Take 10 with... Bartek Ewertowski

1. Describe your research topic to us in 10 words or less.

Parametrised cohomology of Cartan holonomy reductions.

2. Now explain it in everyday terms!

I study how one geometry can transition into another. For example, according to the Hartle-Hawking cosmological model, the Big Bang was not a point-like singularity, but rather a three-dimensional hypersurface marking the metamorphosis of a primordial fourth spatial dimension into the familiar time dimension we now experience.

In my own research, I focus on a broad class of similar geometric transitions called Cartan holonomy reductions. I constructed a differential operator which relates solutions to Bernstein-Gelfand-Gelfand (BGG) equations in the different geometric regions. Each geometry has its own family of BGG equations, and solutions to these equations encode important geometric data - so it was nice to be able to show a relationship between them.

3. Describe some of your day-to-day research activities.

I spend a lot of time reading articles and papers – typically I attempt to write up the proofs myself to make sure that I understand what is going on. I usually write quick notes by hand and then type up my notes more professionally in LaTeX so that I can reference them later. I don’t use any AI tools at all. 

4. What do you enjoy most about your research?

I love to discover connections between mathematical ideas and results which might seem completely unrelated at first glance. The main tool that I use for this is a branch of metamathematics known as category theory.

5. Tell us something that has surprised you in the course of your research.

I needed to teach myself homotopy theory, which is a very abstract and challenging area of modern geometry. While struggling with this, I stumbled upon the work of Vladimir Voevodsky et al.: they had just recently discovered that a functional programming language called MLTT was secretly the language of homotopy theory, giving birth to the new field of study at the intersection of mathematics and computer science which they called homotopy type theory.

To discover that functional programming and geometry are somehow the same thing was truly mind-blowing for me! And yet, as I read more, the impossible became obvious and I learned how to translate ideas back and forth between computer science and geometry. This clarified many of the confusing aspects of homotopy theory for me and also gave me a new-found appreciation for theoretical computer science.

6. How have you approached any challenges you’ve faced in your research

When I struggle to make progress in any pursuit, whether it is mathematics or sports, I often just need to shore up my fundamentals. Each layer of a pyramid requires the layer below to be complete and sturdy. On the other hand, if I feel that I’m just lacking creativity, I’ll either go for a walk or work on something else – this often sparks some good ideas.

7. What questions have emerged as a result?

The aforementioned change of geometry that occurred in our universe according to Hartle and Hawking looks eerily similar to a geometric transition which occurs in projective differential geometry when there exists a certain BGG solution. As far as I am aware, most cosmologists are not very familiar with projective differential geometry and likely haven’t encountered these ideas. I suspect that there are some very interesting discoveries to be made here.

8. What kind of impact do you hope your research will have?

I hope that one day a cosmologist will look into my question above and find some of my geometric results helpful!

9. If you collaborate across the faculty or University, or even outside the University, who do you work with and how does it benefit your research?

In 2019, I had the pleasure of attending the first ever international conference on homotopy type theory at Carnegie-Mellon University in Pittsburgh. It was a really cool environment, where geometers, logicians, and computer scientists came together to unravel the implications of the newfound connection between geometry and computer science. Meeting Emily Riehl was a particular highlight for me, since I’m a big fan of her work. She was even nice enough to give feedback on a proof that I had written for my PhD dissertation!

10. What one piece of advice would you give your younger, less experienced research self?

Most people tend not to venture too far outside of their chosen specialty, so having broad interests can be useful – what might seem obvious to people working in one field might be entirely novel to people in another. David Epstein discusses this in his book Why Generalists Triumph in a Specialized World. If I could, I would mail a copy to my younger self so that I could be confident that my approach would pay off.