Complex projective structures, or PSL( $2, \mathbb{C})$-opers, play a central role in the theory of uniformization of Riemann surfaces. A very natural generalization of this notion is to consider complex projective structures with ramification points. This gives rise to the notion of branched projective structure, which is much more flexible in many aspects. For example, any representation of a surface group with values in $\operatorname{PSL}(2, \mathbb{C})$ is obtained as the holonomy of a branched projective structure. We will show that one of the central properties of complex projective structures, namely the complex analytic structure of their moduli spaces, extends to the branched case.[-]

The Grothendieck-Knudsen moduli space of stable rational curves n markings is arguably one of the simplest moduli spaces: it is a smooth projective variety that can be described explicitly as a blow-up of projective space, with strata corresponding to nodal curves similar to the torus invariant strata of a toric variety. Conjecturally, its Mori cone of curves is generated by strata, but this is known only for n up to 7. In contrast, the cones of effective divisors are not f initely generated, in all characteristics, when n is at least 10. After a general introduction to these topics, I will discuss what we call elliptic pairs and LangTrotter polygons, relating the question of finite generation of effective cones of blow-ups of certain toric surfaces to the arithmetic of elliptic curves. These lectures are based on joint work with Antonio Laface, Jenia Tevelev and Luca Ugaglia.[-]

The Grothendieck-Knudsen moduli space of stable rational curves n markings is arguably one of the simplest moduli spaces: it is a smooth projective variety that can be described explicitly as a blow-up of projective space, with strata corresponding to nodal curves similar to the torus invariant strata of a toric variety. Conjecturally, its Mori cone of curves is generated by strata, but this is known only for n up to 7. In contrast, the cones of effective divisors are not f initely generated, in all characteristics, when n is at least 10. After a general introduction to these topics, I will discuss what we call elliptic pairs and LangTrotter polygons, relating the question of finite generation of effective cones of blow-ups of certain toric surfaces to the arithmetic of elliptic curves. These lectures are based on joint work with Antonio Laface, Jenia Tevelev and Luca Ugaglia.[-]

Real-analytic manifolds are studied very much in the last century until the time when people found the partition of unity on smooth manifolds makes the manifold theory very tractable. The group of real-analytic diffeomorphisms is the natural automorphism group of the real-analytic manifold. Because of the analytic continuation, there are no partition of unity by functions with support in balls. The germ at a point of a real-analytic diffeomorphism determines the diffeomorphism and hence the group of them looks rigid. However, the group of real-analytic diffeomorphisms is dense in the group of smooth diffeomorphisms and diffeomorphisms can exhibit all kinds of smooth stable dynamics. I would like to convince the audience that the group of real-analytic diffeomorphisms is a really interesting object.In the first course, I would like to review the theorem by Herman which says the identity component of the group of real analytic diffeomorphisms of the n-torus is simple, which gives a motivation to study the group for other manifolds. We also review several fundamental facts in the real analytic category.In the second course, we introduce the regimentation lemma which can play in the real analytic category the role of the partition of unity in the smooth category. For manifolds with nontrivial circle actions, we show that any real analytic diffeomorphism isotopic to the identity is homologous to a diffeomorphism which is an orbitwise rotation.In the third course, we state a lemma which says that the multiple actions of the standard action on the plane is a final (terminal) object in the category of circle actions. This lemma would imply that the identity component of the group of real analytic diffeomorphisms is perfect.[-]

In these three lectures on quantum information and complexity, we will (1) review the basic concepts of quantum information processing units (QPUs), (2) prove a version of the claim that almost all quantum circuits are very complex in the sense that they are exponentially expensive to realize in the quantum circuit model of computation and (3) that the quantum complexity of a random quantum circuit grows linearly with the size of the circuit up to exponentially large circuits.

The underlying proof technique uses a versatile proof strategy from high-dimensional probability theory that can (and has been) readily extended to other problems within quantum computing theory and beyond.[-]

I will first explain the joint work with Walter van Suijlekom on a new result about th zeros of the Fourier transform of extremal eigenvectors for quadratic forms associated to distributions on a bounded interval and its relation with the spectral action. Then I will explain how these results allow to advance in the joint work which I am doing with Consani and Moscovici on the zeta spectral triple. Finally, if time permits, I will discuss several ideas in connection with physics and non-commutative geometry.[-]