Yu Zhang

Department of Mathematics
The Ohio State University
Columbus, OH 43202
USA

Web: https://people.math.osu.edu/zhang.4841/
Email: zhang.4841@osu.edu
Office: Math Tower 304

Research Publications Teaching Talks Conferences

About me

I'm currently a graduate student in Math Department at The Ohio State University. My advisor is John E. Harper.

My primary research is in the area of algebraic topology, more specifically in homotopy theory.

Here is my CV (last updated: March 2019).

Research Interests

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

Teaching (as teaching associate at OSU)

Selected Talks

Conference Participation