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  1. ; (, Journal of the American Mathematical Society)
    For any Kawamata log terminal (klt) singularity and any minimizer of its normalized volume function, we prove that the associated graded ring is always finitely generated, as conjectured by Chi Li. As a consequence, we complete the last step of establishing the Stable Degeneration Conjecture proposed by Chi Li and the first named author for an arbitrary klt singularity. 
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    Free, publicly-accessible full text available July 1, 2026
  2. (, Springer Nature Switzerland)
    Free, publicly-accessible full text available March 1, 2026
  3. ; ; ; (, Communications of the American Mathematical Society)
    Over a Cohen-Macaulay local ring, the minimal number of generators of a maximal Cohen-Macaulay module is bounded above by its multiplicity. In 1984 Ulrich [Math. Z. 188 (1984), pp. 23–32] asked whether there always exist modules for which equality holds; such modules are known nowadays as Ulrich modules. We answer this question in the negative by constructing families of two dimensional Cohen-Macaulay local rings that have no Ulrich modules. Some of these examples are Gorenstein normal domains; others are even complete intersection domains, though not normal. 
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    Free, publicly-accessible full text available January 1, 2026
  4. (, Inventiones mathematicae)
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  5. (, Journal of Algebraic Geometry)
    Recent study in K-stability suggests that Kawamata log terminal (klt) singularities whose local volumes are bounded away from zero should be bounded up to special degeneration. We show that this is true in dimension three, or when the minimal log discrepancies of Kollár components are bounded from above. We conjecture that the minimal log discrepancies of Kollár components are always bounded from above, and verify it in dimension three when the local volumes are bounded away from zero. We also answer a question from Han, Liu, and Qi on the relation between log canonical thresholds and local volumes. 
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  6. (, Geometry & Topology)
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  7. ; ; ; (, Forum of Mathematics, Pi)
    Abstract We prove an algebraic version of the Hamilton–Tian conjecture for all log Fano pairs. More precisely, we show that any log Fano pair admits a canonical two-step degeneration to a reduced uniformly Ding stable triple, which admits a Kähler–Ricci soliton when the ground field . 
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