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In this paper, we study the algebraic symplectic geometry of the singular moduli spaces of Higgs bundles of degree 0 and rank n on a compact Riemann surface X of genus g. In particular, we prove that such moduli spaces are symplectic singularities, in the sense of Beauville [Invent. Math. 139 (2000), 541–549], and admit a projective symplectic resolution if and only if g = 1 or (g, n) = (2, 2). These results are an application of a recent paper by Bellamy and Schedler [ArXiv e-print (2016)] via the so-called Isosingularity Theorem. [ABSTRACT FROM AUTHOR]

We study intersection theory for differential algebraic varieties. Particularly, we study families of differential hypersurface sections of arbitrary affine differential algebraic varieties over a differential field. We prove the differential analogue of Bertini's theorem, namely that for an arbitrary geometrically irreducible differential algebraic variety which is not an algebraic curve, generic hypersurface sections are geometrically irreducible and codimension one. Surprisingly, we prove a stronger result in the case that the order of the differential hypersurface is at least one; namely that the generic differential hypersurface sections of an irreducible differential algebraic variety are irreducible and codimension one. We also calculate the Kolchin polynomials of the intersections and prove several other results regarding intersections of differential algebraic varieties. [ABSTRACT FROM AUTHOR]

Flag manifolds are generalizations of projective spaces and other Grassmannians: they parametrize flags, which are nested sequences of subspaces in a given vector space. These are important objects in algebraic and differential geometry, but are also increasingly being used in data science, where many types of data are properly understood as subspaces rather than vectors. In this paper we discuss partially oriented flag manifolds, which parametrize flags in which some of the subspaces may be endowed with an orientation. We compute the expected distance between random points on some low-dimensional examples, which we view as a statistical baseline against which to compare the distances between particular partially oriented flags coming from geometry or data. [ABSTRACT FROM AUTHOR]

The concept of a Turing computable embedding was introduced by Calvert, Cummins, Miller, and Knight as an effective analog of Borel embeddings. However, while the theory of Borel embeddings is often applied to general equivalence relations, research so far in Turing computable embeddings has focused mostly on the isomorphism relation. This paper investigates Turing computable embeddings of general equivalence relations. We first examine a relation on trees which is coarser than isomorphism and can be handled in much the same way as isomorphism. We then examine the computable isomorphism relation on computable structures, a finer relation which requires new techniques. The main result of this paper is that every class of computable structures under computable isomorphism embeds into the class of computable copies of ω as a linear ordering under the computable isomorphism, indicating that Turing computable embeddings are unlikely to produce meaningful distinctions when restricted to computable isomorphism relations. [ABSTRACT FROM AUTHOR]

Let (X,D) be a log smooth pair of dimension n, where D is a reduced effective divisor such that the log canonical divisor KX +D is pseudoeffective. Let G be a connected algebraic subgroup of Aut(X, D). We show that G is a semi-abelian variety of dimension = min{n- (V), n} with V := X \D. In the dimension two, Iitaka claimed in his 1979 Osaka J. Math. paper that dimG = q(V) for a log smooth surface pair with (V) = 0 and pg(V) = 1. We (re-)prove and generalize this classical result for all surfaces with = 0 without assuming Iitaka's classification of logarithmic Iitaka surfaces or logarithmic K3 surfaces. [ABSTRACT FROM AUTHOR]

We are interested in stringy invariants of singular projective algebraic varieties satisfying a strict monotonicity with respect to elementary birational modifications in the Mori program. We conjecture that the algebraic stringy Euler number is one of such invariants. In the present paper, we prove this conjecture for varieties having an action of a connected algebraic group G and admitting equivariant desingularizations with only finitely many G-orbits. In particular, we prove our conjecture for arbitrary projective spherical varieties. [ABSTRACT FROM AUTHOR]

We prove that if a symplectic 4-manifold X becomes a rational 4-manifold after applying rational blow-down surgery, then the symplectic 4-manifold X is originally rational. That is, a symplectic rational blow-up of a rational symplectic 4-manifold is again rational. As an application we show that a degeneration of a family of smooth rational complex surfaces is a rational surface if the degeneration has at most quotient surface singularities, which generalizes slightly a classical result of Bădescu [J. Reine Angew. Math. 367 (1986), pp. 76–89] in algebraic geometry under a mild additional condition. [ABSTRACT FROM AUTHOR]

In recent work, Beukers characterized A-hypergeometric systems having a full set of algebraic solutions. He accomplished this by (1) determining which A-hypergeometric systems have a full set of polynomial solutions modulo p for almost all primes p and (2) showing that these systems come from geometry. He then applied a fundamental theorem of N. Katz, which says that such systems have a full set of algebraic solutions. In this paper we establish some connections between nonresonant A-hypergeometric systems and de Rham-type complexes, which leads to a determination of which Ahypergeometric systems come from geometry. We do not use the fact that the system is irreducible or find integral formulas for its solutions. [ABSTRACT FROM AUTHOR]

We compute the classes of universal theta divisors of degrees zero and g-1 over the Deligne-Mumford compactification Mg,n of the moduli space of curves, with various integer weights on the points, in particular reproving a recent result of Müller. We also obtain a formula for the class in CHg(Mctg,n) (moduli of stable curves of compact type) of the double ramification cycle, given by the condition that a fixed linear combination of the marked points is a principal divisor, reproving a recent result of Hain. Our approach for computing the theta divisor is more direct, via test curves and the geometry of the theta divisor, and works easily over all of Mg,n. We used our extended result in another paper to study the partial compactification of the double ramification cycle. [ABSTRACT FROM AUTHOR]

We obtain an improvement and broad generalisation of a result of N. Ailon and Z. Rudnick (2004) on common zeros of shifted powers of polynomials. Our approach is based on reducing this question to a more general question of counting intersections of level curves of complex functions. We treat this question via classical tools of complex analysis and algebraic geometry. [ABSTRACT FROM AUTHOR]

In this paper we study finite p-solvable groups having irreducible complex characters χ ∈ Irr(G) which take root of unity values on the p-singular elements of G. [ABSTRACT FROM AUTHOR]

In a 2001 paper, S. Montgomery-Smith provides a one-dimensional model for the three-dimensional, incompressible Navier-Stokes equations, for which he proves the blow-up of solutions associated with a class of large initial data, while the same global existence results as for the Navier-Stokes equations hold for small data. In this paper the model is adapted to the cases of two and three space dimensions, with the additional feature that the divergence-free condition is preserved. It is checked that a family of initial data constructed by Chemin and Gallagher, which is arbitrarily large yet generates a global solution to the Navier-Stokes equations in three space dimensions, actually causes blow-up for the toy model --- meaning that the precise structure of the nonlinear term is crucial to understanding the dynamics of large solutions to the Navier-Stokes equations. [ABSTRACT FROM AUTHOR]

Let rk(n) denote the number of representations of the positive integer n as the sum of k squares. In 1934, the Russian mathematician A. I. Popov stated, but did not rigorously prove, a beautiful series transformation involving rk(n) and certain Bessel functions. We provide a proof of this identity for the first time, as well as for another identity, which can be regarded as both an analogue of Popov's identity and an identity involving r2(n) from Ramanujan's lost notebook. [ABSTRACT FROM AUTHOR]

Cross-ratio degrees count configurations of points z_1,\ldots,z_n\in \mathbb {P}^1 satisfying n-3 cross-ratio constraints, up to isomorphism. These numbers arise in multiple contexts in algebraic and tropical geometry, and may be viewed as combinatorial invariants of certain hypergraphs. We prove an upper bound on cross-ratio degrees in terms of the theory of perfect matchings on bipartite graphs. We also discuss several of the many perspectives on cross-ratio degrees—including a connection to Gromov-Witten theory—and give many example computations. [ABSTRACT FROM AUTHOR]

Let l1, … , ll be l lines in P² such that no three lines meet in a point. Let X(l) be the set of points {li ∩lj ∣ 1 ≤ i < j ≤ l} ⊆ P². We call X(l) a star configuration. We describe all pairs (d, l) such that the generic degree d curve in P² contains an X(l). Our proof strategy uses both a theoretical and an explicit algorithmic approach. We also describe how one may extend our algorithmic approach to similar problems. [ABSTRACT FROM AUTHOR]

In this paper we extend some results of Norman and Oort and of de Jong, and give an explicit description of the geometry of the Siegel modular threefold with paramodular level structure. We also discuss advantages and restrictions of three standard methods for studying moduli spaces of abelian varieties. [ABSTRACT FROM AUTHOR]

In this paper we prove linearly repetitive Delone systems have finitely many Delone system factors up to conjugacy. This result is also applicable to linearly repetitive tiling systems. [ABSTRACT FROM AUTHOR]

In the present paper, by making use of the concatenation of $17^2-1=288$ points on the sphere of radius $4$ in $mathbb {R}^{16}$ and subcodes of algebraic geometry codes over $mathbb {F}_{17^2}$, we improve the best-known constructive bound on kissing numbers by A. Vardy. [ABSTRACT FROM AUTHOR]

This paper studies the regularity of certain coherent sheaves that arise naturally from Segre-Veronese embeddings of a product of projective spaces. We give an explicit formula for the regularity of these sheaves and show that their regularity is subadditive. We then apply our results to study the Tate resolutions of these sheaves. [ABSTRACT FROM AUTHOR]

This paper deals with the blow-up properties of the solution to the degenerate nonlinear reaction diffusion equation with nonlocal source $x^{q}u_{t}-(x^{\gamma}u_{x})_{x}=\int_{0}^{a}u^{p}dx$ in $(0,a)\times (0,T)$ subject to the homogeneous Dirichlet boundary conditions. The existence of a unique classical nonnegative solution is established and the sufficient conditions for the solution exists globally or blows up in finite time are obtained. Furthermore, it is proved that under certain conditions the blow-up set of the solution is the whole domain. [ABSTRACT FROM AUTHOR]

In this paper we compute homotopical equivariant bordism for the group ${\mathbb{Z}}/2$, namely $MO_*^{{\mathbb{Z}/2}}$, geometric equivariant bordism ${\mathfrak{N}}^{{\mathbb{Z}/2}}_*$, and their quotient as modules over geometric bordism. This quotient is a module of stable transversality obstructions. We construct these rings from knowledge of their localizations. [ABSTRACT FROM AUTHOR]

This paper contains $1\frac 12$ proofs that the construction of functions on the moduli space of flat $G$-connections from chord diagrams gives all functions in the case of the simple group $G_2$. [ABSTRACT FROM AUTHOR]

Let R denote a 6-dimensional subspace of the ring M4(k) of 4 × 4 matrices over an algebraically closed field k. Fix a vector space isomorphism M4(k) ≅ k4 ⊗ k4. We associate to R a closed subscheme XR of the Grassmannian of 2-dimensional subspaces of k4, where the reduced subscheme of XR is the set of 2-dimensional subspaces Q ⊆ k4 such that (Q ⊗ k4) ∩ R ≠ 0. Our main result is that if XR has minimal dimension (namely, one), then its degree is 20 when it is viewed as a subscheme of P5 via the Plücker embedding. We present several examples of XR that illustrate the wide range of possibilities for it; there are reduced and non-reduced examples. Two examples involve elliptic curves: in one case, XR is a P1-bundle over an elliptic curve the second symmetric power of the curve; in the other, it is a curve having seven irreducible components, three of which are quartic elliptic space curves, and four of which are smooth plane conics. These two examples arise naturally from a problem having its roots in quantum statistical mechanics. The scheme XR appears in non-commutative algebraic geometry: under appropriate hypotheses, it is isomorphic to the line scheme L of a certain graded algebra determined by R. In that context, it has been an open question for several years to describe such L of minimal dimension, i.e., those L of dimension one. Our main result implies that if (L) = 1, then, as a subscheme of P5 under the Plücker embedding, deg (L) = 20. [ABSTRACT FROM AUTHOR]

An explicit construction of closed, orientable, smooth, aspherical 4-manifolds with any odd Euler characteristic greater than 12 is presented. The constructed manifolds are all Haken manifolds in the sense of B. Foozwell and H. Rubinstein and can be systematically reduced to balls by suitably cutting them open along essential codimension-one submanifolds. Euler characteristics divisible by 3 are known to arise from complex algebraic geometry considerations. Examples with Euler characteristic 1, 5, 7, or 11 appear to be unknown. [ABSTRACT FROM AUTHOR]

Let G be a simple algebraic group over the algebraic closure of Fp (p prime), and let G(q) denote a corresponding finite group of Lie-type over Fq, where q is a power of p. Let X be an irreducible subvariety of Gr for some r ≥ 2. We prove a zero-one law for the probability that G(q) is generated by a random r-tuple in X(q) = X ∩ G(q)r: the limit of this probability as q increases (through values of q for which X is stable under the Frobenius morphism defining G(q)) is either 1 or 0. Indeed, to ensure that this limit is 1, one only needs G(q) to be generated by an r-tuple in X(q) for two sufficiently large values of q. We also prove a version of this result where the underlying characteristic is allowed to vary. In our main application, we apply these results to the case where r = 2 and the irreducible subvariety X = C ×D, a product of two conjugacy classes of elements of finite order in G. This leads to new results on random (2, 3)- generation of finite simple groups G(q) of exceptional Lie-type: provided G(q) is not a Suzuki group, we show that the probability that a random involution and a random element of order 3 generate G(q) tends to 1 as q→∞. Combining this with previous results for classical groups, this shows that finite simple groups (apart from Suzuki groups and PSp4(q)) are randomly (2, 3)-generated. Our tools include algebraic geometry, representation theory of algebraic groups, and character theory of finite groups of Lie-type. [ABSTRACT FROM AUTHOR]

We study the uniqueness of complete biconservative surfaces in the Euclidean space ℝ3 and prove that the only complete biconservative regular surfaces inℝ3 are either CMC or certain surfaces of revolution. In particular, any compact biconservative regular surface in R3 is a round sphere. [ABSTRACT FROM AUTHOR]

We show that K(2)-locally, the smash product of the string bordism spectrum and the spectrum T2 splits into copies of Morava E-theories. Here, T2 is related to the Thom spectrum of the canonical bundle over ΩSU(4). [ABSTRACT FROM AUTHOR]

We study the arithmetically Cohen-Macaulay (ACM) property for finite sets of points in multiprojective spaces, especially (P1)n. A combinatorial characterization, the (*s)-property, is known in P1 × P1. We propose a combinatorial property, (s) with 2 ≤ s ≤ n, that directly generalizes the (*)-property to (P1)nn for larger n. We show that X is ACM if and only if it satisfies the (*n)-property. The main tool for several of our results is an extension to the multiprojective setting of certain liaison methods in projective space. [ABSTRACT FROM AUTHOR]

We show that the eigenspaces of the Laplaciank on Î"k-forms on a compact KÂahler manifold carry Hodge and Lefschetz decompositions. Among other consequences, we show that the positive part of the spectrum of Î"k lies in the spectrum of Î"k+1 for k < dimX. [ABSTRACT FROM AUTHOR]

We give new proofs of two results of Stafford, which generalize two famous Theorems of Serre and Bass regarding projective modules. Our techniques are inspired by the theory of basic elements. Using these methods we further generalize Serre’s Splitting Theorem by imposing a condition to the splitting maps, which has an application to the case of Cartier algebras. [ABSTRACT FROM AUTHOR]

Let X be a surface and let L be an ample line bundle on X. We first obtain a lower bound for the Seshadri constant ε(X,L, r), when r ≥ 2. We then assume that X is a ruled surface and study Seshadri constants on X in greater detail. We also make precise computations of Seshadri constants on X in some cases. [ABSTRACT FROM AUTHOR]

The existence of positive radial solution is obtained for a mean curvature equation in Minkowski space of the form ... where Ω is a unit ball in ℝN, f(r, u) has singularities at u = 0, r = 0 and/or r = 1. The main tool is the perturbation technique and nonlinear alternative of Leray-Schauder type. The interesting point is that the nonlinear term f(r, u) at u = 0 may be strongly singular. [ABSTRACT FROM AUTHOR]

There is a method for bounding the cyclicity of non-degenerate monodromic singularities of polynomial planar families of vector fields Xλ which can work even in the case that the Poincaré first return map has associated a non-radical Bautin ideal B. The method is based on the stabilization of the integral closures of an ascending chain of polynomial ideals in the ring of polynomials in the parameters λ of the family that stabilizes at B. In this work we use computational algebra methods to provide an explicit example in which the classical procedure to find the Bautin depth of B fails but the new approach is successful. [ABSTRACT FROM AUTHOR]

By a result of Orlov there always exists an embedding of the derived category of a finite-dimensional algebra of finite global dimension into the derived category of a high-dimensional smooth projective variety. In this article we give some restrictions on those algebras whose derived categories can be embedded into the bounded derived category of a smooth projective surface. This is then applied to obtain explicit results for hereditary algebras [ABSTRACT FROM AUTHOR]

We study proper rational maps from the unit disk to balls in higher dimensions. After gathering some known results, we study the moduli space of unitary equivalence classes of polynomial proper maps from the disk to a ball, and we establish a normal form for these equivalence classes. We also prove that all rational proper maps from the disk to a ball are homotopic in target dimension at least 2. [ABSTRACT FROM AUTHOR]

A collection of sets is intersecting if every two members have nonempty intersection. We describe the structure of intersecting families of rsets of an n-set whose size is quite a bit smaller than the maximum ... given by the Erdős-Ko-Rado Theorem. In particular, this extends the Hilton-Milner theorem on nontrivial intersecting families and answers a recent question of Han and Kohayakawa for large n. In the case r = 3 we describe the structure of all intersecting families with more than 10 edges. We also prove a stability result for the Erdős matching problem. Our short proofs are simple applications of the Delta-system method introduced and extensively used by Frankl since 1977. [ABSTRACT FROM AUTHOR]

Let A be a graded complete intersection over a field and B the monomial complete intersection with the generators of the same degrees as A. The EGH conjecture says that if I is a graded ideal in A, then there should be an ideal J in B such that B/J and A/I have the same Hilbert function. We show that if the EGH conjecture is true, then it can be used to prove that every graded complete intersection over any field has the Sperner property. [ABSTRACT FROM AUTHOR]

We investigate the distribution of the coefficients of a modular form modulo pj where p ≥ 3 is a prime and j is a positive integer. Our work heavily relies on the result of Bellaıche and Soundararajan. [ABSTRACT FROM AUTHOR]

We remove the assumption of the continuum hypothesis from the Akemann-Doner construction of a non-separable C*-algebra A with only separable commutative C*-subalgebras. We also extend a result of Farah and Wofsey's, constructing N1 commuting projections in the Calkin algebra with no commutative lifting. This removes the assumption of the continuum hypothesis from a version of a result of Anderson. Both results are based on Luzin's almost disjoint family construction. [ABSTRACT FROM AUTHOR]

We construct ergodic discrete probability-measure-preserving equivalence relations R that have no proper ergodic normal subequivalence relations and no proper ergodic finite-index subequivalence relations. We show that every treeable equivalence relation satisfying a mild ergodicity condition and cost > 1 surjects onto every countable group with ergodic kernel. Lastly, we provide a simple characterization of normality for subequivalence relations and an algebraic description of the quotient. [ABSTRACT FROM AUTHOR]

Abstract. For a region Ω ⊂ C denote by Ψ(Q;Ω) the number of complex algebraic numbers in Ω of degree ≤ n and naive height ≤ Q. We show that....where ν is the Lebesgue measure on the complex plane and the function ψ will be given explicitly. [ABSTRACT FROM AUTHOR]

We classify the possible Mumford-Tate groups of polarizable rational Hodge structures. [ABSTRACT FROM AUTHOR]

In this note, we justify rigorously the formal method proposed in [M. HILLAIRET, J. Math. Fluid Mech. 2007] to derive viscous and compressible multi-component flow equations. We present here a simpler proof than in [D. BRESCH, X. HUANG, Arch. Ration. Mech. Anal. 2011] to show that the homogenized system may be reduced to a viscous and compressible multi-component flow system (with one velocity-field) getting rid of the no-crossing assumption on the partial densities. We also discuss formally why our multi-component system may be seen as a physically-relevant relaxed system for the well-known bi-fluid system with algebraic closure (pressure equilibrium) in the isothermal case. [ABSTRACT FROM AUTHOR]

It is a well-known result due to Busemann that the intersection body of an origin-symmetric convex body is also convex. Koldobsky introduced the notion of fc-intersection bodies. We show that the k-intersection body of an origin-symmetric convex body is not necessarily convex if k > 1. [ABSTRACT FROM AUTHOR]

We show that on the d-dimensional cube Id ≡ [0, 1]d the discrete moduli of smoothness which use only the values of the function on a diadic mesh are sufficient to determine the moduli of smoothness of that function. As an important special case our result implies for f ∊ C(Id) and a given integer r that when 0 < α < r, the condition ∣Λ2-nen.f(k1/2n,...kd/2n)∣⩽M2-nα for integers 1 ⩽ i ⩽ d, 0 ⩽ ki ⩽ 2n-r, 0 ⩽ kj ⩽ 2n when j ≠= i and n = 1, 2, ... is equivalent to ∣▵ r hef(ξ)∣ ⩽ M1hα for ξ, e ∊ ℝd, h>0 and ∣e∣ = 1 such that ξ, ξ+rhe ∊ Id. [ABSTRACT FROM AUTHOR]

We discuss several aspects of Łojasiewicz inequalities, namely local and global versions, and the relations between the gradient inequality and regular separation of real algebraic sets. We give effective estimates for Łojasiewicz's exponents in the real and complex setting. [ABSTRACT FROM AUTHOR]

Given a finite set of points Γ in Pk-1 not all contained in a hyperplane, the "fitting problem" asks what is the maximum number hyp(Γ) of these points that can fit in some hyperplane and what is (are) the equation(s) of such hyperplane(s)? If Γ has the property that any k - 1 of its points span a hyperplane, then hyp(Γ) = nil(I) + k - 2, where nil(I) is the index of nilpotency of an ideal constructed from the homogeneous coordinates of the points of Γ. Note that in P² any two points span a line, and we find that the maximum number of collinear points of any given set of points Γ ⊂ P² equals the index of nilpotency of the corresponding ideal, plus one. [ABSTRACT FROM AUTHOR]

L. Small asked whether two finite dimensional division algebras containing a common central subfield F are embeddable in a common division algebra. Although we have a counterexample, the question is answered affirmatively for division algebras whose centers are finitely generated over a common perfect subfield. [ABSTRACT FROM AUTHOR]

We describe an explicit semi-algebraic partition for the complement of a real hyperplane arrangement such that each piece is contractible and so that the pieces form a basis of Borel-Moore homology. We also give an explicit correspondence between the de Rham cohomology and the Borel-Moore homology. [ABSTRACT FROM AUTHOR]