2022-2024 (A variant was originally displayed starting in the summer of 2022 to celebrate the 600th anniversary of Henry 5 and Charles 6.)
The piece is centrally about kings Henry 5 and his rival Charles 6 ('The Mad King") during the 100 Years War and some information on the subsequent War of the Roses. Two of the central themes of this period were: the extensive and extremely effective use of the English long bow and conflicts over the English throne by cousins after Edward III's death. In addition, an important story in Henry 's life was his devastating injury at Shrewsbury from a deflected arrow that hit him in the face. When one thinks of the first observation, homological algebra, which is full of math diagrams where arrows (morphisms) abound, comes to mind. When one thinks of homological algebra, the Tohoku paper comes into focus and that points to sheaf cohomology. Thankfully, this tool combines themes related to the two other observations, and the word sheaf has a related meaning: sheaf of arrows. A natural way to define sheaf cohomology is with injective objects. Now the diagram defining an injective resembles the diagram next to Henry 5's face that represented the arrows path. Secondly a classic example of sheaf cohomologys power is to address the Cousin problems. The Wars of the Roses was an extremely volatile time in English history and saw the King of England change 6 times in roughly 30 years. I allude to this with the use of Cech cohomology (pronounced check as in the chess term which means the king is under attack) a tool which is deeply related to sheaf cohomology. The combatants vying for the thrones were cousins (Cousin problems). As mentioned above, the roots for this conflict lie in the death of Edward III’s son 100 years earlier (and the mathematician, who devised Cech cohomology’s first name was Eduard, ihe same root as Edward). Finally, the name Henry relates to a significant figure in the math world related to these ideas Henri Cartan (and his famous A and B theorems).
The central question "what is left?" has many motivations: first, a teenage Henry was shot in the left side of his face at the battle of Shrewsbury (many feel the arrow likely deflected before entering his face). Thankfully, he survived the brutal injury, but the left side of his face was severely damaged, which probably explains why his portrait only shows one side of his face. Hence what is left of his face? The math symbols representing the arrow's path feature mathematical "arrows" (also called morphisms) hints at what is called an injective object. These are integral to the construction of sheaf cohomology. The letter I, which is often the letter to represent an injective object, is missing in this diagram. it should be over his face. This i is hinted at by the I next to the crossed out “s” in “Is” at the top of the work. Ie what is left after the s is removed.
Second, at the historic Battle of Agincourt, Henry's significantly outnumbered army thoroughly defeated the French with the brilliant use of the long bow and arrows. But, tragically, he had French prisoners of war executed (as he feared they might rearm and overwhelm the English), so what is left of the French prisoners?
Third, Henry's military victories forced Charles to name Henry his successor to the French throne (albeit after Charles died). Unfortunately, Henry died young (months before Charles) and could not assume this throne. After his death, the French reclaimed large tracts of the previously conquered land. Hence, what is left of the English lands in France? (on a side note, Henry’s son, Henry the 6's inept reign played a significant role in causing the War of the Roses).
(Finally, the global sections functor of a mathematical sheaf is left exact. This is important for sheaf cohomology)
As this is a short description, it is beyond its scope to go over detail of this work, but some more of the relevant history will. be detailed.
The night before the battle of Agincourt, Henry 5 ordered silence from his troops, threatening to cut off their ears if they failed to obey. The battle was fought on a rain-soaked field that significantly hampered the heavily armored French troops and cavalry. The portrait of van Gogh, with a missing ear, with an "X" over his mouth references Charles 6 insanity and this episode as does the van Gogh picture of rain out his asylum window (on the right of my work).
Just as heavily armored knights did not function well on the rain-soaked field of Agincourt, allowing the English to defeat the French with arrows. To define a “good” cohomology theory for varieties over a finite field, more “open sets” are needed. The way this was accomplished was by defining open sets to be particular “arrows” (etale maps) to the space (as opposed to particular subsets of the space). This idea and the extensive use of the long bow by the English at the beginning of the battle of Agincourt are hinted at by “open with arrows”.
Charles the 6th was known as the mad king as he believed he was made of glass and had iron rods inserted into his clothes to prevent from breaking.
Henry 5th was the first English monarch to primarily use English as his language. This is pointed to with the two spellings of Henry (i).
The 100 years war actually lasted 116 years….100=116
As previously mentioned Charles 6 struggled badly with mental illness and during these bouts he could be completely incoherent for days. This was a far cry from the brilliant military leader Henry 5.
Wars and problems between cousins determined who would rule the English throne directly before and after Henry's reign.
Japan refers to the Japanese print on the wall behind Van Gogh in his original painting, but its also the name of the country where a journal published the revolutionary Tohoku paper.
Long exact sequences play an important role in cohomology and those terms are also relevant to the long bow.
Refer to the description for Stable 1 to get the baseline information about this piece. There is a lot going on here but at minimum the work hints at the Landweber Exact Functor Theorem (LEFT) , Quillen's Theorem, formal group laws, chromatic homotopy theory, all of which are related to or are aspects of stable homotopy. For LEFT to produce a homology theory the sequence next to Bugs need to be regular for all p. This is hinted at by bugs facing left and pressing regular on the gas pump. The BP logo stands for British Petroleum but in this context, it also represents the Brown Peterson Spectrum. illustrated on the right is chromatic tower.
The suspension bridge points to several ideas firstly some of the math displayed points to a bridge between spectra and formal groups, and secondly the fundamental importance of the suspension to spectra.
2023
This is a quick and rough sketch. ill clean it up later. This work addresses the War of the Roses (WOTR) and the 100 Years War. The former helped motivate the Game of Thrones. The WOTR was an extremely volatile time in English history, and the King of England changed 6 times in roughly 30 years. This is alluded to with the use of Cech cohomology (pronounced check as in the chess term, which means the king is under attack). The combatants vying for the throne were cousins which is hinted at by the Cousin problems in math Cech cohomology and its relative sheaf cohomology are deeply tied to the Cousin problems. (On a side note, Edward the 4th was a central Yorkist, and many argue the premature death of Edward 3’s son, the Black Prince, laid the groundwork for the WOTR a century earlier. Eduard Cech developed Cech cohomology (his first is the same root as Edward).
In the following, Ill give a synopsis of some of the interesting elements from the WOTR that are featured here. While Richard Neville was making a deal with the king of France to find Edward 4 (Queen of Spades), Edward 4, instead, married Elizabeth Woodville (Queen of hearts) because of love not politics. Richard of York lost his head, in the early stages of the Wars of the Roses, which was placed on a pike. Richard 3t was knocked off his horse at the Battle of Bosworth (recounted by Shakespeare "My kingdom for a horse") but unfortunately the horse/dragon he seeks is his rival Henry 7's standard. Henry 6 struggled badly and may have been cursed by the genes of his grandfather Charles "the mad king". His impotent leadership led the English to lose most of their lands in France by 1453. 6 is a perfect number but Henry 6 certainly was not but Bobby Fischer in game 6 of his championship math vs Spassky in 1972 was perfect. In the game he played Queens gambit (for the “first” time) and that points to Margaret of Anjou.
Exec Ord 6102 was signed on Apr 5 1933 the same year a theorem important for Bitcoin was proven. Satoshi's bday is Apr 5 (1975 see below). The Fed's inept actions helped worsen the great depression instead of dampening the business cycle. A motif in this work is based on a rat on a cycle...Lets start with Satoshis “birthday”: April 5, 1975. April 5th, 1933 is the day personal gold ownership in America was banned (Exec Ord 6102). Gerald Ford ended this ban starting in 1975. 1933 was also the year a theorem was proven central to Bitcoin. It essentially gives bounds for the number of points on an elliptic curve (over a finite field. I don’t want to go into this technicality but essentially think clock like arithmetic ie 12 plus any number still results in the original number on a clock, but something ill address in later works). Bitcoin is based on a particular elliptic curve. Ill now motivate the central “rat on a cycle“ motif. Warren Buffet famously called Bitcoin rat poison squared, but one might wonder if the Fed is a “rat” that might not be such a bad thing. There is a decent argument that the Fed’s inept actions worsened the Great Depression and more recently with the 2008 Crash iconic Fed Chairman Alan Greenspan admitted he made mistakes. Here he is represented as the famous yet ineffective duck hunter Elmer Fudd. In addition to overseeing a complete collapse of the price of the dollar over the last century these examples may question the effectiveness of the Fed to dampen the business “cycle”. In 1910, an ultra-secret meeting between a handful of the worlds top bankers disguised as duck hunters occurred at Jekyll Island. This led to the creation of the Fed in 1913. The word “fixed” has many interpretations eg the number of Bitcoin is “fixed” (21 million), the price of gold was “fixed” (eg $35/oz), does the Fed need to be “fixed”, some even argue the (economic) “game” itself is fixed eg who gets a Fed bailout and who does not. It also has a mathematical one, “fixed” points, which Ill get to later. The rat imagery is an altered from rats made by the famous street artist Banksy, a name which like Satoshi is a pseudonym (another unnamed pseudonym which ill address in later works relates to the math picture). Now for a tiny bit of the math. A program, undertaken by some of the 20th centuries foremost mathematicians to generalize the aforementioned theorem (Hasse) and revolutionize algebraic geometry, led to Grothendieck’s development of motives (same root as motif). One example of this is a motive which is based on rational equivalence on (algebraic) cycles (the former is usually abbreviated “rat” when writing math) ie “rat on a cycle. For example in the work you’ll see an M with a superscript rat and subscript k, it represents the “universe”, category, of these motives. To tie back to economics the large M should be juxtaposed with another “decorated” M, named M1 which is an important “measure” of the money supply which unlike in bitcoin’s case fluctuates and is impacted by the Fed. I mentioned “fixed” points in relation to the title: a deep observation about the “fixed” points of a particular map (Frobenius) and a "Lefschetz-like" fixed point theorem was important for the initiation of the program to generalize Hasse. Another of Gothendieck’s brilliant innovations (he was responsible for a significant amount of the work on this revolution) was the scheme. This "mathematical" scheme appears to be preferred to a Ponzi scheme, a term that is sometimes disparingly used in the context of economics and finance. The Standard logo has many relevant interpretations eg bitcoin standard, gold standard, a famous conjecture on cycles related to motives and the aforementioned program, and the idea of a monopoly. This can be interpreted in the context of the Fed’s monopoly” on "money printing" and some critic’s concerns about the Fed’s role in helping to determine who “wins” in our economy (eg with bailouts). There is much more to say but brevity is sometimes wit.
notes:
Mantle twisted his knee his rookie year and badly needed surgery.
The statue that makes up Mantle’s body originally held a disc (discus).. And gloves are for catching a ball.
See piece Seven for info about Mantle and Milnor’s exotic 7-sphere.
Henry 7th (inside the dotted square) survived the war of the Roses. Does the element h7² in the Adams Spectral Sequence (7 in subscript) survive?
Exec Ord 6102 was signed on Apr 5 1933 the same year a theorem central to BTC was proven. Satoshi's bday is Apr 5. The Fed's inept actions helped worsen the great depression instead of dampening the business cycle. A motif in this work is a rat on a cycle...
This is a cursory first draft which serves as an incomplete introduction to the work. This art work focuses on Mickey Mantle, spheres, toplogy, and the number 7. Topology's first results come from the great Leonhard Euler: the seven Bridges of Konnisberg problem and his Polyhedron formula for the 2-sphere. John Milnor revolutionized topology with his discovery of the exotic 7-sphere in 1956, for which he won the Fields Medal and essentially every other prestigious math award. Mantle wore number 7, won 7 World Series, and achieved his best year in 1956, winning the Triple Crown. 7 years after discovering his exotic 7-sphere Milnor, jointly with Kervaire (M-K), went on to classify large swaths of these exotic spheres which also led tp the Kervaire invariant problem for framed manifolds. Their work relied heavily on mathematical surgery. Mantle tore his right ACL and badly needed surgery. One of many interesting aspects of M-K’s work is they show a portion of the classification of exotic spheres ties to the Bernoulli numbers. In essence constructing a “bridge” between topology and number theory. Several hundred years earlier, Euler gave an interesting definition of the Bernoulli numbers and he proved one of the first theorems of topology with the 7 bridges of Konnisberg problem.
Mantle's seven was retired in 1969. 1969 was also the year, Browder, who later chaired Princeton's math department, proved a substantial result in the continued classification of exotic spheres in particular the Kervaire invariant problem eg. showing it was necessary that n had to equal 2^k-2 for the Kervaire invariant to be nonzero.
The story continues starting with the hyperbolically named Doomsday Conjecture, an exaggerated name given to an esoteric conjecture in math about framed manifolds having nonzero Kervaire invariant. The horrifying, yet slightly humorous, tone is due to the fact that some important work on problems such as the homotopy groups of spheres, would be nullified if it were confirmed. An important paper written in 2009 shows there exists a spectrum (cohomology theory) with several interesting properties including a Gap theorem. This enabled the authors to resolve a large part of the Kervaire invariant problem, only leaving the case n=2^7-2 open.
Some other details in the artwork include the fact Mantle and Milnor were both born in 1931 and there were two Yankees players who had their number 8 retired.
Hints: Is this a sequence of cursive e's, a loop, Taz's whirlwind, a top, a Cy Twombly scribble, etc?
A smash B? Suspension?
2023
2022-2023
A lot going on in here, but here are 2 quick hints for one of the themes of the work: The math hints at a nontrivial pointless topos (a mathematical object whose name means place in Greek). This is juxtaposed with a place in Gulliver's Travels called Lagado, where pointless experiments are being run (e.g., attempting to extract sunlight from cucumbers).
Wiley Coyote's appearance in this work has many interpretations. Still, the simplest one augments this story as he is known for engineering elaborate plans, but this effort was pointless as he could never catch the Roadrunner.
