Analysis on compact Lie groups of large dimension and on connected compact groups

The study of Gaussian convolution semigroups is a subject at the crossroad between abstract and concrete problems in harmonic analysis. This article suggests selected open problems that are in large part motivated by joint work with Alexander Bendikov. This paper is dedicated to the memory of Andrzej Hulanicki. My first scientific encounter with Andrzej was through his students, which is fitting given the importance they had to him throughout his career. During my Ph.D., I came across the work of Pavel Glowacki (he told me later that he was the referee of my first research paper, published by Studia Mathematica and based on my Ph.D. Thesis). Later, I met Waldemar Hebisch, in Boston, in the late nineteen eighties. In 1991, Andrzej invited me (together with my wife, Cathy) to Wroc law for a month. We met for the first time at the Wroc law train station where he picked us up, carrying a mathematical book so that we could recognized him. This turned out to be a beginning of a very enjoyable personal and scientific relation with Andrzej, his many students and collaborators, and with the Mathematical Institute in Wroc law. As many others, I benefited from attending the conferences and schools organized by Andrzej’s group in Tuczno and, later, in Zakopane. My first Ph.D. student, Andrzej Zuk, wrote a Master thesis under Andrzej’s supervision in Wroc law before following Andrzej’s suggestion to work with me in Toulouse (in fact, Zuk prepared his thesis under the joint supervision of Andrzej and me, spending half his time in Toulouse and the other half in Wroc law, as required by his doctoral fellowship). Throughout his career, Andrzej, in addition to pursuing his own research in mathematics, put a tremendous energy into organizing and supporting mathematics in Poland and in Wroc law. He maintained many long term scientific international relations, including during periods when it was not necessarily an easy thing to do. He, ∗Research partially supported by NSF grant DMS 0603886