# Sergei Yakovenko's blog: on Math and Teaching

## Tuesday, December 30, 2014

### Final announcement

This is to inform the noble audience of the course that the main program of the course is completed. I will stay in Pisa for one more week (till January 8, 2015) and will be happy to discuss any subject (upon request).

Meanwhile one of the subjects discussed in this course was brought to a pre-final form: the manuscript

1. Shira Tanny, Sergei Yakovenko, On local Weyl equivalence of higher order Fucshian equations, arXiv:1412.7830,

was posted on arXiv and submitted to the Arnold Mathematical Journal, a new venue for publications molded in the spirit of  the late V. I. Arnol’d and his seminar.

Any criticism will be most appreciated. Congratulations modestly accepted.

Tanti auguri, carissimi! Buon anno, happy New Year, с наступающим Новым Годом, שנה (אזרחית) טובח, bonne année!

## Tuesday, November 18, 2014

### Lecture 5 (Nov. 17)

Filed under: Analytic ODE course,lecture — Sergei Yakovenko @ 12:27
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## Fuchsian equivalence and Fuchsian classification

Definitions
A (formal, genuine) Fuchsian operator of order $n$ is a (formal, resp., converging) series of the form $L=\sum_{k=0}^\infty t_kp_k(\epsilon)$ with the coefficients $p_k\in\mathbb C[\epsilon]$ from the ring of polynomials in the variable $\epsilon$ with $n=\deg p_0\ge \deg p_k\ \forall k=1,2,\dots$.
The polynomial $p_0\in\mathbb C[\epsilon]$ is called the Euler part of $L$.

Two Fuchsian operators $L,M$ are $\mathscr F$-equivalent (formally or analytically), if there exist two Fuchsian operators $H,K$ such that $MH=KL$ and the Euler parts of $H,L$ are mutually prime in $\mathbb C[\epsilon]$.

Unlike the Weyl algebra $\mathscr M(\mathbb C,0)[\epsilon]$, the collection of Fuchsian operators is not a subalgebra, although it is “multiplicatively” (compositionally) closed.

The Fuchsian equivalence is indeed reflexive, transitive and symmetric. The first two properties are obvious, to prove the last one an additional effort is required. Indeed, for two Fuchsian operators $L,H$ of order $n$ with mutually prime Euler parts, one can construct two operators $U,V$ with holomorphic coefficients so that the identity $UL+VH=1$ holds, but in the leading terms of $U,V$ may well degenerate, thus violating the Fuchsian condition. However, one can always find such pair of operators of order greater by 1, which will still be Fuchsian. The rest is easy.

The following results can be proved by more or less direct computation in the algebra $\mathscr W=\mathbb C[[t]]\otimes\mathbb C[\epsilon]$:

• A Fuchsian operator with a nonresonant Euler part is $\mathscr F$-equivalent to its Euler part.
• Any Fuchsian operator is $\mathscr F$-equivalent to a polynomial operator from $\mathbb C[t]\otimes\mathbb C[\epsilon]$.
• Any Fuchsian operator is $\mathscr F$-equivalent to a polynomial operator of the form $L=(\epsilon -\lambda_1+q_1(t))\cdots(\epsilon -\lambda_n+q_n(t))$ with $q_1(t),\dots,q_n(t)$ being polynomials without free terms, $q_i(0)=0$, which is Liouville integrable.
• A Fuchsian operator has trivial (identical) monodromy if and only if it is $\mathscr F$-equivalent to an Euler operator with pairwise different integer roots. The corresponding equation has an apparent singularity (all solutions are analytic) if and only if all these roots are pairwise different nonnegative integers.

# Stokes phenomenon for irregular singularities of linear systems

1. Irregular singularities: total recall. Formal diagonalizability of non-resonant systems.
2. Sectorial gauge equivalence: formal, holomorphic, asymptotic series.
3. Separation rays. Sibuya theorem on sectorial normalization (statement).
4. Sectorial authomorphisms. Rigidity of the normal form in large sectors.
5. Stokes matrix cochain and Stokes matrix multipliers as complete invariants of holomorphic classification of irregular singularities.
6. Stokes phenomenon. Realization theorem (Birkhoff). Generic divergence of the formal gauge normalizing transformations.

Recommended reading: Sections 20F-20I from the Book

# Bolibruch Impossibility Theorem

Revealing an obstruction for realization of a matrix group as the monodromy of a Fuchsian system on $\mathbb C P^1$.

1. Degree (Chern class) of a complex bundle vs. that of a subbundle. The total trace of residues of a meromorphic connexion.
2. Linear algebra: Monoblock operators and their invariant subspaces.
3. Local theory revisited: local invariant subbundles of a (resonant) Fuchsian singularity in the Poincaré–Dulac–Levelt normal form.
4. Bolibruch connexions on the trivial bundle: theorem on the spectra of residues.
5. Three Matrices $4\times 4$: the Bolibruch Counterexample.

Reading: Section 18E from the book (printing disabled).

Refresh your memory: Sections 16C16D (local theory), 17E-17I (degree of bundles)

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