Mathematics Success

Erik Carlsson and Anton Mellit

Anton Mellit from Ukraine and Erik Carlsson from the United States started collaborating over three years ago at ICTP. Back then, they were both postdocs in the Mathematics section. Their fruitful scientific discussions have now resulted in the publication “A proof of the shuffle conjecture”, recently featured in one of the most prestigious journals in the field, the Journal of the American Mathematical Society (JAMS). We have interviewed the authors of the paper to find out more about this work.

How did you get interested in this specific project?

Erik: Anton and I met very early and began working on our project soon after we became friends and started talking about math.

Anton: During my postdoc at ICTP I collaborated with [mathematics section head] Fernando Rodriguez Villegas and Erik. It was Erik who first mentioned to me a famous conjecture, very hard to achieve, that many experts already tried to solve in the past. He had some ideas on how to tackle it, hoping to get at least a partial result. We started discussing ideas, making experiments, and our understanding improved very quickly. At some point we even joked we would have proven the conjecture by the end of Erik’s visit at ICTP. This is exactly what happened: after a couple of months, we completed the solution.   

What’s exciting about this work for you?

Erik: This paper proves a long-standing open conjecture called the “shuffle conjecture”, important in algebraic geometry, combinatorics, and mathematical physics.

Anton: The best part was when, after making some experiments, we somehow generalized our first observations. We had discovered some new operators!

How long did it take you to finalize this paper?

Anton:  We finished this work while I and Erik were still in Trieste, three years ago. After that, it was just a matter of writing and re-writing the paper and polishing the text. It took almost two years from the first draft to the final publication.

How do new collaborations usually start in your field?

Anton: The easiest way is at conferences, dynamic and various environments where you get to know many researchers with similar interests. It can be harder in your own department, where you are often the only one working on a specific topic.

Are you still in contact with each other?

Anton: Since Erik moved back to California, it has been more difficult for us to talk: our time schedules are not “intersecting”. But I have already visited him there once and he might visit me next spring in Vienna. We might get the chance to discuss more about a project we are still working on together.

What has been your career path after your postdoc at ICTP?

Erik: after ICTP, I went back to the United States, spent one year as a post-doc at Harvard and since then I've been an assistant professor at the University of California Davis.

Anton: I got a postdoctoral position at SISSA, so I stayed in Trieste for another year. After that, I moved to Austria where I obtained a tenure track position in Vienna. Overall, I have spent ten years working and travelling around Europe since my PhD in Bonn, in 2008: now I would really love to settle down here in Vienna. This paper just published will certainly help my future career.

What do you remember most about working at ICTP?

Erik: I really, really enjoyed my time at the ICTP, and I can say that it was the most productive year of my career as a post-doc. The atmosphere was perfect for meeting new collaborators.

Anton: I was very lucky to share the same interests with Erik and Fernando at ICTP, it has been a great working environment indeed. 

--- Anna Lombardi