In this presentation, I will establish the global existence of entropy weak solutions for scalar balance laws with nonlocal singular sources, along with a partial uniqueness result. A detailed description of the solution is provided for a general class of initial data in a neighborhood where two shocks interact. Additionally, some open questions will be discussed.
NB:This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C23000330006

For a long time, quantum field theory (QFT) lacked a clear and consistent measurement framework, a gap that was described as "a major scandal in the foundations of quantum physics" [1]. I will review the recent framework put forward by Verch and myself [2], which is consistent with relativity in flat and curved spacetimes and has resolved the long-standing problem of "impossible measurements" put forward by Sorkin [3]. The central idea in this framework is that the "system" QFT of interest is measured by coupling it to a "probe" QFT, in which the system, probe, and their coupled variant, all obey axioms of AQFT in curved spacetime. It has been shown that every local observable of the free scalar field has an asymptotic measurement scheme, i.e., can be measured to arbitrary accuracy by a sequence of probes and couplings [4]. I will describe new results that (a) show that there are exact measurement schemes for all local observables in a class of free theories, (b) provide a protocol for the construction of Hadamard local product states in curved spacetime. The latter is complementary to a recent existence result of Sanders [5].
[1] Earman, J., and Valente, G. Relativistic Causality in Algebraic Quantum Field Theory, International Studies in the Philosophy of Science, 28:1, 1-48, (2014) [2] Fewster, C.J., Verch, R. Quantum Fields and Local Measurements. Commun. Math. Phys. 378, 851–889 (2020). [3] Bostelmann, H., Fewster, C.J., and Ruep, M.H. Impossible measurements require impossible apparatus Phys. Rev. D 103, 025017 (2021) [4] Fewster, C.J., Jubb, I. & Ruep, M.H. Asymptotic Measurement Schemes for Every Observable of a Quantum Field Theory. Ann. Henri Poincaré 24, 1137–1184 (2023). [5] Sanders, K. On separable states in relativistic quantum field theory, J. Phys. A: Math. Theor. 56 505201 (2023)
Note: This talk is part of the activity of the MUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)

The classical Minkowski problem asks for necessary and sufficient conditions on a non-negative Borel measure on the unit sphere to be the surface area measure of a convex body. In a smooth setting, it reduces to the study of a Monge-Ampere equation on the unit sphere. This problem has been completely solved through the seminal works of Nirenberg, Pogorelov, Cheng-Yau, etc. In this talk, a new Minkowski-type problem will be introduced. The problem asks for the existence of a convex hypersurface with prescribed Gauss-Kronecker curvature and capillary boundary supported on an obstacle, which can be deduced as a Monge-Ampere equation with a Robin (or Neumann) boundary value condition on the spherical cap. Then obtain a necessary and sufficient condition for solving this problem.
NB:This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C23000330006

Stability conditions in algebraic geometry are used to construct moduli spaces. Experience from the theory of vector bundles (and coherent sheaves) suggests it is useful to have many stability conditions, so that one can geometrically understand the birational behaviour of resulting moduli spaces by varying the stability condition. Motivated by this, I will describe a mostly conjectural analogous story for projective varieties with an ample line bundle. Here the classical notion of stability is K-stability, which aims to construct higher dimensional analogues of the moduli space of stable curves, and the main point will be to introduce variants of K-stability defined using extra topological choices. The main results will link these new stability conditions with differential geometry, through the solvability of certain geometric PDEs, and I will try to explain how these links come about and what the general picture should be.

  Let k be a field of positive characteristic p>2. We explain a duality property concerning the kernel of coinduced representations of Lie k-(super)algebras. This property was already proved by M. Duflo for Lie algebras in any characteristic under more restrictive finiteness conditions. It was then generalized to Lie superalgebras in characteristic 0 in previous works.
  In characteristic 0, it is known that the induced representation can be realized as the local cohomology with coefficients in some coinduced representation. In positive characteristic, in the case of a restricted Lie algebra, we prove a similar result for the restricted induced representation.

  Grassmannians and flag varieties are important moduli spaces in algebraic geometry. Quiver Grassmannians are generalizations of these spaces arise in representation theory as the moduli spaces of quiver subrepresentations. These represent arrangements of vector subspaces satisfying linear relations provided by a directed graph.
  The methods of tropical geometry allow us to study these algebraic objects combinatorially and computationally. We introduce matroidal and tropical analoga of quivers and their Grassmannians obtained in joint work with Alessio Borzì and separate joint work in progress with Giulia Iezzi; and describe them as affine morphisms of valuated matroids and linear maps of tropical linear spaces.

In this talk, the relative entropy between states of the CAR algebra will be considered. One of the states (the "reference state") is a KMS state with respect to a 1-parametric automorphism group induced by a unitary group on the 1-particle Hilbert space, and the other is a multi-excitation state relative to the reference state. In the case that the reference state is quasifree, a compact formula for the relative entropy can be derived. The results are taken from joint work with Stefano Galanda and Albert Much (MPAG 26 (2023) 21; arXiv:2305.02788 [math-ph]). Time permitting, results on work in progress (with Harald Grosse and Albert Much) will be mentioned on the relative entropy for coherent states of the Rieffel-Moyal deformed quantized Klein Gordon field on algebras of wedge regions on Minkowski spacetime.
Note: This talk is part of the activity of the MUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)

Let C be a smooth complex curve of genus 2. We construct double Kodaira fibrations with small signature as (branched) Galois covers of C × C, whose Galois group is extra-special of order 32. This is based on joint papers with A. Causin and P. Sabatino.

Given a 2-dimensional closed surface, we will show that the Moser-Trudinger functional has critical points of arbitrarily high energy. Since the functional is too critical to directly apply to it the known variational methods (in particular the Struwe monotonicity trick), we will approximate it by subcritical ones, which in fact interpolate it to a Liouville-type functional from conformal geometry. Hence our result will also unify and give common results for these two apparently unrelated problems. This is a joint work with F. De Marchis, A. Malchiodi and P-D. Thizy.
NB:This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C23000330006

The aim of this talk is to provide a uniform and intuitive description of the cohomology ring of arrangement complements. We introduce complex hyperplane arrangements and state the Orlik-Solomon theorem (1980). Then, we describe the real case and the Gelfand-Varchenko ring (1987). We define toric arrangements and present their cohomology ring (De Concini, Procesi (2005) and Callegaro, D'Adderio, Delucchi, Migliorini, and P. (2020)). Finally, we show a new technique to prove the Orlik-Solomon and De Concini-Procesi relations from the Gelfand-Varchenko ring. The technique applied to abelian arrangements provides a presentation of their cohomology. This is work in progress with Evienia Bazzocchi and Maddalena Pismataro. This talk is part of the activity of the MIUR Excellence Department Project Mat-Mod@TOV (CUP E83C23000330006).