Dipartimento di Matematica -UTV-

Date	Type	Start	End	Room	Speaker	From	Title
27/10/21	Seminario	14:00	14:59	1201 Dal Passo	Erik Tonni	SISSA	Modular Hamiltonians for the massless Dirac field in the presence of a boundary or of a defect - in blended mode - Microsoft Teams link in the abstract. Abstract The reduced density matrix of a spatial subsystem can be written as the exponential of the modular Hamiltonian, hence this operator contains a lot of information about the entanglement of the corresponding spatial bipartition. First we consider the massless Dirac field on the half-line, imposing the most general boundary conditions that ensure the global energy conservation. This leads to two inequivalent phases where either the vector or the axial symmetry is preserved. In these two phases, we discuss the analytic expressions for the modular Hamiltonians of an interval on the half-line when the system is in its ground state, for the corresponding modular flows of the Dirac field and for the corresponding modular correlators. The method allows to obtain analytic expressions also for the modular Hamiltonians, the modular flows and the modular correlators for two disjoint equal intervals at the same distance from a point-like defect characterised by a unitary scattering matrix, that allows both reflection and transmission. Microsoft Teams Link
26/10/21	Seminario	14:30	15:30	1201 Dal Passo	Amos Turchet	University of Roma Tre	Geometry Seminar Campana’s program and special varieties Abstract Campana proposed a series of conjectures relating algebro-geometric and complex-analytic properties of algebraic varieties and their arithmetic. The main ingredient is the definition of the class of special varieties, which is the key for a new functorial classification of algebraic varieties, that is more suitable to answer arithmetic questions. In the talk we will review the main conjectures and constructions, and we will discuss some recent results that give evidence for some of these conjectures. This is joint work with E. Rousseau and J. Wang.
20/10/21	Seminario	16:00	17:00	1201 Dal Passo	Jean-Luc Sauvageot	Institut de Mathématiques de Jussieu	Misurabilità, densità spettrali e tracce residuali in geometria non commutativa Abstract We introduce, in the dual Macaev ideal of compact operators of a Hilbert space, the spectral weight rho(L) of a positive, self-adjoint operator L having discrete spectrum away from zero. We provide criteria for its measurability and unitarity of its Dixmier traces ( rho(L) is then called a spectral density) in terms of the growth of the spectral multiplicities of L and in terms of the asymptotic continuity of the eigenvalue counting function NL. Existence of meromorphic extensions and residues of the zeta-function zeta L of a spectral density are provided, under summability conditions on the spectral multiplicities. The hypertrace property of the states Omega L(·) = Tr omega(· rho(L)) on the norm closure of the Lipschitz algebra AL follows if the relative multiplicities of L vanish faster then its spectral gaps or if, at least, NL is asymptotically regular.
19/10/21	Seminario	14:30	15:30	1201 Dal Passo	Laura Pertusi	University of Milano	Geometry Seminar Serre-invariant stability conditions and cubic threefolds Abstract Stability conditions on the Kuznetsov component of a Fano threefold of Picard rank 1, index 1 and 2 have been constructed by Bayer, Lahoz, Macrì and Stellari, making possible to study moduli spaces of stable objects and their geometric properties. In this talk we investigate the action of the Serre functor on these stability conditions. In the index 2 case and in the case of GM threefolds, we show that they are Serre-invariant. Then we prove a general criterion which ensures the existence of a unique Serre-invariant stability condition and applies to some of these Fano threefolds. Finally, we apply these results to the study of moduli spaces in the case of a cubic threefold X. In particular, we prove the smoothness of moduli spaces of stable objects in the Kuznetsov component of X and the irreducibility of the moduli space of stable Ulrich bundles on X. These results come from joint works with Song Yang and with Soheyla Feyzbakhsh and in preparation with Ethan Robinett. These talks are part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006.
15/10/21	Seminario	16:00	17:00	1201 Dal Passo	Giovanni CERULLI IRELLI	“Sapienza” Università di Roma	Algebra & Representation Theory Seminar (ARTS) "On degeneration and extensions of symplectic and orthogonal quiver representations" - in live & streaming mode - (see the instructions in the abstract) Abstract Motivated by linear degenerations of flag varieties, and the study of 2-nilpotent B-orbits for classical groups, I will review the representation theory of symmetric quivers, initiated by Derksen and Weyman in 2002. I will then focus on the problem of describing the orbit closures in this context and how to relate it to the orbit closures for the underlying quivers. In collaboration with M. Boos we have recently given an answer to this problem for symmetric quivers of finite type. I believe that this result is a very special case of a much deeper and general result that I will mention in the form of conjectures and open problems. The talk is based on the preprint version of my paper with Boos available on the arXiv as 2106.08666. N.B.: please click HERE to attend the talk in streaming
15/10/21	Seminario	14:30	15:30	1201 Dal Passo	Lorenzo GUERRA	Scuola Normale Superiore - Pisa	Algebra & Representation Theory Seminar (ARTS) "Symmetric groups, tensor powers and extended powers of a topological space" - in live & streaming mode - (see the instructions in the abstract) Abstract The n-th cohomology of the symmetric group S_n on n objects with coefficients in the n-th tensor power of a vector space V on a field k, is endowed with an extremely rich algebraic structure. Indeed, their direct sum for all n ∈ N is an example of what goes under the name of "Hopf ring". First I will recall and review the definition of Hopf ring, then I will explicitly describe the cohomology algebras above, and finally I will briefly discuss the link with extended powers and other topological spaces interesting for homotopy theorists. The content of this talk stems from an ongoing collaboration with Paolo Salvatore and Dev Sinha. N.B.: please click HERE to attend the talk in streaming
12/10/21	Seminario	14:30	15:30	1201 Dal Passo	Daniele Agostini	Max Planck Institute for Mathematics in the Sciences in Leipzig	Geometry Seminar Theta functions and tau functions of algebraic curves Abstract The theta function of the Jacobian of a projective curve induces a solution of an infinite series of partial differential equations, the KP hierarchy. These solutions are packaged into the so-called tau function in integrable systems theory. I will recall the well-known picture in the case of smooth curves, and I will present some new results in the case of singular curves, focusing on those curves for which the theta function is actually polynomial. This is joint work with T. Çelik and J. Little. This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006.
06/10/21	Seminario	14:00	15:00	1201 Dal Passo	Edoardo D'Angelo	Universita' di Genova	Role of the relative entropy in the entropy of dynamical black holes Abstract Since the discovery of the Bekenstein-Hawking formula, there had been many attempts to derive the entropy of black holes from the entanglement between the degrees of freedom of matter fields inside and outside the event horizon. The entanglement is usually measured in terms of the entanglement entropy, which is obtained from the von Neumann entropy tracing over the degrees of freedom outside the black hole. However, the entanglement entropy is divergent in the continuum limit, and its regularization-dependence is in contrast with the universality of the Bekenstein-Hawking formula. In a recent paper, Hollands and Ishibashi adopted a different measure for the matter entropy: the relative entropy, which is well-defined also for continuum theories such as QFT. Hollands and Ishibashi showed that it reproduces the Bekenstein-Hawking formula for Schwarzschild black holes. In this talk I present a generalization of the work of Hollands and Ishibashi for the case of dynamical, spherically symmetric black holes. Using the back-reaction of a free, scalar quantum field on the metric, I showed that a variation in the relative entropy between coherent states of the field produces a variation of one-quarter of the black hole horizon area, thus finding that the black hole entropy is naturally defined as S = A/4 also in the dynamical case.
05/10/21	Seminario	14:30	15:30	1101 D'Antoni	Claudio Onorati	Tor Vergata	Geometry Seminar Remarks on sheaves on hyper-Kahler manifolds Abstract The geometry of moduli spaces of sheaves on K3 surfaces is very rich and led to very deep results in the last decades. Moreover, under certain hypotheses, these varieties are smooth projective and have a hyper-Kahler structure, providing non-trivial examples of compact hyper-Kahler manifolds. In higher dimensions the situation is much more complicated, nevertheless in the '90s Verbitsky introduced a set of sheaves on hyper-Kahler manifolds, called hyper-holomorphic, whose moduli spaces are singular hyper-Kahler (but not compact in general). Recently O'Grady proved that such sheaves belong to a larger set of sheaves for which there exists a good wall-and-chamber decomposition of the ample cone. This suggests an analogy between the study of moduli spaces of hyper-holomorphic sheaves on hyper-Kahler manifolds and the study of moduli spaces of sheaves on K3 surfaces. After having recalled the needed definitions and results, in this talk I will face the formality problem for such set of sheaves. In particular, I will extend the notion of hyper-holomorphic to complexes of locally free sheaves, and show how the associated dg Lie algebra of derived endomorphism is formal, namely quasi-isomorphic to its cohomology. As a corollary one gets a different proof of a quadraticity result of Verbitsky. This is a joint work in progress with F. Meazzini (INdAM). This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006.
28/09/21	Seminario	14:30	15:30	1101 D'Antoni	Rick Miranda	Colorado State University	Geometry Seminar Moduli spaces for rational elliptic surfaces (of index 1 and 2) Abstract Elliptic surfaces form an important class of surfaces both from the theoretical perspective (appearing in the classification of surfaces) and the practical perspective (they are fascinating to study, individually and as a class, and are amenable to many particular computations). Elliptic surfaces that are also rational are a special sub-class. The first example is to take a general pencil of plane cubics (with 9 base points) and blow up the base points to obtain an elliptic fibration; these are so-called Jacobian surfaces, since they have a section (the final exceptional curve of the sequence of blowups). Moduli spaces for rational elliptic surfaces with a section were constructed by the speaker, and further studied by Heckman and Looijenga. In general, there may not be a section, but a similar description is possible: all rational elliptic surfaces are obtained by taking a pencil of curves of degree 3k with 9 base points, each of multiplicity k. There will always be the k-fold cubic curve through the 9 points as a member, and the resulting blowup produces a rational elliptic surface with a multiple fiber of multiplicity m (called the index of the fibration). A. Zanardini has recently computed the GIT stability of such pencils for m=2; in joint work with her we have constructed a moduli space for them via toric constructions. I will try to tell this story in this lecture. This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006.