The heck is a modular form?

January 18, 2013 10:47 AM

For a work assignment, I need to come by a conceptual understanding of modular forms that's light on jargon and, ahem, actual math. If such a thing is possible.
posted by Zerowensboring to Science & Nature (12 answers total)

I know modular forms from number theory, algebraic topology and string theory. Maybe you should tell us what kind of work you do and a little bit more about the assignment.
posted by ubiquity at 11:16 AM on January 18, 2013 [2 favorites]

Response by poster: I'm a writer who's done some science writing in the past, so I'm not inimical to technical stuff. This piece is going to be about number theory. Does that help?
posted by Zerowensboring at 11:39 AM on January 18, 2013

As ubiquity notes, more information on your background is needed to give a good answer to this question. Unfortunately, since the very term "modular form" is itself mathematical jargon, there's no possible explanation that totally avoids jargon or actual math. The Wikipedia article gives a good introduction, and there's also a nice readable overview in the Princeton Companion to Mathematics (which you can see on Google books by doing a search for "modular forms" in that book).
posted by Frobenius Twist at 11:44 AM on January 18, 2013

Response by poster: Frobenius, ubiquity, some math and jargon would be fine, but I don't have anything approaching a math degree. Surely there's a way to give a vague idea of their significance through analogy or hand-waving?
posted by Zerowensboring at 12:08 PM on January 18, 2013

You still haven't given a helpful explanation as to what kind of work this is. You are writing a "piece" about number theory? Does that mean you are a journalist? Are you writing about a number theorist who specializes in modular forms? Do you need to explain their significance to the readers of a general magazine so they understand why the number theorist is doing important work? If so, who is the mathematician in question? This is the sort of information that would be helpful.

That's a very different kind of work assignment than if you were supposed to, say, explain the significance of modular forms to calculating the value of financial securities to your company's chief financial officer.
posted by grouse at 12:40 PM on January 18, 2013 [1 favorite]

Response by poster: As I said, I'm a writer and I'm writing an article (we call em pieces) about a number theorist who works with modular forms. It's a general audience. I'm not looking for applications of modular forms, but some way of understanding what these functions do and why they're interesting. For example, if I asked what an integral was, or a cosine, I feel like those would be pretty easy to understand in a vague, non-technical way. Maybe this isn't the case for modular forms...
posted by Zerowensboring at 1:02 PM on January 18, 2013

I'd give a very different explanation of an integral to someone who is trying understand how it is used to calculate the volume of a wine barrel than to someone trying to understand how work is done by a spring. Context is important especially for a non-technical answer, but you don't seem to want to give it.
posted by grouse at 1:12 PM on January 18, 2013

There's a chapter in Simon Singh's book Fermat's Enigma [US title] where he goes into the Taniyama-Shimura Conjecture. Proving the conjecture, that modular forms are related to elliptical equations, was vital to the proof of Fermat. I am no mathematician - though I did take calculus in high school - and I remember the chapter being understandable enough to get the gist. Some info from his site.
posted by expialidocious at 1:24 PM on January 18, 2013

Response by poster: Grouse, I'm not trying to withhold information about context. Thanks for your response, though. I'll look into the Singh book, Expi. Thanks!
posted by Zerowensboring at 1:28 PM on January 18, 2013

This mathoverflow question "Why are Modular Forums interesting?" is quite helpful. You might also look at these two papers [1, 2] (even just look at the first page or two of each, which gives a helpful overview). Also see this list of Applications of Modular Forms.

I somehow missed Modular Forms entirely in my former life as a math grad student, but if trying to get the gist of the concept across to a nontechnical readership, this sentence from the first paper seems to be helpful:

The theory of modular form originates from the work of C.F. Gauss of 1831 in which he gave a geometrical interpretation of some basic notions of number theory.

So if trying to define or explain Modular Forms to a nontechnical audience, it might go something like this:

Modular Forms are a special type of function that allows mathematicians to find deep and useful links between widely different fields of mathematics--complex analysis, number theory, group theory, topology, algebra, geometry, differential equations, string theory, cryptography, and others. They were invented in the 1830s by mathematician Carl Frederick Gauss, who was working to interpret the difficult and abstract concepts of number theory in a geometric way. Modular Forms have been key to the solution of a number of difficult problems in mathematics--most famously, Fermat's Last Theorem.

posted by flug at 4:44 PM on January 18, 2013 [3 favorites]

God, gauss was a different life form! We are still dealing with his great ideas from centuries back. I have nothing to offer, but there are a few lay books on Fermat that might help. As with anything math oriented, a lay explanation will be technically (necessarily?) inaccurate, misunderstood, quickly forgotten. Maybe aim for the gross 10% of it and call it a wrap? Good luck. I'd love to read what you eventually produce.
posted by FauxScot at 9:33 PM on January 18, 2013

Best answer: I'll give a hand-wavey explanation that may or may not be useful.

Suppose you want to build a house. You want to know what size pieces of wood you need. Rather than cutting out lots of wood pieces in various sizes and holding them up next to each other, you would probably draw a picture of a house and measure the lines and angles. Or, equivalently, you'd create a house in a 3d modeling program and use that to derive the sizes of wood you need.

The picture of the house and the 3d model of the house are representations of the real house. The lines on the page, and functions in the 3d modeling program, have some properties that are equivalent to pieces of wood in the real world. Therefore one can be used to represent the other, and it is easier to manipulate lines on a page or pixels than it is to manipulate wood.

Math equations are often used to represent objects in the real world. In that case, there is a mathematical object that represents a real world object. However, you can also use one type of mathematical object to represent another type of mathematical object. For instance, you might have an algebraic equation that represents a geometric shape. You might have a type of polynomial equation that is equivalent in certain ways to a topological group. Etc.

A modular form is one type of mathematical object. Described briefly, it is a function over a lattice of points in the complex plane, a.k.a. a lattice of imaginary numbers. A modular form is not quite that simple... it also has some other various properties about what it does at the boundaries where it reaches infinity, and how quickly its values change, etc.

Modular forms are sometimes used to represent other types of objects in mathematics. Modular forms are actually pretty versatile in that they can represent many other mathematical objects. Why would you represent one mathematical object as another? Sometimes a problem that is difficult to solve in one representation is easy to solve in another. Just like it is easier to move around lines on a page than it is to move around pieces of wood in the real world to build a house, sometimes when you switch a mathematical problem from one representation to another, a solution jumps out at you. One way this can happen is if there are already theorems that people have worked out in the new representation, you can use all of those theorems, like tools in your toolkit, to solve the problem -- as long as you can convert your problem to that new representation.

Modular forms are applicable in a lot of domains. That is, mathematical equations in a lot of domains have representations as modular forms. When you convert some mathematical equation to the corresponding/equivalent modular form, interesting features of the equation (or group of equations...) can jump out at you. There is a good list of examples at that mathoverflow page linked by flug.

Modular forms gained some press in the last decade when they were an essential part of proving Fermat's Last Thereom. Proving Fermat's Last Theorem involved switching the original problem to a few different types of representations at various points in the proof. These types of representations, or mathematical objects, included elliptic curves, modular forms, and Galois spaces. In each of those representations, certain properties of the original equation could be established, and when put together they led to the final proof.

Nobody really thought that modular forms would be essential in proving Fermat's Last Theorem when they were studied in the 1950's and 1960's. It was sort of a surprise that they were an essential part of the proof. Overall, modular forms are considered a somewhat beautiful mathematical structure that has usefulness in a number of different domains.

That is my very hand wavey explanation!
posted by kellybird at 10:22 AM on January 19, 2013 [4 favorites]