About this project

The goal of this project is to present a few results in finite geometry and finite calculus that have been developed in the last two decades. It is part of a larger ``quantum calculus'' project. All is finite. If real numbers are involved, like for functions, one could take a finite subset of real numbers or a finite totally ordered set. The goal is to spot-light some results, proofs, illustration, collect some code and organize the material.

At the moment, the plan is to present 4 results per week and to stop at 12. Definitions and notation are moved into an appendix. The text should be holographic and short. References are moved to a footnotes section. I'm at the moment still mostly interested in exploring new results rather than looking back. To avoid building on sand however, it is important to solidify from time to time results that have already been reached, especially since most of my own work is only available on the ArXiv.

To the reason for this project: for the last 24 years, I have taught the Harvard summer multi-variable calculus course. The summer classes 2026 have just started. The summer course 2025 or and 2024 are still fresh in memory. I did not continue to teach summer this summer, as the school has made substantial changes to the course, like having it team taught, have an online option (eventually scraped) and increase the number of class hours from 42 to 60 hours. I have team taught for 26 years here during the semester (led the large math 21a course last time in 2019 and math 21b course last in Spring 2018). The effort is much higher, due to coordination and meetings etc. The current set-up would have absorbed the summer. In the past, the time effort could be kept under 40 hours per week, allowing an other 40 hours for research. Of course, I'm sad about not teaching this summer as I loved the summer course setup as it had been. I learned several times during my life that seemingly unfortunate circumstances can open up opportunities. In this case, I entertain myself with this ``topic course". Of course, I'm happy if somebody should find this useful. I'm also happy to get feedback.

The picture on the website banner shows the Mischabel massive in the Swiss alps. I took this photo in the summer of 2026. One can see 12 peaks on the horizon. The highest is the Dom, the tallest mountain within Switzerland. Theorems are the mountain peaks in the landscape of mathematics. There are always different ways to reach some peak, some are easier, some less. Also the climbing style matters. I myself mostly enjoyed to climb light and fast, also in the alps. My middle school math teacher Edi Schmid, who was also our neighbor and friend in Uhwiesen, drew once the Mischabel group in 1967 while visiting the alp. If one only looks at the peaks bordering the sky in the picture, one can count 12, like the 12 topics covered in this story.

Edi Schmid was a gentle teacher (and teaching in the same school than my father). He introduced me to infinity, when I was maybe 8 years old: he took a sheet of paper and sketched a bed in which the figure 8 was seeping, so that it looked like the symbol for infinity. He took the pencil and traced the figure eight again and again and again: "This is infinity", he said "It sleeps for ever".

(* On this picture, one can see Edi Schmid showing us how to shoot bow arrows on the alp below Salmenfee. In red are his daughters Barbara and Katherine, in black my brother and me: from left to right: Salmenfee cottage, Barbara, Matthias, Edi, Katherine, Oliver, when I were maybe 7 years old. The picture was taken by my father, close in the line of shot of my bow, trusting that I could aim. )