I'm currently working in the area of homotopy theory and its application to other fields. I'm particularly interested in problems related to localization and completion techniques, model category structures and structured ring spectra.
In my recent work, the main objects of study are structured ring spectra and their Topological Quillen (TQ) homology. Structured ring spectra are spectra with extra algebraic structures which can be described as algebras over an operad O.
TQ homology is weakly equivalent to stabilization in the category of O-algebras. Analogous to topological spaces, homology is easier than homotopy, stable homotopy theory is easier than unstable homotopy theory. Given an O-algebra X, we
would like to extract the part of X that "TQ homology sees". There's two possible ways to do this. One is TQ-completion, the other is TQ-localization. TQ-completion construction is more concrete and easier to compute, however, this construction could
sometimes "go wrong" in the sense that TQ completion of X could look totally different from X. On the other hand, TQ-localization, defined as the universal TQ-local approximation of X, is always be the correct construction.
However, if one just looks at the definition, it is not clear whether TQ-localization would actually exist for a given X. In my recent work joint with John E. Harper, we construct TQ-localization for all cofibrant X. We also set up TQ-local "almost" simplicial model structure
where the functorial TQ-localization is just fibrant replacement functor under such structure.
To have a better understanding of what TQ-localization actually looks like, in my more recent work, we identify a reasonable large class of TQ-local objects. The main result is that homotopy pro-nilpotent objects are TQ-local. Here we say a
structured ring spectra is nilpotent if its m-nary multiplication is trivial for all large enough m. We say an object is homotopy pro-nilpotent if it can be
write as homotopy limit of a tower of nilpotent objects.
Here are some questions I'm currently thinking about: How to get more computational results of TQ homology? How could these results help us study algebraic K-theory?
In the near future, I also plan to work on functor calculus and equivariant homotopy theory.
The story without jargons?
Sure. The following is how I explain my research to my friends.
After almost 20 years of experience in studying mathematics, I finally come to a simple undoubtable conclusion: Math is difficult, at least for me. Therefore, I'm fond of methods that can help me simply problems. Some of my favorites include:
(i) replacing objects by equivalent but better behaved objects (localization methods, cellular approximation...)
(ii) approximating general objects by well behaved objects ((co)simplicial resolutions, functor calculus...)
(iii) decomposing a complicated problem into smaller and easier pieces (skeletal filtration, fracture square...)
(iv) proof by induction (principal Postnikov towers...)
In my research, I use and develop those tools to analyze the questions I care about. I would also love to adapt those tools to other areas and help others attack their targets.
Preprints
Topological Quillen localization of structured ring spectra.
(with J. Harper). Submitted for publication (18 pages), 2018. pdf
(This paper will be updated soon. In the new version we set up the TQ-local semi-model structure where weak equivalences are defined to be TQ homology equivalences and fibrant objects are exactly TQ-local objects.)
Principal Postnikov towers and TQ localization of structured ring spectra.
Available as arXiv:1902.03500 (7 pages), 2019. pdf
(This paper will be updated soon. New version will contain a new result showing homotopy pro-nilpotent objects are TQ-local. Here we say a
structured ring spectra is nilpotent if its m-nary multiplication is trivial for all large enough m. We say an object is homotopy pro-nilpotent if it can be
write as homotopy limit of a tower of nilpotent objects.)
Teaching (as teaching associate at OSU)
Math 2177: Mathematical Topics for Engineers, Autumn 2018.
Math 1172: Engineering Mathematics A, Autumn 2017.
Math 1172: Engineering Mathematics A, Autumn 2016.
Math 1152: Calculus II, Spring 2016.
Selected Talks
An easy proof of the homological Whitehead theorem for nilpotent spaces (lightning talk) slides, University of Illinois at Urbana-Champaign, Graduate Student Topology and Geometry Conference, March 2019.
Induce model structure along adjunctions, The Ohio State University, Student Homotopy Seminar, March 2019.
Reedy model structure, The Ohio State University, Student Homotopy Seminar, February 2019.
Postnikov tower and obstruction theory for ring spectra, The Ohio State University, Student Homotopy Seminar, September 2018.