[Masslessxphoton]

What is the hardest math course one can take?
What is the hardest math problem ever?

[JTG]

I think the highest level math classes are relevant to physics? I'm not sure. I know calculus has many different flavors, and some are among the final classes taught at collegiate level.

I'm not sure if there is a "hardest" so much as there are several very high level maths specialized for use in various fields.

We have a bunch of math fiends here on the forum though, and i'm sure some of them will be able to shed more light.

[Innovate]

Inb4 Monte314 !! Ruff Ruff
To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.


It probably has something to do with a proof relating to quantum mechanics or a structure that is hard to comprehend let alone solve.

[Monte314]

Each area of mathematics has its collection of "hard" problems... e.g., long-standing problems that have not been solved. I'm not sure how one would determine which of several unsolved problems is "hardest", since you don't really understand the difficulty until you have plumbed it by producing a solution.

Interestingly enough, the area of mathematics that has over and over again produced very difficult problems is Number Theory. The motivation for advancement in other areas of mathematics (particularly in real analysis, complex analysis, and abstract algebra) has been to obtain solutions to problems in Number Theory.

Three of the most well-known unsolved problems still around are:

1.) Is the Riemann Conjecture true (Number Theory, complex analysis)?
2.) Is the class P equal to the class NP (computer science)?
3.) Is Goldbach's Conjecture True (Number Theory)?

The Clay Institute offers a $1 Million prize for a solution to each of seven math problems, called the Millennium Prize Problems. Read about them at:


To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.

[roninpro]

I guess I'm always one of the first to make this correction, but there isn't a "hardest math". There are many different areas of math out there, and they all have their demons.

Right now, you can find a list of some of the most important and difficult problems from several fields in mathematics (and one from computer science, if you draw a distinction): this list is called the Millennium Prize Problems. (See Wikipedia:
To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
.) The problems are so difficult and so important that solving one will net a one million dollar prize! (I'm personally interested in the P vs. NP problem and the Riemann Hypothesis, though I don't yet plan to try to resolve them! They have huge consequences for computational number theory.)

[PlatoHagel]

	There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world.— Nikolai Lobachevsky

The
To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
is most interesting in terms of what can be discover by attempts of different individuals in trying to make the proof.


To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.

William Thurston of Cornell, the author of a deeper conjecture that includes Poincaré’s and that is now apparently proved, said, “Math is really about the human mind, about how people can think effectively, and why curiosity is quite a good guide,” explaining that curiosity is tied in some way with intuition. “You don’t see what you’re seeing until you see it,” Dr. Thurston said, “but when you do see it, it lets you see many other things.” To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.

To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts. -For a long time, physicists have struggled with perplexing “meta-questions” (my phrase): Does God play dice with the universe? Does a theory of everything exist? Do parallel universes exist? As the physics community is acutely aware, these are extremely difficult questions and one may despair of ever finding meaningful answers. The mathematical community has had its own meta-questions that are no less daunting: What is “truth”? Do infinitesimals exist? Is there a single set of axioms from which all of mathematics can be derived? In what many consider to be on the short list of great intellectual achievements, Frege, Russell, Tarski, Turing, Godel, and other logicians were able to clear away the fog and sort these questions out. The framework they created, mathematical logic, has put a foundation under mathematics, provided great insights and profound results. After many years of consideration, I have come to believe that mathematical logic, suitably extended and modified (perhaps to include complexity theoretic ideas), has the potential to provide the same benefits to physics. In the following remarks, I will explore this possibility.

[Zodd]

Monte, why couldn't you do what this 16 Indian could? Physics problem of calculating how a ball bounces of a wall, in 350 years no one could do it.


To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.


His formula still hasn't been reviewed, though, as far as I know.

[Monte314]

I'd never heard of the problem before this story appeared. I still haven't seen a formal statement of it.

[InfiniteLoop]

Hardest math class period? I haven't a clue, but most calculus courses are up there. I've heard Non-Finite Calculus is awful.

Hardest class I ever took? Precalculus. My professor was a real hard-ass and it's only because I'm good at trigonometry that I passed. With a D-. I'm a math tutor, and I got A's in Statistics, Algebra, and Geometry. X_X

[Kisai]

Harvard claims to have
To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
.

Its a year long Honors level Abstract Algebra & Real/Complex Analysis course. It takes about 24-60 hours per week for students to do the problem sets.

In the class of 1970, only 20 of the 75 students who began the class finished it due to its difficulty. Similar drop-out rates were true for the class of 1976: "Seventy started it, 20 finished it, and only 10 understood it." and for the class of 2009: "51 students the first day, 31 students the second day, 24 for the next four days, 23 for two more weeks, and then 21 for the rest of the first semester."

Bill Gates managed to pass it in his first year.

[thod]

Maths consists of entities, represented by Greek letters, their relationships and some operators for manipulating them. It is deterministic and regular. You do not need to understand what an operator is for in order to follow the rules of its application.

The hardest area is not the one that throws up the most unsolved problems. It is the one that is hardest to understand overall. Although one mathematician may be just as bright as another, he cannot get his head around certain fields of study. Thus mathematicians tend to talented in one area of mathematics and not another and study does not give that talent. This would seem to suggest that unusual brains are necessary to be at the top of a given field.

Thus there is no hardest field since one can only be 'odd' in a certain way. One could argue that the hardest field is the one which the fewest people are capable of understanding.

[Latro]

	Originally Posted by thod To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	It is deterministic and regular.

This is deceptive--given axioms, the actual theorems implied by the axioms are deterministic and regular, but the choice of the axioms themselves is driven by external factors, and the means by which mathematicians write proofs involve a great deal of intuition and similar "non-deterministic" ways of thinking.

[Doggzilla]

	Originally Posted by Kisai To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	Harvard claims to have To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts. . Its a year long Honors level Abstract Algebra & Real/Complex Analysis course. It takes about 24-60 hours per week for students to do the problem sets. Bill Gates managed to pass it in his first year.

Oh God, Dogg want!

[roninpro]

	Originally Posted by thod To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	Maths consists of entities, represented by Greek letters, their relationships and some operators for manipulating them. It is deterministic and regular. You do not need to understand what an operator is for in order to follow the rules of its application.

I take issue with your statement; it is an incorrect representation of what mathematics is all about and where it comes from. You'd be incredibly hard-pressed to find somebody who produces meaningful work who also works directly from axioms. The fact of the matter is that mathematics came before formal axioms. People originally used (and still use) some kind of physical, geometric, or numerical intuition to produce results in every field.

	Originally Posted by thod To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	Although one mathematician may be just as bright as another, he cannot get his head around certain fields of study. Thus mathematicians tend to talented in one area of mathematics and not another and study does not give that talent.

I don't think that this is true either. The best researchers are the ones who have their hands in many different fields. We need all of the tools we can get to solve the most important problems.

[Latro]

Originally Posted by roninpro To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.

I take issue with your statement; it is an incorrect representation of what mathematics is all about and where it comes from. You'd be incredibly hard-pressed to find somebody who produces meaningful work who also works directly from axioms. The fact of the matter is that mathematics came before formal axioms. People originally used (and still use) some kind of physical, geometric, or numerical intuition to produce results in every field.

There are some exceptions to this now, but yeah, axioms almost never come first in practice. Where the exceptions arise (category theory comes to mind), the external factors come into play in the *objects* under study (in the example of category theory, the objects are of course categories, with most categories of interest being (portions of) classes of mathematical structures like sets or vector spaces).

[Zsych]

I think the working assumption should be that if you're not getting something that can be logically understood, you're not thinking about it right and there's something missing within the knowledge you are using to understand that thing.

... So as long as you followed the right path of learning, stuff shouldn't really be hard.

[Senhor Jose]

As a current undergrad in math... I'd say my first class in real analysis was my hardest math class. The combination of the course's timing (it's often your first really rigorous math class) and inherent difficulty make it, well, tough.

Speaking more hypothetically, algebraic topology and category theory come to mind as some of the more abstract (==> difficult? maybe.) stuff that's out there, but then I have no experience with either so I'm just dropping names
To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.


There are many, many branches of math that are actively being expanded by very smart individuals, so any one of those subjects can be about as hard as you want it to be.

[nacht]

	Originally Posted by InfiniteLoop To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	Hardest math class period? I haven't a clue, but most calculus courses are up there.

Most calculus classes are really not that difficult by any stretch of the imagination. Some advanced topics (e.g., complex analysis) get tricky, but those aren't "calculus courses" so much as "courses that use calculus." Let alone topics such as PDEs, decision theory, probabilistic graphical models, or linear vector spaces.

[stoopidkitty]

	Originally Posted by Zodd To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	Monte, why couldn't you do what this 16 Indian could? Physics problem of calculating how a ball bounces of a wall, in 350 years no one could do it. To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts. His formula still hasn't been reviewed, though, as far as I know.

Upon review, it was determined that the kid did not provide a full solution or discover anything previously unknown to mathematicians. Apparently he does have an exceptional grasp of relevant concepts for his age, but he did overlook some things that are not so commonly taught in all undergrad courses. The formula review article (by two German professors) was posted on reddit a while back. I actually read it, although the math itself went way over my head.

[scorpiomover]

	Originally Posted by Masslessxphoton To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	What is the hardest math course one can take?

For me, it was stats. Still don't understand how so many people find it so easy. Analysis was a breeze. They should teach Hahn-Banach spaces in primary school. Would make everything much easier to understand.

	Originally Posted by Masslessxphoton To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	What is the hardest math problem ever?

At the moment? Probably solving the Continuum Hypothesis. That's supposed to be impossible to prove.

[Latro]

	Originally Posted by scorpiomover To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	At the moment? Probably solving the Continuum Hypothesis. That's supposed to be impossible to prove.

Erm...no, not really. CH is "solved" to the extent that it can be "solved", which as it happens is not very much. You may have been speaking in a tongue-in-cheek manner, however.