Fu Liu


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.)

Publications and preprints

All of my research articles in reverse chronological order:

  1. The permuto-associahedron revisited, with Federico Castillo
    Submitted for publication.

  2. Ehrhart positivity of Tesler polytopes and Berline-Vergne's Valuation, with Yonggyu Lee
    Submitted for publication.

  3. 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 Aprodu-Farkas 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.

  4. On the Todd class of the permutohedral variety, with Federico Castillo
    Algebraic Combinatorics 4 (2021), no. 3, 387--407. 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 Berline-Vergne's construction for an Euler-Maclaurin 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 Xd is not effective for d ≥ 24.

  5. 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, 367--405.
         Uses limit linear series on chains of genus-1 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.

  6. h*-polynomials with roots on the unit circle, with Ben Braun
    Experimental Mathematics 30 (2021), no. 3, 332--348.
         Motivated by a theorem of Rodriguez-Villegas, 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.

  7. 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.

  8. On the relationship between Ehrhart unimodality and Ehrhart positivity, with Liam Solus
    Annals of Combinatorics 23 (2019), no. 2, 347--365.
         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.

  9. Stanley's non-Ehrhart-positive order polytopes, with Akiyoshi Tsuchiya
    Advances in Applied Mathematics 108 (2019), 1-10.
         Stanley provides an example of non-Ehrhart-positivity 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.

  10. On positivity of Ehrhart polynomials
    Recent Trends in Algebraic Combinatorics (2019), pp 189-237. 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.

  11. Berline-Vergne valuation and generalized permutohedra, with Federico Castillo
    Discrete and Computational Geometry 60 (2018), no. 4, 885-908. journal online version
         Continuing work in Ehrhart positivity for generalized permutohedra, we use Berline-Vergne's valuation to study our conjecture on the Ehrhart positivity of generalized permutohedra, and the stronger conjecture that the Berline-Vergne 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.

  12. Smooth polytopes with negative Ehrhart coefficients, with Federico Castillo, Benjamin Nill, and Andreas Paffenholz
    Journal of Combinatorial Theory Ser. A 160 (2018), 316-331.
         Bruns asked whether all smooth integral polytopes are Ehrhart positive. We show the answer is false by presenting counterexamples in dimensions 3 and higher.

  13. Severi degrees on toric surfaces, with Brian Osserman
    Journal fur die reine und angewandte Mathematik (Crelle's journal) 739 (2018), 121-158.
         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 Goettsche-Yau-Zaslow formula, and are used to give combinatorial expressions for the coefficients arising in the latter.

  14. A combinatorial analysis of Severi degrees
    Advances in Mathematics 298 (2016), 1-50. journal online version.
    Extended abstract, DMTCS proceedings BC (2016), 779-790.
         Based on results by Brugalle and Mikhalkin, Fomin and Mikhalkin give formulas for computing the classical Severi degree using long-edge graphs. Motivated by a conjecture of Block-Colley-Kennedy, we consider a special multivariate function associated to long-edge graphs, and show that this function is always linear.

  15. Ehrhart positivity for generalized permutohedra, with Federico Castillo
    DMTCS proceedings FPSAC'15 (2015), 865-876.
         We conjecture that the generalized permutohedra have positive Ehrhart coefficients, generalizing a conjecture by DeLoera-Haws-Koeppe. 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 Berlne-Vergne's valuation is positive on regular permutohedra. We then show our two conjectures hold for small dimension cases.

  16. On bijections between monotone rooted trees and the comb basis
    DMTCS proceedings FPSAC'15 (2015), 453-464.
         Gozalez D'Leon and Wachs, in their study of (co)homology of the poset of weighted partitions, asked whether there are nice bijections between RA,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 RA,i and the comb basis. We confirm the conjecture for the extreme cases where i=0 or n-1.

  17. A distributive lattice connected with arithmetic progressions of length three, with Richard P. Stanley
    Ramanujan Journal 36 (2015), no.1, 203-226.
         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.

  18. Factorizations of cycles and multi-noded rooted trees, with Rosena R.X. Du
    Graphs and Combinatorics 31 (2015), no. 3, 551-575.
         Pure-cycle 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 pure-cycle Hurwitz number is d r-3. We give the first desymmetrized bijective proof of this result by constructing a new class of combinatorial objects, multi-noded rooted trees, which generalize rooted trees.

  19. The lecture hall parallelepiped, with Richard P. Stanley.
    Annals of Combinatorics 18 (2014), no. 3, 473-488.
         We introduce the s-lecture hall parallelepiped, which we show can be used to find the Ehrhart polynomial of an s-lecture hall polytope. We define bijections between the lattice points inside s-lecture hall parallelepiped and fundamental combinatorial sets. Using these, we are able to show that the s-lecture hall polytope has the same Ehrhart polynomial as the unit cube when s = (n, n-1, ... , 1) or s = (1, 2, ... , n). (The latter case was first proved by Savage and Schuster.)

  20. Perturbation of transportation polytopes
    Journal of Combinatorial Theory Ser. A 120 (2013), no. 7, 1539--1561.
         We describe a perturbation method that can be used to compute the multivariate generating function (MGF) of a non-simple 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.

  21. Perturbation of central transportation polytopes of order kn × n
    DMTCS proceedings, FPSAC 24 (2012), 971-984.
         This is an extended abstract of a preliminary and slightly weaker version of this paper.

  22. Higher integrality conditions, volumes and Ehrhart polynomials
    Advances in Mathematics 226 (2011), no. 4, 3467-3494.
         We introduce the definition of k-integral polytopes, and show that the Ehrhart polynomial of a k-integral 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.

  23. Combinatorial bases for multilinear parts of free algebras with two compatible brackets
    Journal of Algebra 323 (2010), no. 1, 132-166.
         Let X be an ordered alphabet. Lie2(n) and P2(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 Lie2(n) and P2(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 Eil2(n) to Lie2(n), give a pairing between them, and show that the pairing is perfect.

  24. Moduli of crude limit linear series
    International Mathematics Research Notices 2009 (2009), no. 21, 4032-4050.
         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 Eisenbud-Harris construction. In the Eisenbud-Harris 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.

  25. A note on lattice-face polytopes and their Ehrhart polynomials
    Proceedings of the American Mathematical Society 137 (2009), no. 10, 3247-3258.
         We redefine lattice-face polytope by removing an unnecessary restriction in the old definition and show that the Ehrhart polynomial of a new lattice-face poltyope has the same simple form as the old ones. Furthermore, we show that the new family of lattice-face polytopes contains all possible combinatorial types of rational polytopes.

  26. A generating function for all semi-magic squares and the volume of the Birkhoff polytope with Jesus A. De Loera and Ruriko Yoshida
    Journal of Algebraic Combinatorics 30 (2009), no. 1, 113-139.
         We provide explicit combinatorial formulas for the Ehrhart polynomial and the volume of the polytope Bn of n × n doubly-stochastic 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 Bn from its MGF.

  27. Hook length polynomials for plane forests of a certain type
    Annals of Combinatorics 13 (2009), no. 3, 315-322.
         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.

  28. The irreducibility of certain pure-cycle Hurwitz spaces, with Brian Osserman
    American Journal of Mathematics 130 (2008), no. 6, 1687-1708.
         We use a combination of limit linear series arguments and group theory to prove the connectedness of genus-0 Hurwitz spaces in the “pure-cycle” 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 higher-genus case, requiring at least 3g simply branched points.

  29. Ehrhart polynomials of lattice-face polytopes
    Transactions of the American Mathematical Society 360 (2008), no. 6, 3041-3069.
         We define a new family of polytopes, lattice-face polytopes, which is a generalization of cyclic polytopes. We show that the Ehrhart polynomial of a lattice-polytope has the same simple form as cyclic polytopes. The main techniques of the proof include developing a way of decomposing a d-simplex 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.

  30. (k,m)-Catalan numbers and hook length polynomials for plane trees, with Rosena R.X. Du
    European Journal of Combinatorics 28 (2007), no. 4, 1312-1321.
         Motivated by a formula of Postnikov relating binary trees and a conjecture by Lascoux, we define the hook length polynomials for m-ary trees and plane forests. By introducing a new generalization of Catalan numbers, we show that our hook length polynomials have a simple binomial expression.

  31. Mochizuki's indigenous bundles and Ehrhart polynomials, with Brian Osserman
    Journal of Algebraic Combinatorics 23 (2006), no. 2, 125-136.
         We study a special case of crys-stable 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 Frobenius-destabilized vector bundles.

  32. Ehrhart polynomials of cyclic polytopes
    Journal of Combinatorial Theory Ser. A 111 (2005), no. 1, 111-127.
         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.