April 29, 2021 (online)
Alexei Stepanov (Immanuel Kant Baltic Federal Unibersity, Kaliningrad, Russia)
On the Borel-Cantelli lemma
In our report, we discuss strong convergence and the Borel-Cantelli lemma. First, we present classical form of this lemma and debate how to derive strong limit results by this lemma. We then consider different generalization of the first and second parts of this lemma. In the report, we also analyze some applications associated with lemma.
May 20, 2021 (online)
Artur Sergyeyev (Silesian University in Opava)
Integrable system in 4D with an algebraic nonisospectral Lax pair
We present a system which is, to the best of our knowledge, the first known example of an integrable dispersionless system in fourindependent variables (4D) with nonisospectral Lax pair involving algebraic, rather than rational, dependence on the spectral parameter.
This result shows inter alia that the class of integrable 4D dispersionless systems with nonisospectral Lax pairs is significantly more diverse than it appeared before. The Lax pair for the system in question is of a novel type, discovered in our earlier work , and is related to contact geometry; we will briefly review the basic aspects of integrable systems in general and the construction in question in order to make the talk more self-contained.
 A. Sergyeyev, Integrable (3+1)-dimensional system with an algebraic Lax pair, Appl. Math. Lett. 92 (2019), 196--200; arXiv:1812.02263
 A. Sergyeyev, New integrable (3+1)-dimensional systems and contact geometry, Lett. Math. Phys. 108 (2018), 359--376; arXiv:1401.2122.