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From Notices of the AMS

A Fruitful Interaction Between Algebra, Geometry, and Topology: Varieties through the Lens of Group Actions

A pink figure with two smaller figures beneath it. $X$ has an open covering $\{U_j\}_{j \in I}$ and homeomorphisms $\{\varphi_j:U_j\to V_j\subset \mathbb{C}^n\}_{j \in I}$ such that for $i,j \in I$, one has either $U_i\cap U_j=\emptyset$, or $\varphi_i$ and $\varphi_j$ are required to satisfy the condition displayed in this picture.

by Rubí E. Rodríguez
Anita M. Rojas
Communicated by Han-Bom Moon

Beautiful theories have emerged from mixing different areas of mathematics, when building bridges between areas provides the right paths to achieve the comprehension of a certain subject. Mathematicians know that if an object has several underlying structures, then the answer to some question considering one aspect of the object may come from another aspect of itself. A remarkable example of this is Descartes's use of coordinates; this milestone was the key to translate questions from geometry to algebra. But there are many other examples: Galois theory, algebraic geometry, mathematical physics, and the list goes on and on.

In these notes we will discuss how the knowledge about group algebras and representation theory has proved fruitful in understanding fundamental questions about complex abelian varieties, compact Riemann surfaces, and their moduli spaces.

Let us begin by roughly introducing these objects. Abelian varieties and compact Riemann surfaces are complex manifolds; that is, (connected, Hausdorff) topological spaces that locally look like open subsets of $\mathbb{C}^n$ for some $n$---plus a technical condition better explained in Figure 1---and that also share the property of being varieties.

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