arXiv subjects

Fatemeh Mohammadi

Publications and source records attributed to Fatemeh Mohammadi.

At least 55 records · Page 3Linked to original sources

arXiv2024

Multigraded algebras and multigraded linear series

This paper is devoted to the study of multigraded algebras and multigraded linear series. For an $\mathbb{N}^s$-graded algebra $A$, we define and study its volume function $F_A:\mathbb{N}_+^s\to \mathbb{R}$, which computes the asymptotics of the Hilbert function of $A$. We relate the volume function $F_A$ to the volume of the fibers of the global Newton-Okounkov body $Δ(A)$ of $A$. Unlike the classical case of standard multigraded algebras, the volume function $F_A$ is not a polynomial in general. However, in the case when the algebra $A$ has a decomposable grading, we show that the volume function $F_A$ is a polynomial with non-negative 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.

math.AC↗

arXiv2024

Identifiability of Points and Rigidity of Hypergraphs under Algebraic Constraints

The identifiability problem arises naturally in a number of contexts in mathematics and computer science. Specific instances include local or global rigidity of graphs and unique completability of partially-filled tensors subject to rank conditions. The identifiability of points on secant varieties has also been a topic of much research in algebraic geometry. It is often formulated as the problem of identifying a set of points satisfying a given set of algebraic relations. A key question then is to prove sufficient conditions for relations to guarantee the identifiability of the points. This paper proposes a new general framework for capturing the identifiability problem when a set of algebraic relations has a combinatorial structure and develops tools to analyse the impact of the underlying combinatorics on the local or global identifiability of points. Our framework is built on the language of graph rigidity, where the measurements are Euclidean distances between two points, but applicable in the generality of hypergraphs with arbitrary algebraic measurements. We establish necessary and sufficient (hyper)graph theoretical conditions for identifiability by exploiting techniques from graph rigidity theory and algebraic geometry of secant varieties. In particular our work analyses combinatorially the effect of non-generic projections of secant varieties.

math.MG↗

arXiv2023

A differential approach to Maxwell-Cremona liftings

In 1864, J. C. Maxwell introduced a link between self-stressed frameworks in the plane and piecewise linear liftings to 3-space. This connection has found numerous applications in areas such as discrete geometry, control theory and structural engineering. While there are some generalisations of this theory to liftings of $d$-complexes in $d$-space, extensions for liftings of frameworks in $d$-space for $d\geq 3$ have been missing. In this paper, we introduce and study differential liftings on general graphs using differential forms associated with the elements of the homotopy groups of the complements to the frameworks. Such liftings play the role of integrands for the classical notion of liftings for planar frameworks. We show that these differential liftings have a natural extension to self-stressed frameworks in higher dimensions. As a result we generalise the notion of classical liftings to both graphs and multidimensional $k$-complexes in $d$-space ($k=2,\ldots, d$). Finally we discuss a natural representation of generalised liftings as real-valued functions on Grassmannians.

math.MG↗

arXiv2023

Global Rigidity of Line Constrained Frameworks

We consider the global rigidity problem for bar-joint frameworks where each vertex is constrained to lie on a particular line in $\mathbb R^d$. In our setting we allow multiple vertices to be constrained to the same line. Under a mild assumption on the given set of lines we give a complete combinatorial characterisation of graphs that are generically globally rigid in this setting. This gives a $d$-dimensional extension of the well-known combinatorial characterisation of 1-dimensional global rigidity.

math.CO↗

arXiv2023

Toric degenerations of partial flag varieties and combinatorial mutations of matching field polytopes

We study toric degenerations arising from Gröbner degenerations or the tropicalization of partial flag varieties. We produce a new family of toric degenerations of partial flag varieties whose combinatorics are governed by matching fields and combinatorial mutations of polytopes. We provide an explicit description of the polytopes associated with the resulting toric varieties in terms of matching field polytopes. These polytopes encode the combinatorial data of monomial degenerations of Plücker forms for the Grassmannians. We give a description of matching field polytopes of flag varieties as Minkowski sums and show that all such polytopes are normal. The polytopes we obtain are examples of Newton-Okounkov bodies for particular full-rank valuations for flag varieties. Furthermore, we study a certain explicitly-defined large family of matching field polytopes and prove that all polytopes in this family are connected by combinatorial mutations. Finally, we apply our methods to explicitly compute toric degenerations of small Grassmannians and flag varieties and obtain new families of toric degenerations.

math.AG↗

arXiv2023

Double Schubert polynomials do have saturated Newton polytopes

We prove that double Schubert polynomials have the Saturated Newton Polytope property. This settles a conjecture by Monical, Tokcan and Yong. Our ideas are motivated by the theory of multidegrees. We introduce a notion of standardization of ideals that enables us to study non-standard multigradings. This allows us to show that the support of the multidegree polynomial of each Cohen-Macaulay prime ideal, and in particular, that of each Schubert determinantal ideal is a discrete polymatroid.

math.AC↗

arXiv2023

A study of nonlinear multiview varieties

We study the nonlinear generalization of the classical multiview variety, which is a fundamental concept in computer vision. In this paper, we take the first comprehensive step to develop the nonlinear analogue of multiview varieties. To this end, we introduce a multigraded version of the saturated special fiber ring. By applying this tool, we are able to compute the multidegrees of several families of nonlinear multiview varieties.

math.AG↗

arXiv2023

On the uniqueness of collections of pennies and marbles

In this note we study the uniqueness problem for collections of pennies and marbles. More generally, consider a collection of unit $d$-spheres that may touch but not overlap. Given the existence of such a collection, one may analyse the contact graph of the collection. In particular we consider the uniqueness of the collection arising from the contact graph. Using the language of graph rigidity theory, we prove a precise characterisation of uniqueness (global rigidity) in dimensions 2 and 3 when the contact graph is additionally chordal. We then illustrate a wide range of examples in these cases. That is, we illustrate collections of marbles and pennies that can be perturbed continuously (flexible), are locally unique (rigid) and are unique (globally rigid). We also contrast these examples with the usual generic setting of graph rigidity.

math.CO↗

arXiv2023

Tensors in statistics and rigidity theory

This is a short report on the discussions of appearance of tensors in algebraic statistics and rigidity theory, during the semester ``AGATES: Algebraic Geometry with Applications to TEnsors and Secants". We briefly survey some of the existing results in the literature and further research directions. We first provide an overview of algebraic and geometric techniques in the study of conditional independence (CI) statistical models. We study different families of algebraic varieties arising in statistics. This includes the determinantal varieties related to CI statements with hidden random variables. Such statements correspond to determinantal conditions on the tensor of joint probabilities of events involving the observed random variables. We show how to compute the irreducible decompositions of the corresponding CI varieties, which leads to finding further conditional dependencies (or independencies) among the involved random variables. As an example, we show how these methods can be applied to extend the classical intersection axiom for CI statements. We then give a brief overview about secant varieties and their appearance in the study of mixture models. We focus on examples and briefly mention the connection to rigidity theory which will appear in the forthcoming paper \cite{rigidity}.

math.AC↗

arXiv2022

Rational tensegrities through the lens of toric geometry

A classical tensegrity model consists of an embedded graph in a vector space with rigid bars representing edges, and an assignment of a stress to every edge such that at every vertex of the graph the stresses sum up to zero. The tensegrity frameworks have been recently extended from the two dimensional graph case to the multidimensional setting. We study the multidimensional tensegrities using tools from toric geometry. For a given rational tensegrity framework $\mathcal{F}$, we construct a glued toric surface $X_\mathcal{F}$. We show that the abelian group of tensegrities on $\mathcal{F}$ is isomorphic to a subgroup of the Chow group $A^1(X_\mathcal{F};\QQ)$. In the case of planar frameworks, we show how to explicitly carry out the computation of tensegrities via classical tools in toric geometry.

math.AG↗

arXiv2022

Conditional independence ideals with hidden variables

We study a class of determinantal ideals that are related to conditional independence (CI) statements with hidden variables. Such CI statements correspond to determinantal conditions on a matrix whose entries are probabilities of events involving the observed random variables. We focus on an example that generalizes the CI ideals of the intersection axiom. In this example, the minimal primes are again determinantal ideals, which is not true in general.

math.AC↗

arXiv2022

Phase transition and fractionalization in superconducting Kondo lattice model

Topology, symmetry, electron correlations, and the interplay between them have formed the cornerstone of our understanding of quantum materials in recent years and are used to identify new emerging phases. While the first two give a fair understanding of noninteracting and, in many cases, weakly interacting wave function of electron systems, the inclusion of strong correlations could change the picture substantially. The Kondo lattice model is a paradigmatic example of the interplay of electron correlations and conduction electrons of a metallic system, describing heavy fermion materials and also fractionalized Fermi liquid pertaining to an underlying gauge symmetry and topological orders. In this work, we study a superconducting Kondo lattice model, a network of 1D Kitaev superconductors Kondo coupled to a lattice of magnetic moments. Using slave-particle representation of spins and exact numerical calculations, we obtain the phase diagram of the model in terms of Kondo coupling $J_K$ and identify a topological order phase for $J_{K} J_{K}^c$, where $J_{K}^c$ is the critical point. Setting the energy scales of electron hopping and pairing to unity, the mean-field theory calculations achives $J_{K}^c=2$ and in exact numerics we found $J_{K}^c\simeq 1.76$, both of which show that the topological order is a robust phase. We argue that in terms of slave particles, the compensated phase corresponds to an invertible phase, and a Mott insulating transition leads to a topological order phase. Furthermore, we show that in the regime $J_{K}<J_{K}^c$ in addition to the low-energy topological states, a branch of subgap states appears inside the superconducting gap.

cond-mat.str-el↗

arXiv2022

Progress on Infinite Cluster Categories Related to Triangulations of the (Punctured) Disk

In this mostly expository paper, we present recent progress on infinite (weak) cluster categories that are related to triangulations of the disk, with and without a puncture. First we recall the notion of a cluster category. Then we move to the infinite setting and survey recent work on infinite cluster categories of types $\mathbb{A}$ and $\mathbb{D}$. We conclude with our contributions, two infinite families of infinite (weak) cluster categories of type $\mathbb{D}$. We first present a discrete, infinite version of Schiffler's combinatorial model of the punctured disk with marked points. We then produce each (weak) cluster category starting with representations of thread quivers, taking the derived category, and then taking the appropriate orbit category. We show that the combinatorics in the (weak) cluster categories match with the corresponding combinatorics of the punctured disk with countably-many marked points. We also state two conjectures concerning weak cluster structures inside our (weak) cluster categories.

math.RT↗

arXiv2022

Matroid stratifications of hypergraph varieties, their realization spaces, and discrete conditional independence models

We study varieties associated to hypergraphs from the point of view of projective geometry and matroid theory. We describe their decompositions into matroid varieties, which may be reducible and can have arbitrary singularities by the Mnëv--Sturmfels universality theorem. We focus on various families of hypergraph varieties for which we explicitly compute an irredundant irreducible decomposition. Our main findings in this direction are threefold: (1) we describe minimal matroids of such hypergraphs; (2) we prove that the varieties of these matroids are irreducible and their union is the hypergraph variety; and (3) we show that every such matroid is realizable over real numbers. As corollaries, we give conceptual decompositions of various, previously-studied, varieties associated with graphs, hypergraphs, and adjacent minors of generic matrices. In particular, our decomposition strategy gives immediate matroid interpretations of the irreducible components of multiple families of varieties associated to conditional independence (CI) models in statistical theory and unravels their symmetric structures which hugely simplifies the computations.

math.CO↗

arXiv2022

Template-based Program Synthesis using Stellensätze

Template-based synthesis, also known as sketching, is a localized approach to program synthesis in which the programmer provides not only a specification, but also a high-level ``sketch'' of the program. The sketch is basically a partial program that models the general intuition of the programmer, while leaving the low-level details as unimplemented ``holes''. The role of the synthesis engine is then to fill in these holes such that the completed program satisfies the desired specification. In this work, we focus on template-based synthesis of polynomial imperative programs with real variables, i.e.~imperative programs in which all expressions appearing in assignments, conditions and guards are polynomials over program variables. While this problem can be solved in a sound and complete manner by a reduction to the first-order theory of the reals, the resulting formulas will contain a quantifier alternation and are extremely hard for modern SMT solvers, even when considering toy programs with a handful of lines. Moreover, the classical algorithms for quantifier elimination are notoriously unscalable and not at all applicable to this use-case. In contrast, our main contribution is an algorithm, based on several well-known theorems in polyhedral and real algebraic geometry, namely Putinar's Positivstellensatz, the Real Nullstellensatz, Handelman's Theorem and Farkas' Lemma, which sidesteps the quantifier elimination difficulty and reduces the problem directly to Quadratic Programming (QP). Alternatively, one can view our algorithm as an efficient way of eliminating quantifiers in the particular formulas that appear in the synthesis problem. The resulting QP instances can then be handled quite easily by SMT solvers. Notably, our reduction to QP is sound and semi-complete, i.e.~it is complete if polynomials of a sufficiently high degree are used in the templates...

cs.PL↗

arXiv2022

Combinatorial Mutations of Gelfand-Tsetlin Polytopes, Feigin-Fourier-Littelmann-Vinberg Polytopes, and Block Diagonal Matching Field Polytopes

The Gelfand-Tsetlin and the Feigin-Fourier-Littelmann-Vinberg polytopes for the Grassmannians are defined, from the perspective of representation theory, to parametrize certain bases for highest weight irreducible modules. These polytopes are Newton-Okounkov bodies for the Grassmannian and, in particular, the GT-polytope is an example of a string polytope. The polytopes admit a combinatorial description as the Stanley's order and chain polytopes of a certain poset, as shown by Ardila, Bliem and Salaza. We prove that these polytopes occur among matching field polytopes. Moreover, we show that they are related by a sequence of combinatorial mutations that passes only through matching field polytopes. As a result, we obtain a family of matching fields that give rise to toric degenerations for the Grassmannians. Moreover, all polytopes in the family are Newton-Okounkov bodies for the Grassmannians.

math.CO↗

arXiv2022

Lattice Conditional Independence Models and Hibi Ideals

Lattice Conditional Independence models are a class of models developed first for the Gaussian case in which a distributive lattice classifies all the conditional independence statements. The main result is that these models can equivalently be described via a transitive directed acyclic graph (TDAG) in which, as is normal for causal models, the conditional independence is in terms of conditioning on ancestors in the graph. We demonstrate that a parallel stream of research in algebra, the theory of Hibi ideals, not only maps directly to the LCI models but gives a vehicle to generalise the theory from the linear Gaussian case. Given a distributive lattice (i) each conditional independence statement is associated with a Hibi relation defined on the lattice, (ii) the directed graph is given by chains in the lattice which correspond to chains of conditional independence, (iii) the elimination ideal of product terms in the chains gives the Hibi ideal and (iv) the TDAG can be recovered from a special bipartite graph constructed via the Alexander dual of the Hibi ideal. It is briefly demonstrated that there are natural applications to statistical log-linear models, time series, and Shannon information flow.

math.ST↗

arXiv2022

Standard monomial theory and toric degenerations of Richardson varieties in flag varieties

We study standard monomial bases for Richardson varieties inside the flag variety. In general, writing down a standard monomial basis for a Richardson variety can be challenging, as it involves computing so-called defining chains or key tableaux. However, for a certain family of Richardson varieties, indexed by compatible permutations, we provide a very direct and straightforward combinatorial rule for writing down a standard monomial basis. We apply this result to the study of toric degenerations of Richardson varieties. In particular, we provide a new family of toric degenerations of Richardson varieties inside flag varieties.

math.AG↗