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Schubert Calculus on Newton–Okounkov Polytopes
A Newton–Okounkov polytope of a complete flag variety can be turned into a convex geometric model for Schubert calculus. Namely, we can represent...
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Newton–Okounkov theory in an abstract setting
We extend the theory of Newton–Okounkov bodies, originally developed by Boucksom–Chen, Kaveh–Khovanskii, and Lazarsfeld–Mustaţă for lattice...
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On the Mori theory and Newton–Okounkov bodies of Bott–Samelson varieties
We prove that on a Bott–Samelson variety X every movable divisor is nef. This enables us to consider Zariski decompositions of effective divisors,...
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Seshadri stratifications and standard monomial theory
We introduce the notion of a Seshadri stratification on an embedded projective variety. Such a structure enables us to construct a Newton-Okounkov...
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The Convex-Set Algebra and Intersection Theory on the Toric Riemann-Zariski Space
In previous work of the author, a top intersection product of toric b -divisors on a smooth complete toric variety is defined. It is shown that a nef...
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Newton–Okounkov bodies on projective bundles over curves
In this article, we study Newton–Okounkov bodies on projective vector bundles over curves. Inspired by Wolfe’s estimates used to compute the volume...
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Polyhedral realizations of crystal bases and convex-geometric Demazure operators
The main object in this paper is a family of rational convex polytopes whose lattice points give a polyhedral realization of a highest weight crystal...
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The Gromov Width of Bott-Samelson Varieties
We prove that the Gromov width of any Bott-Samelson variety associated to a reduced expression and equipped with a rational Kähler form equals the...
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Combinatorial mutations and block diagonal polytopes
Matching fields were introduced by Sturmfels and Zelevinsky to study certain Newton polytopes, and more recently have been shown to give rise to...
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Historical steps of development of convexity as a field
In this chapter we will show historical steps of the development of convexity as a field and, in addition, developments of the relations between...
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High-Dimensional Convex Sets Arising in Algebraic Geometry
We introduce an asymptotic notion of positivity in algebraic geometry that turns out to be related to some high-dimensional convex sets. The...
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Notes on Local Positivity and Newton–Okounkov Bodies
We explore the notion of local numerical equivalence in higher dimension and its relationship with Newton–Okounkov bodies with respect to flags...
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Newton–Okounkov Bodies and Reified Valuations of Higher Rank
We study the shape change of the Newton–Okounkov body of a fixed divisor D with respect to a valuation v moving in a suitable space of (higher-rank)...
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Beyond this book
In this final chapter, we outline some of the natural directions of further study for a reader of this book, and point out a few interesting recent...
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Geometry of Hessenberg varieties with applications to Newton–Okounkov bodies
In this paper, we study the geometry of various Hessenberg varieties in type A, as well as families thereof. Our main results are as follows. We find...
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Birational Maps to Grassmannians, Representations and Poset Polytopes
We study the closure of the graph of the birational map from a projective space to a Grassmannian. We provide explicit description of the graph...
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Generalized flatness constants, spanning lattice polytopes, and the Gromov width
In this paper we motivate some new directions of research regarding the lattice width of convex bodies. We show that convex bodies of sufficiently...
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Newton–Okounkov Bodies of Exceptional Curve Plane Valuations Non-positive at Infinity
In this note we announce a result determining the Newton–Okounkov bodies of the line bundle...
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Seshadri stratification for Schubert varieties and standard monomial theory
The theory of Seshadri stratifications has been developed by the authors with the intention to build up a new geometric approach towards a standard...
