Pieri resolutions for classical groups

We generalize the constructions of Eisenbud, Fl{\o}ystad, and Weyman for equivariant minimal free resolutions over the general linear group, and we construct equivariant resolutions over the orthogonal and symplectic groups. We also conjecture and provide some partial results for the existence of an equivariant analogue of Boij-S\"oderberg decompositions for Betti tables, which were proven to exist in the non-equivariant setting by Eisenbud and Schreyer. Many examples are given.Comment: 40 pages, no figures; v2: corrections to sections 2.2, 3.1, 3.3, and some typos; v3: important corrections to sections 2.2, 2.3 and Prop. 4.9 added, plus other minor corrections; v4: added assumptions to Theorem 3.6 and updated its proof; v5: Older versions misrepresented Peter Olver's results. See "New in this version" at the end of the introduction for more detail

Simplicial Homology of Random Configurations

Given a Poisson process on a $d$-dimensional torus, its random geometric simplicial complex is the complex whose vertices are the points of the Poisson process and simplices are given by the \u{C}ech complex associated to the coverage of each point. By means of Malliavin calculus, we compute explicitly the n$th$ order moment of the number of $k$-simplices. The two first order moments of this quantity allow us to find the mean and the variance of the Euler caracteristic. Also, we show that the number of any connected geometric simplicial complex converges to the Gaussian law when the intensity of the Poisson point process tends to infinity. We use a concentration inequality to find bounds for the for the distribution of the Betti number of first order and the Euler characteristic in such simplicial complex

The type numbers of closed geodesics

A short survey on the type numbers of closed geodesics, on applications of the Morse theory to proving the existence of closed geodesics and on the recent progress in applying variational methods to the periodic problem for Finsler and magnetic geodesicsComment: 29 pages, an appendix to the Russian translation of "The calculus of variations in the large" by M. Mors