chapter 3 I introduce Q-algebraic approximation techniques a là Akbulut-King developed in collaboration with Ghiloni and the main consequences we proved, that are, versions ‘over Q’ of the classical and the relative Nash-Tognoli theorems. Last results can be found in in Chapters 3 & 4, respectively. In particular, we obtained a positive answer to above Open Problem in the case of compact nonsingular algebraic sets. Then, after extending ‘over Q’ the Akbulut-King blowing down lemma, we are in position to give a complete positive answer to above Open Problem also in the case of compact algebraic sets with isolated singularities in Chapter 4. After algebraic Alexandroff compactification, we obtained a positive answer also in the case of non-compact algebraic sets with isolated singularities. Other related topics are investigated in Chapter 4 such as the existence of Q-nonsingular Q-algebraic models of Nash manifolds over every real closed field and an answer to the Q-algebrization problem for germs of an isolated algebraic singularity. Appendices A & B contain results on Nash approximation and an evenness criterion for the degree of global smoothings of subanalytic sets, respectively.

Very few standards exist for tting products to people. Footwear fit is a noteworthy example for consumer consideration when purchasing shoes. As a result, footwear manufacturing industry for achieving commercial success encountered the problem of developing right footwear which is fulfills consumer's requirement better than it's competeries. Mass customization starts with understanding individual customer's requirement and it finishes with fulllment process of satisfying the target customer with near mass production efficiency. Unlike any other consumer product, personalized footwear or the matching of footwear to feet is not easy if delivery of discomfort is predominantly caused by pressure induced by a shoe that has a design unsuitable for that particular shape of foot. Footwear fitter have been using manual measurement for a long time, but the combination of 3D scanning systems with mathematical technique makes possible the development of systems, which can help in the selection of good footwear for a given customer. This thesis, provides new approach for addressing the computerize footwear fit customization in industry problem. The design of new shoes starts with the design of the new shoe last. A shoe last is a wooden or metal model of human foot on which shoes are shaped. Despite the steady increase in accuracy, most available scanning techniques cause some deficiencies in the point cloud and a set of holes in the triangle meshes. Moreover, data resulting from 3D scanning are given in an arbitrary position and orientation in a 3D space. To apply sophisticated modeling operations on these data sets, substantial post-processing is usually required. We described a robust algorithm for filling holes in triangle mesh. First, the advance front mesh technique is used to generate a new triangular mesh to cover the hole. Next, the triangles in initial patch mesh is modified by estimating desirable normals instead of relocating them directly. Finally, the Poisson equation is applied to optimize the new mesh. After obtaining complete 3D model, the result data must be generated and aligned before taking this models for shape analysis such as measuring similarity between foot and shoe last data base for evaluating footwear it. Principle Component Analysis (PCA), aligns a model by considering its center of mass as the coordinate system origin, and its principle axes as the coordinate axes. The purpose of the PCA applied to a 3D model is to make the resulting shape independent to translation and rotation asmuch as possible. In analysis, we applied "weighted" PCA instead of applying the PCA in a classical way (sets of 3D point-clouds) for alignment of 3D models. This approach is based on establishing weights associated to center of gravity of triangles. When all of the models are aligned, an efficient algorithm to cut the model to several sections toward the heel and toe for extracting counters is used. Then the area of each contour is calculated and compared with equal sections in shoe last data base for finding best footwear fit within the shoe last data base.