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 personal page.
What is topology?
Topology is the foundational and formal study of continuity. While geometric are rigid objects characterized by numerical values, two topological spaces are equivalent if you can continuously deform one into the other and then undo this deformation continuously. From here, topology becomes the study of space and shape and how to formally tell when two given spaces are different. In many ways, this is similar to the taxonomic classification of organisms in biology. Biologists use characteristic properties like anatomical structure, behavior, 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.) encodingthe characteristics spaces that we 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.
Wild Algebraic Topology
Though I have interests in many areas of mathematics, my specialty is the algebraic topology of locally complicated (or wild) topological spaces, which requires dealing with techniques from algebra, geometric, and general topology. To be clear, “wild” only means that the spaces under consideration are not locally contractible. In many ways, the main objects studied in this field are better understood than many popular objects in classical algebraic topology. The direction in which classical algebraic topology moves is vertically – to higher dimensions. In wild algebraic topology, we are more focused on expansion – understanding the structure of the wider class of spaces that show up naturally in various areas of mathematics yet which traditional theory can’t handle.
The field of wild algebraic topology is young but has made rapid progress in the past 20 years. Key objects like the fundamental group of the Hawaiian earring are well-understood and have been applied to prove the remarkable homotopy classification of one-dimensional Peano continua (Katsuya Eda). With rich theoretical machinery, we are finding deep connections to linear order theory, infinite non-abelian group theory, and topological algebra. Of course, many important 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 little 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.