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

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.

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.

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.

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.

String-localized quantum field theory (SL QFT) provides an alternative to gauge theoretic approaches to QFT. In the last one-and-a-half decades, many conceptual benefits of SL QFT have been discovered. However, a renormalization recipe for loop graphs with internal SL fields was not at hand until now. In this talk, I present a proof that the problem of renormalization remains a pure short distance problem in SL QFT. This happens in spite of the delocalization of SL fields and the analytic complexity of their propagators – provided that one takes care in how to set up perturbation theory in SL QFT. As a result, the improved short-distance behavior of SL fields remains a meaningful notion, which indicates that there can exist renormalizable models in SL QFT whose point-localized counterparts are non-renormalizable. The talk is based on arXiv:2107.12834.

We study coupled map lattices where the interaction takes place via rare but intense 'collisions' and the dynamics on each site is given by a piecewise uniformly expanding map of the interval. Using transfer operator techniques, we derive an explicit formula for 'first collision rates' per lattice unit. This is joint work with F. Sélley.

Reflection positivity plays an important role both in mathematics and physics. It appears as the Osterwalder--Schrader positivity in Constructive QFT, and more recently, it became relevant in the context of the representation theory of Lie groups. In this talk, we will mainly discuss reflection positive representations for the symmetric semigroups (Z,N,-id_Z) and (R,R_+,-id_R) and our new perspective given by positive Hankel operators, which are nicely characterized by their Carleson measure. In this regard, we showed that positive Henkel representations produce reflection positive representations by a suitable change of scalar product on the reflection positive Hilbert space.

This is joint work with K.-H. Neeb, J. Schober.

This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006.

MS Teams link

Suitably regular self-similar fractals may be defined as fixed points in the category of compact p-pointed spaces, namely in a purely topological setting. Moreover, this procedure may be quantized, producing self-similar C*-algebras that can be considered noncommutative self-similar fractals. We illustrate the mentioned procedure in the case of the commutative and noncommutative Sierpinski Gasket (SG).
After this purely topological definition, we endow the C*-algebra with a noncommutative Dirichlet form, and with a spectral triple. Both constructions parallel analogous construction for the SG. In particular, the spectral triple produces a noncommutative metric (Lip-norm) on the algebra, and allows the reconstruction of a canonical noncommutative integral and of the noncommutative Dirichlet form.

This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006.

MS Teams link

  In the first part of the talk, I will recall some basic results about shifted Yangians (and their trigonometric versions-the shifted quantum affine algebras), which first appeared in the work of Brundan-Kleshchev relating type A Yangians and finite W-algebras and have become a subject of renewed interest over the last five years due to their close relation to quantized Coulomb branches introduced by Braverman-Finkelberg-Nakajima.
  In the second part of the talk, I will try to convince that the case of antidominant shifts (opposite to what was originally studied in the work of Brundan-Kleshchev in type A and of Kamnitzer-Webster-Weekes-Yacobi in general type) is of particular importance as the corresponding algebras admit the RTT realization (at least in the classical types).
  In particular, this provides a conceptual explanation of the coproduct homomorphisms, gives rise to the integral forms of shifted quantum affine algebras, and also yields a family of (conjecturally) integrable systems on the corresponding Coulomb branches. As another application, the GKLO-type homomorphisms used to define truncated version of the above algebras provide a wide class of rational/trigonometric Lax matrices in classical types.
  This talk is based on the joint works with Michael Finkelberg as well as Rouven Frassek and Vasily Pestun.
  N.B.: please click HERE to attend the talk in streaming