Les Houches lecture notes on topological recursion

A major strength of these lecture notes is the clarity of their conceptual organization and
the choice of perspective adopted throughout the text. Rather than introducing topological
recursion directly in its original Eynard--Orantin residue formulation, the author begins with
Airy structures and differential constraints. This modern point of view provides a unifying
algebraic framework that naturally encompasses Virasoro constraints, $W$-constraints, and
topological recursion itself. As a result, the reader is guided toward an understanding of
topological recursion as a structural phenomenon, rather than as a collection of ad hoc
recursive formulas.

The exposition is particularly successful in explaining why Airy ideals represent a notion
of ``maximal'' systems of differential constraints and how such systems lead to uniquely
determined partition functions. The formulation in terms of the Rees Weyl algebra and the
Bernstein filtration is handled with care, and the motivation for each definition is clearly
explained. The central existence and uniqueness result for Airy partition functions
(Theorem~2.11) is presented at the right level for lecture notes: the statement is precise,
while the proof is outlined in a way that conveys the key ideas without overwhelming the
reader with technical details.

From a pedagogical standpoint, the notes are very well written. The author consistently
takes the time to explain why certain constructions are introduced and how they relate to
classical objects such as the Kontsevich--Witten tau-function, the BGW model, and Virasoro
representations.

Another strong point is the breadth of connections highlighted throughout the notes.
The author succeeds in conveying how topological
recursion sits at the intersection of integrable systems, enumerative geometry, representation
theory, and mathematical physics.

The choice of examples is particularly effective. The Kontsevich--Witten and BGW Virasoro
constraints are presented not merely as classical results, but as prototypical instances of
the general Airy framework.

Finally, the notes reflect the current state of research in the field. The references are
up to date, and several recent developments are mentioned, including generalized Airy
structures, new cohomological classes on moduli spaces, and relations with quantum curves
and resurgence. This makes the notes not only a pedagogical introduction but also a useful
gateway to contemporary research directions. Overall, the lecture notes combine clarity,
depth, and modern perspective in a way that makes them a valuable resource for both
newcomers and active researchers.

These lecture notes are of high quality and provide a clear and modern introduction to
topological recursion and Airy structures. I recommend the paper for the publication after correcting minor typographical issues.

\begin{itemize}[leftmargin=2em]
\item I highly recommend adding a table of contents.
\item p.~4 (around Eq.~(2.2)): \emph{``intepretation''} should read
\emph{``interpretation''}.
\item p.~4: \emph{``particulat tau-function''} should read
\emph{``particular tau-function''}.
\item p.~7, Definition~2.7: \emph{``partition fuction''} should read
\emph{``partition function''}.
\item p.~6: \emph{``Our main of object of study''} should read
\emph{``Our main object of study''}.
\item p. ~11. after Equation 2.28: \emph{Of course, $Z$ is nothing but th KW tau functions}. Maybe it would be instructive to add the link or some explanation.
\item p.~23, Remark 3.14: the phrase \emph{``which is a common way of writing in the literature"} is duplicated.
\item p. ~30, Equation 3.56: $dz$ is missed.
\end{itemize}

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