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.

1 aAgrachev, Andrei, A.1 aGentile, Alessandro1 aLerario, Antonio uhttps://www.math.sissa.it/publication/geodesics-and-horizontal-path-spaces-carnot-groups01015nas a2200109 4500008004100000245004300041210004300084260001300127520070800140100002100848856003600869 2012 en d00aConvex pencils of real quadratic forms0 aConvex pencils of real quadratic forms bSpringer3 aWe 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).1 aLerario, Antonio uhttp://hdl.handle.net/1963/709900723nas a2200121 4500008004100000245003800041210003800079260001000117520039200127100002500519700002100544856003600565 2012 en d00aSystems of Quadratic Inequalities0 aSystems of Quadratic Inequalities bSISSA3 aWe 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_2.1 aAgrachev, Andrei, A.1 aLerario, Antonio uhttp://hdl.handle.net/1963/707201189nas a2200109 4500008004100000245004200041210004200083260001000125520088700135100002101022856003601043 2011 en d00aHomology invariants of quadratic maps0 aHomology invariants of quadratic maps bSISSA3 aGiven 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...1 aLerario, Antonio uhttp://hdl.handle.net/1963/6245