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29/10/21Seminario16:0017:001201 Dal Passo
Université Libre de Bruxelles
Algebra & Representation Theory Seminar (ARTS)
"Partial and global representations of finite groups"
- in live & streaming mode -
(see the instructions in the abstract)
N.B.: this talk is part of the activity of the MIUR Excellence Department Project CUP E83C18000100006

  The notions of partial actions and partial representations have been extensively studied in several algebraic contexts in the last 25 years. In this talk we introduce these concepts and give a short overview of the results known for finite groups.
  We will briefly show how this theory extends naturally the classical global theory, in particular in the important case of the symmetric group.
  This is joint work with William Hautekiet, Paolo Saracco and Joost Vercruysse.
N.B.: please click HERE to attend the talk in streaming
29/10/21Seminario14:3015:301201 Dal Passo
University of York
Algebra & Representation Theory Seminar (ARTS)
"Soergel diagrammatics in modular representation theory"
- in live & streaming mode -
(see the instructions in the abstract)

  We provide an elementary introduction to Elias-Williamson’s Soergel diagrammatics and p-Kazhdan-Lusztig theory and discuss the applications in representation theory. In particular we will discuss the recent proof of (generalised versions of) Libedinsky-Patimo’s conjecture, which states that certain simple characters of affine Hecke algebras are given in terms of p-Kazhdan-Lusztig polynomials and of Berkesch-Griffeth-Sam’s conjecture which states that the unitary representations admit cohomological constructions via BGG resolutions.
  This is joint work with Anton Cox, Amit Hazi, Emily Norton, and Jose Simental.
  N.B.: please click HERE to attend the talk in streaming
27/10/21Seminario14:0014:591201 Dal PassoErik TonniSISSAModular 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.

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
26/10/21Seminario14:3015:301201 Dal Passo
Amos Turchet
University of Roma Tre
Geometry Seminar
Campana’s program and special varieties

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.
20/10/21Seminario16:0017:001201 Dal PassoJean-Luc Sauvageot Institut de Mathématiques de JussieuMisurabilità, densità spettrali e tracce residuali in geometria non commutativa

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.
19/10/21Seminario14:3015:301201 Dal PassoLaura PertusiUniversity of Milano
Geometry Seminar
Serre-invariant stability conditions and cubic threefolds

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.
15/10/21Seminario16:0017:001201 Dal Passo
“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)

  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
15/10/21Seminario14:3015:301201 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)

  The n-th cohomology of the symmetric group Sn 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 nN 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
12/10/21Seminario14:3015:301201 Dal PassoDaniele AgostiniMax Planck Institute for Mathematics in the Sciences in Leipzig
Geometry Seminar
Theta functions and tau functions of algebraic curves

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.
06/10/21Seminario14:0015:001201 Dal PassoEdoardo D'AngeloUniversita' di GenovaRole of the relative entropy in the entropy of dynamical black holes

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.

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