a coulpe of home experiments about #inertia
from Science Comics #2
Riemann-Roch Theorem: The final cry: The diagram is commutative! To give an approximate sense to the statement about f: X → Y, I had to abuse the listeners' patience for almost two hours. Black on white (in Springer lecture notes) it probably takes about 400, 500 pages. A gripping example of how our thirst for knowledge and discovery indulges itself more and more in a logical delirium far removed from life, while life itself is going to Hell in a thousand ways and is under the threat of final extermination. High time to change our course!Alexander Grothendieck about the Grothendieck–Riemann–Roch theorem via Math 245
Aidan Dwyer, was one of twelve students to receive the 2011 Young Naturalist Award from the American Museum of Natural History in New York for creating an innovative approach to collecting sunlight in photovoltaic arrays. Dwyer’s investigation into the mathematical relationship of the arrangement of branches and leaves in deciduous trees led to his discovery that these species utilized the Fibonacci Sequence in their branch and leaf design. Dwyer transformed this organic concept into a photovoltaic array based upon the Fibonacci pattern of an oak tree and conducted experiments comparing his design to conventional solar panel arrays. After computer analysis, Dwyer discovered that his Fibonacci tree design surpassed the performance of conventional methods in sunlight collection and utilized the greatest quantity of PV panels within the least amount of physical space, making it a versatile and aesthetically pleasing solution for confined and obstructed urban areas.
A professor of art history from the future travels by time machine some centuries into the past in search of an artist whose works are celebrated in the professor's time. On meeting the artist in the flesh, the professor is surprised to find the artist’s current paintings talentlessly amateurish. The professor happens to have brought with him from the future a catalogue containing reproductions of the paintings later attributed to the artist, which the professor has come to see are far too accomplished to be the artist's work. When he shows this to the artist, the latter quickly grasps the situation, and, by means of a ruse, succeeds in using the time machine to travel into the future (taking the catalogue with him), where he realizes he will be welcomed as a celebrity, so stranding the professor in the "present". To avoid entanglements with authority the critic assumes the artist's identity and later achieves fame for producing what he believes are just copies of the paintings he recalls from the catalogue. This means that he, and not the artist, created the paintings in the catalogue. But he could not have done so without having seen the catalogue in the first place, and so we are faced with a causal loop.
the physical structure of the inhabitants had advanced millions of years compared to ours. Reaching maturity, people of that race earned a titanic force!
Maryam Mirzakhani has made several contributions to the theory of moduli spaces of Riemann surfaces. In her early work, Maryam Mirzakhani discovered a formula expressing the volume of a moduli space with a given genus as a polynomial in the number of boundary components. This led her to obtain a new proof for the conjecture of Edward Witten on the intersection numbers of tautology classes on moduli space as well as an asymptotic formula for the length of simple closed geodesics on a compact hyperbolic surface. Her subsequent work has focused on Teichmüller dynamics of moduli space. In particular, she was able to prove the long-standing conjecture that William Thurston's earthquake flow on Teichmüller space is ergodic.Riemann surfaces are one dimensional complex manifolds introduced by Riemann: in some sense, his approach is a cut-and-paste procedure.
Mirzakhani was awarded the Fields Medal in 2014 for "her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces".
He imagined taking as many copies of the open set as there are branches of the function and joining them together along the branch cuts. To understand how this works, imagine cutting out sheets along the branch curves and stacking them on top of the complex plane. On each sheet, we define one branch of the function. We glue the different sheets to each other in such a way that the branch of the function on one sheet joins continuously at the seam with the branch defined on the other sheet. For instance, in the case of the square root, we join each end of the sheet corresponding to the positive branch with the opposite end of the sheet corresponding to the negative branch. In the case of the logarithm, we join one end of the sheet corresponding to the $2 \pi n$ branch with an end of the $(2n+1) \pi n$ sheet to obtain a spiral structure which looks like a parking garage.A more formal approach to the construction of Riemann surfaces is performed by Hermann Weyl, and the work by Maryam Mirzakhani puts in this line of research.