Lunedì 17 Novembre 2025, dalle ore 14:00 in Aula U1-08 (Edificio U1 - Piazza della Scienza 1, Milano), Zhuang He (Università di Torino) e Gabriele Barbieri (Università degli Studi di Milano-Bicocca) terranno i seguenti interventi
Relatore: Zhuang He
Titolo: Reflexive polytopes and the Picard ranks of Gorenstein Fano toric varieties
Abstract: Reflexive polytopes are lattice polytopes such that every facet is of lattice distance one from the origin. The toric varieties associated with reflexive polytopes are exactly those that are Gorenstein and Fano. Casagrande proved that in the Q-factorial case, the Picard ranks of the associated toric varieties are at most twice the dimension. We show a new result in general, that the sum of the Picard ranks of a polar pair of Gorenstein toric Fano varieties of dimension d>=3 is at most the number of facets of the reflexive polytopes minus (d - 1). This generalizes Eikelberg's theory of affine dependences of Picard groups of non-Q-factorial toric varieties.
Relatore: Gabriele Barbieri
Titolo: Diffeological Symplectic Frobenius Reciprocity and the Co-Moment Map Portrait of Quantum Hydrodynamics
Abstract: In this talk, I will address two problems in symplectic geometry and mathematical physics, connected by the common framework of Hamiltonian spaces.
The first concerns a symplectic analogue of Frobenius reciprocity, which is realized as a bijection between certain symplectically reduced spaces that are not necessarily manifolds. To address the singularities arising in such quotients we employ diffeology, a generalization of classical differential geometry that provides a natural setting for treating non-smooth spaces. Motivated by a conjecture formulated by T. Ratiu and F. Ziegler, it is shown that the symplectic Frobenius reciprocity holds as a diffeomorphism between diffeological spaces, preserving the reduced forms they may carry. This raises the foundational question of when such forms exist, for which new sufficient conditions are established: local freeness, strictness, or properness of the group action ensures their existence. Parallel results are obtained for prequantum spaces. This part of the talk is based on joint work with J. Watts and F. Ziegler.
The second problem originates in quantum hydrodynamics. Recent work has shown that the Madelung transform defines a moment map correspondence between wave functions and the cotangent bundle of densities. This correspondence, however, breaks down on nodal lines, where the probability density vanishes and the phase is undefined. To overcome this limitation, analogous relations are developed using the Clebsch geometry of the probability current, yielding a network of symplectic manifolds naturally associated with both quantum-mechanical and hydrodynamical structures. These manifolds are connected by moment maps, and their relations remain valid when restricted to the nodal line, thereby extending the geometric interplay between quantum mechanics and fluid dynamics into this singular regime. This part of the talk is based on joint work with M. Spera.
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