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.
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.
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.
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.
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?
A quick intro into the piece and then there are some hints below. This work is my first foray into. stable homotopy, First, the Tasmanian Devil’s explosive personality is anything but stable, but more importantly, the whirlwind behind him looks like a sequence of lowercase cursive e's (Cy Twomblyesque) or possibly a long loop. Central to Stable homotopy is an object called a spectrum, which is a sequence of CW complexes (CW, not Cy) with additional properties related to their loop spaces. that are typically denoted with a sequence of capital Es. While there are several interpretations of Taz, this one shows how he opens the conversation into an abstract field of math.
Extra Notes: The only stable portion of the work is a parachute slowly delivering an ACME spectrum. The parachute itself can actually be interpreted as a suspension of the 1 sphere? The suspension is integral to a spectrum? To the right we see Bugs Bunny dropping U-235 an unstable Uranium isotope, an element which would make quite a good bomb shell?
