# Sergei Yakovenko's blog: on Math and Teaching

## Wednesday, November 12, 2014

### Lecture 4 (Nov. 14, 2014)

Filed under: Analytic ODE course,lecture — Sergei Yakovenko @ 5:47
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## Algebraic theory of linear ordinary differential operators

• Differential field $\Bbbk=\mathscr M(\mathbb C^1,0)$ of meromorphic germs of functions of one variable $t\in(\mathbb C^1,0)$ + derivation $\partial =\frac{\mathrm d}{\mathrm dt}$ produce noncommutative polynomials $\Bbbk[\partial]$: a polynomial $L=\sum_{j=0}^n a_j\partial ^{n-j}$ acts on $\Bbbk$ in a natural way.
• The equation $Lu=0$ only exceptionally rarely has a solution in $\Bbbk$, but one can always construct a differential extension of $\Bbbk$ which will contain solutions of this equation.
• Analytically solutions of the equation form a tuple of functions $(u_1,\dots,u_n)$ analytic and multivalued in a punctured neighborhood of the origin. The multivaluedness is very special: the linear span remains the same after the analytic continuation, hence there exists a matrix $M\in\mathrm{GL}(n,\mathbb C)$ such that $\Delta (u_1,\dots,u_n)=(u_1,\dots, u_n)\cdot M$.
• Instead of $\partial$, any other derivation can  be used, in particular, the Euler derivation $\epsilon=t\partial$.
• Example. Equations with constant coefficients have the form $L=\sum c_j \partial^{n-j}$ with constant coefficients $c_j\in\mathbb C$. Such an operator can always be factorized into commuting factors, $L=c_0\,\prod_{\lambda_i\in\mathbb C} (\partial-\lambda_i)^{\nu_i}$ with $\sum\nu_i=n=\deg L$. A fundamental system of solutions consists of quasipolynomials $q_{ik}(t)=\mathrm e^{\lambda_i}t^k$, $0\leqslant k < \nu_i$. In a similar way the Euler operator has the form $L=\sum c_j\epsilon^{n-j}$ and its solutions are functions $u_{ik}=t^{\lambda_i}\ln^k t$, $k=0,1,\dots,\nu_i-1$ (look at the model equation $\epsilon^\nu u=0$).
• Weyl equivalence of of two operators. Two operators $L,M\in\Bbbk[\partial]$ of the same order are called Weyl equivalent, if there exist an operator $H\in\Bbbk[\partial]$ which maps any solution $u$ of the equation $Lu=0$ to a solution  $v=Hu$ of the equation $Mv=0$ isomorphically (i.e., no solution is mapped to zero).
The above definition means that the composition $MH$ vanishes on all solutions of $Lu=0$, hence must be divisible by $L$: $MH=KL$ for some $K\in\Bbbk[\partial]$.Note that the operator represented by each side of the above equality, is a non-commutative analog of the least common multiple of mutually prime polynomials $H,L$: it is divisible by both $L$ and $H$.
• Theorem. The Weyl equivalence is indeed an equivalence relationship: it is reflexive, symmetric and transitive.
The only thing that needs to be proved is the symmetry. Since $H, L$ are mutually prime, there exist two operators $U,V\in\Bbbk[\partial]$ such that $UL+VH=1$,  hence $LUL+LVH=L$. This identity means that $LVH$ is simultaneously divisible by $L$ and by $H$ (immediately). Hence $LVH$ is divisible by their least common multiple $KL=MH$: there exists an operator $W\in\Bbbk[\partial]$ such that $LVH=W\cdot MH=WMH$. But since the algebra $\Bbbk[\partial]$ is without zero divisors, the right factor $H$ can be cancelled, implying $LV=WM$, which means that $V$ maps solutions of $Mv=0$ into those of $Lu=0$.
• Different flavors of Weyl equivalence: regular (nonsingular) requiring $H, K$ be nonsingular or arbitrary.
• Theorem. Any nonsingular operator $L=\partial^n+\sum_1^n a_j \partial^j$ with holomorphic coefficients $a_j\in\mathscr O(\mathbb C,0)$, is regular Weyl equivalent to the operator $M=\partial^n$.
This result is analogous to the rectification theorem reducing any nonsingular system $\mathrm dX=\Omega X$ to $\mathrm dX=0$.
• Theorem. Any Fuchsian operator is Weyl equivalent to an Euler operator.
This is similar to the meromorphic classification of tame systems. The conjugacy $H$ may be non-Fuchsian.
• Missing part: a genuine analog of holomorphic classification of Fuchsian systems.

## Poincare-Dulac-Fuchs classification of Fuchsian operators

Instead of representing operators as non-commutative polynomials in $\partial$ or in $\epsilon$, one can represent them as non-commutative (formal) Taylor series of the form $L=\sum_{k\geqslant 0}t^k p_k(\epsilon)$ with the coefficients $p_k\in\mathbb C[\epsilon]$ from the commutative algebra of univariate polynomials, but not commuting with the “main variable” $t$.

Such an operator is Fuchsian of order $n$, if and only if $\deg p_k\leqslant n$ for all $k=1,2,\dots$, and $\deg p_0=n$. The polynomial $p_0$ is the “eulerization” of $L$, and the series can be considered as a noncommutative perturbation of the Euler operator $L_0=p_0\in\mathbb C[\epsilon]$.

Definition. The operator $L=p_0+tp_1+\cdots$ is non-resonant, if no two roots of $p_0$ differ by a nonzero integer, $\lambda_i-\lambda_j\notin\mathbb Z^*$.

Theorem. A non-resonant Fuchsian operator is Weyl equivalent to its Euler part with the conjugacy $H$ being a Fuchsian operator, $H=h_0+th_1+\cdots$, $\deg h_0\leqslant n-1,\ \gcd(p_0,h_0)=1$.

## In search of the general theory (to be continued)

References.

The classical paper by Ø. Ore (1932) in which the theory of non-commutative polynomials was established, and the draft of the paper by Shira Tanny and S.Y., based on Shira’s M.Sc. thesis (Weizmann Institute of Science, 2014).

## Local theory of Fuchsian systems (cont.)

• Resonant normal form.
Definition. A meromorphic Fuchsian singularity $\dot X=t^{-1}(A_0+tA_1+\cdots+t^k A_k+\cdots)X$, $A_0=\mathrm{diag}(\lambda_1,\dots,\lambda_n)+\mathrm N$, is in the (Poincare-Dulac) normal form, if for all $k=1,2,\dots$, the identities $t^\Lambda A_k t^{-\Lambda}=t^k A_k$ hold.
• Theorem. Any Fuchsian system is holomorphically gauge equivalent to a system in the normal form.
• Integrability of the normal form: let $I=\mathrm N+A_1+\cdots +A_k+\cdots$ (in fact, the sum is finite). Then the solution is given by the (non-commutative) product $X(t)=t^\Lambda t^I$. The monodromy is the (commutative) product, $M=\mathrm e^{2\pi \mathrm i \Lambda}\mathrm e^{2\pi\mathrm i I}$.

References: [IY], section 16.

## Linear high order homogeneous differential equations

• Differential operators as noncommutative polynomials in the variable $\partial=\frac {\mathrm d}{\mathrm dt}$ with coefficients in a differential field $\Bbbk=\mathscr M(\mathbb C^1,0)$ of meromorphic germs at the origin.
• Composition and factorization.
• Reduction of a linear equation $Lu=0$ to a system of linear first order equations and back. Singular and nonsingular equations.
• Euler derivation $\epsilon=t\partial$ and Fuchsian equations (“nonsingular with respect to $\epsilon$“).
• Division with remainder, greatest common divisor of two operators, divisibility and common solutions of two equations.
• Sauvage theorem. Tame equations are Fuchsian.

References: [IY], Section 19.

# Meromorphic flat connexions on holomorphic manifolds: Integrability, monodromy, classification

1. Pfaffian systems and their integrability
2. From local to global solutions: monodromy
3. Geometric language: covariant derivative and its curvature
4. Meromorphic functions, meromorphic forms
5. Example: multidimensional Euler system
6. Regular singularities
7. Flat connexions vs. isomonodromic deformations

Recommended reading: D. Novikov & S.Y., Lectures on meromorphic flat connexions, sect. 1-2.

# Riemann–Hilbert Problem: positive results

1. Formulation of the problem and its tautological solution on an abstract holomorphic vector bundle
2. Meromorhic trivialization and Plemelj theorem (solvability of the problem if one of the monodromies is diagonalizable).
3. Invariant subbundles, (ir)reducibility of a regular connexion.
4. Lemma on too different orders. Bounds on the splitting type of a bundle with irreducible Fuchsian connexion.
5. Bolibruch–Kostov theorem: solvability of the Riemann–Hilbert problem for irreducible representations.

Reading: Sections 18A-18D from the book (printing disabled).

# 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.

# Local theory of regular singular points of linear systems

This lecture, in an exceptional way, will take place on Sunday, 16:00-18:00, in the Room 261.

1. Regular and irregular singularities: growth matters.
2. Local gauge equivalence (holomorphic, meromorphic, formal). Meromorphic classification of regular singularities.
3. Fuchsian singularities as a particular class of regular singularities (Sauvage lemma).
4. Formal classification of Fuchsian singularities (Poincaré-Dulac theorem revisited). Resonances. Levelt upper triangular normal form.
5. Coincidence of formal and holomorphic classification in the Fuchsian case.
6. Integrability of the normal form.
7. Towards global theory of Fuchsian systems on $\mathbb C P^1$: Monopoles as special classes of rational matrix functions.

Reminder: Today (actually, on Friday) was the deadline for submission of the home exam 😦

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