Titles and abstracts

Leandro Arosio, "A Julia-Wolff-Carathéodory Theorem in convex domains of finite type"

The classical Julia-Wolff-Carathéodory shows that, if f is a holomorphic self-map of the disc, the derivative f' admits a positive nontangential limit near any boundary regular fixed point z, and the limit equals the dilation of f at z which can be computed in terms of the Poincaré distance. This result had several generalizations to several variables: in particular Rudin proved a version of it in the ball, Abate in strongly convex domains, Bracci-Saracco-Trapani in strongly pseudoconvex domains and Abate-Tauraso in convex domains of D'Angelo finite type, adding a couple of technical assumptions. In this talk I will show how to prove the full theorem in the context of convex domains of D'Angelo finite type, using the strong asymptoticity of complex geodesics and the existence of horospheres. This result turns out to be related to the pluricomplex Poisson kernel introduced by Bracci-Patrizio-Trapani. This is a joint work with Matteo Fiacchi.

Giuseppe Barbaro, "Pluriclosed Calabi–Yau with torsion metrics and the stability of the pluriclosed flow"

We study special metrics in Hermitian geometry, characterized by classical constraints on the curvature of their Bismut connection. In particular, we describe the Calabi–Yau with torsion metrics of submersion type on toric bundles over Hermitian manifolds, constructing explicit examples. Moreover, we analyze the cohomological properties of compact complex manifolds equipped with a Bismut flat metric. This leads to a better understanding of the evolution of the pluriclosed flow on Bismut flat manifolds. Specifically, we deduce a global stability property for the Bismut flat metrics with respect to the pluriclosed flow, which in turn prevents the existence of non-flat pluriclosed Calabi–Yau with torsion metrics on Wang C-spaces.

Nicola Cavallucci, "Convergence and collapse of CAT(0)-spaces and groups"

I will present the possible Gromov-Hausdorff limits of geodesically complete, CAT(0)-spaces admitting a discrete group of isometries of bounded codiameter and the structure of the possible limit groups. The focus will be on the collapsing case: namely when the injectivity radius of the quotient space goes to zero along the sequence. Joint work with A.Sambusetti.

Chiara de Fabritiis, "Zeroes of slice-regular functions and of their vectorial parts"

Slice-regular functions of one quaternionic variable display a very interesting zero set, which is one of the most striking differences with the theory of holomorphic functions of one complex variable. Similarly to what happens to quaternions, slice regular functions can be represented in the form of two summands: a "real" and a "vectorial" part. Recent results (Altavila-dF, Gentili-Prezelj-Vlacci) show that the zeroes of the vectorial part of a never vanishing slice regular function are related with the eventual existence/non-existence of a suitable analogue of the logarithm in this setting. We will also give some partial results, obtained in collaboration with J. Prezelj, on the zeroes of slice regular functions with zero real part.

Andrea Del Prete, "A Twin correspondence for prescribed mean curvature graphs in Killing submersions"

A Riemannian (resp. Lorentzian) Killing submersion is a Riemannian submersion from a Riemannian (resp. Lorentzian) 3-manifold $E$ onto a Riemannian surface $M$, both connected and orientable, whose fibers are the integral curves of a non-vanishing (temporal) Killing vector field. In this setting we give a suitable definition of the graph of a smooth function defined over an open subset of $M$,  and we prove a generalized Calabi-type duality between (spacelike) graphs of prescribed mean curvature in Riemannian and Lorentzian Killing submersions. Finally we use this result to prove the existence of entire spacelike graph of certain prescribed mean curvature in Lorentz-Minkowski space. It is based on a joint work with H. Lee and J. M. Manzano.

Filippo Fagioli, "On Griffiths' conjecture about the positivity of Chern–Weil forms"

In the last few years there has been a renewed interest around a long-standing conjecture by Griffiths characterizing which should be the positive characteristic differential forms for any Griffiths positive vector bundle. This conjecture can be interpreted as the differential geometric counterpart of the celebrated Fulton-Lazarsfeld theorem on positive polynomials for ample vector bundles. The aim of this talk is to present a result that confirms the above conjecture for several characteristic forms. The positivity of these differential forms is due to a theorem which provides the version at the level of representatives of the universal push-forward formula for flag bundles valid in cohomology.

Giovanni Gentili, "The Monge-Ampère equation in hypercomplex geometry"

Alesker and Verbitsky posed in 2010 a Calabi-Yau type problem in hypercomplex geometry which can be phrased as a quaternionic version of the Monge-Ampère equation. The solvability of such an equation would lead to the existence of "special metrics". We will briefly overview the relevant mathematical framework and describe the current state of the art regarding the -still open- conjecture. This talk is based on joint works with Lucio Bedulli and Luigi Vezzoni.

Debora Impera, "Rigidity and non-existence results for collapsed translators"

Translators for the MCF give rise to a special class of eternal solutions, that besides having their own intrinsic interest, are models of a certain class of singularities that arise along the flow. Recently there has been a great effort in trying to construct examples of translators and in trying to classify them. In particular, many examples have been constructed of translators confined into slabs in space (collapsed translators), assuming that the slab width is greater than or equal to π. In this talk I will discuss some non-existence and rigidity results for collapsed translators when the slab width is sufficiently small. The talk is based on a joint project with N. M. Møller and M. Rimoldi.

Giovanni Manno, "(C-)Projective equivalence and (c-)projective symmetries"

Two metrics are called projectively equivalent if they share the same (unparametrized) geodesics. A projective symmetry is a vector field whose flow preserves geodesics. We shall give an overview of some methods for classifying metrics admitting a projective symmetry, discussing also the Kaehler analogue.

Stefano Marini, "On finitely non-degenerate closed homogeneous CR manifolds"

A complex flag manifold F= G /Q decomposes into finitely many real orbits under the action of a real form of G. Their embedding into F define on them CR manifold structures. I will characterize and list all the closed simple homogeneous CR manifolds which have finitely non-degenerate Levi form.

Tommaso Pacini, "G2 vs Kahler"

I will discuss work in progress, joint with Alberto Raffero, concerning G2 geometry. Our general goal is to investigate and develop certain analogies with Kahler geometry. The seminar will be accessible (enjoyable!) also to the non-G2 community.

Riccardo Piovani, "Harmonic forms on compact almost complex manifolds"

A question of Kodaira and Spencer, appeared as Problem 20 in Hirzebruch’s 1954 problem list, asks if the dimension of the space of Dolbeault harmonic (p,q)-forms, here denoted by h(p,q), depends on the choice of the metric on a given compact almost complex manifold. On a compact complex manifold, it is well known that by Hodge theory this space of harmonic forms is isomorphic to Dolbeault cohomology, which is an invariant of the complex structure. On a general compact almost complex manifold, computing h(p,q) on a compact complex manifold is a difficult analytic task which consists of solving a system of PDE's. Nonetheless, in 2020 Holt and Zhang built a family of almost complex structures on the Kodaira-Thurston 4-manifold where h(0,1) varies with the metric. In this talk we will present the main results obtained in the last few years concerning the spaces of Dolbeault, Bott-Chern and Hodge harmonic (p,q)-forms on a given compact almost complex manifold.

Massimiliano Pontecorvo, "Complex structures on Riemannian Four-Manifolds"

We report on new exixtence and non-existence results for compact bi-Hermitian surfaces.

Filippo Salis, "Kähler immersions of toric Kähler-Einstein manifolds into complex projective spaces"

Even though the existence of holomorphic and isometric immersions of a given Kähler manifold into a complex space form can be considered as a classical topic in complex differential geometry, many interesting questions are still unanswered. The aim of the talk is to provide an overview of the open problem of the classification of Kähler-Einstein manifolds admitting a Kähler immersion into a complex projective space. A special focus will be paid to the toric case.

Mario Santilli, "Rigidity and Compactness with constant mean curvature in warped product manifolds"

In this talk I present a rigidity result for volume-constrained critical points of the area functional in a class of warped product manifolds, which includes important models in General Relativity, like deSitter-Schwarzschild and Reissner-Nordstrom manifolds. This rigidity result extends a celebrated result of of Simon Brendle to singular varieties, and it allows to characterize limits of boundaries whose mean curvatures converge to a constant. Joint work with Francesco Maggi.

Diego Santoro, "L-spaces, taut foliations and fibered hyperbolic two-bridge links"

Heegaard Floer homology was defined by Ozsváth and Szabó in the early 2000’s. It consists of a package of invariants of closed oriented 3-manifolds and it has found many important and profound applications in low dimensional topology. I will introduce the L-space conjecture, that boldly predicts strong connections among properties relating Heegaard Floer homology, foliations and the fundamental group of an irreducible rational homology 3-sphere. I will then state a result concerning this conjecture and manifolds that arise as surgeries on fibered hyperbolic two-bridge links. Time permitting we will see how to apply this result to deduce that all non-meridional surgeries on Whitehead doubles of a non-trivial knot support coorientable taut foliations.

Giulia Sarfatti, "Zero sets and Nullstellensatz type theorems for slice regular polynomials"

Among the effects of the non-commutativity of the quaternions, there is the fact that the zero set of the (regular) product of two slice regular functions, in general, it is not given by the union of the two zero sets. In this talk we will discuss some properties of the vanishing sets of slice regular polynomials in several quaternionic variables, aiming to obtain versions of the classical Hilbert Nullstellensatz in this new setting. Joint work with Anna Gori and Fabio Vlacci.

Sara Scaramuccia, "Topological methods in classification problems"

In the last decades, Topological Data Analysis (TDA) is gaining much attention due to its scale-free and robust-to-noise nature. The basic goal of TDA is to apply topological descriptors to compare and classify data based on their shape. We quickly review the main pipeline of TDA: data, filtrations of simplicial complexes, topological signatures. How topological signatures can impact classification tasks? We quickly review some state-of-the-art use cases where topological signatures are integrated in the classification pipeline. In the final part, we present ongoing work in a real-case domain: classification of medical treatment impacts over patients undergoing a slimming diet. Data are collected and supervised by the Department of Biomedical Sciences of the University of Roma Tor Vergata.

Andrea Tamburelli, "Struttura pseudo-Kahler della componente di Hitchin di SL(3,R)"

La componente di Hitchin di SL(3,R) parametrizza le strutture proiettive convesse su superfici, estendendo in maniera naturale lo spazio di Teichmuller. Usando la teoria delle sfere affini, questo spazio eredita una struttura complessa che coincide con quella classica di Weil-Peterson sul Teichmuller. Vedremo come è possibile definire una metrica pseudo-Riemanniana compatibile con tale struttura complessa per la quale lo spazio di Teichmuller risulta totalmente geodetico. Lavoro in collaborazione con Nicholas Rungi (SISSA).

Michela Zedda, "Projectively induced Ricci flat Kähler manifolds"

It is well-known that a Ricci flat Kähler metric on a compact manifold is not projectively induced. The same is not true for noncompact manifolds, as shown by the flat metric on the complex euclidean space. Although, this is the only known example and a conjecture states it is the only possible one. After giving a brief overview of the problem, the talk focuses on studying the Ricci flat Kähler cone over a Sasakian manifold.