Today, I gave an introductory survey on the Langlands program to a group of mathematicians and physicists at the Simons Center for Geometry and Physics at Stonybrook University.
Many of these connections are well-illustrated on the LMFDB. I highly suggest poking around and clicking on things — on many pages there are links to related objects.
Part of the Langlands program includes the modularity conjecture for elliptic curves over $\mathbb{Q}$. On the LMFDB, this means that we can go look at elliptic curves over $\mathbb{Q}$, take some arbitrary elliptic curve like \begin{equation*} y^2 + xy + y = x^3 - 113x - 469, \end{equation*} (which has this homepage in the LMFDB), and then see that this corresponds to this modular form on the LMFDB. And they have the same L-function.
During the talk, Brian gave an example or an Artin representation of the symmetric group $S_3$ on three symbols. For reference, he pulled data from this representation page on the LMFDB.
From one perspective, the Langlands program is a monolithic wall of imposing, intimidating mathematics. But from another perspective, the Langlands program is most interesting because it organizes and connects seemingly different phenomena.
It's not necessary to understand each detail — instead it's interesting to note that there are many fundamentally different ways of producing highly structured data (like $L$-functions). And remarkably we think that every such $L$-function will behave beautifully, including satisfying their own Riemann Hypothesis.
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