Next great mathematical invention

[mryoussef2012]

Hi , I was thinking about this , as you know calculus was used many years before the explicit work of both Newton and Leibniz , it was used the hard way , unrecognized , scattered and buried among unrelated formulas and topics.
The golden question is ; what if there is another revolutionary mathematical "principle" "tool" "method" (call it what you want) lurking on the orizon and living incognito inside the works of mathematicians ?!
Do you have some insights on what would it be like ?
Everyone who works with math , experiences sometimes the "aha" feeling when meeting with some beautiful pattern or similarity or a connection between unrelated stuffs , that would be a great hint for this subject.

[Daedalus]

The golden question is ; what if there is another revolutionary mathematical "principle" "tool" "method" (call it what you want) lurking on the orizon and living incognito inside the works of mathematicians ?! Do you have some insights on what would it be like ?

 

Hi mryoussef2012. To answer your question, there are equations, algorithms, and proofs yet to be found that will have an impact on mathematics and science. Check out the Clay Mathematics Institute's website to see some of the most highly prized problems that are still unsolved to this day. Dr. Grigoriy Perelman has been the only person so far that solved one of these problems. He was awarded one million dollars and bragging rights for proving The Poincaré Conjecture.

 

Formulated in 1904 by the French mathematician Henri Poincaré, the conjecture is fundamental to achieving an understanding of three-dimensional shapes (compact manifolds)

...

James Carlson, President of CMI, said today, "resolution of the Poincaré conjecture by Grigoriy Perelman brings to a close the century-long quest for the solution. It is a major advance in the history of mathematics that will long be remembered."

 

As for what these undiscovered solutions will be like, well, it really depends on the problem and the area of mathematics in which it resides. It could result in new functions or be a proof that provides deeper insight into a particular problem set. The more complicated areas of mathematics will generally have more unsolved problems. However, a major break through might be found rooted in easier areas, but it is highly unlikey because they are well understood. For instance, I developed equations for nested exponentials, which are understood in mathematics, that lead to discovering the nested root and nested logarithm, which might be new. The chance that I am the first person to write a paper regarding these operations is extremely low, and the chance that nested roots and nested logarithms will lead to some profound understanding in mathematics is astronomically lower. However, you never know what advantages and discoveries your work might lead to down the road.

 

Everyone who works with math , experiences sometimes the "aha" feeling when meeting with some beautiful pattern or similarity or a connection between unrelated stuffs , that would be a great hint for this subject.

 

I experience that feeling every time I discover something new for myself. Whether the problem is new or known, there's nothing like cracking a pattern and putting all the pieces of the puzzle together for a solution. The very first problem I solved for myself was the binomial theorem. It was the tenth grade in high school and I was taking Pre-Algebra. The teacher showed us how to expand binomials the hard way, which can take up a lot of paper if the exponent is particularly large. That night while doing my homework I was able to see the pattern for the coefficents in the expanded result and formulated an easy method to work the problem on a single line of paper. I showed the teacher the next day and he let me teach it to the class. My senior year I discovered Newton's interpolation formula for myself because I wanted to find an equation the predicted the summations of [math]x^a[/math]. It took me a year and three months to work the problem, and I was very surprised to discover that the binomial theorem is rooted in the solution. My work on Newton's interpolation formula, although already known, allowed me to get a $2000 scholarship from the University of Oklahoma. That sparked my love of math, and I have been expanding my knowledge and solving problems ever since.

[SamBridge]

 

Hi mryoussef2012. To answer your question, there are equations, algorithms, and proofs yet to be found that will have an impact on mathematics and science. Check out the Clay Mathematics Institute's website to see some of the most highly prized problems that are still unsolved to this day. Dr. Grigoriy Perelman has been the only person so far that solved one of these problems. He was awarded one million dollars and bragging rights for proving The Poincaré Conjecture.

 

I didn't see that goldbach conjecture or whatever it's called that relates all prime numbers in a predictable equation in Ulam's spiral. Has it been solved? I couldn't find solution for it.

[Daedalus]

I didn't see that goldbach conjecture or whatever it's called that relates all prime numbers in a predictable equation in Ulam's spiral. Has it been solved? I couldn't find solution for it.

 

The Goldbach conjecture isn't part of the mellinnium prizes. For a list containing a few more unsolved problems, check out Wolfram's website.

Edited January 24, 2013 by Daedalus

[vbcob]

hi

it is a great wonder of science to next great mathematics invention

because of that math is the based of science technology computer science is the based on mathematics invention because the most common value generate with maths

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electromagnet

[ajb]

The golden question is ; what if there is another revolutionary mathematical "principle" "tool" "method" (call it what you want) lurking on the orizon and living incognito inside the works of mathematicians ?!

Do you have some insights on what would it be like ?

 

Everyone who works with math , experiences sometimes the "aha" feeling when meeting with some beautiful pattern or similarity or a connection between unrelated stuffs , that would be a great hint for this subject.

 

From my own personal experiences, I see lots of mathematical constructions that would be best phrased in terms of supergeometry. This is not really new as supermanifolds have been about since the late 1970s, but there does seem a general lack of awareness. That is apart from experts.

 

For example, Lie algebras and Lie algebroids can be understood in terms of supergeometry. So can the Cartan calculus, various brackets found in algebra and geometry including standard Poisson brackets on a manifold. Sections of vector bundles including exterior powers thereof have nice formulations in supergeometry.

 

The important features here seem to be

1. The antisymmetry provided by odd coordinates
2. The possibility of homological vector fields
3. Being able to include an extra grading called weight to keep track of "algebra" or "linearity".

All of the above are rather "non-classical". Interestingly, by using supergeometry one finds rather natural generalisations of the classical structures.

 

For example, Lie infinity algebras arise in this way as do N-manifolds (supermanifolds with a non-negative weight) which can be interpreted as "non-linear vector bundles" and higher Poisson structures (higher order versions of Poisson structures).

 

So, not strictly new, but it may become more of a trend in the future.

[mryoussef2012]

" supergeometry" ? I woudn't go that far , we have a huge reservoir of less complicated stuff which could still hide some surprises for us , think about prime numbers , they are tackled by everyone but still no one cracked it , may be it needs some new math .

[ajb]

" supergeometry" ? I woudn't go that far , we have a huge reservoir of less complicated stuff which could still hide some surprises for us , think about prime numbers , they are tackled by everyone but still no one cracked it , may be it needs some new math .

So you are interested in number theory?

 

There are some tough tools employed there like algebraic geometry, Galois theory and the Langlands program. In fact, the Langlands program has been quite influencial in recent years with the development of the geometric Langlands program and the links it has with S-duality.

 

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Added: you may find this link interesting http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/supersymmetry.htm

Edited January 24, 2013 by ajb

[mryoussef2012]

Daedalus , I can see clearly how much you love the math

and that joyment is already the nobel prize of mathematicians , I

remember one of my old teachers telling us that there is nothing anymore

to be discovered in mathematics , his statement came after his

disapointement for rejecting his paper as being already discovered by

someone else ; any way I think he missed the real prize , because a

rejected paper is just a new trip in the amzon and a lot of fun... but

elas ! in our days many search for fortune and fame and they try the break-force with machine like memory and mental calculations ... Of course they forget that math and science in general (better word wisdom) are like poetry , if you are not connected at the very deep level , then you just really can't...

[ajb]

I remember one of my old teachers telling us that there is nothing anymore

to be discovered in mathematics , his statement came after his

disapointement for rejecting his paper as being already discovered by

someone else.

There are plenty of classical unsolved problems as well as new mathematical constructions to explore, which bring with them their own problems. I am not sure it is so easy to always judge is a problem is important or not, meaning citations are not always so forthcoming in mathematics.

 

Anyway, rediscovering works known to a small number of experts is just a hazard of research, especially at an early stage of your career. I imagine just about everybody has done it, I know I have. That is why talking to established resaechers and indeed peer review are so important.

[Acme]

I didn't see that goldbach conjecture or whatever it's called that relates all prime numbers in a predictable equation in Ulam's spiral. Has it been solved? I couldn't find solution for it.

 

The Ulam spiral does not relate all primes in a predictable equation.

Ulam_spiral @Wikipedia

 

The Ulam spiral, or prime spiral (in other languages also called the Ulam Cloth) is a simple method of visualizing the prime numbers that reveals the apparent tendency of certain quadratic polynomials to generate unusually large numbers of primes. ...

 

It's an interesting observation however, inasmuch as I have recently discovered/invented another pair of array spirals and mapped the primes on them. In accord with the OP, I think these are new but I leave it to others to determine how great they are. Feel free to discuss "my" arrays here, but please respect my copyright & do not reproduce them elsewhere.

 

These spirals proceed down & clockwise from the origin cell at 1. As the triangle spiral array of primes is not numbered I have also attached a numbered undifferentiated array for clarification.

[SamBridge]

I think there was also some 300 year old problem that Newton braught up with calculating the trajectory of a projectile considering air resistance To me I don't see why it's so hard exactly, there's air pushin back on the ball reducing it's velocity by what seems like no more than a 3rd degree polynomial with force equal to on an atomic scale equal to how much force the atoms of the ball are carrying in individual collisions at any given second. The way I imagine it on a calculator it would be a parametric equation with

x= k(.dy/dx)(ax^2+bx+c)

y= -ax^2+bx+c

where a is 1/2 the gravitational acceleration ad b is the initial velocity and c is the starting height and k is some kind of friction coefficient that models how the air pushes back on the ball according to the ball's velocity over time probably in some kind of difference formula where the variable has a negative coefficient. Normally you'd want the vertical and horizontal components to be separate so I guess maybe I am on the right track but the real answer would be a 3 or 4 variable parametric equation which would have to be modeled by a computer. Which reminds me: are certain great physics achievements only not figured out simply because it takes a lot of processing power to test them since they involve things that cannot be controlled in reality?

On my calculator in standard view it's almost how I want it to look, I just need it to be flipped around, The way I hypothesize that it looks is that it starts off looking like a normal parabola, but then it falls short before it should have if it were a normal parabola, it's x=-.5t^2 and y=-t^2+6t+3. It's almost the way I want it except I want it to flipped about a vertical axis, I use to be able to make that happen when I was playing around on my calculator but I don't remember how to do it.