14 citations · 56 across the 17 of their papers we have counts for
19 papers
Integer decomposition property of three-dimensional smooth polytopes: a shorter proof
Juliana Curtis, Gaku Liu
Oda's smoothness conjecture asserts that every smooth lattice polytope has the integer decomposition property. The three-dimensional case of this conjecture was recently proven by…
The Gröbner Version of White's Conjecture is False
Spencer Backman, Nathan Cheung, Michał Lasoń +2
We show that the toric ideal of the Fano matroid polytope does not have a quadratic Gröbner basis. This resolves in the negative a strong version of White's conjecture from matroid…
The geometric Merkurjev-Panin Conjecture for the Cox category
Daniel Erman, Andrew Hanlon, Gaku Liu +1
We show that a strong version of the geometric Merkurjev-Panin conjecture holds for the Cox category of a projective toric variety. That is, we prove that the full strong exception…
Mixed Eulerian numbers and beyond
Gaku Liu, Mateusz Michałek, Julian Weigert
We derive explicit formulas for the matroidal mixed Eulerian numbers. We resolve a question posed by Berget, Spink, and Tseng, demonstrating that the invariant defined by matroidal…
A Regular Unimodular Triangulation of the Matroid Base Polytope
Spencer Backman, Gaku Liu
We produce the first regular unimodular triangulation of an arbitrary matroid base polytope. We then extend our triangulation to integral generalized permutahedra. Prior to this wo…
-vector inequalities under weak maps
Gaku Liu, Alexander Mason
We study the behavior of -vectors associated to matroid complexes under weak maps, or inclusions of matroid polytopes. Specifically, we show that the -vector of the order com…
Unimodular triangulations of sufficiently large dilations
Gaku Liu
An integral polytope is a polytope whose vertices have integer coordinates. A unimodular triangulation of an integral polytope in is a triangulation in which all sim…
Normal crossing immersions, cobordisms and flips
Karim Adiprasito, Gaku Liu
We study various analogues of theorems from PL topology for cubical complexes. In particular, we characterize when two PL homeomorphic cubulations are equivalent by Pachner moves b…
Semistable reduction in characteristic 0
Karim Adiprasito, Gaku Liu, Michael Temkin
In 2000 Abramovich and Karu proved that any dominant morphism of varieties of characteristic zero can be made weakly semistable by replacing by a smooth alteration…
Log smoothness and polystability over valuation rings
Karim Adiprasito, Gaku Liu, Igor Pak +1
Let be a valuation ring of height one of residual characteristic exponent and with algebraically closed field of fractions. Our main result provides a best possib…
Zonotopes whose cellular strings are all coherent
Rob Edman, Pakawut Jiradilok, Gaku Liu +1
A cellular string of a polytope is a sequence of faces stacked on top of each other in a given direction. The poset of cellular strings, ordered by refinement, is known to be homot…
A counterexample to the extension space conjecture for realizable oriented matroids
Gaku Liu
The extension space conjecture of oriented matroid theory states that the space of all one-element, non-loop, non-coloop extensions of a realizable oriented matroid of rank has…
A zonotope and a product of two simplices with disconnected flip graphs
Gaku Liu
We give an example of a three-dimensional zonotope whose set of tight zonotopal tilings is not connected by flips. Using this, we show that the set of triangulations of $Δ^4 \times…
Flip-connectivity of triangulations of the product of a tetrahedron and simplex
Gaku Liu
A flip is a minimal move between two triangulations of a polytope. An open question is whether any two triangulations of the product of two simplices can be connected through a ser…
Refined dual stable Grothendieck polynomials and generalized Bender-Knuth involutions
Pavel Galashin, Darij Grinberg, Gaku Liu
The dual stable Grothendieck polynomials are a deformation of the Schur functions, originating in the study of the K-theory of the Grassmannian. We generalize these polynomials by…
Mixed volumes of hypersimplices
Gaku Liu
In this paper we consider mixed volumes of combinations of hypersimplices. These numbers, called "mixed Eulerian numbers", were first considered by A. Postnikov and were shown to s…
Structure theorem for U5-free tournaments
Gaku Liu
Let be the tournament with vertices , ..., such that , and if , and . In th…
Various Theorems on Tournaments
Gaku Liu
In this thesis we prove a variety of theorems on tournaments. A \emph{prime} tournament is a tournament such that there is no , , such that…
Trivariate monomial complete intersections and plane partitions
Charles Chen, Alan Guo, Xin Jin +1
We consider the homogeneous components U_r of the map on R = k[x,y,z]/(x^A, y^B, z^C) that multiplies by x + y + z. We prove a relationship between the Smith normal forms of submat…