Any monkey can do the math, but what does the math mean?
What is this all about ?
Okay, I had a summary of my current works on the front page. I also described all my research interests, and listed all of my publications there. Why do I need to devote another page for such a vague topic, namely, "Research" ?
The thing is, I always find the idea of academic personal webpages a little bit bizarre. The webpages are supposed to be standardized, professional and all. Yet most of the time, the audiences (if there are any at all) are people that we know personally, or undergraduates/ young graduate students that are interested in our research. They come to our websites, see a dry list of things that we do, bullet points about our interests, and that's it.
I like a quote from Abstruse Goose (when he talked about quantum mechanics): Any monkey can do the math, but what does the math mean? Analogically, I saw your lists and your bullet points, but still, what exactly are you wasting years and years of your life for?
The purpose of this page to explain, informally, to audiences (again, if there are any at all) with little background in any of the disciplines I'm working on, what I'm really doing in my research.
My research area
Four years into grad school, when "What is your research about?" has become the most frequent question to initiate a small talk, the tiny pause before my usual response still hints that it is not the answer I'm satisfied with.
To get a clean answer to that question, let me explain the general background by the following picture. My research lies somewhere among three disciplines: math, applied math, and biology. Between applied math and biology/engineering are problems like experimental design, control and optimization; while between pure math and applied math are analyses of algorithms. When I have to give a brief answer to the question, I usually go with "mathbio", "applied math" or more specifically, "experimental design for biological systems". Those do not really explain the situation.
Let's talk a bit about philosophy. I identify myself as a mathematician; my opinion is that math should be the only language science is written in. I also think about math as an art, and (sometimes, not always) I'm okay with the idea of doing math for math itself. But I;m not in favor of doing applied math that applied to nowhere; to me, context-free applied math is content-free. If there is such a thing as applied math at all, it should be a bridge that connect theory and application.
With that view in mind, I enjoy the task of figuring out algorithms/methods to solve problems arising in application contexts, as long as I believe that there is some theoretical foundation to back me up. Similarly, I feel most happy when I can use techniques from different branches of pure math to analyze an algorithm that I know for sure will work in practice. The perception of a flow from math to applications makes me feel hesitate to describe my research area by "local" terms like "math biology" or "experimental design", though those are more or less correct. The most appropriate term should be "applied math"; however, most of my works were directly originated from problems in biomedical engineering, and are not actually associated with the mainstream of applied math (pde, numerical analysis, inverse problems...) in the strict sense.
Recently, my project about behaviour discrimination in biological networks gave rise to a general algorithm that can be applied to other frameworks. While the current works focus on applying the algorithm to experimental design, control and systems biology, the long term plan is to use it to tackle problems in the field of uncertainty quantification. That, I hope, will complete the flow.
For the time being, there is a tunnel under the Appmath Mountain. I'm walking back and forth along the tunnel every day. If by any chance you walk that way, let me know.
Mathematical biology
There are two questions that I usually got when I tell people I work on mathematical biology: a) Why do people need math in biology? b) Do I need to learn about biology a lot for my research?
I got the first question quite often, and not until I talked to one of my roommates this year did I really understand it. I realized that when "biology" was mentioned, people still visualized the image of biology in the first half of the 20th century, with sacrifice of hamsters and Mendel's pea experiments. While some of those practice remain active, they are no longer good representatives for modern biology. Our technology has developed to a degree that we can study life at cell and sub-cell levels. Those studies, which require high accuracy in both qualitative and quantitative manner, faccilitate the use of advanced mathematical models.
The communication between math and biology is accomplished through layers. First, a biological model is proposed, in which each component is modelled is a chemical/physical phenomenon/process. Those components, in turn, are modelled by mathematical concepts. Once a mathematical modelled is constructed, mathematical analysis can be done to analyze the model in the desired ways.
Working in a group with biologists, biomedical engineers and control engineers, I'm not under any pressure to learn much about biology for my research: my main concern is the analysis of the constructed mathematical models. Although I find the idea of dealing with science problems directly extremely cool, I'm well aware that my range of research is already a bit ambitious. For now, I'm learning the language of biology/biomedical engineering, absorb their cultures, but I'm not yet ready to give myself in.
Works
As I indicated before, most of my time, I work as a member of the wonderful Ann Rundell's lab ("wonderful" is the only word that I can think of to describe the group, but that's a different story). We work on various subjects, but mainly, experimental design and control, with focus on the T-cell signaling pathway. While I am still a newbie in control theory, I am quite comfortable discussing about experimental design (after three years of efforts).
The problem has been well-studied for years: which experiments should be made to get more information about something. For most of its history, that "something" is the parameter values of a system, while in our research, it is the dynamics (time course) of some system's variable. The motivation is, for example, in a chemical network, if we can measure the concentration of some chemical at 5 different time points, can we approximate with high accuracy the time course concentration of another chemical ? The answer is maybe, especially if the measurements are chosen by some optimal criteria.
Before I stepped into the project, the optimal criteria was already proposed by two other previous members of my group. I twisted the criteria and algorithm, put them in a probabilistic framework, proposed a different reconstruction method and proved the convergence of the new estimator. Since we used an approximation of the Gibbs sampler, we also worked on the theoretical aspect of the method for a while and were able to provide an error estimate.
At some point, I realized that the probabilistic framework and the new estimator seemed to be extendable to other parametric contexts. We went a little bit beyond the scope of experiemental design and used the technique to address a problem called behaviour discrimination (in biology) or parameter systhesis (in engineering). That has been the center of my research recently.