Chandan Singh

Research

My research develops a unified operadic and homotopy-theoretic framework connecting Grothendieck–Teichmüller (GT) theory and the Kashiwara–Vergne (KV) problem. 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 where GT symmetries and KV solutions emerge from a common topological origin. 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 GT group actions, on knotted objects in 3D and 4D topology.
The operadic and homotopy-theoretic structures underlying KV-type equations.

In addition, I am interested in formality problems of operads and constructing operadic models for knotted objects via homotopy-theoretic methods.

Publications and Preprints

Cyclic reflections and the flip equation for the graded Grothendieck-Teichmüller group (Oct 2026). Preprint available at arXiv.

Summary: This paper establishes the connection between two characterisations of the graded Grothendieck-Teichmüller group: one is mould-theoretic, and the other is cyclic operadic. The key insight is that the flip, introduced by Furusho, Hirose, and Komiyama, is fundamentally a cyclic reflection within the operad of framed Drinfeld–Kohno Lie algebras after removing the framings.

Cyclic Symmetries of Chord Diagrams (Apr 2026). 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 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) (Dec 2025). 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 a metric prop. 2. We construct a GT action on quantum tangles. 3. We give a proof of the rational 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) (July 2025). 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 construct an action of Grothendieck–Teichmüller–module groups on KV solution and characterize when KV associators are Drinfeld's associators.

Theses & and Expository Work

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

Monash University, hosted by Dionne Ibarra, 27 July - 2 August 2026.
Sydney Mathematical Research Institute, University of Sydney, hosted by Zsuzsanna Dancso, June-July 2026.
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, Aug 2021 to Aug 2022.
Australian National University, hosted by Vigleik Angeltveit, May 2019-Aug 2019.

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