Ramanujan, The Langlands Project, and Fermat's Last Theorem [Math Mondays]
Even you non Math nerds don't want to miss this
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Hola my amazing amigos,
Today, I’m going to cover what is probably the most important project in Math. This might be my Math nerd bias speaking, but to me, this is the single biggest idea in the whole world. Progress in this field will quite literally impact every part of the world. When you watch the linked video at the end, you’ll see I’m not exaggerating.
Even if you don’t want to get into math research, it is crucial to know more about this project. Ready to learn more? Let’s go.
Points to note
Here are a few important details-
What is the Langlands Project- For the fancy words, I’ve quoted the oversimplified version below. To the rest of you, the project seeks to connect Geometry with Algebra. This is a really big deal.
Simple Example- Take the following infinite summation-
This can be used to approximate the sin function. Let that sink in. By adding a bunch of simple algebraic functions, we end up with a periodic trigonometric function. This is completely unexpected since none of the individual functions are periodic. Using the properties of the sin function we can actually make a lot of optimizations in our calculations. And our computers use algebraic approximations to compute the values of sin/cos/other periodic functions. If you’re not losing your mind, go over what I just wrote again. We solved an Amazon question using periodic functions here
Why is this so important to mathematicians- For much of math history these fields developed largely independently. However, when mathematicians have noticed links between the two show up a lot. One such link was used to solve Fermat’s Last Theorem. Fun Fact- The man who solved this won almost a million dollars and lots of fame. This is how important it is.
Other links are being applied to Ramanujan’s Conjectures and many other advanced Math ideas that important to this world. The way people are trying to merge algebra and geometry is why this project is called "a kind of grand unified theory of mathematics."
Why you should care- Say you have no aspirations of getting into Math Research. There’s still a big reason that you should know about this. This project will quite literally impact every aspect of this world. Just knowing about the project and keeping tabs on it will allow you to take advantage of the advancements in the field.
How to Proceed- The first step is building your basic knowledge. This post covers all the math topics that everyone must know. That will go over the Math topics you should know, how much you’ll need and the best resources. To those of you that have struggled with learning Math, use this system that I’ve used to teach myself complex Math topics. After that set up your Google Alerts/News platforms to stay updated on this topic. For most of you, this is enough. To those interested in looking into this project more deeply, reach out to me on Instagram. We can discuss the specifics there.
Closing Thoughts- I’ve said it many times in this newsletter and my other content- for the best results, you need to have a strong base in multiple domains. Sometimes, the problem you’re trying to solve can be simplified by techniques/ideas in other fields. This story of who got 5 promotions in 4 years is a good example. This post on Generating Functions also combines ideas in multiple domains. Combining different skills also allows you to skyrocket your earning potential. I was able to gain my first 100 USD per hour contract using that. Read more about that here. The Langlands project is another example of this principle. Combining seemingly unrelated things in Algebra and Geometry leads to extremely valuable insights.
To oversimplify, the fundamental lemma of the project posits a direct connection between the generalized fundamental representation of a finite field with its group extension to the automorphic forms under which it is invariant. This is accomplished through abstraction to higher dimensional integration, by an equivalence to a certain analytical group as an absolute extension of its algebra.
-From the Wikipedia Page. Just in case you were wondering what reading Math Research looks like
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