We describe the Chow homology and cohomology of toric variety bundles, with no restrictions on the singularities of the fibre. We present the ordinary and equivariant homologies as modules over the cohomology of the base, identify the ordinary cohomology with homology-valued Minkowski weights, and identify the equivariant cohomology with cohomology-weighted piecewise polynomial functions. We describe the product structure on Minkowski weights via a fan displacement rule, and the non-equivariant limit via equivariant multiplicities. Along the way we establish relative analogues of the Künneth property and Kronecker duality. Applications include the balancing condition in logarithmic enumerative geometry.

An alcoved polytope is a polytope whose facet normal are all in direction of type A roots. This fundamental class of polytopes has ample applications in for instance tropical geometry, statistics and algebra. The mini-workshop showcased various recent developments related to alcoved polytopes and their generalizations with a particular focus on their connections to physics and optimization.

Given a pair of translation surfaces it is very difficult to determine whether they are supported on the same algebraic curve. In fact, there are very few examples of such pairs. In this note we present infinitely many examples of finite collections of translation surfaces supported on the same algebraic curve. The underlying curves are hyperelliptic curves with many automorphisms. For each curve, the automorphism of maximal order acts on the space of holomorphic 1-forms. We present a translation surface corresponding to each of the eigenforms of this action.

A cosmological polytope is a lattice polytope introduced by Arkani-Hamed, Benincasa, and Postnikov in their study of the wavefunction of the universe in a class of cosmological models. More concretely, they construct a cosmological polytope for any Feynman diagram, i.e. an undirected graph. In this paper, we initiate a combinatorial study of these polytopes. We give a complete description of their faces, identify minimal faces that are not simplices and compute the number of faces in specific instances. In particular, we give a recursive description of the $f$-vector of cosmological polytopes of trees.

Approximating periodic solutions to the coupled Duffing equations amounts to solving a system of polynomial equations. The number of complex solutions measures the algebraic complexity of this approximation problem. Using the theory of Khovanskii bases, we show that this number is given by the volume of a certain polytope. We also show how to compute all solutions using numerical nonlinear algebra.

This paper is devoted to the study of multigraded algebras and multigraded linear series. For an Ns$\mathbb {N}<^>s$-graded algebra A$A$, we define and study its volume function FA:N+s -> R$F_A:\mathbb {N}_+<^>s\rightarrow \mathbb {R}$, which computes the asymptotics of the Hilbert function of A$A$. We relate the volume function FA$F_A$ to the volume of the fibers of the global Newton-Okounkov body Delta(A)$\Delta (A)$ of A$A$. Unlike the classical case of standard multigraded algebras, the volume function FA$F_A$ is not a polynomial in general. However, in the case when the algebra A$A$ has a decomposable grading, we show that the volume function FA$F_A$ is a polynomial with nonnegative coefficients. We then define mixed multiplicities in this case and provide a full characterization for their positivity. Furthermore, we apply our results on multigraded algebras to multigraded linear series. Our work recovers and unifies recent developments on mixed multiplicities. In particular, we recover results on the existence of mixed multiplicities for (not necessarily Noetherian) graded families of ideals and on the positivity of the multidegrees of multiprojective varieties.

We establish connections between: the maximum likelihood degree (ML-degree) for linear concentration models, the algebraic degree of semidefinite programming (SDP), and Schubert calculus for complete quadrics. We prove a conjecture by Sturmfels and Uhler on the polynomiality of the ML-degree. We also prove a conjecture by Nie, Ranestad and Sturmfels providing an explicit formula for the degree of SDP. The interactions between the three fields shed new light on the asymptotic behaviour of enumerative invariants for the variety of complete quadrics. We also extend these results to spaces of general matrices and of skew-symmetric matrices.

In this paper we propose a method that uses Lagrange multipliers and numerical algebraic geometry to find all critical points, and therefore globally solve, polynomial optimization problems. We design a polyhedral homotopy algorithm that explicitly constructs an optimal start system, circumventing the typical bottleneck associated with polyhedral homotopy algorithms. The correctness of our algorithm follows from intersection theoretic computations of the algebraic degree of polynomial optimization programs and relies on explicitly solving the tropicalization of a corresponding Lagrange system. We present experiments that demonstrate the superiority of our algorithm over traditional homotopy continuation algorithms.

In 1992, Pukhlikov and Khovanskii provided a description of the cohomology ring of toric variety as a quotient of the ring of differential operators on spaces of virtual polytopes. Later Kaveh generalized this construction to the case of cohomology rings for full flag varieties. In this paper we extend Pukhlikov-Khovanskii type presentation to the case of K-theory of toric and flag varieties. First, we study the Frobenius algebras obtained as quotients of the group algebra of free abelian group (possibly of infinite rank). Then we apply this construction to define a K-ring associated to a linear family of (virtual) polytopes. We study in detail two examples of such families: the family of integer (virtual) polytopes with a fixed normal fan and the family of (virtual) Gelfand-Zetlin polytopes. We show that the K-theory of toric and flag varieties can be realized as K-rings of the above families and use this to get natural set of relations in the above K-rings. Further, we describe the classes of structure sheaves of toric orbit closures and Schubert varieties in type A flag varieties. Finally, we show that our results also hold true in T-equivariant setting.

A toric variety is called fibered if it can be represented as a total space of fibre bundle over toric base and with toric fiber. Fibered toric varieties form a special case of toric variety bundles. In this note we first give an introduction to the class of fibered toric varieties. Then we use them to illustrate some known and conjectural results on topology and intersection theory of general toric variety bundles. Finally, using the language of fibered toric varieties, we compute the equivariant cohomology rings of smooth complete toric varieties.

In this paper we study a broad class of polynomial optimization problems whose constraints and objective functions exhibit sparsity patterns. We give two characterizations of the number of critical points to these problems, one as a mixed volume and one as an intersection product on a toric variety. As a corollary, we obtain a convex geometric interpretation of polar degrees, a classical invariant of algebraic varieties as well as Euclidean distance degrees. Furthermore, we prove BKK generality of Lagrange systems in many instances. Motivated by our result expressing the algebraic degree of sparse polynomial optimisation problems via Porteus' formula, in the appendix we answer a related question concerning the degree of sparse determinantal varieties.

The classical Bernstein-Kushnirenko-Khovanskii theorem (or, the BKK theorem, for short) computes the intersection number of divisors on toric variety in terms of volumes of corresponding polytopes. In [PK92b], it was observed by Pukhlikov and the first author that the BKK theorem leads to a presentation of the cohomology ring of a toric variety as a quotient of a ring of differential operators with constant coefficients by the annihilator of an explicit polynomial. In this paper we generalize this construction to the case of quasitoric bundles. These are fiber bundles with generalized quasitoric manifolds as fibers. First we obtain a generalization of the BKK theorem to this case. Then we use recently obtained descriptions of the graded-commutative algebras which satisfy Poincar? duality to give a description of cohomology rings of quasitoric bundles.

We develop a theory of volume polynomials of generalized virtual polytopes based on the study of topology of affine subspace arrangements in a real Euclidean space. We apply this theory to obtain a topological version of the Bernstein–Kushnirenko theorem as well as Stanley–Reisner and Pukhlikov–Khovanskii type descriptions for the cohomology rings of generalized quasitoric manifolds.

Any totally positive $$(k+m)\times n$$ matrix induces a map $$\pi _+$$ from the positive Grassmannian $$\mathrm{Gr}_+(k,n)$$ to the Grassmannian $$\mathrm{Gr}(k,k+m)$$ , whose image is the amplituhedron $$\mathcal {A}_{n,k,m}$$ and is endowed with a top-degree form called the canonical form $$\varvec{\Omega }(\mathcal {A}_{n,k,m})$$ . This construction was introduced by Arkani-Hamed and Trnka (J High Energy Phys 2014(10):30, 2014), where they showed that $$\varvec{\Omega }(\mathcal {A}_{n,k,4})$$ encodes scattering amplitudes in $$\mathcal {N}=4$$ super Yang-Mills theory. One way to compute $$\varvec{\Omega }(\mathcal {A}_{n,k,m})$$ is to subdivide $$\mathcal {A}_{n,k,m}$$ into so-called generalized triangles and sum over their associated canonical forms. Hence, the physical computation of scattering amplitudes is reduced to finding the triangulations of $$\mathcal {A}_{n,k,4}$$ . However, while triangulations of polytopes are fully captured by their secondary and fiber polytopes (Billera and Sturmfels in Ann Math 527–549, 1992; Gelfand et al. in Discriminants, resultants, and multidimensional determinants, Birkahuser, Boston, 1994), the study of triangulations of objects beyond polytopes is still underdeveloped. In this work, we initiate the geometric study of subdivisions of $$\mathcal {A}_{n,k,m}$$ in order to establish the notion of secondary amplituhedron. For this purpose, we first extend the projection $$\pi _+$$ to a rational map $$\pi :Gr (k,n)\dashrightarrow Gr (k,k+m)$$ and provide a concrete birational parametrization of the fibers of $$\pi $$ . We then use this to explicitly describe a rational top-degree form $$\omega _{n,k,m}$$ (with simple poles) on the fibers and compute $$\varvec{\Omega }(\mathcal {A}_{n,k,m})$$ as a summation of certain residues of $$\omega _{n,k,m}$$ . As main application of our approach, we develop a well-structured notion of secondary amplituhedra for conjugate to polytopes, i.e. when $$n-k-1=m$$ (even). We show that, in this case, each fiber of $$\pi $$ is parametrized by a projective space and its volume form $$\omega _{n,k,m}$$ has only poles on a hyperplane arrangement. Using such linear structures, for amplituhedra which are cyclic polytopes or conjugate to polytopes, we show that the Jeffrey–Kirwan residue computes $$\varvec{\Omega }(\mathcal {A}_{n,k,m})$$ from the fiber volume form $$\omega _{n,k,m}$$ . In particular, we give conceptual proofs of the statements of Ferro et al. (J Phys A Math Theor 52(4):045201, 2018). Finally, we propose a more general framework of fiber positive geometries and analyze new families of examples such as fiber polytopes and Grassmann polytopes.

We study the maximum likelihood (ML) degree of linear concentration models in algebraic statistics. We relate it to an intersection problem on the variety of complete quadrics. This allows us to provide an explicit, basic, albeit of high computational complexity, formula for the ML-degree. The variety of complete quadrics is an exact analog for symmetric matrices of the permutohedron variety for the diagonal matrices.

We study commutative algebras with Gorenstein duality, i.e. algebras $A$ equipped with a non-degenerate bilinear pairing such that $\langle ac,b\rangle=\langle a,bc\rangle$ for any $a,b,c\in A$. If an algebra $A$ is Artinian, such pairing exists if and only if $A$ is Gorenstein. We give a description of algebras with Gorenstein duality as a quotients of the ring of differential operators by the annihilator of an explicit polynomial (or more generally formal polynomial series). This provides a calculation of Macaulay's inverse systems for (not necessarily Artinian) algebras with Gorenstein duality. Our description generalizes previously know description of graded algebras with Gorenstein duality generated in degree 1. Our main motivation comes from the study of even degree cohomology rings. In particular, we apply our main result to compute the ring of cohomology classes of even degree of toric bundles and the ring of conditions of horospherical homogeneous spaces.

The moduli space $\overline{M}_{0,n}$ may be embedded into the product of projective spaces $\mathbb{P}^1\times \mathbb{P}^2\times \cdots \times \mathbb{P}^{n-3}$, using a combination of the Kapranov map $|\psi_n|:\overline{M}_{0,n}\to \mathbb{P}^{n-3}$ and the forgetful maps $\pi_i:\overline{M}_{0,i}\to \overline{M}_{0,i-1}$. We give an explicit combinatorial formula for the multidegree of this embedding in terms of certain parking functions of height $n-3$. We use this combinatorial interpretation to show that the total degree of the embedding (thought of as the projectivization of its cone in $\mathbb{A}^2\times \mathbb{A}^3\cdots \times \mathbb{A}^{n-2}$) is equal to $(2(n-3)-1)!!=(2n-7)(2n-9) \cdots(5)(3)(1)$. As a consequence, we also obtain a new combinatorial interpretation for the odd double factorial.

Let $E_1,\ldots,E_k$ be a collection of linear series on an algebraic variety $X$ over $\mathbb{C}$. That is, $E_i\subset H^0(X, \mathcal{L}_i)$ is a finite dimensional subspace of the space of regular sections of line bundles $ \mathcal{L}_i$. Such a collection is called overdetermined if the generic system \[ s_1 = \ldots = s_k = 0, \] with $s_i\in E_i$ does not have any roots on $X$. In this paper we study solvable systems which are given by an overdetermined collection of linear series. Generalizing the notion of a resultant hypersurface we define a consistency variety $R\subset \prod_{i=1}^k E_i$ as the closure of the set of all systems which have at least one common root and study general properties of zero sets $Z_{\bf s}$ of a generic consistent system ${\bf s}\in R$. Then, in the case of equivariant linear series on spherical homogeneous spaces we provide a strategy for computing discrete invariants of such generic non-empty set $Z_{\bf s}$. For equivariant linear series on the torus $(\mathbb{C}^*)^n$ this strategy provides explicit calculations and generalizes the theory of Newton polyhedra.

We study the maximum likelihood (ML) degree of linear concentration models in algebraic statistics. We relate it to an intersection problem on a smooth compact moduli space of orbits of a ${\mathbb C}^*$ action on the Lagrangian Grassmannian which we call Gaussian moduli. This allows us to provide an explicit, basic, albeit of high computational complexity, formula for the ML-degree. The Gaussian moduli is an exact analog for symmetric matrices of the permutohedron variety for the diagonal matrices.

The space of measured laminations ML(Σ) associated to a topological surface Σ of genus g with n punctures is an integral piecewise linear manifold of real dimension 6g−6+2n. There is also a natural symplectic structure on ML(Σ) defined by Thurston. The integral and symplectic structures define a pair of measures on ML(Σ) which are known to be proportional. The projective class of these measures on ML(Σ) is called the Thurston measure. In this note we compute the ratio between two normalizations of the Thurston measure.