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  1. ; ; (, Transactions of the American Mathematical Society)
    We introduce a topological intersection number for an ordered pair of SL 3 \operatorname {SL}_3 -webs on a decorated surface. Using this intersection pairing between reduced ( SL 3 , A ) (\operatorname {SL}_3,\mathcal {A}) -webs and a collection of ( SL 3 , X ) (\operatorname {SL}_3,\mathcal {X}) -webs associated with the Fockā€“Goncharov cluster coordinates, we provide a natural combinatorial interpretation of the bijection from the set of reduced ( SL 3 , A ) (\operatorname {SL}_3,\mathcal {A}) -webs to the tropical set A PGL 3 , S ^<#comment/> + ( Z t ) \mathcal {A}^+_{\operatorname {PGL}_3,\hat {S}}(\mathbb {Z}^t) , as established by Douglas and Sun in [Forum Math. Sigma 12 (2024), p. e5, 55]. We provide a new proof of the flip equivariance of the above bijection, which is crucial for proving the Fockā€“Goncharov duality conjecture of higher TeichmĆ¼ller spaces for SL 3 \operatorname {SL}_3 . 
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    Free, publicly-accessible full text available August 1, 2026
  2. ; ; ; ; ; (, Journal of the American Mathematical Society)
    We show the existence of cluster A \mathcal {A} -structures and cluster Poisson structures on any braid variety, for any simple Lie group. The construction is achieved via weave calculus and a tropicalization of Lusztigā€™s coordinates. Several explicit seeds are provided and the quiver and cluster variables are readily computable. We prove that these upper cluster algebras equal their cluster algebras, show local acyclicity, and explicitly determine their DT-transformations as the twist automorphisms of braid varieties. The main result also resolves the conjecture of B. Leclerc [Adv. Math. 300 (2016), pp. 190ā€“228] on the existence of cluster algebra structures on the coordinate rings of open Richardson varieties. 
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    Free, publicly-accessible full text available April 1, 2026
  3. ; ; (, Geometric and Functional Analysis)
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  4. ; ; (, Advances in Mathematics)
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