News

Tyler Kelly joined the Centre for Combinatorics, Algebra and Number Theory as a Professor of Pure Mathematics in January 2025. Tyler works in mirror symmetry; an exciting research area with roots in pure mathematics and theoretical physics. We asked him about his mathematical career and interests

---

Tell us about your academic journey and what brought you to Queen Mary.

I did my PhD at the University of Pennsylvania (Penn) under the supervision of Ron Donagi, graduating in 2014. After, I moved to the UK and was a postdoc at Cambridge for four years in their algebraic geometry group. In 2018, I joined the newly formed Geometry and Mathematical Physics group at Birmingham as a Lecturer and was promoted to chair in 2023.

I moved to Queen Mary for many reasons. I'm excited to be a member of such a robust group in algebraic geometry specialising in two of my favourite approaches: combinatorial and noncommutative. I also look at computational number theory using ideas from complex algebraic geometry, so I am excited to be in a Centre that has specialties across Combinatorics, Algebra and Number Theory. Also, London has a wealth of researchers in mirror symmetry, so I get to be around my research discipline in London while having colleagues at Queen Mary that use similar tools in mathematics as myself.

Lastly, I believe that Queen Mary's aim to be the most inclusive research-intensive University in the world is a noble goal. One of my goal for mathematics is to build an inclusive research community that is ambitious in its research objectives. This aim made me feel the direction of the university matches mine.

What general area is your research in?

My research is predominantly in a field called mirror symmetry. Mirror symmetry is the mathematical manifestation of a duality from string theory that linked symplectic and algebraic geometry. That is, the symplectic geometry of a space is encoded in the algebraic geometry of a so-called 'mirror' space. There's a few questions that this leads to:

1. Given a symplectic space M, can you find its mirror?
2. To what extent does this translation between the symplectic geometry and the algebraic geometry hold?
3. Can one apply this link to answer burning questions in either field (or even adjacent disciplines)?

The line of investigation has ended up having influence across algebraic geometry and symplectic geometry, but also cluster algebras, category theory, tropical geometry, homological algebra, combinatorics, number theory, and commutative algebra. Consequently, I have broad interests in subsets of many areas that has been induced by intuition from mirror symmetry.

And could you tell us something about a specific problem you have worked on?

One problem that I think has had a satisfying storyline in my research was that about mirrors of Calabi-Yau complete intersections in toric varieties. Mirror symmetry originally was developed where, given a Calabi-Yau variety M there exists a mirror Calabi-Yau W so that the symplectic geometry of M is the algebraic geometry of W. Calabi-Yau varieties, roughly, are complex manifolds that are 'flat' in some sense that are modelled as solutions to polynomial equations.

To answer question 1. above, there were many constructions in the mathematics and physics literature to construct a mirror W to a Calabi-Yau complete intersection M in a toric variety. Alas, they were not consistent with each other and often gave multiple answers for the mirror to the same space M!

So the problem we wanted to investigate was: which one is the "correct" mirror? Well, it turns out that, under the lens of category theory, one can prove that they are all equivalent.

The key tool we use is the Landau-Ginzburg model, a primary object in my research. A Landau-Ginzburg model encapsulates the geometry of a variety into the singularity theory of a function. It can be a much more malleable algebraic object than a variety, which allowed us to provide a unification of all mirror constructions for Calabi-Yau complete intersections in toric varieties.

What do you enjoy most about your area of maths?

There are two things I enjoy most. One is that the field invites narratives. Mirror symmetry, at its core is a way to interpolate invariants between symplectic and algebraic geometry. As such, one must explain under this exchange which objects, theory or invariants correspond. This requires exploration and imparting knowledge to practitioners in different subdisciplines. There is a folklore that builds intuition in the field and trading ideas is a beautiful tradition in mirror symmetry that yields progress.

The second is that there are so many tools and approaches to the area. This makes the community truly collaborative and communicative. Contributing and being part of a rich mathematical community makes the experience enjoyable for people with different interests and viewpoints.

What is your approach to teaching?

My goal with teaching is first to give clear expectations of a course to students and what they should learn in the course. After setting this up, I aim to provide them with the tools to learn it and facilitate this learning in ways that work for them. I want to create open dialogue for their exploration of the material so that they can internalise the course material better and achieve those learning goals.

Each course has a treasure that they can learn, and I want to share that treasure while recreating the experience of unveiling it.

My last teaching at Birmingham was topology, and I loved how the course gives such rigid tools that then can be toyed with and explored. It opens a new mathematical world that I enjoy inviting students into.

What are your interests outside maths?

I spend time working on Equality, Diversity, and Inclusion work, as part of the London Mathematical Society's Committee for Women and Diversity of Mathematics and the REF 2029 People and Diversity Advisory Panel.

I also knit.

---

You can find more about Tyler and his research on his website.