Activities This Week

Lara—Srinivas—Stix, building on joint work with Esnalut, have recently shown that the etale fundamental group of a connected proper scheme over an algebraically closed field is topologically finitely presented, thus answering a question raised in SGA. The proof relies on a finite presentation criterion of Lubotzky for profinite groups, resolutions of singularities/alterations, a theorem of Deligne—Ilusie on the Euler characteristic, as well as other modern and classical results in (arithmetic) algebraic geometry.

Temporo-spatial differentiations are ergodic averages on a probabilistic dynamical system $(X, \mu, T)$ taking the form $\left( \frac{1}{\mu(C_k)} \int_{C_k} \frac{1}{k} \sum_{j = 0}^{k - 1} T^j f \mathrm{d} \mu \right)_{k = 1}^\infty $ where $C_k \subseteq X$ are measurable sets of positive measure, and $f \in L^\infty(X, \mu)$. These averages combine both the dynamics of the transformation and the structure of the underlying probability space $(X, \mu)$. We will discuss the motivations behind studying these averages, results concerning the limiting behavior of these averages and, time permitting, discuss generalizations to non-autonomous dynamical systems. Joint work with Idris Assani.

In this talk, we explain how the classical Toda lattice model, one of the earliest examples of nonlinear completely integrable systems, can be used to demonstrate that certain configurations in the classical phase space are symplectic balls in disguise. No background in symplectic geometry is needed. The talk is based on joint work with Vinicius Ramos and Daniele Sepe.