Ajay Kumar Nair

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Research

Research Interests

My research interests lie in Lean, AI, and autoformalisation, and in low-dimensional geometry and topology. Within Lean and AI, I contribute to the project LeanAide, which develops tools for translating mathematics written in natural language into Lean, and to resources such as the Lean Metaprogramming Recipes cookbook. A broader record of my coding work is available on my GitHub profile.

Within low-dimensional geometry and topology, I am interested in geometric structures on surfaces, their holonomies, and the dynamics of mapping class group actions.

  1. Hyperbolic cone surfaces, that is, surfaces obtained by gluing hyperbolic geodesic triangles, together with their holonomies.
  2. Mapping class group actions on relative character varieties of punctured surface groups.
  3. Infinite-type surfaces and their mapping class groups.

Conferences

  1. Hackathon: LeanLang for Verified Autonomy at IISc Bangalore + Online, in association with Emergence India Labs. April-May 2026.
  2. LeanLang for Programmers (LeanLangur workshop) at IISc Bangalore, Bengaluru, India. Dec 2025.
  3. Lean for the Curious Mathematician at ICTS Bangalore, Bengaluru, India. Apr 2025.
  4. New Trends in Teichmüller Theory at ICTS Bangalore, Bengaluru, India. Feb 2025.
  5. Lean Together 2025, Online. Jan 2025.
  6. Geometry in Groups at ICTS Bangalore, Bengaluru, India. July-Aug 2024.
  7. Low-Dimensional Topology at IISER Pune, Pune, India. Sept 2023.
  8. 1,2,3: Curves, Surfaces and 3-Manifolds at Technion - Israel Institute of Technology, Haifa, Israel. May 2023.
  9. Ergodic Theory and Dynamical Systems at ICTS Bangalore, Bengaluru, India. Dec 2022.

Talks

  1. Mapping class group dynamics on $\mathrm{PSL}(2,\mathbb{C})$-character varieties, In House Student-PostDoc Symposium, IISc Bangalore. November 2025.
  2. The Goldman bracket characterises the homeomorphisms between non-compact surfaces (Video), World of GroupCraft IV. Sep 2024.
  3. When is a homotopy equivalence of a surface homotopic to a homeomorphism?, In House Student-PostDoc Symposium, IISc Bangalore. Apr 2024.
  4. On Three Theorems of Surface Homeomorphisms, Prime Time Seminars, IISc Bangalore. Jan 2023.
  5. On Three Theorems of Surface Homeomorphisms (Video), IISER Bhopal Geometry and Topology Seminars. Nov 2022.
  6. Conformal Bijections, Biholomorphisms and Hyperbolic Isometries, Prime Time Seminars, IISc Bangalore. Nov 2021.