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This paper consists of four parts. It begins by using the authors' generalized Schauder formula, [41] , and the algebraic multiplicity, χ , of Esquinas and López-Gómez [15,14,31] to package and sharpening all existing results in local and global bifurcation theory for Fredholm operators through the recent author's axiomatization of the Fitzpatrick–Pejsachowicz–Rabier degree, [42]. This facilitates reformulating and refining all existing results in a compact and unifying way. Then, the local structure of the solution set of analytic nonlinearities F (λ , u) = 0 at a simple degenerate eigenvalue is ascertained by means of some concepts and devices of Algebraic Geometry and Galois Theory, which establishes a bisociation between Bifurcation Theory and Algebraic Geometry. Finally, the unilateral theorems of [31,33] , as well as the refinement of Shi and Wang [53] , are substantially generalized. This paper also analyzes two important examples to illustrate and discuss the relevance of the abstract theory. The second one studies the regular positive solutions of a multidimensional quasilinear boundary value problem of mixed type related to the mean curvature operator. [ABSTRACT FROM AUTHOR]

The text is a summary of the Proceedings of the 17th International Workshop on Real and Complex Singularities, which took place in 2022. The workshop is a renowned event in singularity theory and related fields, bringing together experts and young researchers to share their achievements and discuss research trends. The proceedings contain 29 research papers that have been selected and refereed according to the standards of the journal Research in the Mathematical Sciences. The collection aims to provide readers, including graduate students and researchers, with an opportunity to explore the field of singularities. The text also highlights the contributions and achievements of Professor Osamu Saeki, a leading researcher in singularity theory and topology, who was honored during the workshop. [Extracted from the article]

Recently, Yu et al. derived a factorization procedure for detecting and computing the potential self-intersection of 3D integral Bézier cubics, claiming that their proposal distinctly outperforms existing methodologies. First, we recall that in the 2D case, explicit formulas already exist for the parameter values at the self-intersection (the singularity called crunode in algebraic geometry). Such values are the solutions of a quadratic equation, and affine invariants depend only on the curve hodograph. Also, the factorization procedure for cubics is well known. Second, we note that only planar Bézier cubics can display a self-intersection, so there is no need to address the problem in the more involved 3D setting. Finally, we elucidate the connections with the previous literature and provide a geometric interpretation, in terms of the affine classification of cubics, of the algebraic conditions necessary for the existence of a self-intersection. Cubics with a self-intersection are affine versions of the celebrated Tschirnhausen cubic. [ABSTRACT FROM AUTHOR]

In previous papers, we were able to prove that, much like in classical algebraic geometry, it is possible to recover our monoid scheme X from the topos Qcoh(X). This was achieved using topos points and localisations of Qcoh(X). With this philosophy in mind, the aim of this paper is to study the topos of M-sets for non-commutative monoids, especially their points and localisations. We will classify the points and localisations of M-sets for finite monoids in terms of the idempotent elements of M and idempotent ideals of M, respectively. Some of the results obtained in this paper can already be found in previous works, in direct or indirect forms. See the last part of the introduction. [ABSTRACT FROM AUTHOR]

We apply computations of twisted Hodge diamonds to construct an infinite number of non-Fourier–Mukai functors with well-behaved target and source spaces. To accomplish this, we first study the characteristic morphism introduced in Buchweitz and Flenner [Adv. Math. 217 (2008), 205–242] in order to control it for tilting bundles. Then, we continue by applying twisted Hodge diamonds of hypersurfaces embedded in projective space to compute the Hochschild dimension of these spaces. This allows us to compute the kernel of the embedding into the projective space in Hochschild cohomology. Finally, we use the above computations to apply the construction in Rizzardo, Van den Bergh, and Neeman [Invent. Math. 216 (2019), 927–1004] of non-Fourier–Mukai functors and verify that the constructed functors indeed cannot be Fourier–Mukai for odd-dimensional quadrics. Using this approach, we prove that there are a large number of Hochschild cohomology classes that can be used for the construction of Rizzardo, Van den Bergh, and Neeman [Invent. Math. 216 (2019), 927–1004]. Furthermore, our results allow the application of computer-based calculations to construct candidate functors for arbitrary degree hypersurfaces in arbitrary high dimensions. Verifying that these are not Fourier–Mukai still requires the existence of a tilting bundle. In particular, we prove that there is at least one non-Fourier–Mukai functor for every odddimensional smooth quadric. [ABSTRACT FROM AUTHOR]

Poset-blowdown of subgroup posets of groups is an analog of blowdown in algebraic geometry. It is a poset map obtained by contracting normal subgroups. For finite groups, this is considered as a map between the Hasse diagrams of the subgroup posets. Poset-blowdowns are classified into three types: tame, wild, and hybrid depending on the sizes of their fibers. In this paper we describe the poset-blowdowns for generalized quaternion groups Q2n (n ≥ 3). They have distinguished nature in that all types (tame, wild, and hybrid) appear in the successive poset-blowdowns associated with the three chief series of Q2n. [ABSTRACT FROM AUTHOR]

We equip integral graded-polarized mixed period spaces with a natural Ralg-definable analytic structure, and prove that any period map associated to an admissible variation of integral graded-polarized mixed Hodge structures is definable in Ran,exp with respect to this structure. As a consequence we re-prove that the zero loci of admissible normal functions are algebraic. [ABSTRACT FROM AUTHOR]

Let GRSk(α, υ) be a k-dimensional generalized Reed-Solomon (GRS) code over Fq associated with α = (α1,. . ., αn) and υ = (υ1, . . ., υn). In this paper, we determined the dimension of the Euclidean hull GRSk(α, υ) ∩ GRSk(α, υ)⊥, which addresses an open problem posed in [Chen et al., IEEE-TIT, 2023]. We also presentd a new approach to generating all self-dual RS codes. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove a theorem for holomorphic twisted quiver bundles over a special non-compact Gauduchon manifold, connecting the existence of (σ , τ) -Hermite–Yang–Mills metric in differential geometry and the analytic (σ , τ) -stability in algebraic geometry. The proof of the theorem relies on the flow method and the Uhlenbeck–Yau's continuity method. [ABSTRACT FROM AUTHOR]

The interactions between topology and algebraic geometry expose various interesting properties. This paper proposes the deformations of topological n-manifolds over the automorphic polynomial ring maps and associated isomorphic imbedding of locally flat submanifolds within the n-manifolds. The manifold deformations include topologically homeomorphic bending of submanifolds at multiple directions under algebraic operations. This paper introduces the concept of a topological equivalence class of manifolds and the associated equivalent class of polynomials in a real ring. The concepts of algebraic compositions in a real polynomial ring and the resulting topological properties (homeomorphism, isomorphism and deformation) of manifolds under algebraic compositions are introduced. It is shown that a set of ideals in a polynomial ring generates manifolds retaining topological isomorphism under algebraic compositions. The numerical simulations are presented in this paper to illustrate the interplay of topological properties and the respective real algebraic sets generated by polynomials in a ring within affine 3-spaces. It is shown that the coefficients of polynomials generated by a periodic smooth function can induce mirror symmetry in manifolds. The proposed formulations do not consider the simplectic class of manifolds and associated quantizable deformations. However, the proposed formulations preserve the properties of Nash representations of real algebraic manifolds including Nash isomorphism. [ABSTRACT FROM AUTHOR]

In this note, we study certain sufficient conditions for a set of minimal klt pairs (X, Δ) with κ(X, Δ) = dim(X) - 1 to be bounded. [ABSTRACT FROM AUTHOR]

We classify RDP del Pezzo surfaces with global vector fields over arbitrary algebraically closed fields of characteristic p ≉ 2. In characteristic 0, every RDP del Pezzo surface X is equivariant, that is, AutX = AuteX, where eX is the minimal resolution of X; hence the classification of RDP del Pezzo surfaces with global vector fields is equivalent to the classification of weak del Pezzo surfaces with global vector fields. In this article, we show that if p ≉ 2, 3, 5, 7, then it is still true that every RDP del Pezzo surface is equivariant. We classify the non-equivariant RDP del Pezzo surfaces in characteristic p = 3, 5, 7, giving explicit equations for every such RDP del Pezzo surface in all possible degrees. As an application, we construct regular non-smooth RDP del Pezzo surfaces over imperfect fields of characteristic 7, thereby showing that the known bound p ⩽ 7 for the characteristics, where such a surface can exist, is sharp. [ABSTRACT FROM AUTHOR]

For an arbitrary function field, from the knowledge of the minimal generating set of the Weierstrass semigroup at two rational places, the set of pure gaps is characterized. Furthermore, we determine the Weierstrass semigroup at one and two totally ramified places in a Kummer extension defined by the affine equation y m = ∏ i = 1 r (x - α i) λ i over K, the algebraic closure of F q , where α 1 , ⋯ , α r ∈ K are pairwise distinct elements, 1 ≤ λ i < m , and gcd (m , ∑ i = 1 r λ i) = 1 . We apply these results to construct algebraic geometry codes over certain function fields with many rational places. For one-point codes we obtain families of codes with exact parameters. [ABSTRACT FROM AUTHOR]

The decreasing norm-trace codes are evaluation codes defined by a set of monomials closed under divisibility and the rational points of the extended norm-trace curve. In particular, the decreasing norm-trace codes contain the one-point algebraic geometry (AG) codes over the extended norm-trace curve. We use Gröbner basis theory and find the indicator functions on the rational points of the curve to determine the basic parameters of the decreasing norm-trace codes: length, dimension, and minimum distance. We also obtain their dual codes. We give conditions for a decreasing norm-trace code to be a self-orthogonal or a self-dual code. We provide a linear exact repair scheme to correct single erasures for decreasing norm-trace codes, which applies to higher rate codes than the scheme developed by Jin et al. (IEEE Trans Inf Theory 64(2):900–908, 2018) when applied to the one-point AG codes over the extended norm-trace curve. [ABSTRACT FROM AUTHOR]

In this paper, we study the action of diamond operators on Hilbert modular forms with coefficients in a general commutative ring. In particular, we generalize a result of Chai on the surjectivity of the constant term map for Hilbert modular forms with nebentype to the setting of group ring valued modular forms. As an application, we construct certain Hilbert modular forms required for Dasgupta–Kakde's proof of the Brumer–Stark conjecture at odd primes. Since the forms required for the Brumer–Stark conjecture live on the non-PEL Shimura variety associated to the reductive group G = Res F / Q (GL 2) , as opposed to the PEL Shimura variety associated to the subgroup G ∗ ⊂ G studied by Chai, we give a detailed explanation of theory of algebraic diamond operators for G, as well as how the theory of toroidal and minimal compactifications for G may be deduced from the analogous theory for G ∗ . [ABSTRACT FROM AUTHOR]

The geometry of 4-string contact interaction of closed string field theory is characterized using machine learning. We obtain Strebel quadratic differentials on 4-punctured spheres as a neural network by performing unsupervised learning with a custom-built loss function. This allows us to solve for local coordinates and compute their associated mapping radii numerically. We also train a neural network distinguishing vertex from Feynman region. As a check, 4-tachyon contact term in the tachyon potential is computed and a good agreement with the results in the literature is observed. We argue that our algorithm is manifestly independent of number of punctures and scaling it to characterize the geometry of n-string contact interaction is feasible. [ABSTRACT FROM AUTHOR]

In this study, we aim to prove the generalized Hyers-Ulam-Rassias (GHUR) stability for the following Euler-Lagrange (EL) type cubic functional equation f(bx+y)+f(x+by)=(b+1)(b-1)²[f(x)+f(y)]+b(b+1)f(x+y) in non-Archimedean (n.A) quasi-Banach spaces and n.A (n,β) Banach spaces. In recent decades, the stability of functional equations has emerged as one of the most intriguing and engaging topics, as it leads to the applications of functional equations in various domains such as algebraic geometry, Group theory, Mechanics etc., This study is to investigate the GHUR stability for the above equation using Hyers direct method. Furthermore, we obtain the stability results for the aforementioned equation with an illustrative example for the n.A case. With the study of the example one may easily understand how the stability result of functional equations in n.A case differs from the setting of classical Banach spaces. [ABSTRACT FROM AUTHOR]

We give the explicit equations for a P 3 × P 3 embedding of the Jacobian of a curve of genus 2, which gives a natural analog for abelian surfaces of the Edwards curve model of elliptic curves. This gives a much more succinct description of the Jacobian variety than the standard version in P 15 . We also give a condition under which, as for the Edwards curve, the abelian surfaces have a universal group law. [ABSTRACT FROM AUTHOR]

We define the notion of Y-algebroids, generalising the Lie, Courant, and exceptional algebroids that have been used to capture the local symmetry structure of type II string theory and M-theory compactifications to D ≥ 5 dimensions. Instead of an invariant inner product, or its generalisation arising in exceptional algebroids, Y-algebroids are built around a specific type of tensor, denoted Y , that provides exactly the necessary properties to also describe compactifications to D = 4 dimensions. We classify "M-exact" E7-algebroids and show that this precisely matches the form of the generalised tangent space of E7(7) × ℝ+-generalised geometry, with possible twists due to 1-, 4- and 7-form fluxes, corresponding physically to the derivative of the warp factor and the M-theory fluxes. We translate the notion of generalised Leibniz parallelisable spaces, relevant to consistent truncations, into this language, where they are mapped to so-called exceptional Manin pairs. We also show how to understand Poisson-Lie U-duality and exceptional complex structures using Y-algebroids. [ABSTRACT FROM AUTHOR]

We solve Prill's problem, originally posed by David Prill in the late 1970s and popularized in Arbarello, Cornalba, Griffiths and Harris's "Geometry of Algebraic Curves." That is, for any curve Y of genus 2, we produce a finite 'etale degree 36 connected cover f: X → Y where, for every point y ∈ Y, the preimage f-1(y) moves in a pencil. [ABSTRACT FROM AUTHOR]

In this paper we consider a family F of 2n-dimensional F q -linear rank metric codes in F q n × n arising from polynomials of the form x q s + δ x q n 2 + s ∈ F q n [ x ] . The family F was introduced by Csajbók et al. (JAMA 548:203–220) as a potential source for maximum rank distance (MRD) codes. Indeed, they showed that F contains MRD codes for n = 8 , and other subsequent partial results have been provided in the literature towards the classification of MRD codes in F for any n. In particular, the classification has been reached when n is smaller than 8, and also for n greater than 8 provided that s is small enough with respect to n. In this paper we deal with the open case n = 8 , providing a classification for any large enough odd prime power q. The techniques are from algebraic geometry over finite fields, since our strategy requires the analysis of certain 3-dimensional F q -rational algebraic varieties in a 7-dimensional projective space. We also show that the MRD codes in F are not equivalent to any other MRD codes known so far. [ABSTRACT FROM AUTHOR]

An MDS code is a code which achieves equality in the singleton bound. The defect of a code measures how far it is from an MDS code. Amplifying on the relationship between the weight distribution of a code and its dual code as in the well-known MacWilliams identities, we show in this paper that there are indeed strong lower bounds on the defects of codes or the dual codes. [ABSTRACT FROM AUTHOR]

The formulations of polynomials over a topological simplex combine the elements of topology and algebraic geometry. This paper proposes the formulation of simplicial polynomials and the properties of resulting topological manifolds in two classes, non-degenerate forms and degenerate forms, without imposing the conditions of affine topological spaces. The non-degenerate class maintains the degree preservation principle of the atoms of the polynomials of a topological simplex, which is relaxed in the degenerate class. The concept of hybrid decomposition of a simplicial polynomial in the non-degenerate class is introduced. The decompositions of simplicial polynomial for a large set of simplex vertices generate ideal components from the radical, and the components preserve the topologically isolated origin in all cases within the topological manifolds. Interestingly, the topological manifolds generated by a non-degenerate class of simplicial polynomials do not retain the homeomorphism property under polynomial extension by atom addition if the simplicial condition is violated. However, the topological manifolds generated by the degenerate class always preserve isomorphism with varying rotational orientations. The hybrid decompositions of the non-degenerate class of simplicial polynomials give rise to the formation of simplicial chains. The proposed formulations do not impose strict positivity on simplicial polynomials as a precondition. [ABSTRACT FROM AUTHOR]

What is striking about de Broglie's foundational work on wave–particle dualism is the role played by pseudo-Riemannian geometry in his early thinking. While exploring a fully covariant description of the Klein–Gordon equation, he was led to the revolutionary idea that a variable rest mass was essential. DeWitt later explained that in order to obtain a covariant quantum Hamiltonian, one must supplement the classical Hamiltonian with an additional energy ℏ 2 Q from which the quantum potential emerges, a potential that Berry has recently shown also arises in classical wave optics. In this paper, we show how these ideas emerge from an essentially geometric structure in which the information normally carried by the wave function is contained within the algebraic description of the geometry itself, within an element of a minimal left ideal. We establish the fundamental importance of conformal symmetry, in which rescaling of the rest mass plays a vital role. Thus, we have the basis for a radically new theory of quantum phenomena based on the process of mass-energy flow. [ABSTRACT FROM AUTHOR]

The paper summarizes some results on algebraic geometry presence in the homotopy theory. For the homotopy group πm(Sn ), denote by πmalg(Sn ) its subset of homotopy classes represented by R-entire rational maps Sm → Sn of spheres. The main result of this paper concerns to the study of π n+k alg(Sn) for k = 0, . . ., 7. [ABSTRACT FROM AUTHOR]

Let C be a linear code of length n and dimension k over the finite field F q m . The trace code Tr (C) is a linear code of the same length n over the subfield F q . The obvious upper bound for the dimension of the trace code over F q is m k. If equality holds, then we say that C has maximum trace dimension. The problem of finding the true dimension of trace codes and their duals is relevant for the size of the public key of various code-based cryptographic protocols. Let C a denote the code obtained from C and a multiplier vector a ∈ ( F q m ) n . In this study, we give a lower bound for the probability that a random multiplier vector produces a code C a of maximum trace dimension. We give an interpretation of the bound for the class of algebraic geometry codes in terms of the degree of the defining divisor. The bound explains the experimental fact that random alternant codes have minimal dimension. Our bound holds whenever n ≥ m (k + h) , where h ≥ 0 is the Singleton defect of C. For the extremal case n = m (h + k) , numerical experiments reveal a closed connection between the probability of having maximum trace dimension and the probability that a random matrix has full rank. [ABSTRACT FROM AUTHOR]

Let G be a closed symmetric monoidal concrete Grothendieck category. In this paper, we introduce a model structure on (CN(G),P⊗ dw) the exact category of N-complexes with the degree-wise ⊗-pure exact structure. Our result is based on the Gillespie's Theorem by introducing two compatible cotorsion pairs on this category. [ABSTRACT FROM AUTHOR]

We study the coherent cohomology of generalized flag supervarieties. Our main observation is that these groups are closely related to the free resolutions of (certain generalizations of) determinantal ideals. In the case of super Grassmannians, we completely compute the cohomology of the structure sheaf: it is composed of the singular cohomology of a Grassmannian and the syzygies of a determinantal variety. The majority of the work involves studying the geometry of an analog of the Grothendieck-Springer resolution associated with the super Grassmannian; this takes place in the world of ordinary (non-super) algebraic geometry. Our work gives a conceptual explanation of the result of Pragacz-Weyman that the syzygies of determinantal ideals admit an action of the general linear supergroup. In a subsequent paper, we will treat other flag supervarieties in detail. [ABSTRACT FROM AUTHOR]

We give an explicit description of the deformation theory of the Abelian category of (quasi-)coherent sheaves on any separated Noetherian scheme X via the deformation theory of path algebras of quivers with relations, by using any affine open cover of X, or any tilting bundle on X, if available. We also give sufficient criteria for obtaining algebraizations of formal deformations, in which case the deformation parameters can be evaluated to a constant and the deformations can be compared to the original Abelian category on equal terms. We give concrete examples as well as applications to the study of noncommutative deformations of singularities. [ABSTRACT FROM AUTHOR]

We prove that over any perfect field, the plane Cremona group is generated by involutions. [ABSTRACT FROM AUTHOR]

In this paper, we study an algebraic fiber space in positive characteristic whose generic fiber F has finitely generated canonical ring and sufficiently large Frobenius stable canonical ring. An example of such a case is when F is F-pure and its dualizing sheaf is invertible and ample. We treat a Fujita-type conjecture due to Popa and Schnell concerning direct images of pluricanonical bundles, and prove it under some additional hypotheses. As an application, we show the subadditivity of Kodaira dimensions in some new cases. We also prove an analog of Fujino's result regarding his Fujita-type conjecture. [ABSTRACT FROM AUTHOR]

We study the spectra of operators on periodic graphs using methods from combinatorial algebraic geometry. Our main result is a bound on the number of complex critical points of the Bloch variety, together with an effective criterion for when this bound is attained. We show that this criterion holds for Z²- and Z³-periodic graphs with sufficiently many edges and use our results to establish the spectral edges conjecture for some Z²-periodic graphs. [ABSTRACT FROM AUTHOR]

A translation surface is a multifaceted object that can be studied with the tools of dynamics, analysis, or algebraic geometry. Moduli spaces of translation surfaces exhibit equally rich features. This survey provides an introduction to the subject and describes some developments that make use of Hodge theory to establish algebraization and finiteness statements in moduli spaces of translation surfaces. [ABSTRACT FROM AUTHOR]

We construct a consistent dimer model having the same symmetry as its characteristic polygon. This produces examples of non-commutative crepant resolutions of non-toric non-quotient Gorenstein singularities in dimension 3. [ABSTRACT FROM AUTHOR]

This article, published in Mathematical Notes, discusses Hurwitz numbers for reflection groups. Hurwitz numbers are used to solve problems in combinatorics, topology, and algebraic geometry. The article explores the structure of the reflection groups and , and describes the conjugacy classes within these groups. It also discusses the relationship between Hurwitz numbers and cut-and-join equations. The article concludes with a discussion of the KP hierarchy and its connection to Hurwitz numbers. The research was funded by the National Research University Higher School of Economics and the International Laboratory of Cluster Geometry. [Extracted from the article]

There are two main problems in finding the higher genus superstring measure. The first one is that for g ≥ 5 the super moduli space is not projected. Furthermore, the supermeasure is regular for g ≤ 11, a bound related to the source of singularities due to the divisor in the moduli space of Riemann surfaces with even spin structure having holomorphic sections, such a divisor is called the θ-null divisor. A result of this paper is the characterization of such a divisor. This is done by first extending the Dirac propagator, that is the Szegö kernel, to the case of an arbitrary number of zero modes, that leads to a modification of the Fay trisecant identity, where the determinant of the Dirac propagators is replaced by the product of two determinants of the Dirac zero modes. By taking suitable limits of points on the Riemann surface, this holomorphic Fay trisecant identity leads to identities that include points dependent rank 3 quadrics in ℙg−1. Furthermore, integrating over the homological cycles gives relations for the Riemann period matrix which are satisfied in the presence of Dirac zero modes. Such identities characterize the θ-null divisor. Finally, we provide the geometrical interpretation of the above points dependent quadrics and show, via a new θ-identity, its relation with the Andreotti-Mayer quadric. [ABSTRACT FROM AUTHOR]

We present a method to construct the extended Kähler cone of any Calabi-Yau threefold by using Gopakumar-Vafa invariants to identify all geometric phases that are related by flops or Weyl reflections. In this way we obtain the Kähler moduli spaces of all favorable Calabi-Yau threefold hypersurfaces with h1,1 ≤ 4, including toric and non-toric phases. In this setting we perform an explicit test of the Weak Gravity Conjecture by using the Gopakumar-Vafa invariants to count BPS states. All of our examples satisfy the tower/sublattice WGC, and in fact they even satisfy the stronger lattice WGC. [ABSTRACT FROM AUTHOR]

By applying an inverse Kaluza-Klein procedure, we provide explicit coordinate expressions for Riemannian metrics on two homeomorphic but not diffeomorphic spheres in seven dimensions. We identify Milnor's bundles, among which ten out of the fourteen exotic seven-spheres appear (ignoring orientation), with non-principal bundles having homogeneous fibres. Then, we use the techniques in [1] to obtain a general ansatz for the coordinate expression of a metric on the total space of any Milnor's bundle. The ansatz is given in terms of a metric on S4, a metric on S3 (which can smoothly vary throughout S4), and a connection on the principal SO(4)-bundle over S4. As a concrete example, we present explicit formulae for such metrics for the ordinary sphere and the Gromoll-Meyer exotic sphere. Then, we perform a non-abelian Kaluza-Klein reduction to gravity in seven dimensions, according to (a slightly simplified version of) the metric ansatz above. We obtain the standard four-dimensional Einstein-Yang-Mills system, for which we find solutions associated with the geometries of the ordinary sphere and of the exotic one. The two differ by the winding numbers of the instantons involved. [ABSTRACT FROM AUTHOR]

We consider cones of real forms which are sums of squares and invariant under a (finite) reflection group. Using the representation theory of these groups we are able to use the symmetry inherent in these cones to give more efficient descriptions. We focus especially on the A n , B n , and D n case where we use so-called higher Specht polynomials to give a uniform description of these cones. These descriptions allow us, to deduce that the description of the cones of sums of squares of fixed degree 2 d stabilizes with n > 2 d. Furthermore, in cases of small degree, we are able to analyze these cones more explicitly and compare them to the cones of non-negative forms. [ABSTRACT FROM AUTHOR]

A Fano problem consists of enumerating linear spaces of a fixed dimension on a variety, generalizing the classical problem of 27 lines on a cubic surface. Those Fano problems with finitely many linear spaces have an associated Galois group that acts on these linear spaces and controls the complexity of computing them in suitable coordinates. These Galois groups were first defined and studied by Jordan, who in particular considered the problem of lines in P 3 on a cubic surface. Recently, Hashimoto and Kadets determined the Galois groups for a special family of Fano problems and showed that all other Fano problems have Galois group that contains the alternating group. A complete classification of Galois groups of Fano problems rests on the determination of these Galois groups. We use computational tools to prove that several Fano problems of moderate size have Galois group equal to the symmetric group, each of which were previously unknown. [ABSTRACT FROM AUTHOR]

We study integrable deformations of two-dimensional non-linear σ-models and present a new class of classical solutions to critical bi-Yang-Baxter models for general groups. For the simplest example, namely the SL(2, ℝ) bi-Yang-Baxter model, we show that our solutions can be mapped to the known complex uniton solutions of the SU(2) bi-Yang-Baxter model. In general, our solutions are constructed from so-called Sl(2)-orbits that play a central role in the study of asymptotic Hodge theory. This provides further evidence for a close relation between integrable non-linear σ-models and the mathematical principles underlying Hodge theory. We have also included a basic introduction to the relevant aspects of asymptotic Hodge theory and have provided some simple examples. [ABSTRACT FROM AUTHOR]

Scattering amplitudes in planar N = 4 super Yang-Mills theory exhibit singularities which reflect various aspects of the cluster algebras associated to the Grassmannians Gr(4, n) and their tropical counterparts. Here we investigate the potential origins of such structures and examine the extent to which they can be recovered from the Gröbner structure of the underlying Plücker ideals, focussing on the Grassmannians corresponding to finite cluster algebras. Starting from the Plücker ideal, we describe how the polynomial cluster variables are encoded in non-prime initial ideals associated to certain maximal cones of the positive tropical fan. Following [1] we show that extending the Plücker ideal by such variables leads to a Gröbner fan with a single maximal Gröbner cone spanned by the positive tropical rays. The associated initial ideal encodes the compatibility relations among the full set of cluster variables. Thus we find that the Gröbner structure naturally encodes both the symbol alphabet and the cluster adjacency relations exhibited by scattering amplitudes without invoking the cluster algebra at all. As a potential application of these ideas we then examine the kinematic ideal associated to non-dual conformal massless scattering written in terms of spinor helicity variables. For five-particle scattering we find that the ideal can be identified with the Plücker ideal for Gr(3, 6) and the corresponding tropical fan contains a number of non-prime ideals which encode all additional letters of the two-loop pentagon function alphabet present in various calculations of massless five-point finite remainders. [ABSTRACT FROM AUTHOR]

An excellent ring of prime characteristic for which the Frobenius map is pure is also Frobenius split in many commonly occurring situations in positive characteristic commutative algebra and algebraic geometry. However, using a fundamental construction from rigid geometry, we show that excellent F-pure rings of prime characteristic are not Frobenius split in general, even for Euclidean domains. Our construction uses the existence of a complete non-Archimedean field k of characteristic p with no nonzero continuous k-linear maps K 1/p → k . An explicit example of such a field is given based on ideas of Gabber, and may be of independent interest. Our examples settle a long-standing open question in the theory of F-singularities whose origin can be traced back to when Hochster and Roberts introduced the notion of F-purity. The excellent Euclidean domains we construct also admit no nonzero R-linear maps R1/p → R. These are the first examples that illustrate that F-purity and Frobenius splitting define different classes of singularities for excellent domains, and are also the first examples of excellent domains with no nonzero p−1 -linear maps. The latter is particularly interesting from the perspective of the theory of test ideals. [ABSTRACT FROM AUTHOR]

Inflationary spacetimes have been argued to be past geodesically incomplete in many situations. However, whether the geodesic incompleteness implies the existence of an initial spacetime curvature singularity or whether the spacetime may be extended (potentially into another phase of the universe) is generally unknown. Both questions have important physical implications. In this paper, we take a closer look at the geometrical structure of inflationary spacetimes and investigate these very questions. We first classify which past inflationary histories have a scalar curvature singularity and which might be extendible and/or non-singular in homogeneous and isotropic cosmology with flat spatial sections. Then, we derive rigorous extendibility criteria of various regularity classes for quasi-de Sitter spacetimes that evolve from infinite proper time in the past. Finally, we show that beyond homogeneity and isotropy, special continuous extensions respecting the Einstein field equations with a perfect fluid must have the equation of state of a de Sitter universe asymptotically. An interpretation of our results is that past-eternal inflationary scenarios are most likely physically singular, except in situations with very special initial conditions. [ABSTRACT FROM AUTHOR]

We study tree-level biadjoint scalar amplitudes in the language of D-modules. We construct left ideals in the Weyl algebra D that allow a holonomic representation of n-point amplitudes in terms of the linear partial differential equations they satisfy. The resulting representation encodes the simple pole and recursive properties of the amplitude. [ABSTRACT FROM AUTHOR]