I have put together a geometric method to compute the sum of the power p of the first n integers. Do you know of previous articles? A more general framework? Suggestions of applications?
I carve an hypercube into shells of thickness 1, and I repeat until nothing remains. See here https://www.researchgate.net/publication/363484666_A_Geometric_approach_to_Sums_of_Powers_of_Integers_-Faulhaber's_formula_from_Partitions_of_Squares_Cubes_Hypercubes#fullTextFileContent
https://www.quantamagazine.org/with-a-new-shape-mathematicians-link-geometry-and-numbers-20210719/
The work is a collaboration between Laurent Fargues of the Institute of Mathematics of Jussieu in Paris and Peter Scholze of the University of Bonn. It opens a new front in the long-running “Langlands program,” which seeks to link disparate branches of mathematics — like calculus and geometry — to answer some of the most fundamental questions about numbers.
Their paper realizes that vision, giving mathematicians an entirely new way of thinking about questions that have inspired and confounded them for centuries.
It’s some kind of wormhole between two different worlds,” said Scholze. “They really just become the same thing somehow through a different lens.”
(Citation from the source).
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This might be of interest to you, dear Renaud Di Francesco
One way that algebra and geometry can be related is through the use of equations in graphs. We can plot a set of points (x, y) according to an equation (for example, the line graph on the left!) to form a graph. That's one way that algebra is related to geometry.
https://susanleesensei.weebly.com/the-relationship-between-geometry-and-algebra.html
Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra.
https://en.m.wikipedia.org › wiki
Combinatorics - Wikipedia
https://www.researchgate.net/publication/363484666_A_Geometric_approach_to_Sums_of_Powers_of_Integers_-Faulhaber's_formula_from_Partitions_of_Squares_Cubes_Hypercubes,kindly check this RG link.
In reality the interplay between geometry and numbers dates back to the time Pythagoras and Diophantus. There was a major advancement by Alexander Grothendieck. The Grothendieck program established the close relationship between algebraic number theory, topology, complex analysis and category theory. His construct of a the concept of Scheme provide an environment for this unity and a Scheme requires the concept of a sheaf to be defined. Jean-Paul Serre's famous GAGA theorem which generalized Chow's theorem to Schemes with applications to number theory was one of the crown jewels that came out of Grothendieck's approach.
https://en.wikipedia.org/wiki/Algebraic_geometry_and_analytic_geometry
The geometry of elliptic curves led to the proof of Fermat's last theorem by Wiles in 1992. Again the interplay of algebra, geometry and number theory. The key in most cases is to be able to define the appropriate topology so the concept of limits and continuity make sense. The p-adic topology is such an example. The Zariski topology is another example. The open sets defined by maximal ideas in K[V] is another such where V is a variety over K is another example. In general, however, it is not Euclidean that is important but geometry in the projective space.
Independent if we realize it or not the concept of category has paved the way for transitioning a problem in one domain into a different problem in a different domain domain where the solution may be easier. For example by using sheaf cohomology a problem in say overdetermined systems of differential equations on an analytic space can be transitioned to a question of calculating specific sheaf cohomology group which in many cases on many types of spaces are equivalent to CW or singular cohomology which are known or easy to calculate.
W. L. Chow in our many late night sessions while I was in grad school used to love to quote Paul Erdos - "Problems worthy of attack prove their worth by fighting back." The problems in number theory surly establish that.
Who is Paul Endros - one of the most prolific mathematicians of all times and one of the most unique characters. He was so prolific that there is a number, Endros number, to track his tree of influence.
Many thanks Truman Prevatt for this broad and deep tour of mathematics and its multiple facets. Appreciated!
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