California State University, Fullerton  

Title: From a Dream of Apollonius to the Contemporary Tale of Barbilian Spaces

Abstract: 

The Cayley-Poincaré model of Lobachevsky's non-Euclidean geometry yields naturally a distance that can be represented as a logarithmic oscillation. Inspired by the fundamental properties of this model, Dan Barbilian (1895 -- 1961) established in 1934 a theory of metric spaces endowed with a distance that could still inspire us today. For Barbilian, the main interest in mathematics did not lie in the computational aspects, but in the quest for a unifying principle, a generalization to a higher concept, a construction that gathers together, as sides of the same phenomenon, several various representations of the same mathematical fact. To better appreciate how much this vision could inspire us today, we survey several results obtained in the last decades by M. Vuorinen, W.G. Boskoff, Z. Ibragimov, P. Hästö, and other authors. We plan to conclude by asking whether such constructive mathematical endeavors could support our understanding of other mathematical or physical phenomena.

IIT Kanpur

Title: "Gradient Descent: Old and New Perspective"