Fu Liu
Research
Selected talks
Selected talks that I have given on my work, which hopefully serve as a friendly overview of my research.
(Unfortunately and naturally, I don't have slides for some of my most recent work.)

Permutoassociahedra as deformations of nested permutohedra
MPI for Mathematics in the Sciences: (Polytop)ics: Recent advances on polytopes, Leipzig, Germany, 2021. (Online)
Related paper:
The permutoassociahedron revisited.

Combinatorics of nested Braid fan
MIT: Combinatorics Seminar, Cambridge, MA, 2018.
Related paper:
Deformation cones of nested Braid fans.

Uniqueness of BerlineVergne's valuation
AMS special session: Combinatorial Commutative Algebra and Polytopes, Joint Mathematics Meetings, San Diego, CA, 2018.
Related paper:
BerlineVergne valuation and generalized permutohedra.

Reflexive simplices with Ehrhart h^{*}polynomial roots on the unit circle
MSRI: Farewell Seminar, Berkeley, CA, 2017.
Related paper:
h^{*}polynomials with roots on the unit circle.

Ehrhart positivity and McMullen's formula
Oberwolfach: Miniworkshop: Lattice polytopes: methods, advances, applications, Oberwolfach, Germany, 2017
Related paper:
On positivity of Ehrhart polynomials,
BerlineVergne valuation and generalized permutohedra,
Ehrhart positivity for generalized permutohedra,
Smooth polytopes with negative Ehrhart coefficients.

A combinatorial analysis of Severi degrees
ICCM: The 7th International Congress of Chinese Mathematics, Beijing, China, 2016
Related paper:
A combinatorial analysis of Severi degrees,
Severi degrees on toric surfaces.

Ehrhart positivity for generalized permutohedra
RIMS: Workshop on Computational Commutative Algebra and Convex Polytopes, Kyoto, Japan, 2016.
Related paper:
BerlineVergne valuation and generalized permutohedra,
Ehrhart positivity for generalized permutohedra.

On bijections between rooted trees and the comb basis for the cohomology of the weighted partition poset
AMS special session: Topological Combinatorics and Combinatorial Commutative Algebra, San Francisco, CA, 2014.
Related paper:
On bijections between monotone rooted trees and the comb basis.

The lecture hall parallelopiped
AMS special session: Polyhedral Number Theory, San Francisco, CA, 2014.
Related paper:
The lecture hall parallelopiped.

Perturbation of transportation polytopes
RIMS: Workshop on convex polytope, Kyoto, Japan, 2012.
Related paper:
Perturbation of transportation polytopes,
Perturbation of central transportation polytopes of order kn × n.

Purecycle Hurwitz factorizations and multinoded rooted trees
CMS special session: Discrete Mathematics, Vancouver, British Columbia, Canada, 2010
Related paper:
Factorizations of cycles and multinoded rooted trees.

Higher integrality conditions and volumes of slices
UC Berkeley: Combinatorics Seminar, Berkeley, CA, 2010.
Related paper:
Higher integrality conditions, volumes and Ehrhart polynomials.

Volumes and Ehrhart polynomials of polytopes
East China Normal Universitiy: Combinatorics Seminar, Shanghai, China, 2008.
Related paper:
Ehrhart polynomials of cyclic polytopes,
Ehrhart polynomials of latticeface polytopes,
A note on latticeface polytopes and their Ehrhart polynomials.

Combinatorial bases for multilinear parts of free algebras with two compatible brackets
SMSAMS special session: Combinatorics and Discrete Dynamical Systems, Shanghai, China, 2008.
Related paper:
Combinatorial bases for multilinear parts of free algebras with two compatible brackets.

The volume of the Birkhoff polytope
Bay Area Discrete Math Day: San Francisco State University, San Francisco, CA, 2007.
Related paper:
A generating function for all semimagic squares and the volume of the Birkhoff polytope

On purecycle Hurwitz numbers
AMS special session: Algebraic Combinatorics, Fayetteville, AR, 2006.
Related paper:
The irreducibility of certain purecycle Hurwitz spaces.
Publications and preprints
All of my research articles in reverse chronological order:

The permutoassociahedron revisited, with Federico Castillo
Submitted for publication.

Ehrhart positivity of Tesler polytopes and BerlineVergne's Valuation, with Yonggyu Lee
Submitted for publication.

The strong maximal rank conjecture and moduli spaces of curves, with Brian Osserman, Montserrat Teixidor i Bigas and Naizhen Zhang
Submitted for publication.
Building on our recent work on the maximal rank conjecture, we prove two cases of the AproduFarkas strong maximal rank conjecture, which together with divisor class computations of Farkas implies that the moduli spaces of curves of genus 22 and genus 23 are of general type.

On the Todd class of the permutohedral variety, with Federico Castillo
Algebraic Combinatorics 4 (2021), no. 3, 387407. journal version
Extended abstract,
Proceedings of the 32nd Conference on Formal Power Series and Algebraic Combinatorics (Online), 84B (2020), Article #88, 12 pp.
In the special case of braid fans, we give a combinatorial formula for the BerlineVergne's construction for an EulerMaclaurin type formula that computes number of lattice points in polytopes.
By showing that this formula does not always have positive values, we prove that the Todd class of the permutohedral variety X_{d} is not effective for d ≥ 24.

Limit linear series and ranks of multiplication maps, with Brian Osserman, Montserrat Teixidor i Bigas and Naizhen Zhang
Transactions of the American Mathematical Society 374 (2021), no. 1, 367405.
Uses limit linear series on chains of genus1 curves to study multiplication maps in general, and more specifically to prove an elementary criterion for verifying cases of the Maximal Rank Conjecture. Applies the criterion to give a new proof of the Maximal Rank Conjecture for quadrics, and to prove various other ranges of cases of the conjecture.

h^{*}polynomials with roots on the unit circle, with Ben Braun
Experimental Mathematics 30 (2021), no. 3, 332348.
Motivated by a theorem of RodriguezVillegas, we investigate when the h^{*}polynomial of a special family of lattice simplices can be factored into geometric series, in which case Ehrhart positivity follows. Each lattice simplex is constructed based on a positive integer vector q. One of our main results is that that if q is supported on two distinct integers, there are three families of simplices with the desired property. Based on experimental evidence, we also provide both theoretical results and conjectures for cases when q has two or three distinct entries.

Deformation cones of nested Braid fans, with Federico Castillo
International Mathematics Research Notices (2020), rnaa090. journal version.
Generalized permutohedra are defined as polytoeps obtained from usual permutohedra by moving facets without passing vertices; or equivalently they are polytopes whose normal fan coarsens the Braid fan. We consider a refinement of the Braid fan, called the nested Braid fan, and construct generalized nested permutohedra which have the nested Braid fan refining their normal fan. We extend many results on generalized permutohedra to this new family of polytopes.

On the relationship between Ehrhart unimodality and Ehrhart positivity, with Liam Solus
Annals of Combinatorics 23 (2019), no. 2, 347365.
Answering a question by Postnikov during an MSRI workshop in 2017, we show that there is no general implication between the unimodality of h^{*}polynomials and positivity of Ehrhart polynomials of lattice polytopes in any dimension greater than two.

Stanley's nonEhrhartpositive order polytopes, with Akiyoshi Tsuchiya
Advances in Applied Mathematics 108 (2019), 110.
Stanley provides an example of nonEhrhartpositivity order polytope of dimension 21, answering an question posted on Mathoverflow. Stanley's example comes from a certain family of order polytopes. We determine the sign of each Ehrhart coefficient of each polytope in this family. As a consequence of this result, we conclude that there exists an order polytope that is not Ehrhart positivie for any dimension d ≥ 14, and for any positive integer k, there exists an order polytope whose Ehrhart polynomial has precisely k negative coefficients.

On positivity of Ehrhart polynomials
Recent Trends in Algebraic Combinatorics (2019), pp 189237. Association for Women in Mathematics Series, Vol 16. Springer, Cham.
Ehrhart polynomial counts the number of lattice points inside dilations of integral polytopes. We say a polytope is Ehrhart positive if all the coefficients of its Ehrhart polynomial are positive, and ask when a polytope is Ehrhart positive. In this article, we discuss techniques attacking this problem, and survey known results (positive and negative) on interesting famiies of polytopes.

BerlineVergne valuation and generalized permutohedra, with Federico Castillo
Discrete and Computational Geometry 60 (2018), no. 4, 885908. journal online version
Continuing work in Ehrhart positivity for generalized permutohedra, we use BerlineVergne's valuation to study our conjecture on the Ehrhart positivity of generalized permutohedra, and the stronger conjecture that the BerlineVergne valuation is positive on regular permutohedra. We show our conjectures hold for dimension up to 6, and for faces of codimension up to 3. We also give two equivalent statements to the second conjecture in terms of mixed valuations and Todd class, respectively.

Smooth polytopes with negative Ehrhart coefficients, with Federico Castillo, Benjamin Nill, and Andreas Paffenholz
Journal of Combinatorial Theory Ser. A 160 (2018), 316331.
Bruns asked whether all smooth integral polytopes are Ehrhart positive. We show the answer is false by presenting counterexamples in dimensions 3 and higher.

Severi degrees on toric surfaces, with Brian Osserman
Journal fur die reine und angewandte Mathematik (Crelle's journal) 739 (2018), 121158.
Builds on work of Brugalle and Mikhalkin, Ardila and Block, and the author to give universal formulas for the number of nodal curves in a linear system on a certain family of (possibly singular) toric surfaces. These formulas are explicitly related to the GoettscheYauZaslow formula, and are used to give combinatorial expressions for the coefficients arising in the latter.

A combinatorial analysis of Severi degrees
Advances in Mathematics 298 (2016), 150. journal online version.
Extended abstract,
DMTCS proceedings BC (2016), 779790.
Based on results by Brugalle and Mikhalkin, Fomin and Mikhalkin give formulas for computing the classical Severi degree using longedge graphs. Motivated by a conjecture of BlockColleyKennedy, we consider a special multivariate function associated to longedge graphs, and show that this function is always linear.

Ehrhart positivity for generalized permutohedra, with Federico Castillo
DMTCS proceedings FPSAC'15 (2015), 865876.
We conjecture that the generalized permutohedra have positive Ehrhart coefficients, generalizing a conjecture by DeLoeraHawsKoeppe. Using the combination of perturbation methods and a valuation on the algebra of rational pointed polyhedral cones constructed by Berline and Vergne, we reduce the conjecture to a new conjecture: the BerlneVergne's valuation is positive on regular permutohedra. We then show our two conjectures hold for small dimension cases.

On bijections between monotone rooted trees and the comb basis
DMTCS proceedings FPSAC'15 (2015), 453464.
Gozalez D'Leon and Wachs, in their study of (co)homology of the poset of weighted partitions, asked whether there are nice bijections between R_{A,i}, the set of rooted trees on A with i decreasing edges, and the comb basis or the Lyndon basis (for the cohomology). We give a natural definition for "nice bijections", and conjecture that there is a unique nice bijection between R_{A,i} and the comb basis. We confirm the conjecture for the extreme cases where i=0 or n1.

A distributive lattice connected with arithmetic progressions of length three, with Richard P. Stanley
Ramanujan Journal 36 (2015), no.1, 203226.
Proof of two enumerative conjectures of Noam Elkies arising from a problem contributed by Ron Graham to the Numberplay subblog of the New York Times Wordplay blog.

Factorizations of cycles and multinoded rooted trees, with Rosena R.X. Du
Graphs and Combinatorics 31 (2015), no. 3, 551575.
Purecycle Hurwitz numbers count the number of connected branched covers of the projective line where each branch point has only one ramification point over it. We prove that when the genus is 0 and one of the ramification indices is d, the degree of the covers, the purecycle Hurwitz number is d^{ r3}. We give the first desymmetrized bijective proof of this result by constructing a new class of combinatorial objects, multinoded rooted trees, which generalize rooted trees.

The lecture hall parallelepiped, with Richard P. Stanley.
Annals of Combinatorics 18 (2014), no. 3, 473488.
We introduce the slecture hall parallelepiped, which we show can be used to find the Ehrhart polynomial of an slecture hall polytope. We define bijections between the lattice points inside slecture hall parallelepiped and fundamental combinatorial sets. Using these, we are able to show that the slecture hall polytope has the same Ehrhart polynomial as the unit cube when s = (n, n1, ... , 1) or s = (1, 2, ... , n). (The latter case was first proved by Savage and Schuster.)

Perturbation of transportation polytopes
Journal of Combinatorial Theory Ser. A 120 (2013), no. 7, 15391561.
We describe a perturbation method that can be used to compute the multivariate generating function (MGF) of a nonsimple polyhedron, and then construct a perturbation that works for any transportation polytope. Applying this perturbation, we obtain combinatorial formulas for the MGF of the central transportation polytopes of order kn × n. We also recover the formula for the maximum possible number of vertices of transportation polytopes of order kn × n.

Perturbation of central transportation polytopes of order kn × n
DMTCS proceedings, FPSAC 24 (2012), 971984.
This is an extended abstract of a preliminary and slightly weaker version of this paper.

Higher integrality conditions, volumes and Ehrhart polynomials
Advances in Mathematics 226 (2011), no. 4, 34673494.
We introduce the definition of kintegral polytopes, and show that the Ehrhart polynomial of a kintegral polytope P has the properties that the coefficients in degrees of less than or equal to k are determined by a projection of P, and the coefficients in higher degrees are determined by slices of P. A key step of the proof is that under certain generality conditions, the volume of a polytope is equal to the sum of volumes of slices of the polytope.

Combinatorial bases for multilinear parts of free algebras with two compatible brackets
Journal of Algebra 323 (2010), no. 1, 132166.
Let X be an ordered alphabet. Lie_{2}(n) and P_{2}(n) respectively are the multilinear parts of the free Lie algebra and the free Poisson algebra respectively on X with a pair of compatible Lie brackets. We construct bases for Lie_{2}(n) and P_{2}(n) from combinatorial objects, the set of rooted trees, then prove the dimension formulas for these two algebras conjectured by B. Feigin. We also define a complementary space Eil_{2}(n) to Lie_{2}(n), give a pairing between them, and show that the pairing is perfect.

Moduli of crude limit linear series
International Mathematics Research Notices 2009 (2009), no. 21, 40324050.
Eisenbud and Harris introduced the theory of limit linear series, and constructed a space parametrizing their limit linear series. Osserman introduced a new space which compactifies the EisenbudHarris construction. In the EisenbudHarris space, the set of refined limit linear series is always dense on a general reducible curve. Osserman asks when the same is true for his space. We answer his question by characterizing the situations when the crude limit linear series contain an open subset of his space.

A note on latticeface polytopes and their Ehrhart polynomials
Proceedings of the American Mathematical Society 137 (2009), no. 10, 32473258.
We redefine latticeface polytope by removing an unnecessary restriction in the old definition and show that the Ehrhart polynomial of a new latticeface poltyope has the same simple form as the old ones. Furthermore, we show that the new family of latticeface polytopes contains all possible combinatorial types of rational polytopes.

A generating function for all semimagic squares and the volume of the Birkhoff polytope with Jesus A. De Loera and Ruriko Yoshida
Journal of Algebraic Combinatorics 30 (2009), no. 1, 113139.
We provide explicit combinatorial formulas for the Ehrhart polynomial and the volume of the polytope B_{n} of n × n doublystochastic matrices, also known as the Birkhoff polytope. We do this through finding the multivariate generating function (MGF) for the lattice points of the polytope. We also demonstrate that we can derive formulas for the volume of any face of B_{n} from its MGF.

Hook length polynomials for plane forests of a certain type
Annals of Combinatorics 13 (2009), no. 3, 315322.
We define the hook length polynomial for plane forests of a given degree sequence type and show it can be factored into a product of linear forms. Some other enumerative results on forests are also given.

The irreducibility of certain purecycle Hurwitz spaces, with Brian Osserman
American Journal of Mathematics 130 (2008), no. 6, 16871708.
We use a combination of limit linear series arguments and group theory to prove the connectedness of genus0 Hurwitz spaces in the “purecycle” case, which is to say, having a single ramified point (of any order) over each branch point.
In the case of four branch points, we also compute the associated Hurwitz numbers. Finally, we give a conditional result in the highergenus case, requiring at least 3g simply branched points.

Ehrhart polynomials of latticeface polytopes
Transactions of the American Mathematical Society 360 (2008), no. 6, 30413069.
We define a new family of polytopes, latticeface polytopes, which is a generalization of cyclic polytopes. We show that the Ehrhart polynomial of a latticepolytope has the same simple form as cyclic polytopes. The main techniques of the proof include developing a way of decomposing a dsimplex into d! signed sets, each of which corresponds to a permutation in the symmetric group, and giving an explicit formula for the number of lattice in each set.

(k,m)Catalan numbers and hook length polynomials for plane trees, with Rosena R.X. Du
European Journal of Combinatorics 28 (2007), no. 4, 13121321.
Motivated by a formula of Postnikov relating binary trees and a conjecture by Lascoux, we define the hook length polynomials for mary trees and plane forests. By introducing a new generalization of Catalan numbers, we show that our hook length polynomials have a simple binomial expression.

Mochizuki's indigenous bundles and Ehrhart polynomials, with Brian Osserman
Journal of Algebraic Combinatorics 23 (2006), no. 2, 125136.
We study a special case of crysstable bundles, relating their number to Ehrhart polynomials of certain polytopes. This allows us to obtain results in both algebraic geometry and combinatorics: we produce a family of examples of different polytopes with the same Ehrhart polynomials, and also show that the number of indigenous bundles in characteristic p is always counted by polynomials in p. Special cases of the latter result give application to rational functions, as well as to Frobeniusdestabilized vector bundles.

Ehrhart polynomials of cyclic polytopes
Journal of Combinatorial Theory Ser. A 111 (2005), no. 1, 111127.
We prove a conjecture by Beck, De Loera, Develin, Pfeifle and Stanley stating that the coefficients of the Ehrhart polynomial of an integral cyclic polytope are given by the volume of its lower envelopes.
