We study properties of the space of horizontal paths joining the origin with a vertical point on a generic two-step Carnot group. The energy is a Morse-Bott functional on paths and its critical points (sub-Riemannian geodesics) appear in families (compact critical manifolds) with controlled topology. We study the asymptotic of the number of critical manifolds as the energy grows. The topology of the horizontal-path space is also investigated, and we find asymptotic results for the total Betti number of the sublevels of the energy as it goes to infinity. We interpret these results as local invariants of the sub-Riemannian structure.

PB - Mathematical Sciences Publishers VL - 19 ER - TY - JOUR T1 - Convex pencils of real quadratic forms JF - Discrete and Computational Geometry, Volume 48, Issue 4, December 2012, Pages 1025-1047 Y1 - 2012 A1 - Antonio Lerario AB - 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). PB - Springer UR - http://hdl.handle.net/1963/7099 N1 - Updated version to be published in DCG ; was published in : Discrete and Computational Geometry, Volume 48, Issue 4, December 2012, Pages 1025-1047 U1 - 7097 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - Systems of Quadratic Inequalities JF - Proceedings of the London Mathematical Society, Volume 105, Issue 3, September 2012, Pages 622-660 Y1 - 2012 A1 - Andrei A. Agrachev A1 - Antonio Lerario AB - 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. PB - SISSA UR - http://hdl.handle.net/1963/7072 U1 - 7066 U2 - Mathematics U4 - 1 U5 - MAT/03 GEOMETRIA ER - TY - THES T1 - Homology invariants of quadratic maps Y1 - 2011 A1 - Antonio Lerario AB - 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... PB - SISSA UR - http://hdl.handle.net/1963/6245 U1 - 6145 U2 - Mathematics U3 - Mathematical Physics U4 - -1 ER -