Vacuum structure of a quantum field theory is a crucial property. In theories with extended symmetries, such as supersymmetric gauge theories, the vacuum is typically a continuous manifold, called the vacuum moduli space, parametrized by the expectation values of scalar fields. Starting from the R-parity preserving superpotential at renormalizable order, we use Gröbner bases to determine the explicit structure, as an algebraic variety, of the vacuum geometry of the minimal supersymmetric extension of the Standard Model. Gröbner bases have doubly exponential computational complexity (for this case, 7^2^1023 operations); we exploit symmetry and multigrading to render the computation tractable. This geometry has three irreducible components of complex dimensions 1, 15, and 29, each being a so-called rational variety. The defining equations of the components express the solutions to F-terms and D-terms in terms of the gauge invariant operators and are interpreted in terms of classical geometric constructions.

A starting point in the study of the minimal supersymmetric Standard Model (MSSM) is the vacuum moduli space, which is a highly complicated algebraic variety: it is the image of an affine variety X ⊂ℂ^49 under a symplectic quotient map ϕ to ℂ^973. Previous work computed the vacuum moduli space of the electroweak sector; geometrically this corresponds to studying a restriction of ϕ: ℂ^13ϕ^⟶ℂ^22. We analyze the geometry of the full vacuum moduli space for superpotentials W_ minimal (without neutrinos) and W_ MSSM (with neutrinos) in ℂ^973. In both cases, we prove that X consists of three irreducible components X_1, X_2, and X_3, and determine the images M_i of the X_i under ϕ. For W_ minimal we show they have, respectively, dimensions 1, 15, and 29, and prove that each of the M_i is a rational variety, while for W_ MSSM we show that M_3 is the only component. Restricting the M_i to the electroweak sector, we recover known results. We describe the components of the vacuum moduli space geometrically in terms of incidence varieties to a product of Segre varieties.

The Neighbor-Joining (NJ) algorithm is a widely used method for constructing phylogenetic trees from genetic distances. While NJ is known to perform well with tree-like data, its behavior under admixture remains understudied. In this work, we present a geometric framework for analyzing the NJ algorithm under a linear admixture model. We focus on three key properties related to clustering order, distance, and topological path length in the resulting NJ trees involving five taxa. Our approach leverages polyhedral geometry to define NJ cones, which correspond to distinct cherry-picking orders and partition the space of dissimilarity vectors. We project dissimilarity vectors with admixture into a lower-dimensional space without admixture, defining polyhedral regions induced by NJ cones that satisfy specified properties. We compute the exact probabilities that these properties hold by directly calculating the volumes of the induced NJ cones and compare them with Monte Carlo integration and standard NJ simulation methods. Our results show that the property on clustering order is always satisfied, while the other properties are highly probable but depend on the admixture fraction. We also prove that certain induced NJ cones have zero volume, indicating that the corresponding NJ tree topologies are infeasible under admixture. We have implemented our methods as a publicly available NeighborJoining within Macaulay2, providing an efficient tool for analyzing NJ cones and their properties. This work provides new insights into the geometric structure inherent to the NJ algorithm in the presence of admixture, identifying the conditions under which admixture influences the resulting phylogenetic trees. ### Competing Interest Statement The authors have declared no competing interest.

MOTIVATION:While there are software packages that analyze Boolean, ternary, or other multi-state models, none compute the complete state space of function-based models over any finite set. Results: We propose Cyclone, a simple light-weight software package which simulates the complete state space for a finite dynamical system over any finite set.AVAILABILITY AND IMPLEMENTATION:Source code is freely available at https://github.com/discretedynamics/cyclone under the Apache-2.0 license.

The Standard Model of particle physics has been amazingly successful at explaining interactions between three of the four fundamental forces of physics: the strong force, the weak force, and electro-magnetism. But it is an incomplete theory, failing to incorporate gravity. The Minimal Supersymmetric Standard Model (MSSM) is an approach to including gravity and supersymmetry in the standard model, by adding new particle states and interactions to the standard model. For example, supersymmetry pairs fermions (such as electrons) with bosons (such as photons). A starting point in the study of the MSSM is the Vacuum Moduli Space, which is a highly complicated algebraic variety introduced by Witten: it is the image of an affine variety X ⊆ C 49 under a symplectic quotient map to C 973 . We describe the Macaulay2 software package MSSM, which facilitates computational investigation of the Vacuum Moduli Space.

Consider any network of n identical Kuramoto oscillators in which each oscillator is coupled bidirectionally with unit strength to at least μ(n-1) other oscillators. Then, there is a critical value of μ above which the system is guaranteed to converge to the in-phase synchronous state for almost all initial conditions. The precise value of μ remains unknown. In 2018, Ling, Xu, and Bandeira proved that if each oscillator is coupled to at least 79.29% of all the others, global synchrony is ensured. In 2019, Lu and Steinerberger improved this bound to 78.89%. Here, we find clues that the critical connectivity may be exactly 75%. Our methods yield a slight improvement on the best known lower bound on the critical connectivity from 68.18% to 68.28%. We also consider the opposite end of the connectivity spectrum, where the networks are sparse rather than dense. In this regime, we ask how few edges one needs to add to a ring of n oscillators to turn it into a globally synchronizing network. We prove a partial result: all the twisted states in a ring of size n=2m can be destabilized by adding just O(nlog2⁡n) edges. To finish the proof, one needs to rule out all other candidate attractors. We have done this for n≤8 but the problem remains open for larger n. Thus, even for systems as simple as Kuramoto oscillators, much remains to be learned about dense networks that do not globally synchronize and sparse ones that do.

Studying the set of exact solutions of a system of polynomial equations largely depends on a single iterative algorithm, known as Buchberger's algorithm. Optimized versions of this algorithm are crucial for many computer algebra systems (e.g., Mathematica, Maple, Sage). We introduce a new approach to Buchberger's algorithm that uses reinforcement learning agents to perform S-pair selection, a key step in the algorithm. We then study how the difficulty of the problem depends on the choices of domain and distribution of polynomials, about which little is known. Finally, we train a policy model using proximal policy optimization (PPO) to learn S-pair selection strategies for random systems of binomial equations. In certain domains, the trained model outperforms state-of-the-art selection heuristics in total number of polynomial additions performed, which provides a proof-of-concept that recent developments in machine learning have the potential to improve performance of algorithms in symbolic computation.

We prove a formula for the Hodge numbers of square-free divisors of Calabi-Yau threefold hypersurfaces in toric varieties. Euclidean branes wrapping divisors affect the vacuum structure of Calabi-Yau compactifications of type IIB string theory, M-theory, and F-theory. Determining the nonperturbative couplings due to Euclidean branes on a divisor $D$ requires counting fermion zero modes, which depend on the Hodge numbers $h^i({\cal{O}}_D)$. Suppose that $X$ is a smooth Calabi-Yau threefold hypersurface in a toric variety $V$, and let $D$ be the restriction to $X$ of a square-free divisor of $V$. We give a formula for $h^i({\cal{O}}_D)$ in terms of combinatorial data. Moreover, we construct a CW complex $\mathscr{P}_D$ such that $h^i({\cal{O}}_D)=h_i(\mathscr{P}_D)$. We describe an efficient algorithm that makes possible for the first time the computation of sheaf cohomology for such divisors at large $h^{1,1}$. As an illustration we compute the Hodge numbers of a class of divisors in a threefold with $h^{1,1}=491$. Our results are a step toward a systematic computation of Euclidean brane superpotentials in Calabi-Yau hypersurfaces.

Singular value decompositions of matrices are widely used in numerical linear algebra with many applications. In this paper, we extend the notion of singular value decompositions to finite complexes of real vector spaces. We provide two methods to compute them and present several applications.

Let Δ be a connected, pure 2-dimensional simplicial complex embedded in R 2 and let C r ( Δ ˆ ) be the homogenized spline module of Δ with smoothness r as in [7] . To study C r ( Δ ˆ ) , Schenck and Stillman developed in [7] the quotient complex S • / J • . In [8] , Schenck and Stiller conjectured that the regularity of H 1 ( S • / J • ) is less than 2 r + 1 . In this article, we pose a counterexample to this “ 2 r + 1 ” conjecture. We also propose some modifications to the conjecture.

We present an intriguing and precise interplay between algebraic geometry and the phenomenology of generations of particles. Using the electroweak sector of the MSSM as a testing ground, we compute the moduli space of vacua as an algebraic variety for multiple generations of Standard Model matter and Higgs doublets. The space is shown to have Calabi-Yau, Grassmannian, and toric signatures, which sensitively depend on the number of generations of leptons, as well as inclusion of Majorana mass terms for right-handed neutrinos. We speculate as to why three generations is special.

We report on our experiences exploring state of the art Gröbner basis computation. We investigate signature based algorithms in detail. We also introduce new practical data structures and computational techniques for use in both signature based Gröbner basis algorithms and more traditional variations of the classic Buchberger algorithm. Our conclusions are based on experiments using our new freely available open source standalone C++ library.

Intersection rings of flag varieties and of isotropic flag varieties are generated by Chern classes of the tautological bundles modulo the relations coming from multiplicativity of total Chern classes. In this paper we describe the Groebner bases of the ideals of relations and give applications to computation of intersections, as implemented in Macaulay2.

Evaluating the likelihood function of parameters in highly-structured population genetic models from extant deoxyribonucleic acid (DNA) sequences is computationally prohibitive. In such cases, one may approximately infer the parameters from summary statistics of the data such as the site-frequency-spectrum (SFS) or its linear combinations. Such methods are known as approximate likelihood or Bayesian computations. Using a controlled lumped Markov chain and computational commutative algebraic methods, we compute the exact likelihood of the SFS and many classical linear combinations of it at a non-recombining locus that is neutrally evolving under the infinitely-many-sites mutation model. Using a partially ordered graph of coalescent experiments around the SFS, we provide a decision-theoretic framework for approximate sufficiency. We also extend a family of classical hypothesis tests of standard neutrality at a non-recombining locus based on the SFS to a more powerful version that conditions on the topological information provided by the SFS.

Boolean networks have long been used as models of molecular networks, and they play an increasingly important role in systems biology. This paper describes a software package, Polynome, offered as a web service, that helps users construct Boolean network models based on experimental data and biological input. The key feature is a discrete analog of parameter estimation for continuous models. With only experimental data as input, the software can be used as a tool for reverse-engineering of Boolean network models from experimental time course data.

Evaluating the likelihood function of parameters in complex population genetic models from extant deoxyribonucleic acid (DNA) se- quences is computationally prohibitive. In such cases, one may approxi- mately infer the parameters from various summary statistics of the data. Such method are known as approximate likelihood/Bayesian computa- tions. We employ computational commutative algebraic methods to ob- tain the exact likelihood of a large class of summary statistics that are linear combinations of the site frequency spectrum.

This paper describes a new implementation of an algorithm to find all isolated Nash equilibria in a finite strategic game. The implementation uses the game theory software package Gambit to generate systems of polynomial equations which are necessary conditions for a Nash equilibrium, and polyhedral homotopy continuation via the package PHCpack to compute solutions to the systems. Numerical experiments to characterize the performance of the implementation are reported. In addition, the current and future roles of support enumeration methods in the context of methods for computing Nash equilibria are discussed.

We work over an algebraically closed field K. Let X be a nonsingular irreducible projective curve of genus g ^ 2. Let L be a base point free line bundle on X and let <j)L\ X-*P(H°(L)*) denote the morphism defined by L. L is said to be normally generated (following Mumford [6]) if ij>L embeds X as a projectively normal curve. We recall that for X of genus (so that W\_^^\ the Clifford index, Cliff (X\ of X is defined by

Algorithms in algebraic geometry go hand in hand with software packages that implement them. Together they have established the modern field of computational algebraic geometry which has come to play