1. Why “maths”?

There is something very Taoist about mathematics: The goal of much of the subject is to solve extremely difficult problems using the minimum amount of work. I am partial to the philosophy that there is a natural flow to ideas – a correct expression and organization of concepts – which if understood, allows us to find just the right pressure point for each problem. A mathematician is someone who looks for this perfect way, which makes problems fall apart and answers become simple and obvious. So, perhaps, mathematics appeals to my inner lazy person.

2. What are you working on now? (What is the topic? What is it’s main issue?)

At the moment I am busy typing up a series of papers giving a new way to understand a common and important structure in math and physics called “Lie Algebras”. Recent topological work by a coauthor of mine led to a series of joint papers using related ideas to reinterpret and extend classical structures in algebraic topology. Since early this summer, I’ve been investigating what these ideas can give purely algebraically. I’m very excited about this framework, since it implies all of the classical work understanding Lie algebras, giving new understanding and proofs, and furthermore leads to new methods for Lie algebra computation which greatly improve upon existing algorithms. I presented an early version of this work at a workshop in Lille earlier in the summer, but it is only in the last month that the finer points of the theory have really been coming into focus.

3. What are your research interests? (Specific interests & focus topics...)

I am an algebraic topologist by training and by choice. My early work leans towards topology; however, lately and quite unexpectedly, my research has taken me more towards algebra. My joint work with Dev Sinha at the University of Oregon began with a new model for Lie coalgebras using graphs, originally applying it to rational homotopy theory, developing tools useful for solving certain knot theory problems, and later applying it to define generalized Hopf invariants and give new understanding of some classical structures in algebraic topology. Currently we are applying our Lie coalgebra model to solve the word problem in group theory. Parallel to this, I have been developing further foundations for coalgebras in general and for the algebraic side of our structures in particular. In another joint project with Brian Munson at Wellesley College, we extend recent work estimating chromatic numbers of graphs using algebraic topology by developing a calculus of functors to streamline computations.

ü      What is “Algebraic Topology”? What is the importance of “Algebraic Topology”?

Algebraic topology is the study of connections between abstract algebras and abstract topological spaces. Modern algebraic topology begins with the attempt to classify all possible shapes of any dimension (topological spaces) by algorithmically constructing certain abstract algebraic structures from each space. For example, consider drawing loops on the surface of an object. Two loops can be combined by following first one loop and then the other; so there is some sort of notion of product of loops. A loop which never moves is the identity for this product, and doing loops backwards gives inverses. If the loops are drawn on a ball, then all loops can be squeezed shut. If the loops are drawn on, say, a coffee-cup, then there are two basic kinds of loop which cannot be squeezed shut – the loop could follow along the handle of the cup, or the loop could circle the handle the way your hand holds it. Considering also products of these loops, you get an interesting algebraic structure. There are a number of other ways of making algebraic objects from topological objects, as well as ways of going in the other direction. The goal of algebraic topology is to take advantage of these connections – take a very hard algebraic problem and turn it into a topological problem about shapes; or take a very hard topological problem and turn it into a question about algebraic structures. The most famous example of this is probably the Fundamental Theorem of Algebra: “A degree n polynomial has n roots (with multiplicity) in the complex numbers”. This is extremely hard to prove purely with algebra; however, it becomes very simple once translated to topology.

ü      What is the “Calculus of Functors” ?

The name “calculus of functors” is actually a bit of a joke. Roughly, a “functor” is a method for moving problems from one area of mathematics to another. My Ph.D. advisor made a startling discovery some years ago. It turns out that many of the functors used by algebraic topologists can be divided up and analyzed in bits via a method which looks startlingly reminiscent of Taylor series in undergraduate calculus. There is a thing which acts kind of like a derivative, and something else which is kind of like raising to the power n, and something else which acts like division by n factorial. It is really rather amazing, and the implications are still being digested by algebraic topologists. I wrote my dissertation on calculus of functors, and I still have plans to apply some of my more recent ideas to calculus of functors.

ü      Do you involve students in your research?

Sadly, I have not had any graduate students, nor have I advised any undergraduates. Though, there are a number of interesting problems related to my research which would be suitable for a student to work on...

4. How is TRNC for scientific research?

I am reminded of a famous story about the topologist, Dr. Stephen Smale. In the early 1960’s he took a holiday from the Institute of Advanced Studies at Princeton to visit a friend in Brazil. He was a young mathematician, and spent his days on Copacabana beach in Rio de Janeiro, swimming and thinking about math. According to the story, he was sitting on the beach, thinking about the chaotic behavior of the solutions to Cartwright and Littlewood’s differential equations, when a horse rode by. Looking at the shape of the hoofprint left behind on the sand, Smale suddenly understood a way to give a clear geometric explanation for chaos arising from simple conditions. Smale’s Horseshoe and the ideas it produced led to him receiving a Fields Medal (the highest award in mathematics) in 1966.

I myself prefer to do research in my office – where I have quick access to references and scratch paper, and where I can comfortably prepare my work for publication. In general, mathematics is a subject best done away from distractions; though at times, a weekend stroll along the beach, or hike in the mountains can be exactly what is needed to spur your mind to the next idea. In this respect, the TRNC is ideal for mathematics.

5. Does METU NCC provides an infrastructure for the academic and scientific research? (What kind of infrastructure-labs, scientific or academic freedom,etc..)

The campus infrastructure for research seems robust. However, as a mathematician, I don’t require much. The most important thing for me is to have the opportunity to exchange ideas with other math researchers. The mathematics research and teaching group at METU-NCC is friendly and cohesive. Also I have ample opportunity to travel to conferences to hear leading-edge research and tell of my own work.

6. Why academia as a career?

I enjoy learning, and I love universities. As an academic my job is to learn via studying the work of others and making my own discoveries. Being an effective teacher requires learning as well: Learning new ways of understanding ideas so that you can better teach them.

7. Tell us little about yourself? What hobby do you most enjoy? (What are your hobbies and interests?)

I like to hike and to climb. I like to visit new places and see new things. I also like to sit at home and read a good book or play a good computer game. Most of all, I like to spend time with my two kids; hopefully hiking outside, exploring somewhere new, reading a book, or playing a game with them. After the kids are asleep, I maintain and update the math group website and do some simple computer programming in my spare time.

8. What do you most enjoy about your job?

I enjoy the freedom to follow my research interests wherever they take me. I enjoy teaching students about mathematics, trying to come up with clear ways to communicate the beauty and cleverness of even calculus and differential equations. I also enjoy having a working environment which is open to new ideas. METU-NCC is still young and willing to try new things: The year I arrived, we created course web pages for all mathematics courses. This year, we are experimenting with the online "web work" system to assign and grade mathematics homework and we have opened the Math Help Room. So I enjoy that we can contribute to the university – if you have an idea, then you can make the university a better place. I’m not sure which of these I enjoy most – the research, the teaching, or the university.

9. If you weren’t a mathematic researcher, what would you be doing now?

I would likely be either in computers or operations research. I grew up with computers – my mother was in computer science grad school when I was a kid, and three of my brothers have engineering degrees related computers. As an undergraduate, I scrounged parts from various sources to build a small network of computers in my dorm room, which I used to experiment with parallel programming using MPI. I also did supercomputer summer research at Cornell University and later for the US government.

At the same time I was very interested in statistics. Robust nonparametric (Bayesian) statistics was just becoming possible, so it was a very exciting time in that field; by the end of my final year as an undergraduate, I had already completed all of the graduate coursework required for the statistics Ph.D. program. Applying to graduate schools, operations research seemed the ideal field to combine both statistics and computers… But that would have required me to give up theoretical mathematics. In the end, I couldn’t bear to do that.

I still play around with statistics and computers in my spare time, but only at a non-professional level. For example, I maintain the math group website, which I recently streamlined and modernized the php code for, and I designed and coded the online grade-distribution system used by the math group. Last year, I built an online photo-tagging system using javascript and My SQL for taking attendance in my classes (I took a picture of the class every day; students would tag themselves in the picture online), with the intention of eventually automating the process via face-recognition using a hidden “Markov Model”. Unfortunately, students were lax about tagging themselves, so without face-recognition, the system actually ended up requiring more of my time than old-fashioned methods. I’m still interested in trying the system again, speeding up the tagging using jquery autocomplete and using the google prediction api to handle face recognition, but at the moment I’m busy with other projects.