01749nas a2200145 4500008004100000245005800041210005700099260001000156520126500166653006601431100002501497700002401522700002101546856003601567 2013 en d00aGeodesics and admissible-path spaces in Carnot Groups0 aGeodesics and admissiblepath spaces in Carnot Groups bSISSA3 aWe 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.10aCarnot group, Loop space, Betti number, Morse-Bott functional1 aAgrachev, Andrei, A.1 aGentile, Alessandro1 aLerario, Antonio uhttp://hdl.handle.net/1963/722801015nas 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