%0 Report
%D 2013
%T Geodesics and admissible-path spaces in Carnot Groups
%A Andrei A. Agrachev
%A Alessandro Gentile
%A Antonio Lerario
%K Carnot group, Loop space, Betti number, Morse-Bott functional
%X We study the topology of admissible-loop spaces on a step-two Carnot group G. We use a Morse-Bott theory argument to study the structure and the number of geodesics on G connecting the origin with a 'vertical' point (geodesics are critical points of the 'Energy' functional, defined on the loop space). These geodesics typically appear in families (critical manifolds). Letting the energy grow, we obtain an upper bound on the number of critical manifolds with energy bounded by s: this upper bound is polynomial in s of degree l (the corank of the distribution). Despite this evidence, we show that Morse-Bott inequalities are far from sharp: the topology (i.e. the sum of the Betti numbers) of the loop space filtered by the energy grows at most as a polynomial in s of degree l-1. In the limit for s at infinity, all Betti numbers (except the zeroth) must actually vanish: the admissible-loop space is contractible. In the case the corank l=2 we compute exactly the leading coefficient of the sum of the Betti numbers of the admissible-loop space with energy less than s. This coefficient is expressed by an integral on the unit circle depending only on the coordinates of the final point and the structure constants of the Lie algebra of G.
%I SISSA
%G en
%U http://hdl.handle.net/1963/7228
%1 7262
%2 Mathematics
%4 1
%# MAT/03 GEOMETRIA
%$ Submitted by Andrei Agrachev (agrachev@sissa.it) on 2013-12-03T08:53:56Z
No. of bitstreams: 1
1311.6727v1.pdf: 465119 bytes, checksum: ef7a6bb7d185282ed13b2c73a44aa9ec (MD5)
%0 Journal Article
%J Discrete and Computational Geometry, Volume 48, Issue 4, December 2012, Pages 1025-1047
%D 2012
%T Convex pencils of real quadratic forms
%A Antonio Lerario
%X We study the topology of the set X of the solutions of a system of two quadratic inequalities in the real projective space RP^n (e.g. X is the intersection of two real quadrics). We give explicit formulae for its Betti numbers and for those of its double cover in the sphere S^n; we also give similar formulae for level sets of homogeneous quadratic maps to the plane. We discuss some applications of these results, especially in classical convexity theory. We prove the sharp bound b(X)\leq 2n for the total Betti number of X; we show that for odd n this bound is attained only by a singular X. In the nondegenerate case we also prove the bound on each specific Betti number b_k(X)\leq 2(k+2).
%B Discrete and Computational Geometry, Volume 48, Issue 4, December 2012, Pages 1025-1047
%I Springer
%G en
%U http://hdl.handle.net/1963/7099
%1 7097
%2 Mathematics
%4 1
%$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2013-09-17T11:33:51Z
No. of bitstreams: 1
1106.4678v3.pdf: 250496 bytes, checksum: 3ad88e26b8325e50857d0d9f9aabd6ad (MD5)
%R 10.1007/s00454-012-9460-2
%0 Journal Article
%J Proceedings of the London Mathematical Society, Volume 105, Issue 3, September 2012, Pages 622-660
%D 2012
%T Systems of Quadratic Inequalities
%A Andrei A. Agrachev
%A Antonio Lerario
%X We present a spectral sequence which efficiently computes Betti numbers of a closed semi-algebraic subset of RP^n defined by a system of quadratic inequalities and the image of the homology homomorphism induced by the inclusion of this subset in RP^n. We do not restrict ourselves to the term E_2 of the spectral sequence and give a simple explicit formula for the differential d_2.
%B Proceedings of the London Mathematical Society, Volume 105, Issue 3, September 2012, Pages 622-660
%I SISSA
%G en
%U http://hdl.handle.net/1963/7072
%1 7066
%2 Mathematics
%4 1
%# MAT/03 GEOMETRIA
%$ Submitted by Andrei Agrachev (agrachev@sissa.it) on 2013-09-16T10:02:51Z
No. of bitstreams: 1
1012.5731v2.pdf: 367584 bytes, checksum: 136c7bc2abeeb2a239c15689929eae30 (MD5)
%R DOI: 10.1112/plms/pds010
%0 Thesis
%D 2011
%T Homology invariants of quadratic maps
%A Antonio Lerario
%X Given a real projective algebraic set X we could hope that the equations describing it can give some information on its topology, e.g. on the number of its connected components. Unfortunately in the general case this hope is too vague and there is no direct way to extract such information from the algebraic description of X: Even the problem to decide whether X is empty or not is far from an easy visualization and requires some complicated algebraic machinery. A fi rst step observation is that as long as we are interested only in the topology of X, we can replace, using some Veronese embedding, the original ambient space with a much bigger RPn and assume that X is cut by quadratic equations. The price for this is the increase of the number of equations de ning our set; the advantage is that quadratic polynomials are easier to handle and our hope becomes more concrete...
%I SISSA
%G en
%U http://hdl.handle.net/1963/6245
%1 6145
%2 Mathematics
%3 Mathematical Physics
%4 -1
%$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2012-09-14T09:24:43Z\\nNo. of bitstreams: 1\\nPhd_Thesis_Lerario_Antonio.pdf: 937331 bytes, checksum: f34eea56b610434e63b6f3dd1dded09c (MD5)
%] 1 Basic theory. 2 Nondegeneracy conditions. 3 Spectral sequences. 4 Complex theory. 5 Applications and examples