Testbrett

Dehn answered Hilbert's third problem around 1900 by showing that certain tetrahedra are not scissors-congruent to a cube. I will explain progress towards the problem of parametrizing all tetrahedra that are scissors-congruent to a cube. This is based on joint work with Kiran Kedlaya, Alexander Kolpakov, and Michael Rubinstein, and further work of two former MIT undergraduates, A. Anas Chentouf and Yihang Sun.

The Anti-de-Sitter (AdS) space is the analogue in Lorentzian geometry to the hyperbolic space of negative constant curvature familiar to Riemannian geometers. The study of the Anti-de-Sitter space and more generally, of solutions to the Einstein equations which ressemble the Anti-de-Sitter space has many applications in high energy physics and provides mathematicians with problems lying at the intersection of Lorenzian geometry and hyperbolic pdes. I will start with a general introduction to the study of the Einstein equations, with an emphasis on the evolution problem. I will then briefly discuss the well-posedness for wave equations, including the Einstein equations, in the case of AdS type boundaries. In the second part of the talk, I will present several results concerning the linear or non-linear perturbations of the AdS space for the vacuum Einstein equations and some related toy models.

Consider first a memoryless population model described by the usual branching process with a given mean reproduction matrix on a finite space of types. Motivated by the consequences of atavism in Evolutionary Biology, we are interested in a modification of the dynamics where individuals keep full memory of their forebearers, and procreation involves the reactivation of a gene picked at random on the ancestral lineage.
By comparing the spectral radii of the two mean reproduction matrices (with and without memory), we observe that, on average, the model with memory always grows at least as fast as the model without memory. The proof relies on analyzing a biased Markov chain on the space of memories, and the existence of a unique ergodic law is demonstrated through asymptotic coupling.

We introduce a variation of the classic ballistic deposition model in which vertically falling blocks can only stick to the top or the upper right corner of growing columns. We establish that the fluctuations of the height function at points near the time axis are given by the GUE Tracy-Widom limiting distribution, confirming that the strong KPZ universality conjecture is satisfied in this model. The proof is based on a graphical construction of the process in terms of a directed Last Passage Percolation model. This is a joint work with Pablo Groisman, Santiago Saglietti, and Sebastián Zaninovich.

I shall speak about the correspondence between (locally) countably categorical structures (which are mathematical structures that are particularly easy to describe using formal logic) and their automorphism groups, the (locally) Roelcke precompact Polish groups. The non-local correspondence is by now folklore, and lies at the origin of several interactions between model theory and topological dynamics. Locally Roelcke precompact groups were studied by Zielinski, while local categoricity is close in spirit to Hrushovski's "local logic". With Todor Tsankov we recently showed that these two notions fit together perfectly in an extension of the aforementioned correspondence.

tba

tba

tba