In preparation

• One clean qubit and the analytic torsion of quantum flat bundles Show that estimating the torsion of a family of flat bundles on closed hyperbolic 3-manifolds is DQC1 complete.
    Joint with Filippo Iulianelli, Aaron Lauda, and Alexander Schmidhuber
  
• Categorification of Kekule relations Providing a foam categorification of Kekule relations from Morrison's thesis. Studying the indecomposability in the Karoubi envelope of the web category.
    Joint with Ziyi Lei, and Emmanuel Wagner
  
• 2-action of categorified quantum groups on annular web algebras Extending the action of Ehrig, and Tubbenhauer to the annular case.
    Joint with Ross Akhmechet
  

Publications

• Spectral geometry of Khovanov Laplacians Introducing an inner product on rational Khovanov chain complexes leads to the notion of Khovanov Laplacians. The zero eigenvalues correspond to homology. Higher eigenvalues hold more geometric information. We study the spectral gaps of some knot diagrams, Reidemeister torsion, and recover the Lee spectral sequence.
    (arXiv:2608.24298)
    Joint with Aaron Lauda
  
• 6-valent vertex in the GL(N) web category and its categorification Using Robert and Wagner's GL(N) foam evaluation to N = 4 and N = 5, we decompose the benzene web in the Karoubi envelope of the Web category using categorification. A similar decomposition is conjectured for larger N and we outline the first steps towards the construction of a nice web basis in higher ranks.
    (arXiv:2608.21566)
    Joint with Mikhail Khovanov, Haihan Wu, and Melissa Zhang
  
• Analytic Torsion and Spectral Gap Capture Persistent-Laplacian Performance Persistent Laplacians offer a richer representation of data than persistent homology, utilizing them for learning tasks is often hampered by high dimensionality and the "varying length" of feature vectors. We propose a compact spectral representation that distills the persistent Laplacian into three mathematically grounded invariants: Betti numbers, the spectral gap, and analytic torsion.
    (arXiv:2606.16990)
    Joint with Aaron Lauda
  
• Action of the Witt algebra on categorified quantum groups We construct an action of the positive Witt algebra on the categorified simply-laced quantum groups. This action recovers the action of Qi, Robert, Sussan, and Wagner on foams. This action is compatible with the trace decategorification.
    (arXiv:2507.01877), Quantum Topology (DOI:10.4171/QT/251)
    Joint with Aaron Lauda