A topic that you might enjoy learning yourself, and which you can certainly present in one or two 90 minutes lectures, is the group law on an elliptic curve. This is treated in an elementary fashion in Miles Reid's book "Undergraduate algebraic geometry" (and probably in many other places).

The advantages are that it is concrete and specific, but also not at all obvious. Furthermore, verifying associativity is quite tricky from an elementary viewpoint. (Reid gives a pretty complete discussion, if I remember correctly. In your talk, you would probably not be able to, or want to, cover associativity completely, though, since the details are quite elaborate. On the other hand, you can write down an explicit elliptic curve, find three explicit points, and explictly add them via the group operation in the two different ways necessart for verifying the associativity; this is always pretty entertaining to watch --- but make sure that you do the computations first, since you surely won't be able to work them out at the board; they will be too messy!)

There is also a very strong connection with the theory of elliptic functions, but you probably wouldn't want to try to fit this into the same lecture. But if you want to learn it yourself, you will want to read about Weierstrass's elliptic functions.

Actually, one thing that comes out of the theory of elliptic functions is that the complex-valued points on the elliptic curve, as a topological group, are isomorphic to $S^1\times S^1$ (a product of two circle groups). So the algebraic description of the group law on the elliptic curve gives a very complicated, but very interesting, way of describing $S^1\times S^1$!

Why do I say "very interesting"? Well, the fact that it lives in the world of algebra gives it a dimension of richness that you can't obtain just by talking about $S^1\times S^1$. For example, there are beautiful connections to the theory of Diophantine equations.

As one example, if you had enough time, you could mention that if the elliptic curve is defined over $\mathbb Q$, then the set of $\mathbb Q$-valued points on the curve is closed under the group operation, and is (by a famous theorem of Mordell) a finitely generated abelian group. (Note: trying to develop tools to determine the precise structure of this group remains one of the central problems in modern number theory. To learn about this, try googling "Birch--Swinnerton-Dyer conjecture".) (Also note: number theorists refer to the set of $\mathbb Q$-valued points of an elliptic curve over $\mathbb Q$ as the Mordell--Weil group of the elliptic curve. If you google Mordell--Weil group, you will find a statement of Mordell's theorem mentioned above, and a lot more besides.)

A topic that you might enjoy learning yourself, and which you can certainly present in one or two 90 minutes lectures, is the group law on an elliptic curve. This is treated in an elementary fashion in Miles Reid's book "Undergraduate algebraic geometry" (and probably in many other places).

The advantages are that it is concrete and specific, but also not at all obvious. Furthermore, verifying associativity is quite tricky from an elementary viewpoint. (Reid gives a pretty complete discussion, if I remember correctly. In your talk, you would probably not be able to, or want to, cover associativity completely, though, since the details are quite elaborate. On the other hand, you can write down an explicit elliptic curve, find three explicit points, and explictly add them via the group operation in the two different ways necessary for verifying the associativity; this is always pretty entertaining to watch --- but make sure that you do the computations first, since you surely won't be able to work them out at the board; they will be too messy!)

There is also a very strong connection with the theory of elliptic functions, but you probably wouldn't want to try to fit this into the same lecture. But if you want to learn it yourself, you will want to read about Weierstrass's elliptic functions.

Actually, one thing that comes out of the theory of elliptic functions is that the complex-valued points on the elliptic curve, as a topological group, are isomorphic to $S^1\times S^1$ (a product of two circle groups). So the algebraic description of the group law on the elliptic curve gives a very complicated, but very interesting, way of describing $S^1\times S^1$!

Why do I say "very interesting"? Well, the fact that it lives in the world of algebra gives it a dimension of richness that you can't obtain just by talking about $S^1\times S^1$. For example, there are beautiful connections to the theory of Diophantine equations.

As one example, if you had enough time, you could mention that if the elliptic curve is defined over $\mathbb Q$, then the set of $\mathbb Q$-valued points on the curve is closed under the group operation, and is (by a famous theorem of Mordell) a finitely generated abelian group. (Note: trying to develop tools to determine the precise structure of this group remains one of the central problems in modern number theory. To learn about this, try googling "Birch--Swinnerton-Dyer conjecture".) (Also note: number theorists refer to the set of $\mathbb Q$-valued points of an elliptic curve over $\mathbb Q$ as the Mordell--Weil group of the elliptic curve. If you google Mordell--Weil group, you will find a statement of Mordell's theorem mentioned above, and a lot more besides.)