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25/10/22Seminario16:0017:001201 Dal PassoRoberto FeolaUniversità di Roma Tre
Seminario di Equazioni Differenziali
     Long time NLS approximation for a quasilinear Klein-gordon equation on large compact domains

We consider a class of Klein-Gordon equations with quasilinear, Hamiltonian and quadratic nonlinearities posed on a large box with periodic boundary conditions. We discuss how the cubic NLS equation can be derived to describe, approximately, the evolution of slow modulations in time and space of a spatially and temporarily oscillating wave packet. We show that the approximation is valid over a time scale which goes beyond the natural quadratic lifespan of solutions of cubic equations. We provide error estimates in Sobolev spaces. The proof is based on a combination of normal form techniques and energy methods.
Note: This talk is part of the activity of the MIUR Department of Excellence Project MATH@TOV CUP E83C18000100006
14/10/22Seminario16:0017:001201 Dal Passo
UiT / University of Tromsø
Algebra & Representation Theory Seminar (ARTS)
"Symmetries of supergeometries
related to nonholonomic superdistributions"

  We extend the Tanaka theory to the context of supergeometry and obtain an upper bound on the supersymmetry dimension of geometric structures related to strongly regular bracket-generating distributions on supermanifolds and their structure reductions. Several examples will be demonstrated, including distributions with at most simple Lie superalgebras as maximum symmetry.
  The talk is based on joint works with Andrea Santi, Dennis The and Andreu Llabres.
14/10/22Seminario14:3015:301201 Dal Passo
"Sapienza" Università di Roma
Algebra & Representation Theory Seminar (ARTS)
"Moment polytopes of spherical varieties and applications to multiplicity-free Hamiltonian manifolds"

  Spherical varieties are a generalization of toric, symmetric and flag varieties. They are also relevant in symplectic geometry, in particular in relation with multiplicity-free Hamiltonian manifolds. In this talk we will describe how the combinatorial structures arising in this theory can be used to characterize those multiplicity-free manifolds that admit a Kähler structure, in terms of their moment polytopes.
  This is a joint work with Bart Van Steirteghem.
11/10/22Seminario16:0017:001201 Dal PassoGaetano Siciliano University of Sao Paulo (IME-USP, Brazil)
Seminario di Equazioni Differenziali
      Critical points under the energy constraint  

In the talk we discuss the existence of critical points for a family of abstract and smooth functionals on Banach spaces under the energy constraint. By means of the Ljusternick-Schnirelmann theory and the fibering method of Pohozaev we show, under suitable assumptions, multiplicity results. The abstract framework is then applied to some partial differential equations depending on a parameter for which we obtain multiple solutions as well as some bifurcation results.
NB:This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006
05/10/22Seminario16:0017:001201 Dal PassoDetlev BuchholzUniversity of GoettingenProper condensates and long range order

The usual characterization of Bose-Einstein condensates is based on spectral properties of one-particle density matrices. (Onsager-Penrose criterion). The analysis of their specific properties, such as the occurrence of long-range order between particles and peaks in momentum space densities requires, however, the transition to the thermodynamic limit, where the one-particle density matrices are no longer defined. In the present talk, we will explain a new criterion of "proper condensation" that allows us to establish the properties of bosonic systems occupying fixed bounded regions. Instead of going to the idealization of an infinite volume, one goes to the limit of arbitrarily large densities in the given region. The resulting concepts of regular and singular wave functions can then be used to study the properties of realistic finite bosonic systems, the occurrence of condensates, and their large-distance behavior, with a precise control of accuracy.
04/10/22Seminario16:0017:001201 Dal PassoLei ZhangUniversity of Florida (US)
Seminario di Equazioni Differenziali
      Non-simple Blowup solutions of singular Liouville equations  

The singular Liouville equation is a class of second order elliptic partial differential equations defined in two dimensional spaces: $$Delta u+ H(x)e^{u}=4pi gamma delta_0 $$ where $H$ is a positive function, $gamma>-1$ is a constant and $delta_0$ stands for a singular source placed at the origin. This deceptively simply looking equation has a rich background in geometry, topology and Physics. In particular it interprets the Nirenberg problem in conformal geometry and is the reduction of Toda systems in Lie Algebra, Algebraic Geometry and Gauge Theory. Even if we only focus on the analytical aspects of this equation, it has wonderful and surprising features that attract generations of top mathematicians. The structure of solutions is particular intriguing when $gamma$ is a positive integer. In this talk I will report recent joint works with D’Aprile and Wei that give answers to some important issues of this equation. I will report the most recent results and consequences that our results may lead to.

1) This seminar will be held in presence, although the speaker will be connected remotely via MS Teams. The MS Teams link might be provided upon request to the organizers.
2) This talk is part of the activity of the MIUR Department of Excellence Project MATH@TOV CUP E83C18000100006.
30/09/22Seminario16:0017:001201 Dal Passo
Apoorva KHARE
Indian Institute of Science
Algebra & Representation Theory Seminar (ARTS)
"Higher-order theory for highest weight modules: positive weight-formulas,
resolutions and characters for higher order Verma modules"

  We introduce higher order Verma modules over a Kac-Moody algebra g (one may assume this to be sln throughout the talk, without sacrificing novelty). Using these, we present positive formulas - without cancellations - for the weights of arbitrary highest weight g-modules V. The key ingredient is that of "higher order holes" in the weights, which we introduce and explain.
30/09/22Seminario14:3015:301201 Dal Passo
Daniele VALERI
“Sapienza” Università di Roma
Algebra & Representation Theory Seminar (ARTS)
"Integrable triples in simple Lie algebras"

  We define integrable triples in simple Lie algebras and classify them, up to equivalence. The classification is used to show that all (but few exceptions) classical affine W-algebras W(g,f ), where g is a simple Lie algebra and f a nilpotent element, admit an integrable hierarchy of bi-Hamiltonian PDEs. This integrable hierarchy generalizes the Drinfeld-Sokolov hierarchy which is obtained when f is the sum of negative simple root vectors.
28/09/22Seminario16:0017:001201 Dal PassoJean DolbeautUniversité Paris Dauphine - PSL
Seminario di Equazioni Differenziali
 Stability estimates in some classical functional inequalities

In some classical functional inequalities, optimal functions and optimal constants are known. The next question is to understand which distance to the set of the optimal functions is controlled by the deficit, that is, the difference of the two sides of the inequality written with the optimal constant. In 1991, an answer was given by Bianchi and Egnell in the case of a Sobolev inequality on the Euclidean space, using compactness methods. A major issue with the method is that the new constant is so far unknown. The purpose of this lecture is to review some examples of related functional inequalities in which one can at least give an estimate of the stability constant.

Note: This talk is part of the activity of the MIUR Department of Excellence Project MATH@TOV CUP E83C18000100006
21/09/22Seminario16:1517:451201 Dal PassoFausto Di BiaseUniversità On the differentiation of integrals in measure spaces along filters

In 1936, R. de Possel observed that, in the general setting of a measure space with no metric structure, certain phenomena, relative to the differentiation of integrals, which are familiar in the Euclidean setting precisely because of the presence of a metric, are devoid of actual meaning. In this work, in collaboration with Steven G. Krantz, we show that, in order to clarify these difficulties,it is useful to adopt the language of filters, which has been introduced by H. Cartan just a year after De Possel's contribution.

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