DRP 2025 Spring Recap

I wanted to recap my sixth and final semester as part of UPenn’s Directed Reading Program. During the semester, the general goal was to learn about Mapping Class Groups, a nice tool from geometric topology. Similar to algebraic topology, mapping class groups involve assigning certain algebraic structures (groups) to topological spaces, generating an invariant of the space. A central theme of problems in topology is distinguishing topological spaces apart. In a point-set topology class, we learn some topological properties that do so. For example, properties preserved under continuous mappings, such as compactness or being simply-connected, are fundamentally studying the structure of the open sets of a space.

Spaces in mathematics is actually not a well-defined term. Informally, it refers to some sort of set equipped with a structure. For example, topological spaces capture general notions of points being “near” or “far” away. A more rigid notion of spaces might be metric spaces, which come equipped with a special function that gives the distance between any two elements of the set. An easy first example might be the Euclidean space $\R^{n}$ , the set of ordered $n$ -tuples. We understand this space quite well, since the physical position of objects in the real world are when $n = 3$ .

Once we have a formal notion of spaces, we might need to think about what makes spaces equivalent. Exact equivalence is too restrictive a notion for most spaces, so we might think of weakening what it means for two spaces to be the “same”. This is actually a very common theme across mathematics (i.e. functions being equal $\mu$ -a.e. in measure theory, or homotopy equivalence in algebraic topology). In our case, two spaces are homeomorphic if there exists a continuous, bijective map that has a continuous inverse between them. This special map is called a homeomorphism. Between any two spaces, there can exist multiple homeomorphisms - the map need not be unique.