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\memosubject{AT Reading Group  Updates}
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\color{violet}January 2025\color{black}\\

The fundamental group of a topological space is also known as the "first homotopy group", and accordingly, is one of the first topics a student may encounter in their algebraic topology studies.  However, one can also study topological spaces via their \emph{homology groups}, and indeed, homology has served as a focus for the Hatcher-Dold reading group these past several months.   Like the fundamental group of a space $X$, the homology of a space is a \emph{topological invariant}.  Topological invariants allow us to distinguish spaces, one of the major motivations for topology, and homology groups are often easier to compute than homotopy groups.  The homology group $H_n(X)$ of a topological space $X$ can "count" the number of holes that a space has in dimension $n$, which is information we can use to identify or distinguish the space we are studying.  At the basis of homology (and its dual - cohomology) is the purely algebraic branch of mathematics known as \emph{homological algebra}.  Beyond its reach in algebraic topology, homological algebra has also largely influenced commutative algebra, algebraic geometry and more!  It has also proved useful in the  development of new tools for the computation of algebraic invariants, such as the \emph{Mayer-Vietoris Sequence}.  Indeed, we hope to take many of the new mathematical tools we have discovered in studying homology with us onto the study of cohomology in the coming months of our reading group. 

Reading groups themselves serve as a tool for collaborative learning!  The opportunity to connect with other students in studying a mutual topic of interest is a motivating social as well as academically enriching experience.  How  many times has it occurred to any young mathematician that they \emph{believe} they've understand some topic perfectly well, but when it comes to time to explain it to someone, they falter a bit?  Reading groups allow participants to clarify and catch any faulty theory through weekly discussions on the material designated.  We poke at new questions, collaborate on some problems, and offer complementary perspectives.  Communicating mathematics not only allows to learn more effectively, but is an essential part of being an academic.   Therefore, our reading groups also complement many of TB's greater activities, such as our Feynman Talks series, which provides students with a forum to introduce an expository topic of interest to the greater student community.   

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