In preparation

• 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.
    Joint with Mikhail Khovanov, Haihan Wu, and Melissa Zhang
  
• 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.
    Joint with Aaron Lauda
  

Publications

• 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