Pagina 19

07/02/23Seminario16:0017:001201 Dal PassoDavid RuizUniversidad de Granada
Seminario di Equazioni Differenziali
Symmetry results for compactly supported solutions of the 2D steady Euler equations

In this talk we present some recent results regarding compactly supported solutions of the 2D steady Euler equations. Under some assumptions on the support of the solution, we prove that the streamlines of the flow are circular. The proof uses that the corresponding stream function solves an elliptic semilinear problem -Delta phi = f(phi) with abla phi=0 at the boundary. One of the main difficulties in our study is that f can fail to be Lipschitz continuous near the boundary values. If f(phi) vanishes at the boundary values we can apply a local symmetry result of F. Brock to conclude. Otherwise, we are able to use the moving plane scheme to show symmetry, despite the possible lack of regularity of f. We think that such result is interesting in its own right and will be stated and proved also for higher dimensions. The proof requires the study of maximum principles, Hopf lemma and Serrin corner lemma for elliptic linear operators with singular coefficients.
03/02/23Seminario16:0017:001201 Dal Passo
Francesco ESPOSITO
Università di Padova
Algebra & Representation Theory Seminar (ARTS)
"Cohomology of quiver Grassmannians and Motzkin combinatorics"

  Quiver Grassmannians are projective algebraic varieties generalizing ordinary Grass-mannians and flag varieties. The cohomology of quiver Grassmannians of particular type has appli-cations to the geometric interpretation of various algebraic objects such as quantized universal enveloping algebras and cluster algebras. The variation in the cohomology of families of quiver Grassmannians of equioriented type A has been studied by Lanini-Strickland and Fang-Reineke.
  In this talk, I relate on joint work with Cerulli Irelli-Fang-Fourier and Cerulli Irelli-Marietti, in which we prove an upper semicontinuity statement for the cohomology of quiver Grassmannians of type A and we study the relation with Motzkin combinatorics found in work of Fang-Reineke.
03/02/23Seminario14:3015:301201 Dal Passo
Lorenzo VECCHI
Università di Bologna
Algebra & Representation Theory Seminar (ARTS)
"Categorical valuative invariants of matroids"

  Matroids are combinatorial objects that abstract the notion of linear independence and can be used to describe several structures such as, for example, vector spaces and graphs. Informa-tion on matroids can be encoded in several polynomial invariants, the most famous one being the characteristic polynomial; some of these polynomials can also be upgraded to graded vector spaces via abelian categorification or, when the matroid has a non-trivial group of symmetries, to graded virtual representations.
  Moreover, to each matroid, one can associate a polytope that belongs to the more general class of generalized permutahedra; a matroid invariant is called valuative if it behaves well under subdivi-sions of matroid polytopes.
  After introducing matroids and their invariants, the goal of the talk is to formulate the new notion of categorical valuativity and give some examples.
  This is based on a joint ongoing project with Dane Miyata and Nicholas Proudfoot.
31/01/23Seminario16:0017:001201 Dal PassoGiulio TiozzoUniversity of Toronto
Seminario congiunto di Equazioni Differenziali ed Analisi Complessa
     The harmonic measure for random walks on cocompact Fuchsian groups  

We consider random walks on groups of isometries of the hyperbolic plane, known as Fuchsian groups. It is well-known since Furstenberg that such random walks converge to the boundary at infinity, and the probability to reach a given subset of the boundary defines a hitting, or harmonic, measure on the circle. It has been a long-standing question whether this harmonic measure is absolutely continuous with respect to the Lebesgue measure. Conjecturally, this is never the case for random walks on cocompact, discrete groups. In the talk, based on joint work with Petr Kosenko, we settle the conjecture for nearest neighbour random walks on hyperelliptic groups. In fact, we show that the dimension of the harmonic measure for such walks is strictly less than one. This is also related to an inequality between entropy and drift.
25/01/23Seminario16:0017:001201 Dal Passo Roberto Conti Sapienza University of RomeHeat properties for groups

Somewhat motivated by the original approach of J.-B. Fourier to solve the heat equation on a bounded domain, we formulate some new properties of countable discrete groups involving certain completely positive multipliers of the reduced group C*-algebra and norm-convergence of Fourier series. The stronger "heat property" implies the Haagerup property, while the "weak heat property" is satisfied by a much larger class of groups. Examples will be provided to illustrate the various aspects. In perspective, a challenging goal would be to obtain yet another characterization of groups with Kazhdan's property (T). (Based on joint work with E. Bédos.)
24/01/23Seminario16:0017:001201 Dal PassoPaolo RoselliUniversità di Roma "Tor Vergata"
Seminario di Equazioni Differenziali
Il paradosso del piano tangente e il rapporto incrementale vettoriale di un piano secante

La retta tangente il grafico di una funzione sufficientemente regolare è la posizione limite di rette secanti il grafico. Ci si aspetterebbe che il piano tangente il grafico di una funzione a due variabili sufficientemente regolare sia la posizione limite di piani secanti il grafico in tre punti non collineari (a,f(a)), (b,f(b)) e (c,f(c)), ma così non è. Questo fenomeno paradossale è una versione locale del paradosso dell'area di una superficie curva (detto anche paradosso di Schwarz). In questo seminario visualizzerò il fenomeno paradossale, e mostrerò come il "coefficiente angolare vettoriale" di un piano secante possa esprimersi sia come combinazione vettoriale delle normali esterne al triangolo di vertici a, b e c, sia come rapporto vettoriale incrementale, quando il prodotto vettoriale è quello geometrico di Clifford. Se rimarrà tempo, accennerò anche a come modificare tale rapporto incrementale vettoriale per renderlo sempre convergente al gradiente di f.
17/01/23Seminario16:0017:001201 Dal PassoLiangjun Weng Università di Roma "Tor Vergata"A constrained mean curvature type flow

In this talk, we will discuss the isoperimetric inequality and its high order version -- Alexandrov Fenchel inequality, which dates back to the Queen Dido in ancient Carthage era. We introduce the quermass integrals for compact hypersurfaces with capillary boundary. Then by using a constrained mean curvature type flow, one can obtain the Alexandrov-Fenchel inequality for compact hypersurfaces with capillary boundary.
13/12/22Seminario16:0017:001201 Dal PassoCristiana De FilippisUniversità di Parma
Seminario di Equazioni Differenziali
     Schauder estimates for any taste  

So-called Schauder estimates are a standard tool in the analysis of linear elliptic and parabolic PDE. They have been originally obtained by Hopf (1929, interior case), and by Schauder and Caccioppoli (1934, global estimates). The nonlinear case is a more recent achievement from the ’80s (Giaquinta & Giusti, Ivert, Lieberman, Manfredi). All these classical results hold in the uniformly elliptic framework. I will present the solution to the longstanding problem, open since the ‘70s, of proving estimates of such kind in the nonuniformly elliptic setting. I will also cover the case of nondifferentiable functionals and provide a complete regularity theory for a new double phase model. From joint work with Giuseppe Mingione (University of Parma).
Note: This talk is part of the activity of the MIUR Department of Excellence Project MATH@TOV CUP E83C18000100006
06/12/22Seminario16:0017:001201 Dal PassoMarco GhimentiUniversità di PisaCompactness and blow up for Yamabe boundary problem

In 1992 Escobar extended the well known Yamabe problem to manifolds with boundary. The case of the scalar flat target manifold is particularly interesting since it also represents a generalization to Riemann mapping theorem to higher dimensions. In this talk we discuss when the solutions of the Yamabe boundary problem are a compact set, or when they form a blowing up sequence, underlining the affinities and the differences with the classical Yamabe problem.
NB:This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006
29/11/22Seminario16:0017:001201 Dal Passo Francesco EspositoUniversità della CalabriaSymmetry results for singular solutions to the p-Laplace equation

In this talk we will consider positive singular solutions to semilinear or quasilinear elliptic problems. We will deduce symmetry and monotonicity results of the solutions via a careful adaptation of the moving plane procedure of Alexandrov-Serrin.
NB:This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006

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