Man-Wai Cheung
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View article: Valuative independence and cluster theta reciprocity
Valuative independence and cluster theta reciprocity Open
We prove that theta functions constructed from positive scattering diagrams satisfy valuative independence. That is, for certain valuations $\operatorname{val}_{v}$, we have $\operatorname{val}_v(\sum_u c_u \vartheta_u)=\min_{c_u\neq 0} \o…
View article: Newton--Okounkov bodies and minimal models for cluster varieties
Newton--Okounkov bodies and minimal models for cluster varieties Open
Let $Y$ be a (partial) minimal model of a scheme $V$ with a cluster structure. Under natural assumptions, for every choice of seed we associate a Newton--Okounkov body to every divisor on $Y$ supported on $Y \setminus V$ and show that thes…
View article: Cluster structures for the A∞$A_\infty$ singularity
Cluster structures for the A∞$A_\infty$ singularity Open
We study a category of ‐graded maximal Cohen‐Macaulay (MCM) modules over the curve singularity and demonstrate that it has infinite type cluster combinatorics. In particular, we show that this Frobenius category (or a suitable subcategory)…
View article: Categories for Grassmannian Cluster Algebras of Infinite Rank
Categories for Grassmannian Cluster Algebras of Infinite Rank Open
We construct Grassmannian categories of infinite rank, providing an infinite analogue of the Grassmannian cluster categories introduced by Jensen, King, and Su. Each Grassmannian category of infinite rank is given as the category of graded…
View article: Clustering Cluster Algebras with Clusters
Clustering Cluster Algebras with Clusters Open
Classification of cluster variables in cluster algebras (in particular, Grassmannian cluster algebras) is an important problem, which has direct application to computations of scattering amplitudes in physics. In this paper, we apply the t…
View article: Cluster structures for the $A_{\infty}$ singularity
Cluster structures for the $A_{\infty}$ singularity Open
We study a category $\mathcal{C}_2$ of $\mathbb{Z}$-graded MCM modules over the $A_\infty$ curve singularity and demonstrate it has infinite type $A$ cluster combinatorics. In particular, we show that this Frobenius category (or a suitable…
View article: Scattering Diagrams from Holomorphic Discs in Log Calabi-Yau Surfaces
Scattering Diagrams from Holomorphic Discs in Log Calabi-Yau Surfaces Open
We construct special Lagrangian fibrations for log Calabi-Yau surfaces, and scattering diagrams from Lagrangian Floer theory of the fibres. Then we prove that the scattering diagrams recover the scattering diagrams of Gross-Pandharipande-S…
View article: Cluster scattering diagrams and theta functions for reciprocal generalized cluster algebras
Cluster scattering diagrams and theta functions for reciprocal generalized cluster algebras Open
We give a construction of generalized cluster varieties and generalized cluster scattering diagrams for reciprocal generalized cluster algebras, the latter of which were defined by Chekhov and Shapiro. These constructions are analogous to …
View article: Some examples of family Floer mirror
Some examples of family Floer mirror Open
In this article, we give explicit calculations for the family Floer mirrors of some non-compact Calabi-Yau surfaces. We compare it with the mirror construction of Gross-Hacking-Keel for suitably chosen log
\nCalabi-Yau pairs and the rank t…
View article: Compactifications of Cluster Varieties and Convexity
Compactifications of Cluster Varieties and Convexity Open
Gross–Hacking–Keel–Kontsevich [13] discuss compactifications of cluster varieties from positive subsets in the real tropicalization of the mirror. To be more precise, let ${\mathfrak {D}}$ be the scattering diagram of a cluster variety $V$…
View article: Some Examples of Family Floer Mirrors
Some Examples of Family Floer Mirrors Open
In this article, we give explicit calculations for the family Floer mirrors of some non-compact Calabi-Yau surfaces. We compare it with the mirror construction of Gross-Hacking-Keel for suitably chosen log Calabi-Yau pairs and the rank two…
View article: Categories for Grassmannian cluster algebras of infinite rank
Categories for Grassmannian cluster algebras of infinite rank Open
We construct Grassmannian categories of infinite rank, providing an infinite analogue of the Grassmannian cluster categories introduced by Jensen, King, and Su. Each Grassmannian category of infinite rank is given as the category of graded…
View article: Grassmannian categories of infinite rank
Grassmannian categories of infinite rank Open
We construct Grassmannian categories of infinite rank, providing an infinite analogue of the Grassmannian cluster categories introduced by Jensen, King, and Su. Each Grassmannian category of infinite rank is given as the category of graded…
View article: Quantization of deformed cluster Poisson varieties
Quantization of deformed cluster Poisson varieties Open
Fock and Goncharov described a quantization of cluster $\mathcal{X}$-varieties (also known as cluster Poisson varieties) in [FG09]. Meanwhile, families of deformations of cluster $\mathcal{X}$-varieties were introduced in [BFMNC18]. In thi…
View article: Theta functions and quiver Grassmannians
Theta functions and quiver Grassmannians Open
In this article, we use the relationship between cluster scattering diagrams and stability scattering diagrams to relate quiver representations with these diagrams. With a notion of positive crossing of a path $γ$, we show that if $γ$ has …
View article: The Model Orbit in $G_2$
The Model Orbit in $G_2$ Open
In this article, we decompose the ring of regular functions on the nilpotent orbit of dimension 8 for the complex $G_2$ in which every irreducible representation of $G_2$ appears exactly once. This confirms the predication of McGovern and …
View article: Tropical techniques in cluster theory and enumerative geometry
Tropical techniques in cluster theory and enumerative geometry Open
There are three parts in this thesis. First, we generalize the class of tropicalcurves from trivalent to 3-colorable which can be realized as the tropicalization of an algebraic curve whose non-archimedean skeleton is faithfully represente…