Chandan Singh

Research

My research develops a unified operadic and homotopy-theoretic framework connecting Grothendieck–Teichmüller (GT) theory, the Kashiwara–Vergne (KV) problem, and low-dimensional topology. These structures arise in diverse contexts including deformation quantization, quantum topology, and the study of moduli spaces.

The central objective of my work is to construct explicit geometric and algebraic models in which GT symmetries and KV solutions emerge from a common underlying operadic structure. This provides a conceptual bridge between arithmetic symmetries (via GT theory) and Lie-theoretic structures (via the KV problem).

In particular, my current research focuses on

Galois actions via variants of the GT group actions on knotted objects in 3D and 4D topology.
Operadic and homotopy-theoretic structures underlying KV-type equations
Formality problems in operads.
Operadic models for knotted objects via homotopy-theoretic methods

Publications and Preprints

Cyclic Symmetries of Chord Diagrams. Preprint available at arXiv. (Submitted)

Summary: We give a cyclic operadic characterization of the proalgebraic graded Grothendieck-Teichmüller group GRT(𝕂)- We show that GRT(𝕂) is isomorphic to the group Aut(PaCD(𝕂)) of automorphisms of the prounipotent cyclic operad of parenthesized ribbon chord diagrams. As an application, we describe a GRT(𝕂)-action on the category of framed chord diagrams with self-dual objects, which is closely related to the target category of the Kontsevich integral for framed tangles.

Grothendieck-Teichmüller Symmetries of Cyclic Operads and Tangles (with Marcy Robertson). Preprint available at arXiv. (Submitted)

Summary: We characterise the profinite Grothendieck-Teichmüller group GT as the group of automorphisms of the profinite completion of a cyclic operad of parenthesised ribbon braids. It has three applications. 1. we provide an operadic model for profinite tangles using metric prop. 2. We construct a GT action on quantum tangles. 3. We provide an alternative proof of the formality of the cyclic framed little disks operad using this action.

Genus zero Kashiwara-Vergne solutions from braids (with Zsuzsanna Dancso, Iva Halacheva, Guillaume Laplante-Anfossi and Marcy Robertson). Preprint available at arXiv. (Submitted)

Summary: We develop an operadic framework for constructing genus-zero Kashiwara–Vergne (KV) solutions from Drinfeld associators. It reinterprets the Alekseev–Enriquez–Torossian construction via moperads of parenthesized braids with a frozen strand, showing that homomorphic expansions yield families of KV solutions. As applications, we further establishes actions of Grothendieck–Teichmüller–type groups and characterizes when KV solutions arise from associators.

Contributions to Grothendieck–Teichmüller theory and the genus zero Kashiwara–Vergne problem (Chandan Singh), PhD thesis, University of Melbourne. Available at URL.

Projects

Sheaves and its Cohomology (supervised by Thomas Andrew Ducat), masters thesis, Imperial College London.
Spectral Sequences in Algebraic Topology (supervised by Vigleik Angeltveit), Australian National University.

Research Stays

Institut de Mathématiques Jussieu-Paris Rive Gauche (IMJ-PRG), Université Paris Cité, hosted by Adrien Brochier (Jan 2026) .
Stockholm University, hosted by Dan Petersen, Aug 2025.
Queen's University Belfast, hosted by David Barnes, Feb 2025.
Centre of Quantum Mathematics, University of Southern Denmark, hosted by Guillaume Laplante-Anfossi, Feb 2025.
Indian Institute of Science- IISc, hosted by Phaneendra Yalavarthy, Aug 2021 to Aug 2022.
Australian National University, hosted by Vigleik Angeltveit, May 2019-Aug 2019.

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