( … under construction … )
Get in touch if you are interested.
I directed various PhDs in logic, category theory, and computer science. More recently, I am also interested branching out to other fields such as engineering, economics, and the social sciences. The rationale is that these areas use a substantial amount of mathematics stemming from physics (such as calculus, linear algebra, probability and statistics) but much less of the more recent mathematics developed for computer science (this includes not only algorithms and complexity theory but also logic and category theory). This looks like a research opportunity .
I offer various research projects, see eg 291/491 Student-Faculty Research and Creative Activity for Course Credit for one way of formalizing this.
I maintain a changing lists of topics I am interested in.
Big questions are not easy to turn into concrete research questions, but that only makes them more interesting.
Logic has taken a quantitative turn. Instead of being concerned only with 2 truth values, there are applications of many-valued logics (including fuzzy logic) to pure mathematics, software engineering and control theory, to name just a few. What is the common core underlying both classical and many valued logics? How can we leverage this for the construction of new software tools?
Theorem proving has been a very successful application of computer science to mathematics … but we have not been getting much closer to the point where mathematicians who dislike programming feel comfortable to make use of this technology. Why? What are the next steps needed to close this gap?
Machine learning is great as long as we have, or can generate, large amounts of data. But humans are capable to learn successfully from small numbers of examples. One of the ideas to tackle this gap is to bring together symbolic and statistical AI. While there is work along these lines, there is much more to discover.
What is a proof? What is an algorithm? What is information? After more than a century of research in algorithms, proof theory and information theory, there is still no satisfactory answer to these questions.
Algorithms run on machines, mechanisms run on rational agents. But what are the theoretical foundations for human cooperation intelligence? I put some preliminary thoughts in lectures on Designing Algorithms for Cooperation and Challenges in Software Engineering.