Sergei Yakovenko's blog: on Math and Teaching

Monday, November 3, 2014

Lecture 1 (Nov. 3, 2014)

Filed under: Analytic ODE course — Sergei Yakovenko @ 6:34
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The first lecture was introductory, containing the motivation for the forthcoming subjects.

The world of (real or complex) algebraic sets is tame: any question on the topological complexity admits an algorithmic solution and explicitly bounded answer.In particular, any algebraic set which consists of finitely many isolated points, admits an explicit bound for the number of these points by the product of degrees of equations defining this set (Bézout theorem). All the way around, equations involving nonalgebraic solutions to even simplest algebraic differential equations (sine/cosine), may define infinite sets (integer numbers). We will try to find out how the algebraic universe can be enlarged to include transcendental objects which still admit explicit bounds on their complexity.

It turns out that periods, integrals of rational forms over algebraic cycles, do possess such constructive finiteness, although this is far from easy to see. This finiteness is characteristic for solutions of rational Pfaffian systems with moderate singularities and special monodromy group.

Part 1: General linear systems.

A linear system locally lives on a cylinder, the product of a (complex) linear space $\mathbb C^n$ and an open base $U\subset \mathbb C^k$. If $\Omega=\bigl(\omega_{ij}\bigr)$ is an $n\times n$-matrix of holomorphic 1-forms on the base $U$, then a linear system defined by this matrix 1-form, is a matrix differential equation $\mathrm dX=\Omega\cdot X$, whose solution is a holomorphically invertible matrix function $X=X(t)$, $t\in U$. If the base is one-dimensional, then $\Omega=A(t)\,\mathrm dt$ with a holomorphic matrix function $A(t)$, and the linear system takes the familiar shape $\dot X(t)=A(t)X(t)$ [IY, sect. 15]

A necessary and sufficient condition for a local existence of solution is vanishing of the curvature, which amounts to the  matrix identity $\mathrm d\Omega=\Omega\land\Omega$ (the right hand side is the matrix 2-form with the entries $\sum_{\ell=1}^k \omega_{i\ell}\land\omega_{\ell j}$, $i,j=1,\dots,k$).  See [NY, sect. 1].

Solution of a linear system is defined modulo a right multplicative constant matrix factor: $\mathrm d(XC)=\Omega XC$ for any $C\in\mathrm{GL}(n,\mathbb C)$, and any other solution has such form. Using this observation, any piecewise curve $latex\gamma$ on the base can be covered by small neighborhoods $U_\alpha$ with local solutions $X_\alpha$ in these neighborhoods, which agree on the pairwise intersections $U_\alpha\cap U_\beta$. If this was not the case for the initial choice of local solutions, this can be always achieved by suitably twisting them (replacing by $X_\alpha C_\alpha$ so that $X_\alpha C_\alpha=X_\beta C_\beta$ on the intersections). This explains how solutions can be continued analytically along any simple curve, yet after continuation along a closed path $\gamma$ the solution may acquire a non-trivial monodromy factor.

Piecemeal remarks on rational matrix functions of a complex variable

The global theory of rational linear systems on $\mathbb C P^1$ requires the study of (rational) gauge transformations which are holomorphic and holomorphically invertible except for a single point. If this point is at infinity, then the matrix of such transformation is necessarily polynomial with constant nonzero determinant. Such matrix functions are provisionally referred to as monopoles, $H(t)\in\textrm{GL}(n,\mathbb C[t]),\ \text{det}H=\text{const}\ne 0$.

Multiplication of a rational matrix function $H(t)$ from the left by a monopole matrix $\begin{pmatrix}1 & t\\ & 1\end{pmatrix}$ corresponds to adding the second row of $H$, multiplied by $t$, to the first row. Thus manipulations with rows of $H$, which aim at Gauss-type elimination of certain monomials from matrix elements, can be represented as gauge actions of the monopole group. The principal result that will be used throughout the next few lectures, is the following Bolibruch Permutation Lemma.

Lemma. Let $H(t)$ be the germ of a matrix function, holomorphic and invertible at $t=\infty$. Then for any ordered tuple of integer numbers $D=\{d_1,\dots,d_n\}$ the product $t^D\,H(t)$, $t^D=\text{diag}(t^{d_1},\dots,t^{d_n})$, is monopole equivalent to a product of the form $H'(t)\,t^{D'}$, where $H'(t)$ is also holomorphic and invertible at $t=\infty$, and $D'$ is a permutation of the tuple $D$.

The proof of this result is not difficult, yet is too technical to be delivered in the classroom.

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