13 Febbraio 2019, ore 12, Aula A1.2, Coppito 0
Conferenziere: Emilia Mezzetti, Università degli studi di Trieste
Titolo: Lefschetz properties, Laplace equations, and Togliatti systems
Abstract: In two articles, published in 1929 and 1946, Eugenio Togliatti introduced and studied a rational surface in the 5-dimensional projective space, that satisfies a Laplace equation. In geometric terms, this means that all its osculating spaces have dimension less than expected. In an article in collaboration with Rosa M. Mirò-Roig and Giorgio Ottaviani (Canad. J. Math. 65, 2013), a relation is established, due to apolarity, between Artinian homogeneous ideals of a polynomial ring not satisfying the Weak Lefschetz Property - WLP - and projective varieties that verify a Laplace equation of a certain order s, i.e. such that all the s-osculating spaces have dimension less than expected. Thanks to this relation, the theorem of Togliatti has been extended to various classes of toric varieties verifying Laplace equations. In the seminar, I will introduce these notions and I will speak of some recent results relating them to Galois cyclic coverings and circulant matrices.
10 Maggio 2018, ore 14, Aula A1.4, Coppito 0
Conferenziere: Eugenio Vecchi, Sapienza
Titolo: Gauss Curvature in The Heisenberg Group.
Abstract: The notion of Gauss and mean curvature play a crucial role in the study of differential geometry of smooth Euclidean surfaces embedded in the Euclidean space. In the first Heisenberg group H there is a currently well accepted notion of horizontal mean curvature for smooth Euclidean surfaces, but it is still not understood what could be a reasonable candidate for the notion of horizontal Gauss curvature. In this talk I will suggest a possible definition based on the so called Riemannian approximation scheme. I will also present a Heisenberg version of the well known Gauss-Bonnet theorem for Euclidean C2-smooth, oriented and compact surfaces in H. If time permits, I will also show the connection to a localized Steiner-type formula which holds in H.
The talk is based on a joint work with Z.M. Balogh and J.T. Tyson.
19 Aprile 2018, ore 14, Aula A1.4, Coppito 0
Conferenziere: Joan Pons-Llopis, Università dell’Aquila
Titolo: Ulrich bundles on projective varieties.
Abstract: For a vector bundle E on a projective variety X, the condition of having no intermediate cohomology- namely being "arithmetically Cohen-Macaulay" - imposes very strong restictions on E. For istances, thanks to Horrocks' theorem, we know that any aCM bundle on the projective space should totally split. Ulrich proved that, for aCM bundles, there exists a bound on the dimension of their space of global sections. Therefore, we can even strengthen the requirements and pay attention to those aCM bundles attaining this bound (i.e., Ulrich bundles). Despite all of these constraints, it was asked by Eisenbud and Schreyer wheter any projective variety supports an Ulrich bundle. In this talk I intend to report on the history of this problem as well as to explain the contributions that I have obtained.
20 Febbraio 2018, ore 12,30, Aula A1.3, Coppito 0
Conferenziere: Ricardo Sa Earp, PUC-Rio (RJ-Brasil)
Titolo: On the reflection principle for minimal surfaces.
Abstract: We present a reflection principle for minimal surfaces in smooth three manifolds, joint work with Eric Toubiana, Univ Paris VII.
5 Dicembre, ore 12,30, Aula A1.3, Coppito 0
Conferenziere: Luciano Mari, Scuola Normale Superiore
Titolo: Maximum principles at infinity and the Ahlfors-Khas'minskii duality
Abstract: Maximum principles at infinity (or "almost maximum principles"), such as for instance Ekeland and Omori-Yau principles, are a powerful tool to investigate the geometry of Riemannian manifolds. Their validity is intimately related to the geometry of the underlying space M, and exhibits deep relations with the theory of stochastic processes on M and to potential theory. In the first part of the talk, I will present a survey of a few geometric applications to motivate the study of these principles, and discuss their link with probability. Then, I will discuss a recent underlying duality with the existence of suitable exhaustion functions called Khas'minskii potentials. Indeed, duality holds for a broad class of fully-nonlinear operators of geometric interest.
This is joint work with Leandro F. Pessoa
28 Novembre, ore 12,30 Laboratorio di Matematica Applicata (Coppito 1)
Conferenziere: Luigi Vezzoni, Università Di Torino
Titolo: Alexandrov's theorem and its generalizations
Abstract: The celebrated “Soap Bubble Theorem” of Alexandrov affirms that the spheres are the unique compact embedded hypersurfaces of the Euclidean space having constant mean curvature. Alexandrov proved the theorem introducing the method of moving planes, a very powerful technique which has been the source of many insights in analysis and differential geometry. Furthermore in the statement of the theorem the compactness and the embeddedness assumptions cannot be removed, due to some examples in literature (such as the Wente torus and some surfaces of Kapouleas). The talk aims to give a wide and basic exposition of these topics, as well as an introduction to some recent generalizations which I have obtained in collaboration with Giulio Ciraolo.
24 Ottobre, ore 12:30 aula A1.3, Coppito 0
Conferenziere: Giuseppe Pipoli, Università dell'Aquila
Titolo: Inverse mean curvature flow in complex hyperbolic space
Abstract: During last decades, the study of geometric flows is a very active field in geometry and analysis. The inverse mean curvature flow is perhaps the most important among the expanding flows, with deep meaning in general relativity too. We will discuss the evolution of a star-shaped closed and mean convex hypersurface in the complex hyperbolic space, showing similarities and differences with the previous cases. The main new phenomenon is that, in our case, the induced metric, even after a rescaling, degenerates in a special direction and converges to a sub-Riemannian limit.
3 Ottobre, ore 12:30, aula A1.3, Coppito 0
Conferenziere: Margherita Lelli Chiesa, Università dell'Aquila
Titolo: Curves on K3 surfaces
Abstract: I will report on both classical and more recent results concerning curves lying on K3 surfaces, highlighting their relevance in the study of the birational geometry of the moduli space of algebraic curves of fixed genus. Nikulin surfaces, that is, K3 surfaces endowed with a nontrivial double cover branched along eight disjoint rational curves, play a similar role at the level of the moduli space of Prym curves parametrizing étale double covers of curves of fixed genus. I will mention current work in this direction joint with Knutsen and Verra.
13 Giugno, ore 14.30 aula C1.9 (Coppito 2)
Conferenziere: Francesco Mercuri, Universidade Estadual de Campinas & Università dell'Aquila
Titolo: Una breve storia della congettura generalizzata di Poincaré e questioni connesse
Abstract: La congettura generalizzata di Poincaré puó essere enunciata come segue: Sia M^n una varietà omotopicamente equivalente alla sfera S^n. Allora M^n = S^n. In questo seminario discuteremo alcune risposte note per questa questione ed alcuni problemi aperti,
specialmente relazionati al significato di varietà ed a quello di M^n = S^n.
30 Maggio, ore 14.30 aula C1.10 (Coppito 2)
Conferenziere: Giuseppe Tinaglia, King's College London
Titolo: The geometry of constant mean curvature surfaces in Euclidean space.
Abstract: In this talk I will begin by reviewing classical geometric properties of constant mean curvature surfaces, H>0, in R^3. I will then talk about several geometric results for surfaces embedded in R^3 with constant mean curvature, such as curvature and radius estimates for simply-connected surfaces embedded in R^3 with constant mean curvature. Finally I will show applications of such estimates including a characterisation of the round sphere as the only simply-connected surface embedded in R^3 with constant mean curvature and area estimates for compact surfaces embedded in a flat torus with constant mean curvature and finite genus. This is joint work with Meeks.
16 Maggio 2017, ore 14.30, aula A1.4 (Coppito 0)
Conferenziere: Vlad Moraru, Universita' dell'Aquila
Titolo: Interaction between minimal surfaces and scalar curvature.
Abstract: In the late 1970s R. Schoen and S.T. Yau discovered a deep connection between the sign of the scalar curvature of a 3-manifold M and the topology of stable minimal surfaces contained in M; namely that non-negative ambient scalar curvature is an obstruction to the topological richness of stable minimal surfaces contained in M. I will explain this relationship and survey various old and new rigidity results following from it. Among these, I will discuss recent joint work with O. Chodosh and M. Eichmair. which confirms a conjecture by Fischer-Colbrie-Schoen and Cai-Galloway: A complete 3-manifold with non-negative scalar curvature and containing an area-minimising cylinder is flat.