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.