### Relatori

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### Relatori

- Paolo Antonini: Optimal transport between algebraic hypersurfaces

I will report on a recent work in collaboration with F. Cavalletti and A. Lerario where we study complex projective hypersurfaces seen as probability measures on the projective space. Our guiding question is:

"What is the best way to deform a complex projective hypersurface into another one?"

Here the word best means from the point of view of measure theory and mass optimal transportation. In particular we construct an embedding of the space of complex homogeneous polynomials into the probability measures on the projective space and study its intrinsic Wasserstein metric. The Kähler structure of the projective space plays a fundamental role and we combine different techniques from symplectic geometry to the Benamou-Brenier dynamical approach to optimal transportation to prove several interesting facts. Among them we show that the space of hypersurfaces with the Wasserstein metric is complete and geodesic: any two hypersurfaces (possibly singular) are always joined by a minimizing geodesic. Moreover outside the discriminant locus, the metric is induced by a Kähler structure of Weil-Petersson type. - Andrea Appel: The R-matrix for affine Yangians

The theory of Yangians was introduced by Drinfeld in the 1980s as a systematic approach to solving the Yang-Baxter equation: every irreducible finite-dimensional representation is proved to be equipped with a rational R-matrix obtained by normalising the action of the universal R-matrix. Drinfeld's proof of the existence of the universal R-matrix for the Yangian of finite type was non-constructive and cohomological in nature. In this talk, I will present a generalisation of this result for the Yangian of affine type and its representations in category O. This is based on a joint work with S. Gautam and C. Wendlandt. - Paolo Aschieri: Differential Calculi of principal bundles on projective varieties

We present a sheaf approach to noncommutative principal bundles and extend it to differential calculi. This allows to study the differential geometry of bundles on noncommutative projective varieties. In this talk we focus in particular on the examples of the quantum principal bundles SL_{q}(2) → CP(1) and GL_{q}(2) → CP(1) and their associated (exact) sequences of horizontal forms, forms on the total space algebra and vertical forms. - Alberto Cattaneo: BV pushforward ad applications

In the BV formalism the spaces of fields are presented as complexes whose cohomology returns the physical content. Different but equivalent complexes may be used, which turns out to be important conceptually and in practice. One useful operation is that of a partial integration (BV pushforward). This produces a chain map which, under some assumptions, is a quasi-isomorphism. This has several applications: construction of observables (often as L_{∞}representations), renormalization à la Wilson, highly nontrivial equivalences of theories. I will discuss some examples. - Francesco D'Andrea: On morphisms of Hopf-Galois extensions

The ring structure of the K-theory of a compact space can be reconstructed from the knowledge of compact principal bundles over the space. The K-theory of a noncommutative C*-algebra has no obvious ring structure, but one can use Hopf-Galois extensions to construct a replacement of the K-theory ring. I will present some results about morphisms of Hopf-Galois extensions that are a preparation for the study of the "multiplicative K-theory" of a C*-algebra. This is a joint work with T. Maszczyk. - Marvin Dippell: Hochschild Cohomology for Coisotropic Reduction in Deformation Quantization

I will present a framework for incorporating coisotropic reduction into deformation quantization. To this end we will consider so-called constraint algebras and explore their deformation theory using a modification of Hochschild cohomology. In particular, I will present first results on the way to a Hochschild-Kostant-Rosenberg theorem which is compatible with reduction. - Giovanni Landi: Solutions to the quantum Yang Baxter equation and related deformations

We report on a class of noncommutative products of finite-dimensional Euclidean spaces. They are described by families of coordinate algebras A_{R}which are quadratic and are associated with an R-matrix which is involutive and satisfies the Yang-Baxter equations. The algebras enjoy a list of nice properties, being regular of finite global dimension. Notably, we have eight-dimensional noncommutative euclidean spaces R^{4}⊗_{R}R^{4}with particularly well behaved ones having deformation parameter u in S^{2}. Quotients include seven-spheres S^{7}_{u}and noncommutative quaternionic tori T^{H}_{u}=S^{3}⊗_{u}S^{3}. There is invariance for an action of SU(2)×SU(2) on this torus in parallel with the action of U(1)×U(1) on a "complex" noncommutative torus T^{2}_{θ}. This allows one to construct quaternionic toric noncommutative manifolds. There are also noncommutative principal fibration. - Antonio Miti: Multisymplectic observables and higher Courant algebroids

Multisimplectic manifolds are a straightforward generalization of symplectic manifolds where one considers closed non-degenerate k-forms in place of 2-forms. Recent works by Rogers and Zambon showed how one could associate to such a geometric structure two higher algebraic structures: an L_{∞}-algebra of observables and an L_{∞}-algebra of sections of the higher Courant algebroid twisted by $\omega$. Our main result is proving the existence of an L_{∞}-embedding between the above two L_{∞}-algebras generalizing a construction already found by Rogers around 2012 valid for multisymplectic 3-forms only. Moreover, we display explicit formulae for the sought morphism involving the Bernoulli numbers. Although this construction is essentially algebraic, it also admits a geometric interpretation when declined to the particular case of pre-quantizable symplectic forms. The latter case provides some evidence that this construction may be related to the higher analogue of geometric quantization for integral multisymplectic forms. - Margherita Paolini: Yangians and W-algebras

A finite W-algebra W(g,e) is an associative algebra associated with a pair (g,e) where g is a finite-dimensional reductive Lie algebra, and e is a nilpotent element of g. If g = gl_{N}and e is a rectangular nilpotent (i.e. all the Jordan blocks of e are of the same size), Ragoucy and Sorba noticed that W(g,e) is isomorphic to a quotient of the Yangian of dimension N. The above result is the prototypical example of the striking relationship there exists between the finite W-algebras and the Yangians. Such relation that has been investigated by several authors (see Brown, Brundan, Kleshchev). The aim of this talk is to recount the developments obtained in this direction. - Luca Vitagliano: The A
_{∞}-algebra of a Lie pair

A Lie pair is an inclusion A⊂L of Lie algebroids. Standard examples come from i) Lie subalgebras, ii) foliations, iii) Lie algebra actions and iv) complex structures. We show that every Lie pair (A,L) determines an A_{∞}-algebra U_{L/A}. The latter is canonical up to A_{∞}-isomorphisms. In the examples mentioned above U_{L/A}is a "resolution" of the algebra of i) invariant differential operators (DO) on a homogeneous space, ii) DOs on the leaf space, iii) invariant DOs under a Lie algebra action and iv) holomorphic DOs respectively. This is joint work with Mathieu Stiénon and Ping Xu