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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