Giuseppe Barbaro (Università di Aarhus)

Jaroslaw Buczynski (IMPAN, Varsavia)

Title: Three stories of Riemannian and holomorphic manifolds 

Abstract: Compact holomorphic manifolds and Riemannian manifolds invite you all to participate in their three epic stories. In the first tale, the main character is going to be a compact holomorphic manifold, and as in every story, there will be some action going on. More specifically, the group of invertible complex numbers, or even better, several copies of those, act on the manifold. The spirit of late Andrzej Białynicki-Birula until this day helps us to comprehend what is going on. The second story is a tale of holonomies, it begins with "a long time ago,..." and concludes with "... and the last missing piece of this mystery is undiscovered til this day". The protagonist of this part is a quaternion-Kahler manifold, while the legacy of Marcel Berger is in the background all the time. In the third part we meet legendary distributions, which are subbundles in the tangent bundle of one of our main characters. Among others, distributions can be foliations, or contact distributions, which like yin and yang live on the opposite sides of the world, yet they strongly interact with one another. Ferdinand Georg Frobenius is supervising this third part. Finally, in the epilogue, all the threads and characters so far connect in an exquisite theorem on classification of low dimensional complex contact manifolds. In any dimension the analogous classification is conjectured by Claude LeBrun and Simon Salamon, while in low dimensions it is proved by Jarosław Wiśniewski, Andrzej Weber, in a joint work with the narrator. 

Gian Maria Dall’Ara (SNS Pisa)

Sara Angela Filippini (Università degli Studi del Salento)

Alice Garbagnati (Università degli Studi di Milano)

Title: Singular symplectic surfaces

Abstract: There are several possible generalizations of the definition of irreducible holomorphic symplectic manifold to the singular context (primitive symplectic variety, irreducible symplectic orbifold, irreducible symplectic variety). We review these definitions and we consider them in the lowest dimensional case, i.e. the case of the surfaces. We classify the surfaces which satisfy these definitions, showing that all of them are contractions of rational curves on K3 surfaces, but that not all the possible contractions of rational curves on K3 surfaces satisfy all these definitions. We observe that the Hilbert scheme of points of the singular surfaces considered are irreducible symplectic orbifolds, under certain conditions. The talk is based on a joint work with M. Penegini and A. Perego.

Simone Gutt (ULB, Bruxelles)

Eva Miranda (UPC and CRM – Barcellona e Università di Colonia)

Title: Symplectic Geometry in Flow

Abstract: This lecture explores how contact and cosymplectic structures provide a geometric lens for understanding stationary solutions of the Euler and Navier–Stokes equations. We discuss applications to embedding theorems, the search for periodic orbits, and the construction of universal Turing machines encoded within the flows of Reeb vector fields.

Gabriele Mondello (Università degli Studi di Roma La Sapienza)

Title: Unique hyperbolization of relatively maximal PSL(2,R)-representations of punctured surface groups

Abstract: A hyperbolic metric on a punctured surface comes equipped with a PSL(2, ℝ)-valued monodromy representation. If such a metric has conical singularities at the punctures, then the conjugacy class of the monodromy around the i-th peripheral loop is uniquely determined by the angle at the i-th conical point.

In the opposite direction, it is not generally understood when a PSL(2, ℝ) representation of the fundamental group of a punctured surface (with prescribed conjugacy classes around the peripheral loops) arises as the monodromy of a hyperbolic metric with conical points of given angles. A necessary condition is that the Euler number of the representation be relatively maximal.

In joint work with Nicolas Tholozan (ENS Paris), we show that such relatively maximal representations arise as monodromies of a unique hyperbolic metric if and only if the angles satisfy a precise smallness condition.

After introducing the main objects, I will describe the key ingredients of the proof, which include Teichmüller space, the Weil–Petersson metric, and the gradient flow of the energy functional associated with an equivariant map into the hyperbolic plane. 

Francesco Polizzi (Università degli Studi di Napoli, Federico II)

Silvia Sabatini (Università di Colonia)

Title: Positive monotone symplectic manifolds with symmetries and reflexive GKM graphs

Abstract: Positive monotone symplectic manifolds are the symplectic analogues of Fano varieties, namely they are compact symplectic manifolds for which the first Chern class equals the cohomology class of the symplectic form. In dimension 6, if the positive monotone symplectic manifold is acted on by a circle in a Hamiltonian way, a conjecture of Fine and Panov asserts that it is diffeomorphic to a Fano variety. In this talk I will report on recent classification results of positive monotone symplectic manifolds endowed with some special Hamiltonian actions of a torus, showing some evidence that they are indeed (homotopy equivalent/homeomorphic/diffeomorphic to) Fano varieties.  Moreover I will introduce reflexive GKM graphs, which are a generalization of reflexive polytopes, and illustrate their relation to some positive monotone symplectic manifolds as well as some of their properties.

Enrico Schlesinger (Politecnico di Milano)

Title:  Hilbert schemes of curves in projective space and the maximum genus problem.

Abstract: After a brief introduction to Hilbert schemes, I will present results on the the maximum genus problem. The classical maximum genus problem asks what is the maximum genus of a smooth curve of degree d in projective space that is not contained in a surface of degree < s (this is still open!). I will review what is known on the analogous question for the larger class of locally Cohen-Macaulay curves: these are curves that may have several irreducible components and a nonreduced algebra of regular functions, but do not have isolated or embedded points.  Finally, I will discuss results that settle the problem when d=s and when  d > 2s-2. These results were obtained in collaboration with V. Beorchia and P. Lella first, and then with A. Sammartano.