Lo studio dei morfismi birazionali e quindi dei vari modelli proiettivi di una varietà algebrica è uno dei problemi centrali in Geometria Algebrica. Il caso dei fibrati lineari è classicamente più trattato, quindi più standard rispetto a quello dei fibrati vettoriali, che presentano invece numerose questioni aperte. Alcune di queste questioni, presentate come obiettivi di ricerca nel Progetto PRIN 2015 "Geometry of Algebraic Varieties" (PE1 2015EYPTSB_011, Resp. Scientifico F. Flamini), sono state trattate in alcuni recenti risultati (già pubblicati od in progress) ottenuti, nell'ambito del progetto menzionato, in collaborazione con Flaminio Flamini (Università di Roma Tor Vergata), Samuel Boissiére e Alessandra Sarti (Université de Poitiers) e che sono oggetto di questo seminario. Per partecipare al seminario on-line su Zoom, scrivere all'oratore Gilberto Bini entro VENERDI' 19 MARZO 2021 all'indirizzo di posta elettronica gilberto.bini@unipa.it

  In 1985 Zamolodchikov constructed a "non-linear" extension of the Virasoro algebra known as <em>W</em>₃ algebra. This is one of the first appearances of a rich class of algebraic structures, known as <em>W</em>-algebras, which are intimately related to physical theories with symmetries and revealed many applications in mathematics. In the talk I will review some basic facts about the general theory of <em>W</em>-algebras and provide a description using Lax operators. This approach shows the deep connection of the theory of <em>W</em>-algebras with Yangians and integrable Hamiltonian hierarchies of Lax type equations. <br>   This is a joint work with A. De Sole and V. G. Kac. <br>   <strong>N.B.:</strong> please <a href="https://teams.microsoft.com/l/meetup-join/19%3a342c23eced2540e4bcbb6f938e999db6%40thread.tacv2/1614163430980?context=%7b%22Tid%22%3a%2224c5be2a-d764-40c5-9975-82d08ae47d0e%22%2c%22Oid%22%3a%22cf31dbee-7758-4272-af72-503d7694f2ea%22%7d" target="Teams">click <strong>HERE</strong> to attend the talk</a> in streaming

&nbsp; I will discuss results linking the study of certain highly symmetric arrangements of linear subspaces in affine space to representation theory, describing how to obtain qualitative information (e.g., is the arrangement Cohen-Macaulay?) and quantitative information (e.g., what is the Hilbert series of the ideal of the arrangement?) using techniques from the representation theory of Cherednik algebras. <br> &nbsp; <strong>N.B.:</strong> please <a href="https://teams.microsoft.com/l/meetup-join/19%3a342c23eced2540e4bcbb6f938e999db6%40thread.tacv2/1614163430980?context=%7b%22Tid%22%3a%2224c5be2a-d764-40c5-9975-82d08ae47d0e%22%2c%22Oid%22%3a%22cf31dbee-7758-4272-af72-503d7694f2ea%22%7d" target="Teams">click <strong>HERE</strong> to attend the talk</a> in streaming

In this talk I will report on   a refined blow up analysis of  finite  energy approximated solutions to the Nirenberg's problem on half  spheres. The later   consists of prescribing under minimal boundary  conditions the scalar curvature to be a given function. In particular  we give a precise location of blow up points and blow up rates. Such  an analysis shows that the blow up picture of the Nirenberg's problem  on half spheres is far more complicated that in the case of closed  spheres. Indeed besides the combination of interior and boundary blow  ups, there are "non simples blow ups"  for subcritical solutions  having zero or  nonzero weak limit. The formation  of such  non simple  blow ups is governed by a vortex problem, unveiling an  unexpected  connection with  Euler equations in fluid dynamic and mean fields type  equations in mathematical physics. This is joint work with Mohamed Ben Ayed (Sfax University).<br> <b>NB</b>:<i>This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006</i></div>

The symbolic action of a substitution on the binary expansion generates naturally an interval map. In case of erasing substitutions, the map is typically Baire-1 and not Darboux. As a model case, we consider what is arguably the simplest erasing substitution, mapping 0s to the empty word and 1s to 0 or 1 depending on the parity. The corresponding interval map is shown to have fractal properties (bounds are given on the Hausdorff dimension of the fibers) and to display rich dynamical behavior, including Devaney chaos, uniform distributional chaos of type 1, infinite topological entropy as well as the presence of cycles attracting in a finite time every rational. <br> <br> <strong> Note: </strong> <em> </strong> <em> The zoom link to the seminar will be posted on the <a href="https://www.dinamici.org/dai-seminar/"> DinAmicI website </a> and on <a href="https://mathseminars.org/seminar/DinAmicI"> Mathseminars.org</a>. Moreover, it will be also streamed live via the youtube <a href="https://www.youtube.com/channel/UCyNNg155G3iLS7l-qZjboyg"> DinAmicI channel</a>. </em> </em>

&nbsp; To a (reasonable) CY category <strong>C</strong> of global dimension two one can attach an associative algebra - its cohomological Hall algebra (COHA) - which is an algebra structure on the Borel-Moore homology of the stack of objects in <strong>C</strong>. In examples related to quivers (i.e. when <strong>C</strong> is the category of representations of the preprojective algebra of a quiver <em>Q</em>) this yields (positive halves) of Kac-Moody Yangians. In ongoing joint work with Diaconescu, Sala and Vasserot, we consider the case of the category of coherent sheaves supported on the exceptional locus of a Kleinian surface singularity. This is related to the above quiver case by a '2d-COHA' version of Cramer's theorem relating the (usual) Hall algebras of two hereditary categories which are derived equivalent. <br> &nbsp; <strong>N.B.:</strong> please <a href="https://teams.microsoft.com/l/meetup-join/19%3a342c23eced2540e4bcbb6f938e999db6%40thread.tacv2/1614163430980?context=%7b%22Tid%22%3a%2224c5be2a-d764-40c5-9975-82d08ae47d0e%22%2c%22Oid%22%3a%22cf31dbee-7758-4272-af72-503d7694f2ea%22%7d" target="Teams">click <strong>HERE</strong> to attend the talk</a> in streaming

In this talk, I will discuss about a fractional version of the Caffarelli-Kohn-Nirenberg inequality. We first study the existence and nonexistence of extremal solutions. We also show some results for the symmetry and symmetry breaking region for the minimizers. In order to get these result we reformulate the fractional Caffarelli-Kohn-Nirenberg inequality in cylindrical variables. We also get the non-degeneracy and uniqueness of minimizers in the radial symmetry class. This is joint work with Azahara DelaTorre and Maria del Mar Gonzalez.<br> <b>NB</b>:<i>This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006</i>

Aim of the talk is to discuss qualitative properties of the nonlinear Schr\"odinger equation with combined nonlinearities, where the leading term is an intracritical focusing power-type nonlinearity, and the perturbation is given by a power-type defocusing one. Fixed the mass of the problem, we completely answer the question wether the ground state energy , which is a threshold between global existence and formation of singularities, is achieved. As a byproduct of the variational characterization of the ground state energy, we show the existence of blowing-up solutions in finite time, for any initial data with energy below the ground state energy threshold in case of cylindrical symmetry. <br> <b>N.B.</b>: &nbsp;<i>this talk is part of the activity of the MIUR Excellence Department Project CUP E83C18000100006</i>

I will present some decay of correlations results on skew products which are locally accessible. The results rely on the study of a twisted transfer operator and could be generalized to many other situations. I will also present numerical counterparts to such results. <br> <br> <strong> Note: </strong> <em> </strong> <em> The zoom link to the seminar will be posted on the <a href="https://www.dinamici.org/dai-seminar/"> DinAmicI website </a> and on <a href="https://mathseminars.org/seminar/DinAmicI"> Mathseminars.org</a>. Moreover, it will be also streamed live via the youtube <a href="https://www.youtube.com/channel/UCyNNg155G3iLS7l-qZjboyg"> DinAmicI channel</a>. </em> </em>

Symmetry of geometry is inherited by symmetry of function spaces, called the regular representation. From this viewpoint, the classical theory of expansions such as Fourier series or spherical harmonics may be interpreted as "analysis and synthesis" of the regular representation. In this talk, we address the following fundamental questions about the regular representation on manifolds X acted algebraically by reductive Lie groups G such as GL(n,R). A.  Does the group G "control well"  the space of function on X? B.  What can we say about "spectrum" for L^2(X)?  We highlight "multiplicity" for A and "temperdness" for B, and explain some geometric ideas of the solution. <b>N.B.</b>:  <i>this talk is part of the activity of the MIUR Excellence Department Project CUP E83C18000100006</i>