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). <br> <strong> Note: </strong> This talk is part of the activity of the MIUR Department of Excellence Project MATH@TOV CUP E83C18000100006

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. <br> <b>NB</b>:<i>This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006</i>

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. <br> <b>NB</b>:<i>This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006</i>

  I will describe how to construct monomial bases for the integer cohomology rings of compact wonderful models of toric arrangements. In the description of the monomials various combinatorial objects come into play: building sets, nested sets, and the fan of a suitable toric variety. In particular, I will focus on the case of the toric arrangements associated with root systems of type <em>A</em>. Here the combinatorial description of these basis offers a geometrical point of view on the relation between some eulerian statistics on the symmetric group. <br>   This is a joint work with Oscar Papini and Viola Siconolfi.

&nbsp; Delta and Theta operators are two families of operators on symmetric functions that show remarkable combinatorial properties. Delta operators generalise the famous nabla operator by Bergeron and Garsia, and have been used to state the Delta conjecture, an extension of the famous shuffle theorem proved by Carlsson and Mellit. Theta operators have been introduced in order to state a compositional version of the Delta conjecture, with the idea, later proved successful, that this would have led to a proof via the Carlsson-Mellit Dyck path algebra. We are going to give an explicit expansion of certain instances of Delta and Theta operators when <em>t</em>=1 in terms of what we call gamma Dyck paths, generalising several results including the Delta conjecture itself, using interesting combinatorial properties of the forgotten basis of the symmetric functions.

Argomenti: Grafi e matrici: concetti di connettività, matrice di adiacenza, matrici non negative, matrici primitive, teoria di Perron-Frobenius. Matrici stocastiche e substocastiche, problema del consenso. Matrici Laplaciane e di Metzler, punti di equilibrio e consenso nel caso continuo, cenno ai sistemi compartimentali. Il seminario fa parte delle attività del Progetto MIUR Dipartimento d'Eccellenza CUP E83C18000100006 e del centro RoMaDS.

Breve panoramica sulla scienza delle reti. Concetti classici di centralità basati su cammini minimi. Misure di centralità, somiglianza e distanza tra nodi basate su tecniche spettrali e funzioni di matrici. Catene di Markov a tempo discreto: Percorsi casuali classici e non-retrocedenti. Tecniche matriciali per la localizzazione di clusters, strutture core-periphery o quasi-bipartite. Introduzione ai percorsi casuali del secondo ordine: Tensori stocastici, PageRank nonlineare. Il seminario fa parte delle attività del Progetto MIUR Dipartimento d'Eccellenza CUP E83C18000100006 e del centro RoMaDS.

A classical subject in fluid mechanics regards the spectral instability of traveling periodic water waves, called Stokes waves. Benjamin, Feir, Whitam and Zhakarov predicted, through experiments and formal arguments, that Stokes waves in sufficiently deep water are unstable, finding unstable eigenvalues near the origin of the complex plane, corresponding to small Floquet exponents $mu$ or equivalently to long-wave perturbations. The first rigorous mathematical results have been given by Bridges-Mielke (’95) in finite depth and by Nguyen-Strauss (’20) in infinite depth. On the other hand, it has been found numerically that when the Floquet number $mu$ varies, two eigenvalues trace an entire figure-eight. I will present a novel approach to prove this conjecture fully describing the unstable spectrum. This is joint work with A. Maspero and P. Ventura. <br> <strong> Note: </strong> This talk is part of the activity of the MIUR Department of Excellence Project MATH@TOV CUP E83C18000100006.

&nbsp; I will present my attempt to categorify cluster algebras of type <em>B</em> and <em>C</em> using the theory of symmetric quivers in the sense of Derksen and Weyman.

  Linear Degenerations of the Flag Variety arise as very natural generalizations of the Complete Flag Variety and their geometrical properties very often appear to be linked with interesting combinatorial patterns. <br>   The talk will focus on a special class of linear degenerations, the Flat Degenerations, that have the remarkable property of being equidimensional algebraic varieties of the same dimension as the Complete Flag Variety. In some very recent works of M. Lanini and A. Pütz it is proved that Linear Degenerations of the Flag Variety can be endowed with a structure of GKM variety, under the action of a suitable algebraic torus <em>T</em>. <br>   The aim of the talk is to show how GKM Theory can be applied in this setting to prove some new results about the smooth locus in Flat Degenerations, generalizing a smoothness criterion proved by G. Cerulli Irelli, E. Feigin and M. Reineke for Feigin Degeneration. <br>   Finally, we provide a different combinatorial criterion, linking the smoothness property of a <em>T</em>-fixed point to the complete graph and to its orientations.