Research
I study symmetry beyond the classical setting of matrices and vector spaces. The central objects are algebraic and categorical, but the questions naturally spill into knots, diagrams, combinatorics, computation, data, and algorithms.
A rough translation is: I try to understand complicated mathematical worlds by looking at their symmetries, the diagrams that encode them, and the data they generate. Sometimes this means proving structural theorems; sometimes it means building computations that reveal a pattern before we know how to prove it.
A formula is worth a thousand words. A picture is worth a thousand formulas.
Symmetry is not just about pretty pictures. It is a way of organizing information. My research studies what happens when the objects carrying symmetry become richer than points, vectors, or shapes.
Classical representation theory turns symmetry into linear algebra: one studies a group by looking at how it acts on vector spaces. This is already a huge simplification, and it is one reason representation theory appears throughout mathematics and physics. My work starts from this idea and then moves one level up: vector spaces are replaced by categories, linear maps by functors, and equations by diagrams.
This leads to applications inside pure mathematics, especially topology, combinatorics, and geometry, but also to computational projects: big-data comparisons of knot invariants, machine-learning approaches to knot recognition, and reinforcement-learning experiments for simplifying knot diagrams. The point is not to turn pure mathematics into software engineering. The point is to use every available lens on difficult structure.
Mathematical translation. The home base is categorical representation theory: monoidal categories, 2-categories, Soergel bimodules, Hecke categories, diagrammatic algebra, tensor categories, quantum topology, and their computational and asymptotic aspects.
This is the home base: categorification, 2-representation theory, monoidal and tensor categories, and categorical actions. The guiding question is what representation theory becomes when algebras acting on vector spaces are replaced by categories acting on categories.
Kazhdan–Lusztig theory is one of the central machines connecting combinatorics, representation theory, geometry, and topology. I study both its structural and its asymptotic sides, including Hecke categories, Soergel bimodules, cells, traces, dimensions, and growth phenomena.
Many of the structures I care about are best seen, computed, and explained by diagrams. This includes diagram algebras and monoids, web categories, link invariants, knot homologies, and the representation-theoretic origins of quantum topology.
I increasingly use computation as a mathematical microscope. The aim is not to replace proof by data, but to use large-scale experiments to find structure, test conjectures, and expose patterns that are difficult to see by hand.
A current thread asks what modern computational tools can do for hard topological and algebraic problems. This includes image-based approaches to knots and reinforcement learning pipelines for simplifying diagrams and improving unknotting-number bounds.
Another direction studies representations of monoids and diagrammatic categories, including representation gaps and growth questions. Part of the motivation comes from understanding how much information linear representations retain, lose, or hide.
The common theme is structure. Sometimes the structure is algebraic, such as a monoidal category or a Hecke category. Sometimes it is topological, such as a knot invariant. Sometimes it is combinatorial, such as a cell, a diagram basis, or a polynomial with mysterious coefficients. And sometimes it is computational, appearing only after one generates enough data to see the shape of the problem.
Growth in affine Hecke categories. Asymptotic and analytic questions in Hecke categories, with representation-theoretic and categorical input.
RL unknotter, hard unknots and unknotting number. Reinforcement learning and diagrammatic moves for unknotting problems.
On knot detection via picture recognition. Using visual and machine-learning methods to probe knot-theoretic information.
Big data comparison of quantum invariants. Large-scale comparison of quantum invariants and their detection power.
Big data approach to Kazhdan–Lusztig polynomials. Experimental and computational exploration of KL polynomials.
Equivariant neural networks and piecewise linear representation theory. A bridge between representation theory and machine-learning structures.
For the full list, see my research papers or the publication list.
The mathematical core is 2-representation theory and categorical representation theory: monoidal categories and 2-categories acting on categorical analogues of vector spaces. This includes Soergel bimodules, Hecke categories, quantum groups, tensor categories, diagram algebras, and their links to topology and combinatorics.
A second, increasingly visible thread is experimental and computational: using large data sets, computer algebra, machine learning, and reinforcement learning to probe structures such as Kazhdan–Lusztig polynomials, quantum knot invariants, and unknotting phenomena.
