Metric geometry of locally compact groups
... [Mack–57]; see also Appendix B in [GlTW–05]. Gelfand and Raikov (1943) showed that LC-groups have “sufficiently many” irreducible continuous unitary representations [Dixm–69, Corollary 13.6.6]; this does not carry over to topological groups (examples of topological groups that are abelian, locally h ...
... [Mack–57]; see also Appendix B in [GlTW–05]. Gelfand and Raikov (1943) showed that LC-groups have “sufficiently many” irreducible continuous unitary representations [Dixm–69, Corollary 13.6.6]; this does not carry over to topological groups (examples of topological groups that are abelian, locally h ...
VSPs of cubic fourfolds and the Gorenstein locus of the Hilbert
... spaces, but their geometry is still poorly understood. For example, they are believed to possess arbitrary singularities and some progress in this direction is made, especially by Emsalem and Iarrobino [IE78, IK99] and Erman [Erm12]. Nevertheless no examples of non-reduced points are actually known. ...
... spaces, but their geometry is still poorly understood. For example, they are believed to possess arbitrary singularities and some progress in this direction is made, especially by Emsalem and Iarrobino [IE78, IK99] and Erman [Erm12]. Nevertheless no examples of non-reduced points are actually known. ...
Lie groups, lecture notes
... translation by x: We leave it to the reader to verify that x 7! lx is a group homomorphism from G to Sym.G/: Likewise, if x 2 G; then rx W G ! G; y 7! yx; is called right translation by x: We leave it to the reader to verify that x 7! .rx / 1 is a group homomorphism from G to Sym.G/: If x 2 G; then ...
... translation by x: We leave it to the reader to verify that x 7! lx is a group homomorphism from G to Sym.G/: Likewise, if x 2 G; then rx W G ! G; y 7! yx; is called right translation by x: We leave it to the reader to verify that x 7! .rx / 1 is a group homomorphism from G to Sym.G/: If x 2 G; then ...
The derived category of sheaves and the Poincare-Verdier duality
... every distinguished triangle pX, Y, Z, f, g, hq we obtain a long exact sequence in B q f rns n p1q grns n p1q hrns n 1 Ñ F npX q p1ÝÑ F pY q ÝÑ F pZ q ÝÑ F pX q Ñ (1.5) In particular, a homological functor associates to each short exact sequence in C pAq 0 Ñ X Ñ Y Ñ Z Ñ 0 ...
... every distinguished triangle pX, Y, Z, f, g, hq we obtain a long exact sequence in B q f rns n p1q grns n p1q hrns n 1 Ñ F npX q p1ÝÑ F pY q ÝÑ F pZ q ÝÑ F pX q Ñ (1.5) In particular, a homological functor associates to each short exact sequence in C pAq 0 Ñ X Ñ Y Ñ Z Ñ 0 ...
Covering space
In mathematics, more specifically algebraic topology, a covering map (also covering projection) is a continuous function p from a topological space, C, to a topological space, X, such that each point in X has an open neighbourhood evenly covered by p (as shown in the image); the precise definition is given below. In this case, C is called a covering space and X the base space of the covering projection. The definition implies that every covering map is a local homeomorphism.Covering spaces play an important role in homotopy theory, harmonic analysis, Riemannian geometry and differential topology. In Riemannian geometry for example, ramification is a generalization of the notion of covering maps. Covering spaces are also deeply intertwined with the study of homotopy groups and, in particular, the fundamental group. An important application comes from the result that, if X is a ""sufficiently good"" topological space, there is a bijection between the collection of all isomorphism classes of connected coverings of X and the conjugacy classes of subgroups of the fundamental group of X.