Notes

A graveyard/dump for work in progress or just set of notes that deserve to be out there.

• Notes on categorical systems theory
  Categorical systems theory (CST) is a doubly-categorical framework for approaching and developing structural theories of systems and behaviour of any kind. These notes are meant to be (1) a basic reference for CST and (2) a source of examples and working developments. They’re under active development, so quite far from camera-ready, and possibly never complete.
• The universal property of external choice
  External choice of parameterised maps feels like a coproduct but it’s not. It turns out it’s a ‘soft coproduct’, which is a 2-dimensional version of the coproduct whose universal property (which is a representability property) is weakened to an adjunction.
  In the future, I plan to further explore this idea of embracing the second dimension of Para fully, for instance by working out how Kan extensions/lifts can be used to characterize goal-seeking behaviour.
• Bayesian open games, diegetically
  A draft on extending the ideas of Diegetic Representation of Feedback in Open Games to the Bayesian case, basically working out the definition of Nashator for this setting.
• Formal Observability and Behavioural Equivalence
  A theory of observability in the formal framework of CST.
• A fibrational vocabulary for goal-directed systems
  Marrying categorical logic and CST to talk about goals over systems, with some thoughts about how to express regulation goals.

Notes from university courses

I happen to have typeset a number of notes in the last three years, for both graduate and undergraduate courses. As they may be useful to someone else, I decided to publish them here. They haven’t been checked by anyone besides me, let alone by my teachers. I take full responsibility for the errors appearing in them, and in case you found any of them, please let me know.

I’m happy to share LaTeX code as well, upon request.

Graduate courses

The following are notes I took in Padua during 2019. Where due, credit is given to my teachers in each set of notes.

• Analisi Funzionale
  Sbobine delle videolezioni.
  Le note sono su GitHub per chi volesse contribuire o anche solo segnalare correzioni e refusi.
• Hamiltonian Mechanics
  A course in symplectic and Poisson geometry, with some theory of integrability plus a digression on deformation quantization. Red parts are still unclear.
  These notes were started by me and then got some contributions by other people. They are now on GitHub to better handle the process and making them future-proof.
• Representation theory of groups
  Minimal set of notes of the very first part of a course in representation theory (definitions and elementary theory for finite abelian groups).
• L’assioma di scelta: dalle origini a ZFC
  Relazione per il corso di MEPVS. Si tratta di un breve resoconto della parabola dell’assioma di scelta. Ho cercato di riferirmi il più possible alle fonti primarie, quindi è ricco di citazioni da articoli di un centinaio di anni fa.

Bachelor thesis

This is the thesis I wrote for my Bachelor. I wrote about groupoids and monoidal categories, citing applications and developing the theory from the very base.

• Categorie monoidali, gruppoidi e loro prime applicazioni.
  

Undegraduate courses

The following are notes I took in Modena during the years 2016-2018. Due credit is given to my teachers in each set of notes. All of them are in Italian.

• Analisi Matematica C
  Il corso è diviso in una parte sulla teoria delle serie di potenze e delle serie trigonometriche (serie di Fourier elementari) ed una seconda più consistente sull’analisi complessa.
• Teoria della Misura
  Dalla costruzione della misura di Lebsegue fino al teorema di Radon-Nikodym.
• Topologia
  Note di topologia generale, concise ma esaurienti.
• Fisica Matematica B
  Appunti su un corso di meccanica razionale, in particolare accenni di studio dei moti alla Weierstrass, meccanica lagrangiana e hamiltoniana, e approccio perturbativo e variazionale.
• Analisi Numerica A
  Excursus sulla teoria elementare dell’integrazione numerica e dell’approssimazione polinomiale.
• Ottimizzazione numerica
  Metodi del gradiente, SVM, problemi QP semplici.
• Teoria delle funzioni
  Argomenti scelti di analisi funzionale: spazi di Banch, grado topologico di Brouwer e teoremi del punto fisso, trasformata di Laplace, misure di Hausdorff.