The study of constant scalar curvature Kähler (cscK) metrics on compact complex manifolds is a classical topic that has attracted enormous interest since the 1950s. However, detecting the existence of cscK metrics is a difficult task, which in the projective integral case conjecturally amounts to proving an important algebro-geometric stability notion (K-stability). Recent significant advancements have established that the existence of unique cscK metric in a Kähler class is equivalent to the coercivity of the so-called Mabuchi functional. I will extend the notion of cscK metrics to singular varieties, and I will show the existence of these special metrics on Q-Gorenstein smoothable klt varieties when the Mabuchi functional is coercive. A key point in this variational approach is the lower semicontinuity of the coercivity threshold of Mabuchi functional along a degenerate family of normal compact Kähler varieties with klt singularities. The latter strengthens evidence supporting the openness of (uniform) K-stability for general families of normal compact Kähler varieties with klt singularities. This is a joint work with Chung-Ming Pan and Tat Dat Tô.

Interpolation of differential forms is a challenging aspect of modern approximation theory. Not only does it shed new light on some classical concepts of interpolation theory, such as the Lagrange interpolation and the Lebesgue constant, but it also suggests that they can be extended to a very general framework. As an extent of that, it is worth pointing out that the majority of classical shape functions commonly used in finite element methods, such as those involved in Nedelec or Raviart-Thomas elements, can be seen as a specialisation of this theory. Of course, this generality brings along the evident downside of an unfriendly level of abstraction. The scope of this series of two seminars is thus twofold: presenting the main challenges of this branch of approximation theory but in a concrete manner. The first seminar will hinge on a development of a convenient one dimensional toy model that enlightens parallelisms and differences with usual nodal interpolation. In the second seminar will extend these techniques to the multi-dimensional framework, motivating our choices by a geometrical flavour. This talk is part of the activity of the MUR Excellence Department Project MatMod@TOV (CUP E83C23000330006).

A result due to De Commer implies that an important source of locally compact quantum groups (I will explain this notion) is constituted by the unitary dual 2-cocycles on (classical) locally compact groups. I will present geometric methods to explicitly construct such 2-cocycles for (classical) Lie groups of Frobenius type i.e. Lie groups that admit open co-adjoint orbits.
Note: This talk is part of the activity of the MUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)

  Projective representations of a group G with assigned 2-cocycle α are equivalent to (certain) representations of the central extension of G associated with α. This classical result can be seen as a piece of 2-category theory fallen into the realm of 1-categories, and in this perspective it admits natural generalizations relevant to the context of anomalous topological or Euclidean QFTs. In particular, Stolz-Teichner's Clifford field theories naturally emerge as a particular example of this construction.

  Nakajima quiver varieties, originally defined in '94 by H. Nakajima, form an interesting class of algebraic varieties with many applications in algebraic geometry (e.g. resolution of singularities), representation theory (e.g. Kac-Moody algebras), and string theory (e.g. Coulomb and Higgs branches). In this talk, we will begin introducing the general setting of Hamiltonian reductions via GIT, highlighting how this technique produces Poisson quasi-projective varieties in a canonical way, and, in some cases, resolutions of symplectic singularities. We will then apply this theory to quiver representations, defining Nakajima quiver varieties and illustrating how the combinatorial nature of quivers is reflected in the geometry of these varieties.

I will describe ongoing work with Peter Koymans and Mark Shusterman, showing that for fixed q congruent to 3 modulo 4, one has non-vanishing of L(1/2,chi) for 100% of imaginary quadratic characters chi of Fq(T) (ordered by discriminant). This result, predicted by the Katz-Sarnak heuristics, is the probabilistic version of Chowla's non vanishing conjecture: it is known that over function fields one cannot hope for a deterministic statement, as shown in a fairly robust way by Wanlin Li in 2018. I will explain how this result sits into a web of methods aimed at controlling the distribution of 2^{infty}-Selmer groups in quadratic twists families.

In joint work with Chan, Koymans and Pagano we prove matching upper and lower bounds for multiplicative functions when averaged over general integer sequences. We give applications to Cohen—Lenstra conjecture and Manin’s conjecture for counting solutions of Diophantine equations in a small number of variables.

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