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. I feel that it is still too early for a book. I'm still 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.

The reason for this project is the following: during the last 24 years, since 2002, I have taught the Harvard summer multi-variable calculus course. The summer classes 2026 have just started and the ever younger kids are already roaming the campus. I still remember a great time from last year 2025 or 2 years ago. I did not continue to teach summer School this summer, as the Summer school has made substantial changes to the course, like having it team taught, have an online option and increase the number of class hours from 42 hours 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 the large math 21b course last time in Spring 2018). The effort is much higher, in particular for coordinating the different parts, requiring regular meetings for example. It was also structured so that the course head decides about the material, the level of difficulty for homework and exams and make the call for grades. And our semester course reach about 42 hours class-time, not 60.

Of course, I'm sad about not having a summer class this summer as I loved the summer course setup. But I learned several times during my life that more unfortunate circumstances always also open up opportunities. In this case, I can entertain myself with this topic ``course", even so I'm the only student. 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 this summer 2026. One can make up a about 12 peaks, the highest being the Dom, the tallest mountain that is located entirely within Switzerland. Theorems for me are the mountain peaks in the landscape of mathematics. There are usually different ways to reach the top, some are easier, some less. Also the climbing style matters. 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 lying, 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 was 8 years old. The picture was taken by my father, close in the line of shot of my bow, trusting that I could aim. )