Recent works have initiated the study of open r--spin and open Fan-Jarvis-Ruan-Witten (FJRW) intersection theories, and related them to integrable hierarchies and mirror symmetry. This paper uses a new technique, the point insertion technique, to define new open r--spin and open FJRW intersection theories. These new constructions provide candidates for theories whose existence was conjectured before: 1. Hori predicted the existence of open r--spin theory with [ r r2 j types of boundary states. The previously constructed open r--spin theory (Buryak-Clader-Tessler 2018) has only boundary states of one type. In this work we describe [ [r [r2 j open r--spin theories, labelled by Cj E {0, ... , [ r r2 j-1}, where the Cjth one has Cj + 1 types of boundary states. We prove that the Cj = 0 theory is equivalent to the previous construction, and calculate all intersection numbers for all these theories. 2. Aleshkin and Liu conjectured the existence of a quintic Fermat FJRW theory. We construct such an FJRW theory, and provide evidence that this is the conjectured theory. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

We survey the recent progress in defining open enumerative theories for Landau-Ginzburg models. We illustrate the ideas required to develop these new foundations. In particular, we describe how to define the open enumerative invariants as integrals of multisections of certain vector bundles over a moduli space that is a real orbifold with corners, after prescribing boundary conditions for the multisections. We then explain the known situations where the open invariants satisfy certain forms of topological recursion relations, integrable hierarchies, or mirror symmetry. We end with a list of open questions and problems.

The article constructs the Lagrangian analog of the amplituhedron, studies its geometric properties and BCFW decompositions.

We construct the g=1 sector of the open r-spin theory, that is, an open r-spin theory on the moduli space of cylinders. This is the second construction of a g>0 open intersection theory, which includes descendents (the first is the all genus construction of the intersection theory on moduli of open Riemann surfaces with boundaries [23,30], whose g=1 case equals to the r=2 case of our construction). Unlike the construction of [30], in order to construct the r-spin cylinder theory we had to overcome the foundational problem of dimension jump loci, which in analogous closed theories has been treated using virtual fundamental class techniques, that are currently absent in the open setting. For this reason our construction is much more involved, and relies on the point insertion technique developed in [31,32]. We prove that the open g=1 potential equals, after a coordinate change, to the g=1 part of the Gelfand-Dikii wave function, thus confirming a conjecture of [7]. We also prove that our g=1 intersection numbers satisfy a g=1 recursion, also predicted in [7,15]. This recursion is the g=1 analogue of Solomon's famous g=0 Open WDVV equation [25], with descendents, and is also the universal g=1 recursion for F-Cohomological field theories [1]. Again, this is first geometric construction which is not the g=1 sector of [23,30], proven to satisfy this universal recursion.

We give a natural definition of open Hurwitz numbers, where the weight of each ramified covering includes an integer parameter N taken to the power that is equal to the number of boundary components of a Riemann surface with boundary mapping to CP1.We prove that the resulting sequence of partition functions, depending on N is an element of Z, is a tau-sequence of the mKP hierarchy, or in other words it is a sequence of tau-functions of the KP hierarchy where each tau-function is obtained from the previous one by a B & auml;cklund-Darboux transformation. Our result is motivated by a previous observation of Alexandrov and the first two authors that the refined intersection numbers on the moduli spaces of Riemann surfaces with boundary give a tau-sequence of the mKP hierarchy. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

The Amplituhedron is a subspace of the Grassmannian that was recently defined by Arkani-Hamed and Trnka in their study of scattering amplitudes in planar $\mathcal{N}=4$ super Yang Mills theory (arXiv:1312.2007), and was the subject of many papers in the last decade. In this work we define a tropical analog of the amplituhedron, and develop techniques to address it. We prove that many of the key properties of the amplituhedron hold also in this simpler, piecewise linear, model.

We construct an open enumerative theory for the Landau-Ginzburg (LG) model $(\mathbb{C}^2, \mu_r\times \mu_s, x^r+y^s)$. The invariants are defined as integrals of multisections of a Witten bundle with descendents over a moduli space that is a real orbifold with corners. In turn, a generating function for these open invariants yields the mirror LG model and a versal deformation of it with flat coordinates. After establishing an open topological recursion result, we prove an LG/LG open mirror symmetry theorem in dimension two with all descendents. The open invariants we define are not unique but depend on boundary conditions that, when altered, exhibit wall-crossing phenomena for the invariants. We describe an LG wall-crossing group classifying the wall-crossing transformations that can occur.

Local-to-global machinery plays an important role in the study of simplicial complexes, since the seminal work of Garland [G] to our days. In this work we develop a local-to-global machinery for more general posets. We show that the high-dimensional expansion notions and many recent expansion results have a generalization to posets. Examples are fast convergence of high-dimensional random walks generalizing [KO, AL], an equivalence with a global random walk definition, generalizing [DDFH] and a trickling down theorem, generalizing [O]. In particular, we show that some posets, such as the Grassmannian poset, exhibit a qualitatively stronger trickling down effect than simplicial complexes. We use these methods, and a novel idea of posetification to the Ramanujan complexes [LSV1, LSV2], to construct a constant degree expanding Grassmannian poset, and analyze its expansion. This is the first construction of such an object, whose existence was conjectured in [DDFH].

The amplituhedron Ank4 is a geometric object, introduced by Arkani-Hamed and Trnka (2013) in the study of scattering amplitudes in quantum field theories. They conjecture that Ank4 admits a decomposition into images of BCFW positroid cells, arising from the Britto--Cachazo--Feng--Witten recurrence (2005). We prove that this conjecture is true.

The orthogonal momentum amplituhedron O_k was introduced simultaneously in 2021 by Huang, Kojima, Wen, and Zhang, and by He, Kuo, and Zhang, in the study of scattering amplitudes of ABJM theory. It was conjectured that it admits a decomposition into BCFW cells. We prove this conjecture.

A major research area in discrete geometry is to consider the best way to partition the d -dimensional Euclidean space $$\mathbb {R}^d$$ R d under various quality criteria. In this paper we introduce a new type of space partitioning that is motivated by the problem of rounding noisy measurements from the continuous space $$\mathbb {R}^d$$ R d to a discrete subset of representative values. Specifically, we study partitions of $$\mathbb {R}^d$$ R d into bounded-size tiles colored by one of k colors, such that tiles of the same color have a distance of at least t from each other. Such tilings allow for error-resilient rounding, as two points of the same color and distance less than t from each other are guaranteed to belong to the same tile, and thus, to be rounded to the same point. The main problem we study in this paper is characterizing the achievable tradeoffs between the number of colors k and the distance t , for various dimensions d . On the qualitative side, we show that in $$\mathbb {R}^d$$ R d , using $$k=d+1$$ k = d + 1 colors is both sufficient and necessary to achieve $$t>0$$ t > 0 . On the quantitative side, we achieve numerous upper and lower bounds on t as a function of k . In particular, for $$d=3,4,8,24$$ d = 3 , 4 , 8 , 24 , we obtain sharp asymptotic bounds on t , as $$k \rightarrow \infty $$ k → ∞ . We obtain our results with a variety of techniques including isoperimetric inequalities, the Brunn-Minkowski theorem, sphere packing bounds, Bapat’s connector-free lemma, and Čech cohomology.

These notes are based on talks I gave in the seminar "Mathematical structures in scattering amplitudes in quantum field theories" I organized in Weizmann Institute on Fall 24'. They study amplituhedra, and extend the proof of of the BCFW conjecture for tree amplituhedra to one loop.

We define a theory of descendent integration on the moduli spaces of stable pointed disks. The descendent integrals are proved to be coefficients of the�-function of an open KdV hierar- chy. A relation between the integrals and a representation of half the Virasoro algebra is also proved. The construction of the the- ory requires an in depth study of homotopy classes of multivalued boundary conditions. Geometric recursions based on the combined structure of the boundary conditions and the moduli space are used to compute the integrals. We also provide a detailed analysis of orientations. Our open KdV and Virasoro constraints uniquely specify a the- ory of higher genus open descendent integrals. As a result, we ob- tain an open analog (governing all genera) of Witten's conjectures concerning descendent integrals on the Deligne-Mumford space of stable curves.

The amplituhedron is a mathematical object which was introduced to provide a geometric origin of scattering amplitudes in N = 4 super Yang-Mills theory. It generalizes cyclic polytopes and the 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 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 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 Gr 4 , n . This paper is a companion to our previous paper "Cluster algebras and tilings for the m = 4 amplituhedron."

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 k and the known tilings of 𝒜_n,k,4 have cardinality the Narayana number 1/n-3n-3 k+1n-3 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 × (n-k-m) ×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 k. We prove this by showing that all positroid tilings of the hypersimplex Δ_k+1,n have cardinality n-2 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.

We conclude the construction of $r$-spin theory in genus zero for Riemann surfaces with boundary. In particular, we define open $r$-spin intersection numbers, and we prove that their generating function is closely related to the wave function of the $r$th Gelfand--Dickey integrable hierarchy. This provides an analogue of Witten's $r$-spin conjecture in the open setting and a first step toward the construction of an open version of Fan--Jarvis--Ruan--Witten theory. As an unexpected consequence, we establish a mysterious relationship between open $r$-spin theory and an extension of Witten's closed theory.

We show that a generating function for open r -spin enumerative invariants produces a universal unfolding of the polynomial x (R). Further, the coordinates parametrizing this universal unfolding are flat coordinates on the Frobenius manifold associated to the Landau-Ginzburg model (C, x (R)) via Saito-Givental theory. This result provides evidence for the same phenomenon to occur in higher dimension, proven in the sequel [GKT22].

In our previous two papers, we constructed an r-spin theory in genus zero for Riemann surfaces with boundary and fully determined the corresponding intersection numbers, providing an analogue of Witten's r-spin conjecture in genus zero in the open setting. In particular, we proved that the generating series of open r-spin intersection numbers is determined by the genus-zero part of a special solution of a certain extension of the Gelfand–Dickey hierarchy, and we conjectured that the whole solution controls the open r-spin intersection numbers in all genera, which do not yet have a geometric definition. In this paper, we provide geometric and algebraic evidence for the correctness of this conjecture.

In recent works, [20],[21], descendent integrals on the moduli space of Riemann surfaces with boundary were defined. It was conjectured in [20] that the generating function of these integrals satisfies the open KdV equations. In this paper we develop the notions of symmetric Strebel-Jenkins differentials and of Kasteleyn orientations for graphs embedded in open surfaces. In addition we write an explicit expression for the angular form of the sum of line bundles. Using these tools we prove a formula for the descendent integrals in terms of sums over weighted graphs. Based on this formula, the conjecture of [20] was proved in [5].