Christopher Manon

Bio: Christopher Manon is an academic researcher from University of Kentucky. The author has contributed to research in topics: Tropical geometry & Affine variety. The author has an hindex of 11, co-authored 54 publications receiving 480 citations. Previous affiliations of Christopher Manon include George Mason University & University of California, Berkeley.

Khovanskii bases, higher rank valuations and tropical geometry

Abstract: Given a finitely generated algebra $A$, it is a fundamental question whether $A$ has a full rank discrete (Krull) valuation $\mathfrak{v}$ with finitely generated value semigroup. We give a necessary and sufficient condition for this, in terms of tropical geometry of $A$. In the course of this we introduce the notion of a Khovanskii basis for $(A, \mathfrak{v})$ which provides a framework for far extending Grobner theory on polynomial algebras to general finitely generated algebras. In particular, this makes a direct connection between the theory of Newton-Okounkov bodies and tropical geometry, and toric degenerations arising in both contexts. We also construct an associated compactification of $Spec(A)$. Our approach includes many familiar examples such as the Gel'fand-Zetlin degenerations of coordinate rings of flag varieties as well as wonderful compactifications of reductive groups. We expect that many examples coming from cluster algebras naturally fit into our framework.

Khovanskii Bases, Higher Rank Valuations, and Tropical Geometry

Abstract: Given a finitely generated algebra $A$, it is a fundamental question whether $A$ has a full rank discrete (Krull) valuation $\mathfrak{v}$ with finitely generated value semigroup. We give a necessa...

The Toric Geometry of Triangulated Polygons in Euclidean Space

Abstract: Speyer and Sturmfels (SpSt) associated Grobner toric degenerations Gr2(C n ) T of Gr2(C n ) to each trivalent tree T with n leaves. These degenerations induce toric degenerations M T r of Mr, the space of n ordered, weighted (by r) points on the projective line. Our goal in this paper is to give a geometric (Euclidean polygon) description of the toric fibers as stratified symplectic spaces and describe the action of the compact part of the torus as "bendings of polygons." We prove the conjecture of Foth and Hu (FH) that the toric fibers are homeomorphic to the spaces defined by Kamiyama and Yoshida (KY).

The Algebra of Conformal Blocks

Abstract: For each simply connected, simple complex group $G$ we show that the direct sum of all vector bundles of conformal blocks on the moduli stack $\bar{\mathcal{M}}_{g, n}$ of stable marked curves carries the structure of a flat sheaf of commutative algebras. The fiber of this sheaf over a smooth marked curve $(C, \vec{p})$ agrees with the Cox ring of the moduli of quasi-parabolic principal $G-$bundles on $(C, \vec{p})$. We use the factorization rules on conformal blocks to produce flat degenerations of these algebras. These degenerations are toric in the case $G = SL_2(\mathbb{C}),$ and the resulting toric varieties are shown to be isomorphic to phylogenetic algebraic varieties from mathematical biology. We conclude with a proof that the Cox ring of the moduli stack of qausi-parabolic $SL_2(\mathbb{C})$ principal bundles over a generic curve is generated by conformal blocks of levels 1 and 2 with relations generated in degrees $2, 3,$ and 4.

The Algebra of Conformal Blocks

Abstract: For each simply connected, simple complex group $G$ we show that the direct sum of all vector bundles of conformal blocks on the moduli stack $\bar{\mathcal{M}}_{g, n}$ of stable marked curves carries the structure of a flat sheaf of commutative algebras. The fiber of this sheaf over a smooth marked curve $(C, \vec{p})$ agrees with the Cox ring of the moduli of quasi-parabolic principal $G-$bundles on $(C, \vec{p})$. We use the factorization rules on conformal blocks to produce flat degenerations of these algebras. These degenerations are toric in the case $G = SL_2(\mathbb{C}),$ and the resulting toric varieties are shown to be isomorphic to phylogenetic algebraic varieties from mathematical biology. We conclude with a proof that the Cox ring of the moduli stack of qausi-parabolic $SL_2(\mathbb{C})$ principal bundles over a generic curve is generated by conformal blocks of levels 1 and 2 with relations generated in degrees $2, 3,$ and 4.

Phd by thesis

Abstract: Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most ancient clastic carbonate slope deposits fit the submarine fan systems better. Calciclastic submarine fans are consequently rarely described and are poorly understood. Subsequently, very little is known especially in mud-dominated calciclastic submarine fan systems. Presented in this study are a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) that reveals a >250 m thick calciturbidite complex deposited in a calciclastic submarine fan setting. Seven facies are recognised from core and thin section characterisation and are grouped into three carbonate turbidite sequences. They include: 1) Calciturbidites, comprising mostly of highto low-density, wavy-laminated bioclast-rich facies; 2) low-density densite mudstones which are characterised by planar laminated and unlaminated muddominated facies; and 3) Calcidebrites which are muddy or hyper-concentrated debrisflow deposits occurring as poorly-sorted, chaotic, mud-supported floatstones. These

Integrable systems, toric degenerations and Okounkov bodies

Abstract: Let \(X\) be a smooth projective variety of dimension \(n\) over \(\mathbb {C}\) equipped with a very ample Hermitian line bundle \(\mathcal {L}\). In the first part of the paper, we show that if there exists a toric degeneration of \(X\) satisfying some natural hypotheses (which are satisfied in many settings), then there exists a surjective continuous map from \(X\) to the special fiber \(X_0\) which is a symplectomorphism on an open dense subset \(U\). From this we are then able to construct a completely integrable system on \(X\) in the sense of symplectic geometry. More precisely, we construct a collection of real-valued functions \(\{H_1, \ldots , H_n\}\) on \(X\) which are continuous on all of \(X\), smooth on an open dense subset \(U\) of \(X\), and pairwise Poisson-commute on \(U\). Moreover, our integrable system in fact generates a Hamiltonian torus action on \(U\). In the second part, we show that the toric degenerations arising in the theory of Newton-Okounkov bodies satisfy all the hypotheses of the first part of the paper. In this case the image of the ‘moment map’ \(\mu = (H_1, \ldots , H_n): X \rightarrow \mathbb {R}^n\) is precisely the Newton-Okounkov body \(\Delta = \Delta (R, v)\) associated to the homogeneous coordinate ring \(R\) of \(X\), and an appropriate choice of a valuation \(v\) on \(R\). Our main technical tools come from algebraic geometry, differential (Kahler) geometry, and analysis. Specifically, we use the gradient-Hamiltonian vector field, and a subtle generalization of the famous Łojasiewicz gradient inequality for real-valued analytic functions. Since our construction is valid for a large class of projective varieties \(X\), this manuscript provides a rich source of new examples of integrable systems. We discuss concrete examples, including elliptic curves, flag varieties of arbitrary connected complex reductive groups, spherical varieties, and weight varieties.

Crystal bases and Newton–Okounkov bodies

Abstract: Let G be a connected reductive algebraic group. We prove that the string parameterization of a crystal basis for a finite-dimensional irreducible representation of G extends to a natural valuation on the field of rational functions on the flag variety G/B, which is a highest-term valuation corresponding to a coordinate system on a Bott–Samelson variety. This shows that the string polytopes associated to irreducible representations, can be realized as Newton–Okounkov bodies for the flag variety. This is closely related to an earlier result of Okounkov for the Gelfand–Cetlin polytopes of the symplectic group. As a corollary, we recover a multiplicativity property of the canonical basis due to Caldero. We generalize the results to spherical varieties. From these the existence of SAGBI bases for the homogeneous coordinate rings of flag and spherical varieties, as well as their toric degenerations, follow recovering results by Alexeev and Brion, Caldero, and the author.