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Andrius: I am writing

A Pilgrim's Account of Bott Periodicity and the Tenfold Way

I imagine myself a traveler, trekker, mountain climber, returning from a fantastic mathematical realm, yearning to tell of the metaphysical wonders of Bott periodicity and the tenfold way. I rest in the foothills, at Plato's Café, a literal cave, adorned with Platonic solids - tetrahedra, cubes, octahedra, dodecahedra, icosahedra - which are but shadows compared with the Platonic forms I brought back with me, not in my pack but my mentality.

As a youth, in my quest to know everything, my search for absolute truth, my study of the limits of my imagination, I had discerned these forms - the twosome, threesome, foursome - all the way up to the eightsome, whereupon they collapse back into the nullsome where all things are true. Decades later, when I heard of eightfold Bott periodicity, when I learned that it was the pattern of the spheres, and as I supposed, of all perspectives, then I journeyed to these mathematical mountains. I wished to trace with my eyes what I had grappled with blindly in the prison of my mind. I hoped to free my tongue to make comprehensible what must be told, what I was born to tell.

The place was quite full, with climbers and their gear, and a couple kindly asked to sit with me. They had driven a long day's road, in conversation and in silence, back and forth, laughing and sighing, Sophia and Phil. She was a young graduate student, searching for her topic, ambitious to climb as high and far as anyone ever could. He would stay back at camp, a high school teacher preparing his geometry lessons, aspiring to inspire. She sought my advice. The young climb up, the old climb down, and they exchange some words about the weather, and if they like, the mountains that shape the weather. He was fascinated, and they indulged me that I indulge myself.

Had they heard of Grothendieck? Attiyah? Bott? She knew of Grothendieck, and also the Attiyah-Singer Index theorem. But who was Bott? She had taken a year of algebraic topology and likewise algebraic geometry. Ah, then you know more than me, but I will tell you my tale of Bott periodicity, and when you climb your climb, you may see it all in a new light.

I would mention how - Bott, Attiyah, Grothendieck - approached the subject using the most abstract mathematics (K-theory, etc.), but I would focus on the concrete math by which I myself found my way, including facts about matrices, and also Pascal's triangle, which Clifford algebras encode. For example, there is a very important pattern regarding the squares of products of generators e_1, e_2, e_3... of a Clifford algebra. Suppose e_i^2=-1 and e_ie_j=-e_je_i for all i,j. The pattern is that (e_i)^2=-1, (e_ie_j)^2=-1 but then (e_ie_je_k)^2=+1, (e_ie_je_ke_l)^2=+1 and in general it is a four-fold pattern -1, -1, +1, +1, -1, -1, +1, +1 .... And this pattern comes from summing the numbers 1+2+3+4...+ n. For 1 is Odd but 2 is Even so Odd+Even=Odd. Then 3 is Odd and we have Odd+Odd=Even. Then 4 is Ęven and we continue Even+Even=Even. This fact becomes important later on because once we have three operators e_1, e_2, e_3, then we can define an operator, namely K= e_1e_2e_3 which squares to +1, and the fact that K^2-1=0 means its eigenvalues must be +1 or -1, and if we are representing this with real matrices acting on a real vector space, then the vector space splits accordingly. There are quite a few such concrete pieces in this puzzle that I would point to.