These are the notes of the course given in Autumn 2007. Two good books (among many): Adams: Lectures on Lie groups (U. Chicago Press) Fulton and Harris: Representation Theory (Springer) Also various writings of Atiyah, Segal , Bott, Guillemin and Sternberg . . . .

We derive the 2-component Camassa–Holm equation and corresponding N = 1 super generalization as geodesic flows with respect to the H metric on the extended Bott-Virasoro and superconformal groups, respectively. Mathematics Subject Classifications (2000): 53A07, 53B50.

Abstract. Let G be a Banach Lie group modeled on the Banach space, possibly infinite dimensional, E. In this paper first we introduce Euler-Lagrange equations on the Lie group G with potential and right invariant metric. Euler-Lagrange equations are natural extensions of the geodesic equations on manifolds and Lie groups. In the second part, we study the geometry of the mechanical system of a r...

We investigate families of minimal rational curves on Schubert varieties, their Bott–Samelson desingularizations, and generalizations constructed by Nicolas Perrin in the minuscule case. In particular, we describe small resolutions varieties.

Idiopathic neonatal priapism is a rare pathology with an estimated incidence of 15/10,000 births. Despite the distressing clinical picture, it relatively benign pathology. We report case idiopathic treated in our department, newborn on day 1 life no known pathological family history. The picture was characterized by permanent, painless erection penis associated signs or change skin color. No et...

We compute the K-theory of complex projective spaces. There are three major ingredients: the exact sequence of K-groups, the theory of Chern character and the Bott Periodicity Theorem.

The Virasoro-Bott group endowed with the right-invariant L2metric (which is a weak Riemannian metric) has the KdV-equation as geodesic equation. We prove that this metric space has vanishing geodesic distance.