Seminars, for informal dissemination of research results, exploratory work by research teams, outreach activities, etc., constitute the simplest form of meetings at a Mathematics research centre.

CAMGSD has recorded the calendar of its seminars for a long time, this page serving both as a means of public announcement of forthcoming activities but also as a historic record.

We consider the classical 2-opinion dynamics known as the voter model on finite graphs. It is well known that this interacting particle system is dual to a system of coalescing random walkers and that under so-called mean-field geometrical assumptions, as the graph size increases, the characterization of the time to reach consensus can be reduced to the study of the first meeting time of two independent random walks starting from equilibrium.

As a consequence, several recent contributions in the literature have been devoted to making this picture precise in certain graph ensembles for which the above mentioned meeting time can be explicitly studied. I will first review this type of results and then focus on the specific geometrical setting of random regular graphs, both static and dynamic (i.e. edges of the graphs are rewired at random over time), where in recent works we study precise first order behaviour of the involved observables. We will in particular show a quasi-stationary-like evolution for the discordant edges (i.e. with different opinions at their end vertices) which clarify what happens before the consensus time scale both in the static and in the dynamic graph setting. Further, in the dynamic geometrical setting we can see how consensus is affected as a function of the graph dynamics.

Based on recent and ongoing joint works with Rangel Baldasso, Rajat Hazra, Frank den Hollander and Matteo Quattropani.

We will talk about existence of Einstein metrics on manifolds with boundary, while prescribing the induced conformal metric and mean curvature of the boundary. In dimension 3, this becomes the existence of conformal embeddings of surfaces into constant sectional curvature space forms, with prescribed mean curvature. We will show existence of such conformal emebeddings near generic Einstein background. We will also discuss the existence question in higher dimensions, where things become more subtle and a non-degenerate boundary condition is used to construct metrics with nonpositive Einstein constant.

Kaluza-Klein reduction of 11-dimensional supergravity on $G_2$ manifolds yields a 4-dimensional effective field theory (EFT) with $N=1$ supersymmetry. $G_2$ manifolds are therefore the analog of Calabi-Yau (CY) threefolds in heterotic string theory. Since 2017 machine-learning techniques have been applied extensively to study CY manifolds but until 2024 no similar work had been carried out on $G_2$ manifolds. We first show how topological properties of these manifolds can be learnt using simple neural networks. We then discuss how one may try to learn Ricci-flat $G_2$ metrics with machine-learning.

Special Lagrangians form an important class of minimal submanifolds in Calabi-Yau manifolds. In this talk, we will consider the Calabi-Yau $3$-folds with a K3-fibration and the size of the K3-fibres are small. Motivated by tropical geometry, Donaldson-Scaduto conjectured that special Lagrangian collapse to ``gradient cycles" when the K3-fibres collapse. This phenomenon is similar to holomorphic curves in Calabi-Yau manifolds with collapsing special Lagrangian fibrations converging to tropical curves. Similar to the realization problem in tropical geometry, one might expect to reconstruct special Lagrangians from gradient cycles. In this talk, I will report the first theorem of this kind based on a joint work with Shih-Kai Chiu.

A crucial problem in geometric quantization is to understand the relationship among quantum spaces associated to different polarizations. Two types of polarizations on toric varieties, Kähler and real, have been studied extensively. This talk will focus on the quantum spaces associated with mixed polarizations and explore their relationships with those associated with Kähler polarizations on toric varieties.