Teaching and materials

Notes on Biomathematics, Differential Equations, and Numerics 

Scientific Dissemination through Dialogue and Teaching

In my early academic years, my formation was strongly shaped by continuous participation in seminars, workshops, and informal academic discussions, where conversations with experienced researchers played a central role in consolidating my interests and perspectives in mathematics. These formative exchanges were fundamental in developing a critical and reflective approach to mathematical thinking, teaching, and interdisciplinary work.

Over the years, this early exposure to academic dialogue evolved into active participation in a wide range of scientific and academic events, where I have shared ideas, presented research, and engaged with diverse communities. These activities reflect a sustained commitment to scientific dissemination, teaching, and the training of new generations, as well as to fostering interdisciplinary dialogue between mathematics and related fields. Together, these experiences highlight the role of communication and collaboration as essential components of mathematical practice and its applications.

Academic Presentations and Publications

The academic trajectory of William Campillay-Llanos began in the early 2000s, with a strong emphasis on mathematics education, mathematical proof, and the epistemological foundations of the discipline.

In 2007, he actively participated as a lecturer at UMCE and the University of Chile, addressing key issues in the teaching of mathematics in Chile. During this period, he delivered talks such as “The Use of Cabri: Is It Always Appropriate? Three Problems for Reflection” and “Proposals to Overcome Shortcomings and Remedy Frequent Errors in Chilean School Mathematics”, both focused on the role of proof in the classroom. That same year, he published the article “A Proposal for Guiding Mathematical Proof in Chilean Curricula and the Idea of Academic Rigour” in the Revista Chilena de Educación Científica.

In 2009, his work gained broader national and international visibility. He was a speaker at the Ibero-American Congress on Mathematics Education (CIBEM) with the presentation “Proportional Calculus: The Multiplicative World”. He also participated in the 23rd Southern Zone Mathematics Conference (University of Magallanes) with “An Introduction to Infinite Calculi: Proportional Calculus”, and presented “To Prove Is to Do Mathematics, and to Do Mathematics Is to Prove” at the meeting of the Chilean Mathematical Society. In the same year, he contributed as a reviewer to the monograph “Algebra as a Quality of the Student”, within the framework of a FONDEF project.

In 2010, he published the article “Mathematical Proof in Chilean Secondary School Curricula” in the Revista Iberoamericana de Educación Matemática.

Between 2011 and 2012, his academic production took on a more interdisciplinary character, integrating mathematics, biology, epistemology, and education. In 2011, he presented “Sensitivity and Rigour in the Study of Mathematics: From Multiplicative Calculus to the Biology of Love” at AEPMAT, and co-authored with L. Espinoza the paper “Didactical Situations Theory in Chile: Does It Work?” published in Acta Latinoamericana de Matemática Educativa.

In 2012, he presented “The Relationship between Galois Theory and the Reflections of Humberto Maturana” at the Southern Zone Mathematics Conference, and “The Autopoiesis of Galois Theory” at the National Mathematics Meeting in Chile.

From 2013 onwards, his academic work increasingly focused on theoretical and applied mathematics, particularly functional analysis and differential equations. That year, he presented “Proportional Differential Equations”, jointly with M. Pinto, at the Annual Meeting of the Group on Evolution Differential Equations and Functional Analysis (USACH).

In 2014, he delivered the talk “Structure and Form” at the Institute of Philosophy and Complexity Sciences (IFICC), and “Question or Conjecture” at the Seminar on Evolution Equations of the University of La Frontera, strengthening the dialogue between mathematics, philosophy, and complexity.

Finally, in 2015, he participated as a speaker in high-level academic settings, delivering the lecture “What Is an Arithmetic?” at the Postgraduate Mathematics Seminar of USACH (August 2015), and presenting, together with F. Guevara-Morales, “Gelfond’s Constant and Proportional Arithmetic” at the Southern Zone Mathematics Conference (April 2015).

Participation and Attendance at Academic Events (2004–2014)

Alongside his work as a speaker, William Campillay-Llanos has maintained continuous participation in schools, conferences, and seminars, strengthening both his academic training and professional networks.

Between 2004 and 2006, he participated in the Colloquium on Challenges and Perspectives in Mathematics Education (UMCE), the First and Second National Schools of Biomathematics (UMCE), the Mathematics Week organised by PUCV, and the Capricorn Mathematics Congress (COMCA 2006).

In 2010, he attended the Seminar on Functional Analysis and Evolution Equations organised by the GAFEVOL research group (USACH).

During 2011, he expanded his international exposure by participating in the IMPA Summer School (Rio de Janeiro) and the First Workshop in Game Theory and Evolutionary Dynamics at the Fluminense Federal University, Brazil.

In 2014, he attended the 7th International Conference on Functional Equations, held at the University of Limoges, France, further consolidating his training in functional equations and mathematical analysis at an international level.

The passage belongs to the poem “Au Lecteur” (“To the Reader”), which opens Les Fleurs du mal by Charles Baudelaire and serves as a true prologue to the book, setting from the outset its moral, aesthetic, and philosophical tone. Written around 1855 and first published in Paris in 1857, Les Fleurs du mal caused a major public and judicial scandal upon its release: Baudelaire was prosecuted for “offence against public morality and good morals,” and several poems were censored. This context underscores the provocative nature of “Au Lecteur”, which introduces the reader to a work that deliberately confronts the anxieties, vices, and spiritual malaise of modern life. 

Français (original)

Mais parmi les chacals, les panthères, les lices,
Les singes, les scorpions, les vautours, les serpents,
Les monstres glapissants, hurlants, grognants, rampants,
Dans la ménagerie infâme de nos vices,

Español

Pero entre los chacales, las panteras, las perras,
Los simios, los escorpiones, los buitres, las serpientes,
Los monstruos chillones, aullantes, gruñones, reptantes,
En la infame jauría de nuestros vicios,

English

But among the jackals, panthers, hounds,
Apes, scorpions, vultures, serpents,
The yelping, howling, growling, crawling monsters,
In the infamous menagerie of our vices,

Français (original)

Il en est un plus laid, plus méchant, plus immonde !
Quoiqu’il ne pousse ni grands gestes, ni grands cris,
Il ferait volontiers de la terre un débris
Et dans un bâillement avalerait le monde ;

Español

¡Hay uno más feo, más malo, más inmundo!
Aunque no haga grandes gestos ni lance grandes gritos,
Haría gustoso de la tierra un despojo
Y en un bostezo se tragaría el mundo;

English

There is one more ugly, more wicked, more foul!
Though he makes neither grand gestures nor loud cries,
He would gladly turn the earth into rubble
And in a yawn swallow the world;

Français (original)

C’est l’Ennui ! — l’œil chargé d’un pleur involontaire,
Il rêve d’échafauds en fumant son houka.
Tu le connais, lecteur, ce monstre délicat,

Español

¡Es el Tedio! — con los ojos cargados de un llanto involuntario,
Sueña con patíbulos mientras fuma su pipa.
Tú conoces, lector, a este monstruo delicado,

English

It is Boredom! — eyes heavy with involuntary tears,
He dreams of scaffolds while smoking his pipe.
You know him, reader, this delicate monster,

Français (original)

— Hypocrite lecteur, — mon semblable, — mon frère !

Español

— ¡Hipócrita lector, — mi semejante, — mi hermano!

English

— Hypocritical reader, — my likeness, — my brother!