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This work investigates the noncommutative affine geometry of matrix polynomial algebra extensions of a coefficient algebra by (elementary) matrix variables. A precise description of the spectrum (of maximal one-sided or bilateral ideals) of general matrix algebras is required. It results that the Zariski space of the irreducible representations of a matrix algebra is obtained by a natural gluing of the Zariski spaces of the irreducible representations of its diagonal components. An important step for the geometry of matrix polynomial algebras in commuting variables is achieved by a generalization of the Amitsur–Small Nullstellensatz, from which follows a precise description of their primitive quotients. We also characterize which of them are geometric algebras (in the sense of noncommutative deformation theory), reconstructible as algebras of observables from the scheme of irreducible representations. We then prove that each diagonal component of a matrix polynomial algebra in commuting variables is a Jacobson ring, whose non-Noetherian commutative geometry is efficiently described by the geometry of an affine essential subextension. And in the spirit of nonlocal algebraic geometry and addressing an open question by Charlie Beil, we obtain a class of non-Noetherian commutative monoid rings admitting closed points with positive geometric dimension. [ABSTRACT FROM AUTHOR]

In this paper, we develop an extended space-time geometry that includes an internal space built with division octonions with realization via Pauli matrices and Zorn (vectorial) matrices. Here, we extend a former field theory that used split octonion algebra to a division octonion algebra on a non-Riemannian manifold. The octonionic algebra is the last possible algebra allowed by the Hurwitz theorem. The interpretation of the “octonionic field” in this division octonionic algebra is not straightforward, however it behaves in a similar way to the quaternionic Yang–Mills fields. Former work suggests that this Yang–Mills-like octonionic field is associated with the field governing quarks within nucleons. In this work, we discover that the inclusion of octonionic fields in the geometry of an internal space necessarily excludes the quaternionic (Yang–Mills) fields in an extended non-Riemannian geometry. This is not what is expected from the standpoint of a “unified” field theory, which leads us to propose a different approach to Einstein’s unified field theory. [ABSTRACT FROM AUTHOR]

Quantum Entanglement is one of the key manifestations of quantum mechanics that separate the quantum realm from the classical one. Characterization of entanglement as a physical resource for quantum technology became of uppermost importance. While the entanglement of bipartite systems is already well understood, the ultimate goal to cope with the properties of entanglement of multipartite systems is still far from being realized. This paper covers characterization of multipartite entanglement using algebraic-geometric tools. First, we establish an algorithm to classify multipartite entanglement by k -secant varieties of the Segre variety and ℓ -multilinear ranks that are invariant under Stochastic Local Operations with Classical Communication (SLOCC). We present a fine-structure classification of multiqubit and tripartite entanglement based on this algorithm. Another fundamental problem in quantum information theory is entanglement transformation that is quite challenging regarding to multipartite systems. It is captivating that the proposed entanglement classification by algebraic geometry can be considered as a reference to study SLOCC and asymptotic SLOCC interconversions among different resources based on tensor rank and border rank, respectively. In this regard, we also introduce a new class of tensors that we call persistent tensors and construct a lower bound for their tensor rank. We further cover SLOCC convertibility of multipartite systems considering several families of persistent tensors. [ABSTRACT FROM AUTHOR]

We introduce the Cox homotopy algorithm for solving a sparse system of polynomial equations on a compact toric variety X Σ . The algorithm lends its name from a construction, described by Cox, of X Σ as a GIT quotient X Σ = (ℂ k ∖ Z) / / G of a quasi-affine variety by the action of a reductive group. Our algorithm tracks paths in the total coordinate space ℂ k of X Σ and can be seen as a homogeneous version of the standard polyhedral homotopy, which works on the dense torus of X Σ . 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 X Σ . 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]

An alternative proof of the necessary conditions on A , B ∈ q 2 * for f (X) = X + A X 1 + q 2 (q − 1) / 4 + B X 1 + 3 q 2 (q − 1) / 4 to be a permutation polynomial in q 2 , q even, is given. This proof involves standard arguments from algebraic geometry over finite fields and fast symbolic computations. [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]

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]

Let R be a commutative ring with 1 and n a natural number. We say that a submodule N of R n is semiprime if for every f = (f 1 , ... , f n) ∈ R n such that f i f ∈ N for i = 1 , ... , n we have f ∈ N. Our main result is that every semiprime submodule of R n is equal to the intersection of all prime submodules containing it. It follows that every semiprime left ideal of M n (R) is equal to the intersection of all prime left ideals that contain it. For R = k [ x 1 , ... , x d ] where k is an algebraically closed field we can rephrase this result as a Nullstellensatz for M n (R) : For every G 1 , ... , G m , F ∈ M n (R) , F belongs to the smallest semiprime left ideal of M n (R) that contains G 1 , ... , G m iff for every a ∈ k d and v ∈ k n such that G 1 (a) v = ⋯ = G m (a) v = 0 we have F (a) v = 0. [ABSTRACT FROM AUTHOR]

This is an overview on certain higher structural constructions in physics. Main motivations of our current attempt are as follows: (i) to provide a brief introduction to the basics of derived algebraic geometry, (ii) to understand how certain derived objects naturally appear in physics and give rise to a formal mathematical treatment, and (iii) to investigate how the notion of a factorization algebra together with certain higher categorical structures come into play to encode the structure of observables in physics. Adopting such a heavy and relatively enriched language allows us to formalize the notions of quantization and observables in quantum field theory as well. This document is organized to explain the underlying mathematical treatment for each task in an expository manner. [ABSTRACT FROM AUTHOR]

In this paper, we examine the large- g asymptotic Weil–Petersson volume formulas deduced in the previous literature. The volume formulas have application to computing the partition functions and the correlation functions in Jackiw–Teitelboim gravity. We utilize two approaches to assess the validity of the formulas. The first approach is to examine the asymptotic volume formulas from the perspective of the Witten conjecture. If the volume formulas are correct, the generating function of the intersection indices deduced from the asymptotic volume formulas would satisfy constraints that are analogous to the Virasoro constraints. We confirmed that the intersection indices computed from the large- g asymptotic volume formulas satisfy variants of the string and dilaton equations. This implies that the generating function of the intersection indices deduced from the asymptotic volume formulas indeed satisfies constraints analogous to first two of the Virasoro constraints. As another approach, we also examined the asymptotic volume formulas by studying the behavior of the higher-order spectral form factors. Our analyses suggest that the large- g asymptotic Weil–Petersson volume formulas yield plausible estimations. [ABSTRACT FROM AUTHOR]

In this paper we compare the performance of two algebraic geometry codes (Suzuki and Hermitian codes) constructed using maximal algebraic curves over q 4 with large automorphism groups by choosing specific divisors. We discuss their parameters, compare the rate of these codes as well as their relative minimum distances, and we show that both codes are asymptotically good in terms of the rate which is in contrast to their behavior in terms of the relative minimum distance. [ABSTRACT FROM AUTHOR]

A well-known combinatorial algorithm can decide generic rigidity in the plane by determining if the graph is of Pollaczek–Geiringer–Laman type. Methods from matroid theory have been used to prove other interesting results, again under the assumption of generic configurations. However, configurations arising in applications may not be generic. We present Theorem 4.2 and its corresponding Algorithm 1 which decide if a configuration is -locally rigid, a notion we define. A configuration which is -locally rigid may be locally rigid or flexible, but any continuous deformations remain within a sphere of radius in configuration space. Deciding -local rigidity is possible for configurations which are smooth or singular, generic or non-generic. We also present Algorithms 2 and 3 which use numerical algebraic geometry to compute a discrete-time sample of a continuous flex, providing useful visual information for the scientist. [ABSTRACT FROM AUTHOR]

We provide a classification and an explicit realization of all simple Gelfand-Tsetlin modules of the complex Lie algebra sl(3). The realization of these modules, including those with infinite-dimensional weight spaces, is given via regular and derivative Gelfand-Tsetlin tableaux. Also, we show that all simple Gelfand-Tsetlin sl(3)-modules can be obtained as subquotients of localized Gelfand-Tsetlin E21-injective modules. [ABSTRACT FROM AUTHOR]

Differential geometry, especially the use of curvature, plays a central role in modern Hodge theory. The vector bundles that occur in the theory (Hodge bundles) have metrics given by the polarizations of the Hodge structures, and the sign and singularity properties of the resulting curvatures have far reaching implications in the geometry of families of algebraic varieties. A special property of the curvatures is that they are 1 st order invariants expressed in terms of the norms of algebro-geometric bundle mappings. This partly expository paper will explain some of the positivity and singularity properties of the curvature invariants that arise in the Hodge theory with special emphasis on the norm property. [ABSTRACT FROM AUTHOR]

In this paper, we briefly overview how, historically, string theory led theoretical physics first to precise problems in algebraic and differential geometry, and thence to computational geometry in the last decade or so, and now, in the last few years, to data science. Using the Calabi–Yau landscape — accumulated by the collaboration of physicists, mathematicians and computer scientists over the last four decades — as a starting-point and concrete playground, we review some recent progress in machine-learning applied to the sifting through of possible universes from compactification, as well as wider problems in geometrical engineering of quantum field theories. In parallel, we discuss the program in machine-learning mathematical structures and address the tantalizing question of how it helps doing mathematics, ranging from mathematical physics, to geometry, to representation theory, to combinatorics and to number theory. [ABSTRACT FROM AUTHOR]

In this study, we construct four-dimensional F-theory models with 3 to 8 U(1) factors on products of K3 surfaces. We provide explicit Weierstrass equations of elliptic K3 surfaces with Mordell–Weil ranks of 3 to 8. We utilize the method of quadratic base change to glue pairs of rational elliptic surfaces together to yield the aforementioned types of K3 surfaces. The moduli of elliptic K3 surfaces constructed in the study include Kummer surfaces of specific complex structures. We show that the tadpole cancels in F-theory compactifications with flux when these Kummer surfaces are paired with appropriately selected attractive K3 surfaces. We determine the matter spectra on F-theory on the pairs. [ABSTRACT FROM AUTHOR]

The moduli space of Gieseker vector bundles is a compactification of moduli of vector bundles on a nodal curve. This moduli space has only normal-crossing singularities and it provides flat degeneration of the moduli of vector bundles over a smooth projective curve. We prove a Torelli type theorem for a nodal curve using the moduli space of stable Gieseker vector bundles of fixed rank (strictly greater than 1) and fixed degree such that rank and degree are co-prime. [ABSTRACT FROM AUTHOR]

We consider the parameter space 𝒰 d of smooth plane curves of degree d. The universal smooth plane curve of degree d is a fiber bundle ℰ d → 𝒰 d with fiber diffeomorphic to a surface Σ g . This bundle gives rise to a monodromy homomorphism ρ d : π 1 (𝒰 d) → Mod (Σ g) , where Mod (Σ g) : = π 0 ( Diff + (Σ g)) is the mapping class group of Σ g . The main result of this paper is that the kernel of ρ 4 : π 1 (𝒰 4) → Mod (Σ 3) is isomorphic to F ∞ × ℤ / 3 ℤ , where F ∞ is a free group of countably infinite rank. In the process of proving this theorem, we show that the complement Teich (Σ g) ∖ ℋ g of the hyperelliptic locus ℋ g in Teichmüller space Teich (Σ g) has the homotopy type of an infinite wedge of spheres. As a corollary, we obtain that the moduli space of plane quartic curves is aspherical. The proofs use results from the Weil–Petersson geometry of Teichmüller space together with results from algebraic geometry. [ABSTRACT FROM AUTHOR]

We compute all relative dynamical degrees of equivariant dominant rational maps on toric varieties. We use the intersection theory on toric varieties using Minkowski weights. As a result, we see that the relative dynamical degrees can be reduced to the dynamical degrees on toric varieties. Hence these are all algebraic integers. [ABSTRACT FROM AUTHOR]

In this paper, we analyze gauge groups in six-dimensional N = 1 F-theory models. We construct elliptic Calabi–Yau 3-folds possessing various singularity types as double covers of "1/2 Calabi–Yau 3-folds," a class of rational elliptic 3-folds, by applying the method discussed in a previous study to classify the singularity types of the 1/2 Calabi–Yau 3-folds. One to three U(1) factors are formed in six-dimensional F-theory on the constructed Calabi–Yau 3-folds. The singularity types of the constructed Calabi–Yau 3-folds corresponding to the non-Abelian gauge group factors in six-dimensional F-theory are deduced. The singularity types of the Calabi–Yau 3-folds constructed in this work consist of A - and D -type singularities. [ABSTRACT FROM AUTHOR]

In this paper, we introduce weak (Σ , Δ) -compatible rings and study skew Poincaré–Birkhoff–Witt extensions over these rings. We characterize the weak notion of compatibility for several noncommutative rings appearing in noncommutative algebraic geometry and some quantum algebras of theoretical physics. As a consequence of our treatment, we unify and extend results in the literature about Ore extensions and skew PBW extensions over compatible rings. [ABSTRACT FROM AUTHOR]

In this paper, we present explicit toric construction of moduli space of quasi maps from ℙ 1 with two marked points to ℙ 1 × ℙ 1 , which was first proposed by Jinzenji and prove that it is a compact orbifold. We also determine its Chow ring and compute its Poincaré polynomial for some lower degree cases. [ABSTRACT FROM AUTHOR]

Using Gaussian mixture models for clustering is a statistically mature method for clustering in data science with numerous successful applications in science and engineering. The parameters for a Gaussian mixture model (GMM) are typically estimated from training data using the iterative expectation-maximization algorithm, which requires the number of Gaussian components a priori. In this study, we propose two algorithms rooted in numerical algebraic geometry (NAG), namely, an area-based algorithm and a local maxima algorithm, to identify the optimal number of components. The area-based algorithm transforms several GMM with varying number of components into sets of equivalent polynomial regression splines. Next, it uses homotopy continuation methods for evaluating the resulting splines to identify the number of components that is most compatible with the gradient data. The local maxima algorithm forms a set of polynomials by fitting a smoothing spline to a dataset. Next, it uses NAG to solve the system of the first derivatives for finding the local maxima of the resulting smoothing spline, which represent the number of mixture components. The local maxima algorithm also identifies the location of the centers of Gaussian components. Using a real-world case study in automotive manufacturing and extensive simulations, we demonstrate that the performance of the proposed algorithms is comparable with that of Akaike information criterion (AIC) and Bayesian information criterion (BIC), which are popular methods in the literature. We also show the proposed algorithms are more robust than AIC and BIC when the Gaussian assumption is violated. [ABSTRACT FROM AUTHOR]

The local sum conjecture is a variant of some of Igusa's questions on exponential sums put forward by Denef and Sperber. In a remarkable paper by Cluckers, Mustata and Nguyen, this conjecture has been established in all dimensions, using sophisticated, powerful techniques from a research area blending algebraic geometry with ideas from logic. The purpose of this paper is to give an elementary proof of this conjecture in two dimensions which follows Varčenko's treatment of Euclidean oscillatory integrals based on Newton polyhedra for good coordinate choices. Another elementary proof is given by Veys from an algebraic geometric perspective. [ABSTRACT FROM AUTHOR]

We observe a new puzzling physical phenomenon in F-theory on the multisection geometry, wherein a model without a gauge group transitions to another model with a discrete ℤ n gauge group via Higgsing. This phenomenon may suggest an unknown aspect of F-theory compactification on multisection geometry lacking a global section. A possible interpretation of this puzzling physical phenomenon is proposed in this note. We also propose a possible interpretation of another unnatural physical phenomenon observed for F-theory on four-section geometry, wherein a discrete ℤ 2 gauge group transitions to a discrete ℤ 4 gauge group via Higgsing as described in the previous literature. [ABSTRACT FROM AUTHOR]

There are two types of Belyi's Theorems for curves defined over finite fields of characteristic p , namely the Wild and the Tame p -Belyi Theorems. In this paper, we discuss them in the language of function fields. In particular, we provide a constructive proof for the existence of a pseudo-tame element introduced in [Y. Sugiyama and S. Yasuda, Belyi's theorem in characteristic two, Compos. Math. 156(2) (2020) 325–339], which leads to a self-contained proof for the Tame 2 -Belyi Theorem. Moreover, we provide unified and simple proofs for Belyi's Theorems unlike the known ones that use technical results from Algebraic Geometry. [ABSTRACT FROM AUTHOR]

For compact complex manifolds with vanishing first Chern class that are compact torus principal bundles over Kähler manifolds, we prove that all holomorphic geometric structures on them, of affine type, are locally homogeneous. For a compact simply connected complex manifold in Fujiki class 𝒞 , whose dimension is strictly larger than the algebraic dimension, we prove that it does not admit any holomorphic rigid geometric structure, and also it does not admit any holomorphic Cartan geometry of algebraic type. We prove that compact complex simply connected manifolds in Fujiki class 𝒞 and with vanishing first Chern class do not admit any holomorphic Cartan geometry of algebraic type. [ABSTRACT FROM AUTHOR]

In this paper, we develop the theory of residually finite rationally p (RFR p) groups, where p is a prime. We first prove a series of results about the structure of finitely generated RFR p groups (either for a single prime p , or for infinitely many primes), including torsion-freeness, a Tits alternative, and a restriction on the BNS invariant. Furthermore, we show that many groups which occur naturally in group theory, algebraic geometry, and in 3-manifold topology enjoy this residual property. We then prove a combination theorem for RFR p groups, which we use to study the boundary manifolds of algebraic curves ℂ ℙ 2 and in ℂ 2 . We show that boundary manifolds of a large class of curves in ℂ 2 (which includes all line arrangements) have RFR p fundamental groups, whereas boundary manifolds of curves in ℂ ℙ 2 may fail to do so. [ABSTRACT FROM AUTHOR]

Suppose that X A ⊂ ℙ n − 1 is a toric variety of codimension two defined by an (n − 2) × n integer matrix A , and let B be a Gale dual of A. In this paper, we compute the Euclidean distance degree and polar degrees of X A (along with other associated invariants) combinatorially working from the matrix B. Our approach allows for the consideration of examples that would be impractical using algebraic or geometric methods. It also yields considerably simpler computational formulas for these invariants, allowing much larger examples to be computed much more quickly than the analogous combinatorial methods using the matrix A in the codimension two case. [ABSTRACT FROM AUTHOR]

We introduce and study a notion of Castelnuovo–Mumford regularity suitable for weighted projective spaces. [ABSTRACT FROM AUTHOR]

In rough set theory (RST), the notion of decision table plays a fundamental role. In this paper, we develop a purely mathematical investigation of this notion to show that several basic aspects of RST can be of interest also for mathematicians who work with algebraic and discrete methods. In this abstract perspective, we call decision system a sextuple 𝒮 = 〈 U 𝒮 , Ω 𝒮 , C 𝒮 , D 𝒮 , F 𝒮 , Λ 𝒮 〉 , where U 𝒮 , C 𝒮 , Λ 𝒮 are non-empty sets whose elements are called, respectively, objects, condition attributes, values, D 𝒮 is a (possibly empty) set whose elements are called decision attributes, Ω 𝒮 : = C 𝒮 ∪ D 𝒮 and F 𝒮 : U 𝒮 × (C 𝒮 ∪ D 𝒮) → Λ 𝒮 is a map. The basic tool of our analysis is the equivalence relation ≡ A on U 𝒮 , depending on the choice of a condition attribute subset A ⊆ C 𝒮 ∪ D 𝒮 and defined as follows: u ≡ A u ′ : ⇔ F 𝒮 (u , a) = F 𝒮 (u ′ , a) ∀ a ∈ A. We denote by [ u ] A the equivalence class of u ∈ U 𝒮 with respect to ≡ A . We interpret the classical RST notions of consistency and inconsistency for a decision table in an abstract algebraic set operatorial perspective and, in such a context, we introduce and investigate a kind of local consistency in any decision system 𝒮. More specifically, we fix W ⊆ U 𝒮 , A ⊆ C 𝒮 and try to determine in what cases all objects u ∈ W satisfy the condition [ u ] A ∩ W ⊆ [ u ] D 𝒮 ∩ W. Then, we build a formal general framework whose basic tools are two local consistency set operators and a global closure operator, the condition attribute set C 𝒮 . This paper provides a detailed study of these set operators, of the induced set systems and of the most relevant links between them. [ABSTRACT FROM AUTHOR]

We introduce Niebrzydowski algebras, algebraic structures with a ternary operation and a partially defined multiplication, with axioms motivated by the Reidemeister moves for Y -oriented trivalent spatial graphs and handlebody-links. As part of this definition, we identify generating sets of Y -oriented Reidemeister moves. We give some examples to demonstrate that the counting invariant can distinguish some Y -oriented trivalent spatial graphs and handlebody-links. [ABSTRACT FROM AUTHOR]

We classify all projective manifolds with flat holomorphic conformal structure. The Kähler–Einstein case was treated by Kobayashi and Ochiai, there exists a very short list of possible manifolds. In the non-Kähler–Einstein case a classification was only known in small dimensions: by Kobayashi and Ochiai for complex surfaces and by the authors for projective threefolds. This paper completes the general case. [ABSTRACT FROM AUTHOR]

This is an Erratum for our paper "Singular Knots and Involutive Quandles" in [1]. It replaces Sec. 5 in the published version of the paper. [ABSTRACT FROM AUTHOR]

We establish a large sieve inequality for power moduli in ℤ [ i ] , extending earlier work by Zhao and the first-named author on the large sieve for power moduli for the classical case of moduli in ℤ. Our method starts with a version of the large sieve for ℝ 2 . We convert the resulting counting problem back into one for ℤ [ i ] which we then attack using Weyl differencing and Poisson summation. [ABSTRACT FROM AUTHOR]

An arithmetic method of proving the irrationality of smooth projective 3-folds is described, using reduction modulo p. It is illustrated by an application to a cubic threefold, for which the hypothesis that its intermediate Jacobian is isomorphic to the Jacobian of a curve is contradicted by reducing modulo 3 and counting points over appropriate extensions of 𝔽 3 . As a spin-off, it is shown that the 5-dimensional Prym varieties arising as intermediate Jacobians of certain cubic 3-folds have the maximal number of points over 𝔽 q which attains Perret's and Weil's upper bounds. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a generalization of the ordinary ideal transform, denoted by D I , J , which is called the ideal transform with respect to a pair of ideals (I , J) and has an apparent algebraic structure. Then we study its various properties and explore the connection with the ordinary ideal transform. Also, we discuss the associated primes of local cohomology modules with respect to a pair of ideals. In particular, we give a characterization for the associated primes of the nonvanishing generalized local cohomology modules. [ABSTRACT FROM AUTHOR]

Let 𝒢 = { g 1 , ... , g n } be a set of elements in the polynomial ring R = ℂ [ z 1 , ... , z N ] , let I ⊂ R denote the ideal generated by the elements of 𝒢 , and let I denote the radical of I. There is a unique decomposition I = P 1 ∩ ⋯ ∩ P k with each P i a prime ideal corresponding to a minimal associated prime of I over R. Let V (𝒢) = V (I) denote the reduced algebraic set corresponding to the common zeroes of the elements of 𝒢. Techniques from numerical algebraic geometry can be used to determine the numerical irreducible decomposition of V (𝒢) over ℂ. This corresponds to producing a witness set for V (P i) for each i = 1 , ... , k together with the degree and dimension of V (P i) (a point in a witness set for V (P i) can be considered as a numerical approximation for a general point on V (P i)). The purpose of this paper is to show how to extend these results taking into account the field of definition for the polynomial system. In particular, let F be a number field (i.e. a finite field extension of ℚ) and let 𝒢 = { g 1 , ... , g n } be a set of elements in S = F [ z 1 , ... , z N ]. We show how to extend techniques from numerical algebraic geometry to determine the numerical irreducible decomposition of V (𝒢) over F. [ABSTRACT FROM AUTHOR]

Hyperbolic programming is the problem of computing the infimum of a linear function when restricted to the hyperbolicity cone of a hyperbolic polynomial, a generalization of semidefinite programming (SDP). We propose an approach based on symbolic computation, relying on the multiplicity structure of the algebraic boundary of the cone, without the assumption of determinantal representability. This allows us to design exact algorithms able to certify the multiplicity of the solution and the optimal value of the linear function. [ABSTRACT FROM AUTHOR]

We prove an equivariant version of Beilinson's conjecture on non-critical L -values of strongly modular abelian varieties over number fields. The proof builds on Beilinson's theorem on modular curves as well as a modularity result for endomorphism algebras. As an application, we prove a weak version of Zagier's conjecture on L (E , 2) and Deninger's conjecture on L (E , 3) for non-CM strongly modular Q -curves. [ABSTRACT FROM AUTHOR]

We show that, for a closed orientable n -manifold, with n not congruent to 3 modulo 4, the existence of a CR-regular embedding into complex (n − 1) -space ensures the existence of a totally real embedding into complex n -space. This implies that a closed orientable (4 k + 1) -manifold with non-vanishing Kervaire semi-characteristic possesses no CR-regular embedding into complex 4 k -space. We also pay special attention to the cases of CR-regular embeddings of spheres and of simply-connected 5-manifolds. [ABSTRACT FROM AUTHOR]

We consider a category ℬ whose morphisms are n -dimensional pseudomanifolds equipped with certain additional structures (coloring and labeling of some cells), multiplication of morphisms is similar to a concatenation of cobordisms. On the other hand, we consider the product G of (n + 1) copies of infinite symmetric group. We construct a correspondence between the sets of morphisms of ℬ and double coset spaces of G with respect to certain subgroups. We show that unitary representations of G produce functors from the category of ℬ to the category of Hilbert spaces and bounded linear operators. [ABSTRACT FROM AUTHOR]

This paper studies the associativity of gluing of trajectories in Morse theory. We show that the associativity of gluing follows from the existence of compatible manifold with face structures on the compactified moduli spaces. Using our previous work, we obtain the associativity of gluing in certain cases. In particular, associativity holds when the ambient manifold is compact and the vector field is Morse–Smale. [ABSTRACT FROM AUTHOR]

In this paper, we give partial answers to the following questions: Which contact manifolds are contactomorphic to links of isolated complex singularities? Which symplectic manifolds are symplectomorphic to smooth affine varieties? The invariant that we will use to distinguish such manifolds is called the growth rate of wrapped Floer cohomology. Using this invariant we show that if Q is a simply connected manifold whose unit cotangent bundle is contactomorphic to the link of an isolated singularity or whose cotangent bundle is symplectomorphic to a smooth affine variety then M must be rationally elliptic and so it must have certain bounds on its Betti numbers. [ABSTRACT FROM AUTHOR]

Equivariant twisted K theory classes on compact Lie groups G can be realized as families of Fredholm operators acting in a tensor product of a fermionic Fock space and a representation space of a central extension of the loop algebra L G using a supersymmetric Wess–Zumino–Witten model. The aim of the present paper is to extend the construction to higher loop algebras using an abelian extension of a 3D current algebra. We have only partial success: Instead of true Fredholm operators we have formal algebraic expressions in terms of the generators of the current algebra and an infinite dimensional Clifford algebra. These give rise to sesquilinear forms in a Hilbert bundle which transform in the expected way with respect to 3D gauge transformations but do not define true Hilbert space operators. [ABSTRACT FROM AUTHOR]

We give a version of Cartan’s Theorem A for nonsingular real affine varieties after blowing up. [ABSTRACT FROM AUTHOR]

We prove that a weak -Fano -fold with terminal singularities has unobstructed deformations. By using this result and computing some invariants of a terminal singularity, we provide two results on global deformation of a weak -Fano -fold. We also treat a stacky proof of the unobstructedness of deformations of a -Fano -fold. [ABSTRACT FROM AUTHOR]

We call a group FJ if it satisfies the K- and L-theoretic Farrell–Jones conjecture with coefficients in ℤ. We show that if G is FJ, then the simple Borel conjecture (in dimensions ≥5) holds for every group of the form G⋊ℤ. If in addition Wh(G×ℤ)=0, which is true for all known torsion-free FJ groups, then the bordism Borel conjecture (in dimensions n≥5) holds for G⋊ℤ. One of the key ingredients in proving these rigidity results is another main result, which says that if a torsion-free group G satisfies the L-theoretic Farrell–Jones conjecture with coefficients in ℤ, then any semi-direct product G⋊ℤ also satisfies the L-theoretic Farrell–Jones conjecture with coefficients in ℤ. Our result is indeed more general and implies the L-theoretic Farrell–Jones conjecture with coefficients in additive categories is closed under extensions of torsion-free groups. This enables us to extend the class of groups which satisfy the Novikov conjecture. [ABSTRACT FROM AUTHOR]

Recently, Chang and Li generalized the theory of virtual fundamental class to the setting of semi-perfect obstruction theory. A semi-perfect obstruction theory requires only the local existence of a perfect obstruction theory with compatibility conditions. In this paper, we generalize the torus localization of Graber-Pandharipande [T. Graber and R. Pandharipande, Localization of virtual cycles, Invent. Math. 135(2) (1999) 487-518], the cosection localization [Y.-H. Kiem and J. Li, Localizing virtual cycles by cosections, J. Amer. Math. Soc. 26(4) (2013) 1025-1050] and their combination [H.-L. Chang, Y.-H. Kiem and J. Li, Torus localization and wall crossing for cosection localized virtual cycles, Adv. Math. 308 (2017) 964-986], to the setting of semi-perfect obstruction theory. As an application, we show that the Jiang-Thomas theory [Y. Jiang and R. Thomas, Virtual signed Euler characteristics, preprint (2014), arXiv:1408.2541] of virtual signed Euler characteristic works without the technical quasi-smoothness assumption from derived algebraic geometry. [ABSTRACT FROM AUTHOR]