Hello and welcome to my blog!
I am a mathematician working at West Chester University near Philadelphia, PA where I teach a variety of classes and am involved in many research projects, most of which involve topology. Here is a link to my research page.
What is topology?
Topology is the foundational and rigorous study of continuity. Two topological spaces are equivalent if you can continuously deform one space into the other and then undo this deformation continuously. From here, topology becomes the study of space and shape and how to formally decide when two given spaces are different or the same. In many ways, this is similar to the taxonomic classification of organisms in biology. Biologists use characteristic properties like anatomical structures, behaviors, genetic code, etc. to place organisms into various levels of taxonomic rank. In the same way, algebraic topologists use algebraic structures (groups, rings, modules, etc.) encoding the characteristics of spaces they wish to remember. This process helps us classify spaces up to various levels. For instance, the fundamental group, in some sense, remembers the one-dimensional holes in a space (like the hole in a donut). So if the fundamental groups of two spaces are different, then the two spaces must be different since they don’t have the same number/types of holes.
What is “wild” topology?
Though I have interests in many areas of mathematics, my expertise is the algebraic topology of locally complicated (or wild) topological spaces, which requires dealing with a big mix of techniques from algebraic, geometric, and general topology. In fact, the spaces of interest in wild topology are natural to many areas of mathematics, arising frequently in continuum theory, dynamics, geometric group theory (as boundaries of groups), as compactifications of manifolds, etc. Actually, I’ve grown to like the word “wild.” For me, it doesn’t mean “intractable.” To me, the word “wild” emphasizes that the primary algebraic objects of interest are, individually, more highly structured and intricate than the ones in standard algebraic topology (like a regular group). For example, the earring group can be considered as a regular group but it is more natural to consider it as a group with certain infinite product operations. Like any progressive field of mathematics, the theory can be fairly technical on some days but, in many ways, the objects studied here are better understood than many difficult-to-compute objects in traditional algebraic topology.
Lately, I have been interested in algebraic invariants of spaces that admit natural (convergence-based) infinite product operations akin to infinite sums in the real line or complex plane. This includes fundamental groups, higher homotopy groups, and certain homology-like groups. In this setting, the interactions between the algebraic and topological structures are much richer than in classical algebraic topology. Sometimes, I think of this as infinitary algebraic topology because when the most remarkable things happen, infinitary operations are almost always hanging around.
The area of wild topology has seen a great deal of progress in the past 20 years. Key infinitary objects like the fundamental group of the shrinking wedge of circles (the earring space) have been extensively studied and are well-understood. Moreover, the earring group played a key role in one of the areas major achievements, Katsuya Eda’s homotopy classification of one-dimensional Peano continua. Now, there is a push for a deeper understanding of the one-dimensional setting and beyond into higher dimensions. Of course, many open problems remain in this challenging and beautiful area of mathematics!
This blog is about my interests in mathematics: things I’m thinking/reading/writing about and issues/news in my own mathematical world. Blog posts are not meant to be written like research articles so I will often try to expand on ideas in the middle of a definition or proof. Sometimes the content will be friendly to someone who isn’t a topology expert but often I’ll be writing expository posts about interesting mathematics that could be considered “research level.”
I will strive for a decent quality of exposition and, to this end, encourage constructive comments, questions, and/or corrections. If you do have such a reason to contact me, please feel free to post a comment.