I'm looking to find out what active areas of research there are in topological graph theory, particularly those that interface strongly with other areas of math (say, group theory, algebraic topology, Gromov-Witten theory, etc).
Assuming a background given by, say, Topological Graph Theory by Gross and Tucker (a standard reference):
(1) How does topological graph theory fit into and interface with 21st century mathematics, and where is the field going?
(2) I'm particularly interested in applications to algebraic topology and algebraic geometry – can anyone summarize these or give a feel for what the most active areas are?
Best Answer
I wouldn't describe it as an "area" of math, exactly, but there are certainly mathematicians who study embeddings of graphs into surfaces and related objects. Let me recommend two sources to learn more about what people are doing. I am not familiar with the book you listed, so I don't know if it sufficient preparation for reading them; however, I am a big fan of jumping in to an unknown field and then learning background information "as you go".
The first is the long book
MR2036721 (2005b:14068) Lando, Sergei K.(RS-IUM-M); Zvonkin, Alexander K.(F-BORD-LB) Graphs on surfaces and their applications. With an appendix by Don B. Zagier. Encyclopaedia of Mathematical Sciences, 141. Low-Dimensional Topology, II. Springer-Verlag, Berlin, 2004. xvi+455 pp. ISBN: 3-540-00203-0
This might be the closest to what you are looking for in that it focuses on connections with algebraic geometry and number theory, especially Grothendieck's theory of dessins d'enfants and questions related to the moduli space of curves.
The second is a very different direction, namely the study of "spatial graphs"; ie embeddings of graphs into 3-space and other 3-manifolds. This is basically a generalization of knot theory. One possible way of getting into this subject is to start with the survey
MR2179645 (2006e:57009) Ramírez Alfonsín, J. L.(F-PARIS6-CM) Knots and links in spatial graphs: a survey. (English summary) Discrete Math. 302 (2005), no. 1-3, 225--242.