Seminários, para a disseminação informal de resultados de investigação, trabalho exploratório de equipas de investigação, actividades de difusão, etc., constituem a forma mais simples de encontros num centro de investigação de matemática.

O CAMGSD regista e publica o calendário dos seus seminários há bastante tempo, servindo páginas como esta não só como um método de anúncio dessas actividades mas também como um registo histórico.

This talk will survey aspects of mirror symmetry for ten families of non-compact hyperkähler manifolds on which the dynamics of one of the Painlevé equations is naturally defined. They each have a pair of natural realisations: one as the complement of a singular fibre of a rational elliptic surface and another as the complement of a triangle of lines in a (singular) cubic surface. The two realisations relate closely to a space of stability conditions and a cluster variety of a quiver respectively, providing a perspective on SYZ mirror symmetry for these manifolds. I will discuss joint work in progress with Helge Ruddat studying the canonical basis of theta functions on these cubic surfaces.

The moduli of $(C,f)$ where $C$ is a curve and $f$ is a rational function leads to the well-developed theory of Hurwitz spaces. The study of the moduli of $(C,\omega)$ where $C$ is a curve and $\omega$ is a meromorphic differential is a younger subject. I will discuss recent developments in the study of the moduli spaces of holomorphic/meromorphic differentials on curves. Many of the basic questions about cycle classes and integrals have now been solved (through the work of many people) — but there are also several interesting open directions.

Novel bulk-edge dualities have recently emerged in topological materials from the observation of some phenomenological correspondences. The similarity of these dualities with string theory dualities is very appealing and has boosted a quite significant number of cross field studies.

We analyze the bulk-edge dualities in the integer quantum Hall effect, where due to the simpler nature of planar systems the duality can be analyzed by powerful analytic techniques. The results show that the correspondence is less robust than expected. In particular, it is highly dependent of the type of boundary conditions of the topological material. We introduce a formal proof of the equivalence of bulk and edge approaches to the quantization of Hall conductivity for metallic plates with local boundary conditions. However, the proof does not work for non-local boundary conditions, like the Atiyah-Patodi-Singer boundary conditions, due to the appearance of gaps between the bulk and edge states.

I will present several integral representation results for certain functionals arising in the context of optimal design and damage models, in presence of a perimeter penalization term. I will consider several frameworks, and I will also discuss the case with non-standard growth conditions.

Given a topological space, how much of its homotopy type is captured by its algebra of singular cochains? The experienced rational homotopy theorist will argue that one should consider instead a commutative algebra of forms. This raises the more algebraic question

Given a dg commutative algebra, how much of its homotopy type (quasi-isomorphism type) is contained in its associative part?

Despite its elementary formulation, this question turns out to be surprisingly subtle and has important consequences.

In this talk, I will show how one can use operadic deformation theory to give an affirmative answer in characteristic zero.

We will also see how the Koszul duality between Lie algebras and commutative algebras allows us to use similar arguments to deduce that under good conditions Lie algebras are determined by the (associative algebra structure of) their universal enveloping algebras.

Joint with Dan Petersen, Daniel Robert-Nicoud and Felix Wierstra and based on arXiv:1904.03585.

Quantum anomalies offer a useful guide for the exploration of transport phenomena in topological semimetals. A prominent example is provided by the chiral magnetic effect in three-dimensional Weyl semimetals, which stems from the chiral anomaly. Here, we reveal a distinct quantum effect, coined parity magnetic effect, which is induced by the parity anomaly in a four-dimensional topological semimetal. Upon preserving time-reversal symmetry, the spectrum of our model is doubly degenerate and the nodal (Dirac) points behave like $\mathbb{Z}_2$ monopoles. When time-reversal symmetry is broken, while preserving the sublattice (chiral) symmetry, our system supports spin-3/2 quasiparticles and the corresponding Dirac-like cones host tensor monopoles characterized by a $\mathbb{Z}$ number, the Dixmier-Douady invariant. In both cases, the semimetal exhibits topologically protected Fermi arcs on its boundary. Besides its theoretical implications in both condensed matter and quantum field theory, the peculiar 4D magnetic effect revealed by our model could be measured by simulating higher-dimensional semimetals in synthetic matter.

The talk focuses on the influence of the vertex coupling on spectral properties of periodic quantum graphs. Specifically, two questions will be addressed. The first concerns the number of open spectral gaps; it is shown that graphs with a nontrivial $\delta$ coupling can have finite but nonzero number of them. Secondly, motivated by recent attempts to model the anomalous Hall effect, we investigate a class of vertex couplings that violate the time reversal invariance. For the simplest coupling of this type we show that its high-energy properties depend on the parity of the lattice vertices, and discuss various consequences of this property.

We give characterizations of a finite group $G$ acting symplectically on a rational surface ($\mathbb{CP}^2$ blown up at two or more points). In particular, we obtain a symplectic version of the dichotomy of $G$-conic bundles versus $G$-del Pezzo surfaces for the corresponding $G$-rational surfaces, analogous to the one in algebraic geometry. The connection with the symplectic mapping class group will be mentioned.

This is a joint work with Weimin Chen and Weiwei Wu (and partly with Jun Li).