Dipartimento di Matematica -UTV-

Date	Type	Start	End	Room	Speaker	From	Title
28/10/22	Seminario	16:00	17:00	1201 Dal Passo	Salvatore STELLA	Università de L'Aquila	Algebra & Representation Theory Seminar (ARTS) "Dominance order and pointed bases for cluster algebras" Abstract Cluster algebras are a class of commutative rings endowed with a partial canonical basis whose elements are called cluster monomials. They are defined recursively through the combinatorial machinery of seeds and mutations. Cluster monomials have a particularly nice property: they are pointed, i.e. they can be written as the product of a Laurent monomial with a monic polynomial with respect to any seed. One of the main problems in the theory has been to extend the set of cluster monomials to a full basis consisting only of pointed elements. This has been achieved in a variety of generalities using approaches deriving, for example, from representation of associative algebras, Teichmüller theory, and mirror symmetry. Recently Qin introduced a dominance order on the tropical points of the associated cluster variety and showed that this order can be used to parametrize all possible pointed bases. In this talk we will explicitly describe the dominance order in rank two using a simple geometric construction. We will then connect it to certain representations of SL₃ . Time permitting we will conclude discussing how this construction generalizes to higher rank.
28/10/22	Seminario	14:30	15:30	1201 Dal Passo	Carolina VALLEJO RODRIGUEZ	Università di Firenze	Algebra & Representation Theory Seminar (ARTS) "On character conductors" N.B.: This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006 Abstract Given a character χ of a finite group G, there is a minimal positive integer f_χ such that all the values of χ belong to the f_χ-th cyclotomic field over the rationals. This number is often referred to as the conductor of χ. I will discuss some features of irreducible character conductors and their behavior with respect to factor groups.
28/10/22	Seminario	11:30	12:30	1201 Dal Passo	Fabio Cipriani	Politecnico di Milano	Spectral triples, Dirichlet spaces, and discrete groups. Abstract We study natural conditions on extended spectral triples $(A,h,D)$ by which the quantum differentials $da$ of elements $a in A$, belong to the ideal generated by the line element $ds = D^{-1}$. We also study upper and lower bounds on the singular values of the $da$'s to form a conformally invariant energy functional. We apply the general framework to study natural spectral triples of Dirichlet spaces and in particular those on duals of discrete groups arising on negative definite functions.
25/10/22	Seminario	16:00	17:00	1201 Dal Passo	Roberto Feola	Università di Roma Tre	Seminario di Equazioni Differenziali Long time NLS approximation for a quasilinear Klein-gordon equation on large compact domains Abstract 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
14/10/22	Seminario	16:00	17:00	1201 Dal Passo	Boris KRUGLIKOV	UiT / University of Tromsø	Algebra & Representation Theory Seminar (ARTS) "Symmetries of supergeometries related to nonholonomic superdistributions" Abstract 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.
14/10/22	Seminario	14:30	15:30	1201 Dal Passo	Guido PEZZINI	"Sapienza" Università di Roma	Algebra & Representation Theory Seminar (ARTS) "Moment polytopes of spherical varieties and applications to multiplicity-free Hamiltonian manifolds" Abstract 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.
11/10/22	Seminario	16:00	17:00	1201 Dal Passo	Gaetano Siciliano	University of Sao Paulo (IME-USP, Brazil)	Seminario di Equazioni Differenziali Critical points under the energy constraint Abstract 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
05/10/22	Seminario	16:00	17:00	1201 Dal Passo	Detlev Buchholz	University of Goettingen	Proper condensates and long range order Abstract 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.
04/10/22	Seminario	16:00	17:00	1201 Dal Passo	Lei Zhang	University of Florida (US)	Seminario di Equazioni Differenziali Non-simple Blowup solutions of singular Liouville equations Abstract 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. Note: 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.
30/09/22	Seminario	16:00	17:00	1201 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" Abstract We introduce higher order Verma modules over a Kac-Moody algebra g (one may assume this to be sl_n 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.