IMPA - The Institute for Pure and Applied Mathematics

Superalgebraic and supergeometric structures generalize classical nongraded structures to graded ones. This extension introduces an interplay between commuting and anticommuting elements, enriching tools of commutative algebra, differential geometry, and algebraic geometry. Although originally motivated by theoretical physics, supergeometry has evolved into a robust mathematical framework for studying objects such as supermanifolds and superschemes.

In this talk, we introduce $C^\infty$-superrings and their associated geometric spaces, which we term $C^\infty$-superschemes. We establish an equivalence of categories between these algebraic and geometric structures. Furthermore, we present a generalization of Batchelor’s theorem to $C^\infty$-superspaces extending the classical result that every smooth supermanifold is (noncanonically) isomorphic to the sheaf of sections of a vector bundle over its underlying smooth manifold.

Conceptually, $C^\infty$-superrings synthesize the features of $C^\infty$-rings and superrings. Since smooth supermanifolds form a core subclass of $C^\infty$-superscheme, this framework provides a natural scheme-theoretic generalization of supergeometry in the smooth setting. 

This is a joint work with Danilo Olarte and Alexander Torres-Gómez (both at Universidad de Antioquia) and based on the preprints: arXiv:2508.09900v3 and arXiv:2605.07169v2

In this talk, I will discuss the growth rate of Reeb orbits with respect to their periods on certain contact manifolds. We focus on fiberwise star-shaped hypersurfaces in the cotangent bundle of a closed manifold. This class of dynamical systems includes, for example, geodesic flows. I will explain how the string topology of the base manifold implies that the number of Reeb orbits with period at most grows at least like the prime numbers, that is, on the order of . The topological condition is the existence of a certain non-nilpotent homology class of the free loop space (with respect to the Chas–Sullivan product). This is joint work with João Pering.

We prove a lower bound for the Birkhoff min-max invariant of a Riemannian sphere in terms of the min-max width of its embedded circles. The main tool is a method to induce a sweepout by pairs of points in an embedded circle from a given sweepout of the sphere by closed curves, so that the points of each pair of the induced sweepout lie close to two curves of the ambient sweepout that are close to each other. We prove related rigidity results characterizing the round spheres among Zoll spheres. Moreover, considering a non-compact complete Riemannian manifold, we relate upper bounds for the min-max width of its embedded circles to the existence of continuous functions from the manifold to finite graphs with level sets that have uniformly bounded diameters, thus giving an estimate for its 1-width in Urysohn's quantitative dimension theory. This is joint work with Lucas Ambrozio and Rafael Montezuma.
 

Despite the intrinsic complexity of turbulent flows, large-scale coherent structures are present in turbulence, and are related to quantities of interest, such as aerodynamic drag and sound radiation. This presentation shows methods to model such coherent structures, and approaches to control flows in order to reduce drag or noise radiation. Linear models will be considered first, with coherent structures modelled as dominant responses from the linearised Navier-Stokes system. Such dominant modes compare favourably with coherent structures educed from experimental or numerical data, and allow the proposition of changes to the system aiming at drag or noise reduction. Non-linear reduced-order models (ROMs) are then obtained by the Galerkin projection of the full governing equations onto modes obtained from the linearised system. Dynamical systems so obtained typically have dimensions of O(10) to O(10^3) degrees of freedom, and quantitative agreement with reference statistics is obtained for a number of canonical flow configurations. State estimation from a few sensors is explored using an extended Kálmán filter coupled with ROMs, reproducing accurately numerical results. Finally, a ROM for turbulent Couette flow is used to devise a control method aiming at turbulence suppression. The obtained strategy is seen to relaminarise turbulence in direct numerical simulations, showing that ROMs so obtained have promising applications in optimisation and control.