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Register a new userGradient-like flows and self-indexing in stratified Morse theory
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Authors
Mikhail Grinberg
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We develop the idea of self-indexing and the technology of gradient-like vector fields in the setting of Morse theory on a complex algebraic stratification. Our main result is the local existence, near a Morse critical point, of gradient-like vector fields satisfying certain ``stratified dimension bounds up to fuzz for the ascending and descending sets. As a global consequence of this, we derive the existence of self-indexing Morse functions.
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We present a new construction of gradient-like vector fields in the setting of Morse theory on a complex analytic stratification. We prove that the ascending and descending sets for these vector fields possess cell decompositions satisfying the dimension bounds conjectured by M. Goresky and R. MacPherson. Similar results by C.-H. Cho and G. Marelli have recently appeared in arXiv:0908.1862.
We prove the Lefschetz hyperplane section theorem using a simpler machinery by making the observation that we can compose the Lefschetz Pencil with a Real Morse function to get a map from the variety to $mathbb{R}$ which is close to being a Real Morse function. The proof uses a new method unlike the conventional one which uses vanishing cycles, thimbles and monodromy. We prove the genus formula for plane curves using Morse theory, Lefschetz pencil and Bezouts theorem. And then we prove the Riemann Hurwitz formula for ramified maps between curves by employing techniques from deformation theory. Lastly, we prove the Lefschetz Hyperplane Section Theorem solely using Real Morse Theory and exact sequences.
The paper is devoted to finding conditions to the existence of a self-indexing energy function for Morse-Smale diffeomorphisms on a 3-manifold. These conditions involve how the stable and unstable manifolds of saddle points are embedded in the ambient manifold. We also show that the existence of a self-indexing energy function is equivalent to the existence of a Heegaard splitting of a special type with respect to the considered diffeomorphism.
We develop the theory of fractional gradient flows: an evolution aimed at the minimization of a convex, l.s.c.~energy, with memory effects. This memory is characterized by the fact that the negative of the (sub)gradient of the energy equals the so-called Caputo derivative of the state. We introduce the notion of energy solutions, for which we provide existence, uniqueness and certain regularizing effects. We also consider Lipschitz perturbations of this energy. For these problems we provide an a posteriori error estimate and show its reliability. This estimate depends only on the problem data, and imposes no constraints between consecutive time-steps. On the basis of this estimate we provide an a priori error analysis that makes no assumptions on the smoothness of the solution.
The flex locus parameterizes plane cubics with three collinear cocritical points under a projection, and the gothic locus arises from quadratic differentials with zeros at a fiber of the projection and with poles at the cocritical points. The flex and gothic loci provide the first example of a primitive, totally geodesic subvariety of moduli space and new ${rm SL}_2(mathbb{R})$-invariant varieties in Teichmuller dynamics, as discovered by McMullen-Mukamel-Wright. In this paper we determine the divisor class of the flex locus as well as various tautological intersection numbers on the gothic locus. For the case of the gothic locus our result confirms numerically a conjecture of Chen-Moller-Sauvaget about computing sums of Lyapunov exponents for ${rm SL}_2(mathbb{R})$-invariant varieties via intersection theory.
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