गणित का ऐसा सवाल जिसको हल करने पर मिलेंगे 1,000,000 $ Hodge Conjecture million dollar question

Mathematicians' compulsion to make things more and more complex is both a blessing and a curse. Their urge to take an idea and stretch it as far as possible can yield fascinating new insights. The downside is that as the mathematics becomes more abstract and gains power to describe huge swathes of conceptual knowledge, it becomes harder and harder to describe in words.

So it is with a heavy head that I turn the focus of this series on the Millennium Prize Problems to the Hodge Conjecture. It's an amazing intersection of various fields of mathematics, but a pain in the torus to summarise.
Humans have been studying the mathematics of shapes since well before a triangle first caught Pythagoras's eye around 500 BC. Over the generations, more and more complicated shapes were studied until, about two thousand years later, it looked like they were running out of steam. Mathematicians had done all they could think of with shapes, and along the way provided the basis for everything from engineering to perspective painting. Then, in 1637, a bright young mathematician-philosopher realised that if you abstracted it one step further, geometry was actually the same as algebra.

Using the the Cartesian coordinate system that now bears his name, Descartes did a lot of thinking about how a geometrical line was just a set of numbers. Equations can also produce a set of numbers as their solutions. If both of these sets of numbers were exactly the same, then a line drawn on a piece of paper could be considered to be the same thing as the solution to an equation.

This was a watershed moment in mathematics that allowed all the tools developed in algebra to be applied to geometry. It's why your school mathematics teacher got so excited about converting linear graphs into equations: any random line can be thought of as the set of solutions to an equation like y = mx + c. Any circle is the set of solutions to (x - a)2 + (y - b)2 = r2. Now if you want to see where a certain line crosses a particular circle you could either draw out the shapes geometrically or just compare the equations algebraically. Both methods will give the same answer.

Mathematicians were not content to stop at lines and they quickly found that more complicated equations, or even sets of equations all working together, could produce amazing shapes in all sorts of dimensions. Some could still be visualised as shapes – such as the equations whose set of solutions map out the surface of a ring, known as a torus – but many of them were beyond what we can picture and only accessible by algebra and a very stretched imagination.

As mathematicians were now dealing with objects beyond what we can visualise, these "shapes" became known in general as "algebraic cycles". If an algebraic cycle was a nice smooth and generally well-behaved shape, it also earned the title of "manifold".

Two things then happened at once. First: a group of mathematicians know as topologists started looking at what happens if you draw shapes on a manifold. You could imagine that you have a ring doughnut and you draw a triangle right around the top (see picture above). Or maybe a pentagon.

Actually, do you need both? If the shape could slide and stretch then the triangle could be distorted into the pentagon. Topologists grouped all of the shapes that could be distorted from one to the other (without being lifted off the manifold surface) into a "homology class" – a kind of generalised shape. All the shapes that go through the "hole" of the doughnut would form a different homology class.

Second, a group of mathematicians who called themselves algebraists started taking sets of equations that already produced nice tidy manifolds and adding more equations. These additional equations produced new algebraic cycles within those manifolds.
It wasn't long before people realised that topologists drawing homology classes onto manifolds and algebraists embedding algebraic cycles into manifolds was actually the same thing. It was a repeat of when geometric shapes first met algebraic equations. The difficulty was that no one knew for sure when a homology class on a manifold contained at least one shape that was also describable as an algebraic cycle.

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गणित का ऐसा सवाल जिसको हल करने पर मिलेंगे 1,000,000 $ Hodge Conjecture million dollar question

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