Narrow Results

We compare Lagrangian thimbles for the potential of a Landau–Ginzburg model to the Morse theory of its real part. We explore Landau–Ginzburg models defined using Lie theory, constructing their real Lagrangian thimbles explicitly and comparing them to the stable and unstable manifolds of the real gradient flow.

Suppose that $\Sigma$ is a closed and oriented surface equipped with a Riemannian metric. In the literature, there are three seemingly distinct constructions of open books on the unit (co)tangent bundle of $\Sigma$, having complex, contact and dynamical flavors, respectively. Each one of these constructions is based on either an admissible divide or an ordered Morse function on $\Sigma$. We show that the resulting open books are pairwise isotopic provided that the ordered Morse function is adapted to the admissible divide on $\Sigma$. Moreover, we observe that if $\Sigma$ has positive genus, then none of these open books are planar and furthermore, we determine the only cases when they have genus one pages.

We discuss in the present paper the following natural question: is the space of all Morse functions with fixed number of minima and maxima on a closed surface linearly connected? We give an algorithm for reduction of any Morse function on a closed orientable surface to some canonical form. We apply this result to the new representation for the inversion of 2-sphere in Euclidean 3-space, in terms of Reeb graph of the height function.

A right invariant Riemannian metric is defined on a pinned path group over a compact Lie group G. The energy function of the path is a Morse function and the critical points are geodesics. We calculate the eigenvalues of the Hessian at the critical points when G=SU(n). On the other hand, there exists a pinned Brownian motion measure ν_λ with a variance parameter 1/λ on the pinned path group and we can define a Hodge-Kodaira-Witten type operator □_λ on L²(ν_λ)-space of p-forms on the pinned path group. By using the explicit expression of eigenvalues of the Hessian of the energy function, we discuss the asymptotic behavior of the botton of the spectrum of □_λ as λ→∞ by a formal semiclassical analysis.

Let H be a real Hilbert space and Φ:H ↦ R a continuously differentiable function, whose gradient is Lipschitz continuous on bounded sets. We study the nonlinear dissipative dynamical system: , plus Cauchy data, mainly in view of the unconstrained minimization of the function Φ. New results concerning the convergence of a solution to a critical point are given in various situations, including when Φ is convex (possibly with multiple minima) or is a Morse function (the critical point being then generically a local minimum); a counterexample shows that, without peculiar assumptions, a trajectory may not converge. By following the trajectories, we obtain a method for exploring local minima of Φ. A singular perturbation analysis links our results with those concerning gradient systems.

We consider the dynamical system given by an algebraic ergodic automorphism T on a torus. We study a Central Limit Theorem for the empirical process associated to the stationary process (f◦Tⁱ)_i∈ℕ, where f is a given ℝ-valued function. We give a sufficient condition on f for this Central Limit Theorem to hold.

In the second part, we prove that the distribution function of a Morse function is continuously differentiable if the dimension of the manifold is at least three and Hölder continuous if the dimension is one or two. As a consequence, the Morse functions satisfy the empirical invariance principle, which is therefore generically verified.

We give some local moves of the Stein factorization of the product map of two Morse functions on a closed orientable smooth 3-manifold which can be realized by isotopies of the functions.