arXiv subjects

Lauren Williams

Publications and source records attributed to Lauren Williams.

At least 19 recordsLinked to original sources

arXiv2026

Cluster algebras and tilings for the m=4 amplituhedron

The amplituhedron $A_{n,k,m}(Z)$ is the image of the positive Grassmannian $Gr_{k,n}^{\geq 0}$ under the map ${Z}: Gr_{k,n}^{\geq 0} \to Gr_{k,k+m}$ induced by a positive linear map $Z:\mathbb{R}^n \to \mathbb{R}^{k+m}$. Motivated by a question of Hodges, Arkani-Hamed and Trnka introduced the amplituhedron in 2013 as a geometric object whose tilings conjecturally encode the BCFW recursion for computing scattering amplitudes. More specifically, the expectation was that one can compute scattering amplitudes in ${N}=4$ SYM by tiling the $m=4$ amplituhedron $A_{n,k,4}(Z)$ - that is, decomposing the amplituhedron into 'tiles' (closures of images of $4k$-dimensional cells of $Gr_{k,n}^{\geq 0}$ on which ${Z}$ is injective) - and summing the 'volumes' of the tiles. Also in 2013, Golden-Goncharov-Spradlin-Vergu-Volovich gave the first link between scattering amplitudes and cluster algebras, with Drummond-Foster-Gurdogan subsequently formulating the {cluster adjacency conjecture}. In this article we reveal and prove the deep mechanism behind `cluster phenomena' in tree-level scattering amplitudes. By connecting the BCFW recursion to a new cluster quasi-homomorphism on the Grassmannian $\Gr_{4,n}$, we prove the {cluster adjacency conjecture} for BCFW tiles, which says that each tile is a semialgebraic subset of the amplituhedron where a collection of compatible cluster variables take on definite signs. In particular, the facets of these tiles are cut out by compatible cluster variables. We also use the cluster description of BCFW tiles to prove the {BCFW tiling conjecture}, resolving the main original conjecture for the $m=4$ amplituhedron.

math.CO↗

arXiv2026

The m=2 amplituhedron and the hypersimplex: signs, clusters, triangulations, Eulerian numbers

The hypersimplex $Δ_{k+1,n}$ is the image of the positive Grassmannian $Gr^{\geq 0}_{k+1,n}$ under the moment map. It is a polytope of dimension $n-1$ in $\mathbb{R}^n$. Meanwhile, the amplituhedron ${A}_{n,k,2}(Z)$ is the projection of the positive Grassmannian $Gr^{\geq 0}_{k,n}$ into $Gr_{k,k+2}$ under a map $\tilde{Z}$ induced by a matrix $Z\in \text{Mat}_{n,k+2}^{>0}$. Introduced in the context of scattering amplitudes, it is not a polytope, and has dimension $2k$. Nevertheless, there seem to be remarkable connections between these two objects via T-duality, as was first noted by Lukowski--Parisi--Williams (LPW). In this paper we use ideas from oriented matroid theory, total positivity, and the geometry of the hypersimplex and positroid polytopes to obtain a deeper understanding of the amplituhedron. We show that the inequalities cutting out positroid polytopes -- images of positroid cells of $Gr^{\geq 0}_{k+1,n}$ under the moment map -- translate into sign conditions characterizing the T-dual Grasstopes -- images of positroid cells of $Gr^{\geq 0}_{k,n}$ under $\tilde{Z}$. Moreover, we subdivide the amplituhedron into chambers, just as the hypersimplex can be subdivided into simplices, with both chambers and simplices enumerated by the Eulerian numbers. We prove the main conjecture of (LPW): a collection of positroid polytopes is a triangulation of $Δ_{k+1, n}$ if and only if the collection of T-dual Grasstopes is a triangulation of ${A}_{n,k,2}(Z)$ for all $Z$. Moreover, we prove Arkani-Hamed--Thomas--Trnka's conjectural sign-flip characterization of ${A}_{n,k,2}(Z)$, and Lukowski--Parisi--Spradlin--Volovich's conjectures on $m=2$ cluster adjacency and on generalized triangles (images of $2k$-dimensional positroid cells which map injectively into ${A}_{n,k,2}(Z)$). Finally, we introduce new cluster structures in the amplituhedron.

math.CO↗

arXiv2026

First Proof Second Batch

To assess the ability of current AI systems to correctly solve research-level mathematics problems, we tested several AI systems on a set of ten problems in a broad range of mathematical fields; these problems arose naturally in the research process of the contributors. This document includes the problems, our methodology, and the results of our testing. We provide links to supplementary documents including the human solutions, the AI-generated solutions, and the referee reports and logs for the AI-generated solutions. The ten problems were contributed by the following mathematicians: (1) Dariusz Kalociński and Theodore A. Slaman, (2) Richard Schwartz, (3) Aleksa Milojevic and Benny Sudakov, (4) Larry Guth, (5) Oleg Butkovsky, Jonathan Mattingly, and Lorenzo Zambotti, (6) Joshua Evan Greene and Duncan McCoy, (7) Sucharit Sarkar, (8) Sam Payne and Jidong (Jayden) Wang, (9) Sylvie Corteel and John Lentfer, (10) Srivatsav Kunnawalkam Elayavalli.

cs.AI↗

arXiv2026

First Proof

To assess the ability of current AI systems to correctly answer research-level mathematics questions, we share a set of ten math questions which have arisen naturally in the research process of the authors. The questions had not been shared publicly until now; the answers are known to the authors of the questions but will remain encrypted for a short time.

cs.AI↗

arXiv2026

Plabic Tangles and Cluster Promotion Maps

Inspired by the BCFW recurrence for tilings of the amplituhedron, we introduce the general framework of `plabic tangles' that utilizes plabic graphs to define rational maps between products of Grassmannians called `promotions'. The central conjecture of the paper is that promotion maps are quasi-cluster homomorphisms, which we prove for several classes of promotions. In order to define promotion maps, we utilize $m$-vector-relation configurations ($m$-VRCs) on plabic graphs. We relate $m$-VRCs to the degree (a.k.a `intersection number') of the amplituhedron map on positroid varieties and characterize all plabic trees with intersection number one and their VRCs. Finally, we show that promotion maps admit an operad structure and, supported by the class of `$4$-mass box' promotions, we point at new positivity properties for non-rational maps beyond cluster algebras. Promotion maps have important connections to the geometry and cluster structure of the amplituhedron and singularities of scattering amplitudes in planar $\mathcal{N}=4$ super Yang-Mills theory.

math.CO↗

arXiv2026

A probabilistic interpretation for interpolation Macdonald polynomials

Previous work of Ayyer, Martin, and Williams gave a probabilistic interpretation of the Macdonald polynomials $P_λ(x_1,\dots,x_n;1,t)$ at $q=1$ in terms of a Markov chain called the multispecies $t$-Push TASEP, a Markov chain involving particles of types $λ_1,\dots,λ_n$ hopping around a ring. In particular, they showed that for each composition $η$ obtained by permuting the parts of $λ$, the stationary probability of being in state $η$ is proportional to the ASEP polynomial $F_η(x_1,\dots,x_n; 1,t)$, and the normalizing constant (or partition function) is $P_λ(x_1,\dots,x_n; 1,t)$. There is an inhomogeneous generalization of Macdonald polynomials due to Knop and Sahi called interpolation Macdonald polynomials $P^*_λ(x_1,\dots,x_n;q,t)$, as well as an inhomogeneous generalization of ASEP polynomials called interpolation ASEP polynomials $F^*_η(x_1,\dots,x_n;q,t)$ that we introduced in previous work. In this article we introduce a new Markov chain called the interpolation $t$-Push TASEP, and show that its steady state probabilities and partition function are given by the interpolation ASEP polynomials and the interpolation Macdonald polynomial, evaluated at $q=1$. This generalizes the previous result of Ayyer, Martin, and Williams.

math.CO↗

arXiv2025

A combinatorial formula for Interpolation Macdonald polynomials

In 1996, Knop and Sahi introduced a remarkable family of inhomogeneous symmetric polynomials, defined via vanishing conditions, whose top homogeneous parts are exactly the Macdonald polynomials. Like the Macdonald polynomials, these interpolation Macdonald polynomials are closely connected to the Hecke algebra, and admit nonsymmetric versions, which generalize the nonsymmetric Macdonald polynomials. In this paper we give a combinatorial formula for interpolation Macdonald polynomials in terms of signed multiline queues; this formula generalizes the combinatorial formula for Macdonald polynomials in terms of multiline queues given by Corteel-Mandelshtam-Williams.

math.CO↗

arXiv2025

A superpotential for Grassmannian Schubert varieties

While mirror symmetry for flag varieties and Grassmannians has been extensively studied, Schubert varieties in the Grassmannian are singular, and hence standard mirror symmetry statements are not well-defined. Nevertheless, in this article we introduce a ``superpotential'' $W^λ$ for each Grassmannian Schubert variety $X_λ$, generalizing the Marsh-Rietsch superpotential for Grassmannians, and we show that $W^λ$ governs many toric degenerations of $X_λ$. We also generalize the ``polytopal mirror theorem'' for Grassmannians from our previous work: namely, for any cluster seed $G$ for $X_λ$, we construct a corresponding Newton-Okounkov convex body $Δ_G^λ$, and show that it coincides with the superpotential polytope $Γ_G^λ$, that is, it is cut out by the inequalities obtained by tropicalizing an associated Laurent expansion of $W^λ$. This gives us a toric degeneration of the Schubert variety $X_λ$ to the (singular) toric variety $Y(\mathcal{N}_λ)$ of the Newton-Okounkov body. Finally, for a particular cluster seed $G=G^λ_{\mathrm{rec}}$ we show that the toric variety $Y(\mathcal{N}_λ)$ has a small toric desingularisation, and we describe an intermediate partial desingularisation $Y(\mathcal{F}_λ)$ that is Gorenstein Fano. Many of our results extend to more general varieties in the Grassmannian.

math.AG↗

arXiv2025

Root polytopes, flow polytopes, and order polytopes

In this paper we study the class of polytopes which can be obtained by taking the convex hull of some subset of the points $\{e_i-e_j \ \vert \ i \neq j\} \cup \{\pm e_i\}$ in $\mathbb{R}^n$, where $e_1,\dots,e_n$ is the standard basis of $\mathbb{R}^n$. Such a polytope can be encoded by a quiver $Q$ with vertices $V \subseteq \{v_1,\dots,v_n\} \cup \{\star\}$, where each edge $v_j\to v_i$ or $\star \to v_i$ or $v_i\to \star$ gives rise to the point $e_i-e_j$ or $e_i$ or $-e_i$, respectively; we denote the corresponding polytope as $\operatorname{Root}(Q)$. These polytopes have been studied extensively under names such as edge polytope and root polytope. We show that if the quiver $Q$ is strongly-connected then the root polytope $\operatorname{Root}(Q)$ is reflexive and terminal; we moreover give a combinatorial description of the facets of $\operatorname{Root}(Q)$. We also show that if $Q$ is planar, then $\operatorname{Root}(Q)$ is (integrally equivalent to the) polar dual of the flow polytope of the dual quiver. Finally we consider the case that $Q$ comes from a ranked poset $P$, and show that $\operatorname{Root}(Q)$ is polar dual to (a translation of) a marked poset polytope. We then study the toric variety $Y(\mathcal{F}_Q)$ associated to the face fan $\mathcal{F}_Q$ of $\operatorname{Root}(Q)$. If $Q$ comes from a ranked poset $P$ we give a combinatorial description of the Picard group of $Y(\mathcal{F}_Q)$, and we show that $Y(\mathcal{F}_Q)$ is a small partial desingularisation of the Hibi toric variety $Y_{\mathcal{O}(P)}$ of the order polytope $\mathcal{O}(P)$. We show that $Y(\mathcal{F}_Q)$ has a small crepant toric resolution of singularities $Y(\widehat{\mathcal{F}}_Q)$, and as a consequence that the Hibi toric variety $Y_{\mathcal{O}(P)}$ has a small resolution of singularities for any ranked poset $P$. These results have applications to mirror symmetry.

math.CO↗

arXiv2025

BCFW tilings and cluster adjacency for the amplituhedron

In 2005, Britto, Cachazo, Feng and Witten gave a recurrence (now known as the BCFW recurrence) for computing scattering amplitudes in N=4 super Yang Mills theory. Arkani-Hamed and Trnka subsequently introduced the amplituhedron to give a geometric interpretation of the BCFW recurrence. Arkani-Hamed and Trnka conjectured that each way of iterating the BCFW recurrence gives a "triangulation" or "tiling" of the m=4 amplituhedron. In this article we prove the BCFW tiling conjecture of Arkani-Hamed and Trnka. We also prove the cluster adjacency conjecture for BCFW tiles of the amplituhedron, which says that facets of tiles are cut out by collections of compatible cluster variables for the Grassmannian Gr(4,n). Moreover we show that each BCFW tile is the subset of the Grassmannian where certain cluster variables have particular signs.

math.CO↗

arXiv2025

Introduction to Cluster Algebras. Chapter 7

This is a preliminary draft of Chapter 7 of our forthcoming textbook "Introduction to Cluster Algebras." Chapters 1-3 have been posted as arXiv:1608.05735. Chapters 4-5 have been posted as arXiv:1707.07190. Chapter 6 has been posted as arXiv:2008.09189. This installment contains: Chapter 7. Plabic graphs

math.CO↗

arXiv2025

Polyhedral and Tropical Geometry of Flag Positroids

A flag positroid of ranks $\boldsymbol{r}:=(r_1<\dots <r_k)$ on $[n]$ is a flag matroid that can be realized by a real $r_k \times n$ matrix $A$ such that the $r_i \times r_i$ minors of $A$ involving rows $1,2,\dots,r_i$ are nonnegative for all $1\leq i \leq k$. In this paper we explore the polyhedral and tropical geometry of flag positroids, particularly when $\boldsymbol{r}:=(a, a+1,\dots,b)$ is a sequence of consecutive numbers. In this case we show that the nonnegative tropical flag variety TrFl$_{\boldsymbol{r},n}^{\geq 0}$ equals the nonnegative flag Dressian FlDr$_{\boldsymbol{r},n}^{\geq 0}$, and that the points $\boldsymbolμ = (μ_a,\ldots, μ_b)$ of TrFl$_{\boldsymbol{r},n}^{\geq 0} =$ FlDr$_{\boldsymbol{r},n}^{\geq 0}$ give rise to coherent subdivisions of the flag positroid polytope $P(\underline{\boldsymbolμ})$ into flag positroid polytopes. Our results have applications to Bruhat interval polytopes: for example, we show that a complete flag matroid polytope is a Bruhat interval polytope if and only if its $(\leq 2)$-dimensional faces are Bruhat interval polytopes. Our results also have applications to realizability questions. We define a positively oriented flag matroid to be a sequence of positively oriented matroids $(χ_1,\dots,χ_k)$ which is also an oriented flag matroid. We then prove that every positively oriented flag matroid of ranks $\boldsymbol{r}=(a,a+1,\dots,b)$ is realizable.

math.CO↗

arXiv2024

Introduction to Cluster Algebras. Chapters 1-3

This is a preliminary draft of Chapters 1-3 of our forthcoming textbook "Introduction to Cluster Algebras." This installment contains: Chapter 1. Total positivity Chapter 2. Mutations of quivers and matrices Chapter 3. Clusters and seeds

math.CO↗

arXiv2024

The Magic Number Conjecture for the $m=2$ amplituhedron and Parke-Taylor identities

The amplituhedron $A_{n,k,m}$ is a geometric object introduced in the context of scattering amplitudes in $N=4$ super Yang Mills. It generalizes the positive Grassmannian (when $n=k+m$), cyclic polytopes (when $k=1$), and the bounded complex of the cyclic hyperplane arrangement (when $m=1$). Of substantial interest are the tilings of the amplituhedron, which are analogous to triangulations of a polytope. Karp, Williams and Zhang (2020) observed that the known tilings of $A_{n,k,2}$ have cardinality ${n-2 \choose k}$ and the known tilings of $A_{n,k,4}$ have cardinality the Narayana number $\frac{1}{n-3}{n-3 \choose k+1}{n-3 \choose k}$; generalizing these observations, they conjectured that for even $m$ the tilings of $A_{n, k,m}$ have cardinality the MacMahon number, the number of plane partitions which fit inside a $k \times (n-k-m) \times \frac{m}{2}$ box. We refer to this prediction as the `Magic Number Conjecture'. In this paper we prove the Magic Number Conjecture for the $m=2$ amplituhedron: that is, we show that each tiling of $A_{n,k,2}$ has cardinality ${n-2 \choose k}$. We prove this by showing that all positroid tilings of the hypersimplex $Δ_{k+1,n}$ have cardinality ${n-2 \choose k}$, then applying T-duality. In addition, we give combinatorial necessary conditions for tiles to form a tiling of $A_{n,k,2}$; we give volume formulas for Parke-Taylor polytopes and certain positroid polytopes in terms of circular extensions of cyclic partial orders; and we prove new variants of the classical Parke-Taylor identities.

math.CO↗

arXiv2024

Rhombic staircase tableaux and Koornwinder polynomials

In this article we give a combinatorial formula for a certain class of Koornwinder polynomials, also known as Macdonald polynomials of type $\tilde{C}$. In particular, we give a combinatorial formula for the Koornwinder polynomials $K_λ = K_λ(z_1,\dots,z_N; a,b,c,d; q,t)$, where $λ= (1,\dots,1,0,\dots,0)$. We also give combinatorial formulas for all ``open boundary ASEP polynomials'' $F_μ$, where $μ$ is a composition in $\{-1,0,1\}^N$; these polynomials are related to the nonsymmetric Koornwinder polynomials $E_μ$ up to a triangular change of basis. Our formulas are in terms of rhombic staircase tableaux, certain tableaux that we introduced in previous work to give a formula for the stationary distribution of the two-species asymmetric simple exclusion process (ASEP) on a line with open boundaries.

math.CO↗

arXiv2024

The inhomogeneous $t$-PushTASEP and Macdonald polynomials

We study a multispecies $t$-PushTASEP system on a finite ring of $n$ sites with site-dependent rates $x_1,\dots,x_n$. Let $λ=(λ_1,\dots,λ_n)$ be a partition whose parts represent the species of the $n$ particles on the ring. We show that for each composition $η$ obtained by permuting the parts of $λ$, the stationary probability of being in state $η$ is proportional to the ASEP polynomial $F_η(x_1,\dots,x_n; q,t)$ at $q=1$; the normalizing constant (or partition function) is the Macdonald polynomial $P_λ(x_1,\dots,x_n;q,t)$ at $q=1$. Our approach involves new relations between the families of ASEP polynomials and of non-symmetric Macdonald polynomials at $q=1$. We also use multiline diagrams, showing that a single jump of the PushTASEP system is closely related to the operation of moving from one line to the next in a multiline diagram. We derive symmetry properties for the system under permutation of its jump rates, as well as a formula for the current of a single-species system.

math.CO↗

arXiv2024

A cluster of results on amplituhedron tiles

The amplituhedron is a mathematical object which was introduced to provide a geometric origin of scattering amplitudes in $\mathcal{N}=4$ super Yang Mills theory. It generalizes \emph{cyclic polytopes} and the \emph{positive Grassmannian}, and has a very rich combinatorics with connections to cluster algebras. In this article we provide a series of results about tiles and tilings of the $m=4$ amplituhedron. Firstly, we provide a full characterization of facets of BCFW tiles in terms of cluster variables for $\mbox{Gr}_{4,n}$. Secondly, we exhibit a tiling of the $m=4$ amplituhedron which involves a tile which does not come from the BCFW recurrence -- the \emph{spurion} tile, which also satisfies all cluster properties. Finally, strengthening the connection with cluster algebras, we show that each standard BCFW tile is the positive part of a cluster variety, which allows us to compute the canonical form of each such tile explicitly in terms of cluster variables for $\mbox{Gr}_{4,n}$. This paper is a companion to our previous paper ``Cluster algebras and tilings for the $m=4$ amplituhedron''.

math.CO↗

arXiv2022

Schubert polynomials, the inhomogeneous TASEP, and evil-avoiding permutations

Consider a lattice of n sites arranged around a ring, with the $n$ sites occupied by particles of weights $\{1,2,\dots,n\}$; the possible arrangements of particles in sites thus corresponds to the $n!$ permutations in $S_n$. The inhomogeneous totally asymmetric simple exclusion process (or TASEP) is a Markov chain on the set of permutations, in which two adjacent particles of weights $i<j$ swap places at rate $x_i - y_{n+1-j}$ if the particle of weight $j$ is to the right of the particle of weight $i$. (Otherwise nothing happens.) In the case that $y_i=0$ for all $i$, the stationary distribution was conjecturally linked to Schubert polynomials by Lam-Williams, and explicit formulas for steady state probabilities were subsequently given in terms of multiline queues by Ayyer-Linusson and Arita-Mallick. In the case of general $y_i$, Cantini showed that $n$ of the $n!$ states have probabilities proportional to products of double Schubert polynomials. In this paper we introduce the class of evil-avoiding permutations, which are the permutations avoiding the patterns $2413, 4132, 4213$ and $3214$. We show that there are $\frac{(2+\sqrt{2})^{n-1}+(2-\sqrt{2})^{n-1}}{2}$ evil-avoiding permutations in $S_n$, and for each evil-avoiding permutation $w$, we give an explicit formula for the steady state probability $ψ_w$ as a product of double Schubert polynomials. We also show that the Schubert polynomials that arise in these formulas are flagged Schur functions, and give a bijection in this case between multiline queues and semistandard Young tableaux.

math.CO↗