Pagina 27

08/02/22Seminario16:0017:001201 Dal PassoAzahara DelaTorreUniversità di Roma "La Sapienza"The fractional Yamabe problem with singularities
( MS Teams Link for the streaming )

The so called Yamabe problem in Conformal Geometry consists in finding a metric conformal to a given one and which has constant scalar curvature. From the analytic point of view, this problem becomes a semilinear elliptic PDE with critical (for the Sobolev embedding) power non-linearity. If we study the problem in the Euclidean space, allowing the presence of nonzero-dimensional singularities can be transformed into reducing the non-linearity to a Sobolev-subcritical power. A quite recent notion of non-local curvature gives rise to a parallel study which weakens the geometric assumptions giving rise to a non-local semilinear elliptic PDE. In this talk, we will focus on metrics which are singular along nonzero-dimensional singularities. In collaboration with Ao, Chan, Fontelos, González and Wei, we covered the construction of solutions which are singular along (zero and positive dimensional) smooth submanifolds in this fractional setting. This was done through the development of new methods coming from conformal geometry and Scattering theory for the study of non-local ODEs. Due to the limitations of the techniques we used, the particular case of ''maximal’’ dimension for the singularity was not covered. In a recent work, in collaboration with H. Chan, we cover this specific dimension constructing and studying singular solutions of critical dimension.
NB:This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006
08/02/22Seminario14:3015:301201 Dal PassoPaolo Stellari
Università di Milano
Geometry Seminar
Stability conditions in the trivial canonical bundle case: Hilbert schemes of points.

The construction of stability conditions on the bounded derived category of coherent sheaves on smooth projective varieties is notoriously a difficult problem, especially when the canonical bundle is trivial. In this talk, I will review some results and techniques related to the latter setting. I will specifically concentrate on the case of Hilbert scheme of points on K3 surfaces and (as a work in progress) on generic abelian varieties of any dimension. This is joint work in progress with C. Li, E. Macri' and X. Zhao.
01/02/22Seminario14:3015:301201 Dal PassoGiovanni Mongardi
Università di Bologna
Geometry Seminar
Deforming rational curves

In this talk, we will survey known results on deformations of rational curves inside hyperkähler manifolds, and then provide suitable generalizations for singular hyperkähler varieties. As an application, we will construct uniruled divisors in many hyperkähler varieties. This is joint work with Ch. Lehn and G. Pacienza.
28/01/22Seminario16:0017:001201 Dal Passo
Università di Roma
Algebra & Representation Theory Seminar (ARTS)
"Class numbers of cyclotomic fields"
- in live & streaming mode -
( please click HERE to attend the talk in streaming )

  It is notoriously difficult to compute class numbers of cyclotomic fields.
  In this expository lecture we describe an experimental approach to this problem.
  N.B.: please click HERE to attend the talk in streaming.
28/01/22Seminario14:3015:301201 Dal Passo
SNS Pisa
Algebra & Representation Theory Seminar (ARTS)
"Abelian dynamical Galois groups"
- in live & streaming mode -
( please click HERE to attend the talk in streaming )

  Dynamical Galois groups are invariants associated to dynamical systems generated by the iteration of a self-rational map of P1. These are still very mysterious objects, and it is conjectured that abelian groups only appear in very special cases. We will show how the problem is deeply related to a dynamical property of these rational maps (namely that of being post-critically finite) and we will explain how to approach and prove certain non-trivial cases of the conjecture.
  This is based on joint works with A. Ostafe, C. Pagano and U. Zannier.
25/01/22Seminario16:0017:001201 Dal PassoRafael RuggieroPUC Rio De Janeiro
Seminario di Equazioni Differenziali
      On the graph property for totally irrational Lagrangian invariant tori  
     (MS Teams link for the streaming at the end of the abstract)  

We show that every C2 Lagrangian invariant torus W of a Tonelli Hamiltonian defined in the n-torus containing an orbit with totally irrational homology class is a graph of the canonical projection. This result extends the graph property obtained by Bangert and Bialy-Polterovich for Lagrangian minimizing tori without periodic orbits in the unit tangent bundle of a Riemannian metric in the two torus. Motivated by the famous Hedlund's examples of Riemannian metrics in the n-torus with n closed, homology independent, minimizing geodesics having minimizing tunnels, we also show the C1-generic nonexistence of Lagrangian invariant tori with "large" homology.

MS Teams Link for the streaming
Note: This talk is part of the activity of the MIUR Department of Excellence Project MATH@TOV CUP E83C18000100006
25/01/22Seminario14:3015:301201 Dal PassoSimone Diverio
La Sapienza
Geometry Seminar
Universal Gysin formulae and positivity of some characteristic

In the last years there have been a renewed interest for a conjecture by Griffiths stated in 1969. The conjecture characterises the positive characteristic forms for positive (in the sense of Griffiths) holomorphic Hermitian vector bundles: those should be the exactly the forms belonging to the positive cone spanned by Schur forms. After recalling the various definitions of positivity for holomorphic Hermitian vector bundles and (p,p)-forms, we shall explain a recent result, obtained in collaboration with my PhD student F. Fagioli, which partially confirms Griffiths' conjecture. The result is obtained as an application of a pointwise, differential-geometric version of a Gysin type formula for the push-forward of the curvature of tautological bundles over the flag bundle.
18/01/22Seminario14:3015:301201 Dal PassoEleonora di Nezza
Ecole Polytechnique de Paris
Geometry Seminar
Families of Kähler-Einstein metrics.

In a lot of geometric situations we need to work with families of varieties. In this talk we focus on families of singular Kähler-Einstein metric. In particular we study the case of a family of Kähler varieties and we develop the first steps of pluripotential theory in family, which will allow us to have a control on the C^0 estimate when the complex structure varies. This type of result will be applied in different geometric contexts. This is a joint work with V. Guedj and H. Guenancia.
14/01/22Seminario14:3015:301201 Dal Passo
University of Bonn
Algebra & Representation Theory Seminar (ARTS)
"Motivic Springer theory"
- in live & streaming mode -
( please click HERE to attend the talk in streaming )
N.B.: this talk is part of the activity of the MIUR Excellence Department Project CUP E83C18000100006

  Algebras and their representations can often be constructed geometrically in terms of convolution of cycles. For example, the Springer correspondence describes how irreducible repre-sentations of a Weyl group can be realised in terms of a convolution action on the vector spaces of irreducible components of Springer fibers. Similar situations yield the affine Hecke algebra, quiver Hecke algebra (KLR algebra), quiver Schur algebra or Soergel bimodules.
  N.B.: please click HERE to attend the talk in streaming.
16/12/21Colloquium16:0017:301201 Dal PassoNicola GigliSISSA TriesteDifferentiating in a non-differentiable environment 
( MS Teams Link for the streaming )

We all know what the differential of a smooth map from R to R is. By looking at coordinates and then at charts, we also know what it is the differential of a smooth map between differentiable manifolds. With a little bit of work, we can also define a (weak) differential for Sobolev/BV maps in this setting (but the case of manifold-valued maps presents challenges already at this level). In this talk I will discuss how it is possible to differentiate maps between spaces that have no underlying differentiable structure at all. The concepts of Sobolev/BV maps in this setting will also be discussed. 
NB:This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006

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