SISSA

The synthetic theory of Ricci curvature lower bounds introduced more than 15 years ago by Lott-Sturm-Villani has been largely succesful in describing the geometry of metric measure spaces. However, this theory fails to include sub-Riemannian manifolds (an important class of metric spaces, the simplest example being the so-called Heisenberg group). Motivated by Villani's ``great unification'' program, in this talk we propose an extension of Lott-Sturm-Villani's theory, which includes sub-Riemannian geometry. This is a joint work with Barilari (Padua) and Mondino (Oxford). The talk is intended for a general audience, no previous knowledge of optimal transport or sub-Riemannian geometry is required.

Supersymmetric gauge theories in five dimensions, although power counting non-renormalizable, are known to be in some cases UV completed by a superconformal field theory. Many tools, such as M-theory compactification and pq-web constructions, were used in recent years in order to deepen our understanding of these theories. This framework gives us a concrete way in which we can try to search for additional IR conformal field theory via deformations of these well-known superconformal fixed points. Recently, the authors of 2001.00023 proposed a supersymmetry breaking mass deformation of the E_1theory which, at weak gauge coupling, leads to pure SU(2) Yang-Mills and which was conjectured to lead to an interacting CFT at strong coupling. During this talk, I will provide an explicit geometric construction of the deformation using brane-web techniques and show that for large enough gauge coupling a global symmetry is spontaneously broken and the theory enters a new phase which, at infinite coupling, displays an instability. The Yang-Mills and the symmetry broken phases are separated by a phase transition. Quantum corrections to this analysis are discussed, as well as possible outlooks. Based on arXiv: 2109.02662.

Localization technique permits to reduce full dimensional problems to possibly easier lower dimensional ones. During the last years a new approach to localization has been obtained using the powerful tools of optimal transport. Following this approach, we obtain quantitative versions of two relevant geometric inequalities  in comparison geometry as Levy-Gromov isoperimetric inequality (joint with F. Maggi and A. Mondino) and the spectral gap inequality (joint with A. Mondino and D. Semola). Both results are also valid in the more general setting of metric measure spaces verifying the so-called curvature dimension condition.

Donaldson-Thomas invariants of a Calabi-Yau 3-fold Y are related to Pandharipande-Thomas invariants via a wall-crossing formula known as the DT/PT correspondence, proved by Bridgeland and Toda. The same relation holds for the “local invariants”, those encoding the contribution of a fixed smooth curve in Y. We show how to lift the local DT/PT correspondence to the motivic level and provide an explicit formula for the local motivic invariants, exploiting the critical structure on certain Quot schemes acting as our local models. Our strategy is parallel to the one used by Behrend, Bryan and Szendroi in their definition and computation of degree zero motivic DT invariants. If time permits, we discuss a further (conjectural) cohomological upgrade of the local DT/PT correspondence.
Joint work with Ben Davison.
 

Locomotion strategies employed by unicellular organism are a rich source of inspiration for studying mechanisms for shape control. They are particularly interesting because they are invisible to the naked eye, and offer surprising new solutions to the question of how shape can be controlled.

In recent years, we have studied locomotion and shape control in Euglena gracilis. This unicellular protist is particularly intriguing because it can adopt different motility strategies: swimming by flagellar propulsion, or crawling thanks to large amplitude shape changes of the whole body (a behavior known as metaboly). We will survey our most recent findings within this stream of research.