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Many algorithms for determining properties of real algebraic or semi-algebraic sets rely upon the ability to compute smooth points. In this paper, we present a procedure based on computing the critical points of some well-chosen function that guarantees the computation of smooth points in each bounded connected component of a (real) atomic semi-algebraic set. Our technique is intuitive in principal, performs well on previously difficult examples, and is straightforward to implement using existing numerical algebraic geometry software. The practical efficiency of our approach is demonstrated by solving a conjecture on the number of equilibria of the Kuramoto model for the n = 4 case. We also apply our method to design an algorithm to compute the real dimension of algebraic sets, the original motivation for this research. We compare the efficiency of our method to existing methods to compute the real dimension on a benchmark family. [ABSTRACT FROM AUTHOR]

We apply methods of machine-learning, such as neural networks, manifold learning and image processing, in order to study 2-dimensional amoebae in algebraic geometry and string theory. With the help of embedding manifold projection, we recover complicated conditions obtained from so-called lopsidedness. For certain cases it could even reach ∼ 99 % accuracy, in particular for the lopsided amoeba of F 0 with positive coefficients which we place primary focus. Using weights and biases, we also find good approximations to determine the genus for an amoeba at lower computational cost. In general, the models could easily predict the genus with over 90% accuracies. With similar techniques, we also investigate the membership problem, and image processing of the amoebae directly. [ABSTRACT FROM AUTHOR]

We analyze the bit complexity of an algorithm for the computation of at least one point in each connected component of a smooth real algebraic set. This work is a continuation of our analysis of the hypersurface case (On the bit complexity of finding points in connected components of a smooth real hypersurface , ISSAC'20). In this paper, we extend the analysis to more general cases. Let F = (f 1 , ... , f p) in Z [ X 1 , ... , X n ] p be a sequence of polynomials with V = V (F) ⊂ C n a smooth and equidimensional variety and 〈 F 〉 ⊂ C [ X 1 , ... , X n ] a radical ideal. To compute at least one point in each connected component of V ∩ R n , our starting point is an algorithm by Safey El Din and Schost (Polar varieties and computation of one point in each connected component of a smooth real algebraic set , ISSAC'03). This algorithm uses random changes of variables that are proven to generically ensure certain desirable geometric properties. The cost of the algorithm was given in an algebraic complexity model; here, we analyze the bit complexity and the error probability, and we provide a quantitative analysis of the genericity statements. In particular, we are led to use Lagrange systems to describe polar varieties, as they make it simpler to rely on techniques such as weak transversality and an effective Nullstellensatz. [ABSTRACT FROM AUTHOR]

We identify two families of ten-point Feynman diagrams that generalize the elliptic double box, and show that they can be expressed in terms of the same class of elliptic multiple polylogarithms to all loop orders. Interestingly, one of these families can also be written as a dlog form. For both families of diagrams, we provide new 2ℓ-fold integral representations that are linearly reducible in all but one variable and that make the above properties manifest. We illustrate the simplicity of this integral representation by directly integrating the three-loop representative of both families of diagrams. These families also satisfy a pair of second-order differential equations, making them ideal examples on which to develop bootstrap techniques involving elliptic symbol letters at high loop orders. [ABSTRACT FROM AUTHOR]

In this paper we discuss the construction of conserved tensors (currents) involving conformal Killing-Yano tensors (CKYTs), generalising the corresponding constructions for Killing-Yano tensors (KYTs). As a useful preparation for this, but also of intrinsic interest, we derive identities relating CKYTs and geometric quantities. The behaviour of CKYTs under conformal transformations is also given, correcting formulae in the literature. We then use the identities derived to construct covariantly conserved "currents". We find several new CKYT currents and also include a known one by Penrose which shows that "trivial" currents are also useful. We further find that rank-n currents based on rank-n CKYTs k must have a simple form in terms of dk. By construction, these currents are covariant under a general conformal rescaling of the metric. How currents lead to conserved charges is then illustrated using the Kerr-Newman and the C-metric in four dimensions. Separately, we study a rank-1 current, construct its charge and discuss its relation to the recently constructed Cotton current for the Kerr-Newman black hole. [ABSTRACT FROM AUTHOR]

We investigate the reduction of Feynman integrals to master integrals using Gröbner bases in a rational double-shift algebra Y in which the integration-by-parts (IBP) relations form a left ideal. The problem of reducing a given family of integrals to master integrals can then be solved once and for all by computing the Gröbner basis of the left ideal formed by the IBP relations. We demonstrate this explicitly for several examples. We introduce so-called first-order normal-form IBP relations which we obtain by reducing the shift operators in Y modulo the Gröbner basis of the left ideal of IBP relations. For more complicated cases, where the Gröbner basis is computationally expensive, we develop an ansatz based on linear algebra over a function field to obtain the normal-form IBP relations. [ABSTRACT FROM AUTHOR]

The way spacetime behaves as one approaches a spacelike singularity is reinvestigated. We find a simple twistorial presentation that includes and simplifies the classic work of Belinskii, Khalatnikov and Lifshitz as well as the more recent results of Damour, Henneaux and Nicolai. We speculate on the application of our technique to the E10 programme of M-theory. [ABSTRACT FROM AUTHOR]

We present the Gel'fand-Kapranov-Zelevinsky (GKZ) hypergeometric systems of the Feynman integrals of the three-loop vacuum diagrams with arbitrary masses, basing on Mellin-Barnes representations and Miller's transformation. The codimension of derived GKZ hypergeometric systems equals the number of independent dimensionless ratios among the virtual masses squared. Through GKZ hypergeometric systems, the analytical hypergeometric series solutions can be obtained in neighborhoods of origin including infinity. The linear independent hypergeometric series solutions whose convergent regions have non-empty intersection can constitute a fundamental solution system in a proper subset of the whole parameter space. The analytical expression of the vacuum integral can be formulated as a linear combination of the corresponding fundamental solution system in certain convergent region. [ABSTRACT FROM AUTHOR]

We formalise the teleparallel version of extended geometry (including gravity) by the introduction of a complex, the differential of which provides the linearised dynamics. The main point is the natural replacement of the two-derivative equations of motion by a differential which only contains terms of order 0 and 1 in derivatives. Second derivatives arise from homotopy transfer (elimination of fields with algebraic equations of motion). The formalism has the advantage of providing a clear consistency relation for the algebraic part of the differential, the "dualisation", which then defines the dynamics of physical fields. It remains unmodified in the interacting BV theory, and the full non-linear models arise from covariantisation. A consequence of the use of the complex is that symmetry under local rotations becomes as good as manifest, instead of arising for a specific combination of tensorial terms, for less obvious reasons. We illustrate with a derivation of teleparallel Ehlers geometry, where the extended coordinate module is the adjoint module of a finite-dimensional simple Lie group. [ABSTRACT FROM AUTHOR]

We consider various notions of Mayer–Vietoris squares in algebraic geometry. We use these to generalize a number of gluing and push out results of Moret‐Bailly, Ferrand–Raynaud, Joyet and Bhatt. An important intermediate step is Gabber's rigidity theorem for henselian pairs, which our methods give a new proof of. [ABSTRACT FROM AUTHOR]

The theory of commutative rings of differential operators relating the completely integrable systems with the geometry of algebraic curves was constructed several decades ago. It was especially complete in the case of rings, generated by two operators of the coprime order, usually of order 2 and of some odd order; the theory of such rings turned out to be equivalent to the theory of KdV hierarchy. However, the corresponding algebraic curves were always hyperelliptic. In order to handle the general (canonical curves), one should consider the rings, generated by more than two operators. In the previous paper of 1980, the author considered the simplest possible case of this kind—that of generators of orders 3, 4, 5. The goal of the present paper is to give the details of the calculations in that paper and to explain the conjectural geometry underlying some enigmatic phenomena that were used in 1980 to complete the calculations and give some algebro-geometric applications. [ABSTRACT FROM AUTHOR]

In noncommutative algebraic geometry, noncommutative quadric hypersurfaces are major objects of study. In this paper, we focus on studying noncommutative conics Pro j nc A embedded into Calabi-Yau quantum projective planes. In particular, we give complete classifications of homogeneous coordinate algebras A of noncommutative conics up to isomorphism of graded algebras, and of noncommutive conics Pro j nc A up to isomorphism of noncommutative schemes. [ABSTRACT FROM AUTHOR]

Let K be an algebraically closed field. Building upon previous work, we classify, up to isomorphism of algebras, quadratic graded twisted tensor products of K [ x , y ] and K [ z ]. When such an algebra is Artin-Schelter regular, we identify its point scheme and type, in the sense of [6]. We also describe which three-dimensional Sklyanin algebras contain a subalgebra isomorphic to a quantum P 1 , and we show that every algebra in this family is a graded twisted tensor product of K − 1 [ x , y ] and K [ z ]. [ABSTRACT FROM AUTHOR]

A Teichmüller curve V \subset \mathcal {M}_g is an isometrically immersed algebraic curve in the moduli space of Riemann surfaces. These rare, extremal objects are related to billiards in polygons, Hodge theory, algebraic geometry and surface topology. This paper presents the six known families of primitive Teichmüller curves that have been discovered over the past 30 years, and a selection of open problems. [ABSTRACT FROM AUTHOR]

Let n ≥ 1 be an integer, and L ⊂ ℝn be a compact smooth affine real hypersurface, not necessarily connected. We prove that there exist c > 0 and d0 ≥ 1 such that for any d ≥ d0, any smooth complex projective hypersurface Z in ℂ Pn of degree d contains at least c dim H*(Z, ℝ) disjoint Lagrangian submanifolds diffeomorphic to L, where Z is equipped with the restriction of the Fubini-Study symplectic form (Theorem 1.1). If moreover all connected components of L have non-vanishing Euler characteristic, which implies that n is odd, the latter Lagrangian submanifolds form an independent family in Hn-1 (Z, ℝ) (Corollary 1.2). These deterministic results are consequences of a more precise probabilistic theorem (Theorem 1.23) inspired by a 2014 result by J.-Y. Welschinger and the author on random real algebraic geometry, together with quantitative Moser-type constructions (Theorem 3.4). For n = 2, the method provides a uniform positive lower bound for the probability that a projective complex curve in ℂ P² of given degree equipped with the restriction of the ambient metric has a systole of small size (Theorem 1.6), which is an analog of a similar bound for hyperbolic curves given by M. Mirzakhani (2013). In higher dimensions, we pro-vide a similar result for the (n - 1)-systole introduced by M. Berger (1972) (Corollary 1.14). Our results hold in the more general setting of vanishing loci of holomorphic sections of vector bundles of rank between 1 and n tensored by a large power of an ample line bundle over a projective complex n-manifold (Theorem 1.20). [ABSTRACT FROM AUTHOR]

In this article, we generalize the Hodge-Tate decomposition of p-adic étale cohomology to non-smooth rigid spaces. Our strategy is to study pro-étale cohomology of rigid spaces introduced by Scholze, using the resolution of singularities and the simplicial method. [ABSTRACT FROM AUTHOR]

We construct Chern classes of vector bundles in the universal precobordism theory of Annala-Yokura over an arbitrary Noetherian base ring of finite Krull dimension. As an immediate corollary, we show that the Grothendieck ring of vector bundles can be recovered from the universal precobordism ring, and that we can construct candidates for Chow rings satisfying an analogue of the classical Grothendieck-Riemann-Roch theorem. We also strengthen the weak projective bundle formula of Annala-Yokura to the case of arbitrary projective bundles. [ABSTRACT FROM AUTHOR]

We derive a residue formula and as a consequence a recurrence relation for the instanton partition function in N = 2 supersymmetric theory on ℂ2 with SU(N) gauge group. The particular cases of SU(2) and SU(3) gauge groups were considered in the literature before. The recurrence relation with SU(2) gauge group is long well known and was found as the Alday-Gaiotto-Tachikawa (AGT) counterpart of the Zamolodchikov relation for the Virasoro conformal blocks. In the SU(3) case a residue formula for the term with the minimal number of instantons was found and basing on it a recurrence relation was conjectured. We give a complete proof of the residue formula in all instanton orders in presence of any number of matter hypermultiplets in the adjoint and fundamental representations. The recurrence relation however describes only theories with not too much matter hypermultiplets so that the behaviour at infinity is moderate. The guideline of the proof is an algebro-geometric interpretation of the N = 2 supersymmetric gauge theory partition function in terms of the framed torsion-free sheaves. Lead by this interpretation we formulate a refined version of the residue formula and prove it by direct algebraic manipulations. [ABSTRACT FROM AUTHOR]

We construct Poisson bracket relations between the operators which generate the chiral ring of the Coulomb branch of certain 3d N = 4 quiver gauge theories. In the case where the Coulomb branch is a free space, ADE Klein singularity, or the minimal A2 nilpotent orbit, we explicitly compute the Poisson brackets between the generators using either inherited properties of the abstract Coulomb branch variety, or the expected charges of these operators under the global symmetry (known through use of the monopole formula). We also conjecture Poisson brackets for Higgs branches that originate from 6d theories with tensionless strings or 5d theories with massless instantons for which the HWG is known, based on representation theoretic and operator content constraints known from the study of their magnetic quiver. [ABSTRACT FROM AUTHOR]

The 2d (0, 2) supersymmetric gauge theories corresponding to the classes of Yp,k(ℂℙ1× ℂℙ1) and Yp,k(ℂℙ2) manifolds are identified. The complex cones over these Sasaki-Einstein 7-manifolds are non-compact toric Calabi-Yau 4-folds. These infinite families of geometries are the largest ones for Sasaki-Einstein 7-manifolds whose metrics, toric diagrams, and volume functions are known explicitly. This work therefore presents the largest list of 2d (0, 2) supersymmetric gauge theories corresponding to Calabi-Yau 4-folds with known metrics. [ABSTRACT FROM AUTHOR]

Let X be a smooth affine variety over a field \mathbf k of characteristic 0 and T(X) be the Lie algebra of regular vector fields on X. We compute the Lie algebra cohomology of T(X) with coefficients in \mathbf k. The answer is given in topological terms relative to any embedding \mathbf k\subset \mathbb {C} and is analogous to the classical Gel′fand-Fuchs computation for smooth vector fields on a C^\infty-manifold. Unlike the C^\infty-case, our setup is purely algebraic: no topology on T(X) is present. The proof is based on the techniques of factorization algebras, both in algebro-geometric and topological contexts. [ABSTRACT FROM AUTHOR]

The eigenspectrum of the Koopman operator enables the decomposition of nonlinear dynamics into a sum of nonlinear functions of the state space with purely exponential and sinusoidal time dependence. For a limited number of dynamical systems, it is possible to find these Koopman eigenfunctions exactly and analytically. Here, this is done for the Korteweg–de Vries equation on a periodic interval using the periodic inverse scattering transform and some concepts of algebraic geometry. To the authors' knowledge, this is the first complete Koopman analysis of a partial differential equation, which does not have a trivial global attractor. The results are shown to match the frequencies computed by the data-driven method of dynamic mode decomposition (DMD). We demonstrate that in general, DMD gives a large number of eigenvalues near the imaginary axis and show how these should be interpreted in this setting. [ABSTRACT FROM AUTHOR]

Generic initial ideals (gins for short) were systematically introduced by Galligo in 1974 under the name of Grauert invariants since they appeared apparently first in works of Grauert and Hironaka. Ever since they have been of interest in commutative algebra and indirectly in algebraic geometry. Recently, Mayes in a series of articles associated with gins of graded families of ideals geometric objects called limiting shapes. The construction resembles that of Okunkov bodies but there are some differences as well. This work is motivated by Mayes articles and explores the connections between gins, limiting shapes, and some asymptotic invariants of homogeneous ideals which are associated with the gins, for example, asymptotic regularity, Waldschmidt constant and some new invariants, which seem relevant from geometric point of view. [ABSTRACT FROM AUTHOR]

The article focuses on the set-theoretic complete intersection property, which examines the intersection of an irreducible curve with a divisor on a normal, projective variety. The topics covered include the behavior of abelian varieties over different types of fields (finite fields, number fields, and geometric fields), providing insights into the qualitative properties of these fields based on the topology of the variety.

The article focuses on complements, counterexamples, and conjectures in the context of a topological form of the Gabriel-Rosenberg theorem for varieties characterized by constructible étale sheaves. Topics include the determination of the Zariski topological space of a scheme based on the category of constructible étale sheaves, analogues of Gabriel's reconstruction theorem, and examples illustrating inability to recover a field from Zariski topology of a projective space over a finite field.

The article focuses on the definition and properties of algebraic varieties. Topics discussed include the classical definition of algebraic varieties as subsets of projective spaces and introduces the concept of irreducibility and dimension; the morphisms between varieties, quasi-projective algebraic sets; and mentions the sheaf of regular functions.

The article introduces various geometric concepts, including affine and projective geometries, as well as spherical and conic geometries, and discusses the relationship between these geometries and fields. It also poses a question about bijections between points in projective spaces that map algebraic curves to algebraic curves and explores different answers depending on the field involved, such as finite fields, real fields, and algebraically closed fields of characteristic 0.

We present a generalization of the notion of neighborliness to non-polyhedral convex cones. Although a definition of neighborliness is available in the non-polyhedral case in the literature, it is fairly restrictive as it requires all the low-dimensional faces to be polyhedral. Our approach is more flexible and includes, for example, the cone of positive-semidefinite matrices as a special case (this cone is not neighborly in general). We term our generalization Terracini convexity due to its conceptual similarity with the conclusion of Terracini's lemma from algebraic geometry. Polyhedral cones are Terracini convex if and only if they are neighborly. More broadly, we derive many families of non-polyhedral Terracini convex cones based on neighborly cones, linear images of cones of positive-semidefinite matrices, and derivative relaxations of Terracini convex hyperbolicity cones. As a demonstration of the utility of our framework in the non-polyhedral case, we give a characterization based on Terracini convexity of the tightness of semidefinite relaxations for certain inverse problems. [ABSTRACT FROM AUTHOR]

The condition-based complexity analysis framework is one of the gems of modern numerical algebraic geometry and theoretical computer science. Among the challenges that it poses is to expand the currently limited range of random polynomials that we can handle. Despite important recent progress, the available tools cannot handle random sparse polynomials and Gaussian polynomials, that is polynomials whose coefficients are i.i.d. Gaussian random variables. We initiate a condition-based complexity framework based on the norm of the cube that is a step in this direction. We present this framework for real hypersurfaces and univariate polynomials. We demonstrate its capabilities in two problems, under very mild probabilistic assumptions. On the one hand, we show that the average run-time of the Plantinga-Vegter algorithm is polynomial in the degree for random sparse (alas a restricted sparseness structure) polynomials and random Gaussian polynomials. On the other hand, we study the size of the subdivision tree for Descartes' solver and run-time of the solver by Jindal and Sagraloff (2017). In both cases, we provide a bound that is polynomial in the size of the input (size of the support plus the logarithm of the degree) not only for the average but also for all higher moments. [ABSTRACT FROM AUTHOR]

Tropical Differential Algebraic Geometry considers difficult or even intractable problems in Differential Equations and tries to extract information on their solutions from a restricted structure of the input. The fundamental theorem of Tropical Differential Algebraic Geometry and its extensions state that the support of power series solutions of systems of ordinary differential equations (with formal power series coefficients over an uncountable algebraically closed field of characteristic zero) can be obtained either, by solving a so-called tropicalized differential system, or by testing monomial-freeness of the associated initial ideals. Tropicalized differential equations work on a completely different algebraic structure which may help in theoretical and computational questions, particularly on the existence of solutions. We show here that both of these methods can be generalized to the case of systems of partial differential equations, this is, one can go either with the solution of tropicalized systems, or test monomial-freeness of the ideal generated by the initials when looking for supports of power series solutions of systems of differential equations, regardless the (finite) number of derivatives. The key are the vertex sets of Newton polytopes, upon which relies the definition of both tropical vanishing condition and the initial of a differential polynomial. [ABSTRACT FROM AUTHOR]

We apply numerical algebraic geometry to the invariant-theoretic problem of detecting symmetries between two plane algebraic curves. We describe an efficient equality test which determines, with "probability-one", whether or not two rational maps have the same image up to Zariski closure. The application to invariant theory is based on the construction of suitable signature maps associated to a group acting linearly on the respective curves. We consider two versions of this construction: differential and joint signature maps. In our examples and computational experiments, we focus on the complex Euclidean group, and introduce an algebraic joint signature that we prove determines equivalence of curves under this action and the size of a curve's symmetry group. We demonstrate that the test is efficient and use it to empirically compare the sensitivity of differential and joint signatures to different types of noise. [ABSTRACT FROM AUTHOR]

Parameterized systems of polynomial equations arise in many applications in science and engineering with the real solutions describing, for example, equilibria of a dynamical system, linkages satisfying design constraints, and scene reconstruction in computer vision. Since different parameter values can have a different number of real solutions, the parameter space is decomposed into regions whose boundary forms the real discriminant locus. This article views locating the real discriminant locus as a supervised classification problem in machine learning where the goal is to determine classification boundaries over the parameter space, with the classes being the number of real solutions. This article presents a novel sampling method which carefully samples a multidimensional parameter space. At each sample point, homotopy continuation is used to obtain the number of real solutions to the corresponding polynomial system. Machine learning techniques including nearest neighbors, support vector classifiers, and neural networks are used to efficiently approximate the real discriminant locus. One application of having learned the real discriminant locus is to develop a real homotopy method that only tracks real solution paths unlike traditional methods which track all complex solution paths. Examples show that the proposed approach can efficiently approximate complicated solution boundaries such as those arising from the equilibria of the N = 4 Kuramoto model which was previously intractable using traditional methods. [ABSTRACT FROM AUTHOR]

We establish interval arithmetic as a practical tool for certification in numerical algebraic geometry. Our software HomotopyContinuation.jl now has a built-in function certify, which proves the correctness of an isolated nonsingular solution to a square system of polynomial equations. The implementation rests on Krawczyk's method. We demonstrate that it dramatically outperforms earlier approaches to certification. We see this contribution as a powerful new tool in numerical algebraic geometry, which can make certification the default and not just an option. [ABSTRACT FROM AUTHOR]

As is well known, self-similar solutions to the mean curvature flow, including self-shrinkers, translating solitons and self-expanders, arise naturally in the singularity analysis of the mean curvature flow. In this paper we give complete classifications of codimension one complete self-shrinkers with nonnegative constant scalar curvature. We also give alternative proofs of the classifications of codimension one translating solitons and self-expanders with nonnegative constant scalar curvature. [ABSTRACT FROM AUTHOR]

Transcription factors (TFs) and microRNAs (miRNAs) are co-actors in genome-scale decoding and regulatory networks, often targeting common genes. To discover the symmetries and invariants of the transcription and regulation at the scale of the genome, in this paper, we introduce tools of infinite group theory and of algebraic geometry to describe both TFs and miRNAs. In TFs, the generator of the group is a DNA-binding domain while, in miRNAs, the generator is the seed of the sequence. For such a generated (infinite) group π , we compute the S L (2 , C) character variety, where S L (2 , C) is simultaneously a 'space-time' (a Lorentz group) and a 'quantum' (a spin) group. A noteworthy result of our approach is to recognize that optimal regulation occurs when π looks similar to a free group F r ( r = 1 to 3) in the cardinality sequence of its subgroups, a result obtained in our previous papers. A non-free group structure features a potential disease. A second noteworthy result is about the structure of the Groebner basis G of the variety. A surface with simple singularities (such as the well known Cayley cubic) within G is a signature of a potential disease even when π looks similar to a free group F r in its structure of subgroups. Our methods apply to groups with a generating sequence made of two to four distinct DNA/RNA bases in { A , T / U , G , C } . We produce a few tables of human TFs and miRNAs showing that a disease may occur when either π is away from a free group or G contains surfaces with isolated singularities. [ABSTRACT FROM AUTHOR]

Let $f : X \to S$ be a family of smooth projective algebraic varieties over a smooth connected quasi-projective base $S$ , and let $\mathbb {V} = R^{2k} f_{*} \mathbb {Z}(k)$ be the integral variation of Hodge structure coming from degree $2k$ cohomology it induces. Associated to $\mathbb {V}$ one has the so-called Hodge locus $\textrm {HL}(S) \subset S$ , which is a countable union of 'special' algebraic subvarieties of $S$ parametrizing those fibres of $\mathbb {V}$ possessing extra Hodge tensors (and so, conjecturally, those fibres of $f$ possessing extra algebraic cycles). The special subvarieties belong to a larger class of so-called weakly special subvarieties, which are subvarieties of $S$ maximal for their algebraic monodromy groups. For each positive integer $d$ , we give an algorithm to compute the set of all weakly special subvarieties $Z \subset S$ of degree at most $d$ (with the degree taken relative to a choice of projective compactification $S \subset \overline {S}$ and very ample line bundle $\mathcal {L}$ on $\overline {S}$). As a corollary of our algorithm we prove conjectures of Daw–Ren and Daw–Javanpeykar–Kühne on the finiteness of sets of special and weakly special subvarieties of bounded degree. [ABSTRACT FROM AUTHOR]

One fundamental consequence of a scheme $X$ being proper is that the functor classifying maps from $X$ to any other suitably nice scheme or algebraic stack is representable by an algebraic stack. This result has been generalized by replacing $X$ with a proper algebraic stack. We show, however, that it also holds when $X$ is replaced by many examples of algebraic stacks which are not proper, including many global quotient stacks. This leads us to revisit the definition of properness for stacks. We introduce the notion of a formally proper morphism of stacks and study its properties. We develop methods for establishing formal properness in a large class of examples. Along the way, we prove strong $h$ -descent results which hold in the setting of derived algebraic geometry but not in classical algebraic geometry. Our main applications are algebraicity results for mapping stacks and the stack of coherent sheaves on a flat and formally proper stack. [ABSTRACT FROM AUTHOR]

We study the Rouquier dimension of wrapped Fukaya categories of Liouville manifolds and pairs, and apply this invariant to various problems in algebraic and symplectic geometry. On the algebro-geometric side, we introduce a new method based on symplectic flexibility and mirror symmetry to bound the Rouquier dimension of derived categories of coherent sheaves on certain complex algebraic varieties and stacks. These bounds are sharp in dimension at most $3$. As an application, we resolve a well-known conjecture of Orlov for new classes of examples (e.g. toric $3$ -folds, certain log Calabi–Yau surfaces). We also discuss applications to non-commutative motives on partially wrapped Fukaya categories. On the symplectic side, we study various quantitative questions including the following. (1) Given a Weinstein manifold, what is the minimal number of intersection points between the skeleton and its image under a generic compactly supported Hamiltonian diffeomorphism? (2) What is the minimal number of critical points of a Lefschetz fibration on a Liouville manifold with Weinstein fibers? We give lower bounds for these quantities which are to the best of the authors' knowledge the first to go beyond the basic flexible/rigid dichotomy. [ABSTRACT FROM AUTHOR]

This work is devoted to simplifying the formulation and solution of the closure equations associated with the forward kinematic problem (FKP) of the 5-RPUR parallel manipulator, a limited-DOF robot able to perform 3T2R motion. The analysis yields a set of eighteen nonlinear equations that are solved numerically through a combination of the homotopy continuation method and the usual Newton-Raphson technique. Unlike existing methods, the proposed approach is easy to follow and can be easily translated into computer codes. Numerical examples are provided with the aim to illustrate the potential and correctness of the proposed method. [ABSTRACT FROM AUTHOR]

One way to study a hypergraph is to attach to it a tensor. Tensors are a generalization of matrices, and they are an efficient way to encode information in a compact form. In this paper, we study how properties of weighted hypergraphs are reflected on eigenvalues and eigenvectors of their associated tensors. We also show how to efficiently compute eigenvalues with some techniques from numerical algebraic geometry. [ABSTRACT FROM AUTHOR]

Technology of Ancient Greece As Nicos pointed out "automaton" (Graph HT ht ) is a Homeric word that first appeared in the Iliad (Vasileiadou et al., [14]). Assigning the roots of Graph HT ht is much more difficult. Likewise the Greek word Graph HT ht "system" comes from the ancient verb Graph HT ht which occurs frequently in ancient Greek written sources in many contexts with a basic meaning that the whole is made from parts and that the whole may have properties not always attributed to the individual parts. [Extracted from the article]

The algebraic geometry of a universal algebra A is defined as the collection of solution sets of systems of term equations. Two algebras A 1 and A 2 are called algebraically equivalent if they have the same algebraic geometry. We prove that on a finite set A with | A | there are countably many algebraically inequivalent Mal'cev algebras and that on a finite set A with | A | there are continuously many algebraically inequivalent algebras. [ABSTRACT FROM AUTHOR]

We study the derived critical locus of a function f : [ X / G ] → 1 on the quotient stack of a smooth affine scheme X by the action of a smooth affine group scheme G. It is shown that dCrit (f) ≃ [ Z / G ] is a derived quotient stack for a derived affine scheme Z, whose dg-algebra of functions is described explicitly. Our results generalize the classical BV formalism in finite dimensions from Lie algebra to group actions. [ABSTRACT FROM AUTHOR]

We introduce the Cox homotopy algorithm for solving a sparse system of polynomial equations on a compact toric variety XΣ. The algorithm lends its name from a construction, described by Cox, of XΣ as a GIT quotient XΣ = (ℂk∖Z)//G of a quasi-affine variety by the action of a reductive group. Our algorithm tracks paths in the total coordinate space ℂk of XΣ and can be seen as a homogeneous version of the standard polyhedral homotopy, which works on the dense torus of XΣ. It furthermore generalizes the commonly used path tracking algorithms in (multi)projective spaces in that it tracks a set of homogeneous coordinates contained in the G-orbit corresponding to each solution. The Cox homotopy combines the advantages of polyhedral homotopies and (multi)homogeneous homotopies, tracking only mixed volume many solutions and providing an elegant way to deal with solutions on or near the special divisors of XΣ. In addition, the strategy may help to understand the deficiency of the root count for certain families of systems with respect to the BKK bound. [ABSTRACT FROM AUTHOR]

Given a bivariate system of polynomial equations with fixed support sets A , B it is natural to ask which multiplicities its solutions can have. We prove that there exists a system with a solution of multiplicity i for all i in the range { 0 , 1 , ... , | A | − | conv (A) ⊖ B | − 1 } , where A ⊖ B is the set of all integral vectors that shift B to a subset of A. As an application, we classify all pairs (A , B) such that the system supported at (A , B) does not have a solution of multiplicity higher than 2. [ABSTRACT FROM AUTHOR]

We use a correspondence between Ulrich bundles on a projective variety and quiver representations to prove that certain del Pezzo surfaces satisfy the Ulrich trichotomy, for any given polarization. [ABSTRACT FROM AUTHOR]

In this paper, we explain how the abstract notion of a differential bundle in a tangent category provides a new way of thinking about the category of modules over a commutative ring and its opposite category. MacAdam previously showed that differential bundles in the tangent category of smooth manifolds are precisely smooth vector bundles. Here we provide characterizations of differential bundles in the tangent categories of commutative rings and (affine) schemes. For commutative rings, the category of differential bundles over a commutative ring is equivalent to the category of modules over that ring. For affine schemes, the category of differential bundles over the Spec of a commutative ring is equivalent to the opposite category of modules over said ring. Finally, for schemes, the category of differential bundles over a scheme is equivalent to the opposite category of quasi-coherent sheaves of modules over that scheme. [ABSTRACT FROM AUTHOR]

Motivated by classical functional analysis results over the complex numbers and results in the bornological setting over the complex numbers of R. Meyer, we study several aspects of the study of Ind-Banach modules over Banach rings. This allows for a synthesis of some aspects of homological algebra and functional analysis. This includes a study of nuclear modules and of modules which are flat with respect to the projective tensor product. We also study metrizable and Fréchet Ind-Banach modules. We give explicit descriptions of projective limits of Banach rings as ind-objects. We study exactness properties of the projective tensor product with respect to kernels and countable products. As applications, we describe a theory of quasi-coherent modules in Banach algebraic geometry. We prove descent theorems for quasi-coherent modules in various analytic and arithmetic contexts and relate them to well known complexes of modules coming from covers. [ABSTRACT FROM AUTHOR]

Given a bounded-above cochain complex of modules over a ring, it is standard to replace it by a projective resolution, and it is classical that doing so can be very useful. Recently, a modified version of this was introduced in triangulated categories other than the derived category of a ring. A triangulated category is approximable if this modified procedure is possible. Not surprisingly this has proved a powerful tool. For example: the fact that Dqc (X) is approximable when X is a quasi compact, separated scheme led to major improvements on old theorems due to Bondal, Van den Bergh and Rouquier. In this article, we prove that, underweak hypotheses, the recollement of two approximable triangulated categories is approximable. In particular, this shows many of the triangulated categories that arise in noncommutative algebraic geometry are approximable. Furthermore, the lemmas and techniques developed in this article form a powerful toolbox which, in conjunction with the groundwork laid in [16], has interesting applications in existing and forthcoming work by the authors. [ABSTRACT FROM AUTHOR]

In this paper, we give a counterexample to Zariski's cancellation problem for principal ${\mathbb{G}}_{a}$ G a -bundles over certain open subvarieties of non- $\mathbb{A}^{1}$ A 1 -uniruled affine varieties. [ABSTRACT FROM AUTHOR]