4.2 Harmonic Analysis Review

summary

We can observe symmetry in harmonic analysis.

The basic harmonics are $\sin(x)$ and $\cos(x)$. They are invariant under the shift$$x\longrightarrow x+2\pi$$
The group of symmetries is thus$$x\longrightarrow x+2\pi M$$where $M$ is an arbitrary integer.
The same is true for $\sin(nx)$ and $\cos(nx)$, where $n$ is an integer.
More generally, we have a group acting on a space (such as the group of translations acting on a line).
We can express general functions as superpositions of harmonics, like the sound of a symphony.
Extending harmonic analysis beyond a group acting on a line, we can imagine a group acting on a unit disk on a plane. The special linear group that acts here is called $\textrm{SL}(2,\mathbb{Z})$—characterized as the group of linear transformations that preserves area and orientation in two-dimensions.
The harmonics are called modular forms in this case.

The Langlands program has made concrete connections between continents of mathematics.

A large class of questions about the Galois groups can be answered using harmonic analysis.
A problem in number theory might involve counting numbers of solutions of some equations. We can solve them on the computer, generating a sequence of numbers that at first seem completely random and chaotic.
Langlands reformulated such problems in the language of harmonic analysis, where we can begin to see patterns emerge.

The Langlands program was essential in proving Fermat's Last Theorem.

First conjectured by French mathematician Pierre de Fermat in 1637, the theorem quickly became one of the greatest unsolved problems in mathematics.
Fermat’s Last Theorem states that no positive integers $a$, $b$, and $c$ can satisfy$$a^n+b^n=c^n$$for any integer value of $n$ greater than 2.
For 358 years, the theorem remained unproved, until Andrew Wiles published a complete and successful proof in 1995.
Wiles built off a link between the modularity theorem and Fermat’s Last Theorem, established by Ken Ribet in 1986. Ribet showed that Fermat’s Last Theorem followed from a special case of the Langlands program involving modular forms.

Many similar unsolved conjectures still remain.

Initially, Langlands was interested in connecting number theory and harmonic analysis, to find deep insights to both fields. The Langlands program has since propagated to other areas of math and physics, including geometry and quantum physics.
Electromagnetic duality turns out to be closely connected to the Langlands program. This duality concept shows how an expression of the electric field will have a directly analogous expression of the magnetic field. With special relativity, the electric field is a Lorentz transformation of the magnetic field, and vice versa.
This is yet another example of a unifying connection in mathematics, one that emerges from the mathematical framework of electromagnetism.
While we have made tremendous progress in math and physics, we still search for a unified theory of everything. The Langlands program is one way that unifications can be established, and the veil of the unknown can be lifted slightly further.