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Let X \subset \mathbb {C}^{2n} be an n-dimensional algebraic variety. We define the algebraic version of the generic symmetry defect set (Wigner caustic) of X. Moreover, we compute its singularities as well as degree, genus and Euler characteristic for X_d being a generic (smooth and transversal to the line at infinity) curve of degree d in \mathbb {C}^2. [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]

We study cohomologies of a curve with an action of a finite p-group over a field of characteristic p. Assuming the existence of a certain "magical element" in the function field of the curve, we compute the equivariant structure of the module of holomorphic differentials and the de Rham cohomology, up to certain local terms. We show that a generic p-group cover has a "magical element". As an application we compute the de Rham cohomology of a curve with an action of a finite cyclic group of prime order. [ABSTRACT FROM AUTHOR]

Francis Sowerby Macaulay began his career working on Brill and Noether's theory of algebraic plane curves and their interpretation of the Riemann–Roch and Cayley–Bacharach theorems; in fact it is Macaulay who first stated and proved the modern form of the Cayley–Bacharach theorem. Later in his career Macaulay developed ideas and results that have become important in modern commutative algebra, such as the notions of unmixedness, perfection (the Cohen–Macaulay property), and super-perfection (the Gorenstein property), work that was appreciated by Emmy Noether and the people around her. He also discovered results that are now fundamental in the theory of linkage, but this work was forgotten and independently rediscovered much later. The name of a computer algebra program (now Macaulay2) recognizes that much of his work is based on examples created by refined computation. Though he never spoke of the connection, the threads of Macaulay's work lead directly from the problems on plane curves to his later theorems. In this paper we will explain what Macaulay did, and how his results are connected. [ABSTRACT FROM AUTHOR]

We study one variable meromorphic functions mapping a planar real algebraic set A to another real algebraic set in the complex plane. By using the theory of Schwarz reflection functions, we show that for certain A, these meromorphic functions must be rational. In particular, when A is the standard unit circle, we obtain a one dimensional analog of Poincaré [Acta Math. 2 (1883), pp. 97–113], Tanaka [J. Math. Soc. Japan 14 (1962), pp. 397–429] and Alexander's [Math. Ann. 209 (1974), pp. 249–256] rationality results for 2m-1 dimensional sphere in \mathbb {C}^m when m\ge 2. [ABSTRACT FROM AUTHOR]

We give effective lower bounds on caliber numbers of the parametric family of real quadratic fields \mathbb {Q}(\sqrt {t^4-n^2}) as t varies over positive integers for a congruent number n. Furthermore, we provide lower bounds on class numbers of Richaud-Degert type real quadratic fields of the form \mathbb {Q}(\sqrt {n^2k^4-1}) for positive integers k and congruent numbers n whose elliptic curves have algebraic rank greater than 2. [ABSTRACT FROM AUTHOR]

We consider Abel maps for regular smoothing of nodal curves with values in the Esteves compactified Jacobian. In general, these maps are just rational, and an interesting question is to find an explicit resolution. We translate this problem into an explicit combinatorial problem by means of tropical and toric geometry. We show that the solution of the combinatorial problem gives rise to an explicit resolution of the Abel map. We are able to use this technique to construct and study all the Abel maps of degree one. Finally, we write an algorithm, which we implemented in SageMath to compute explicitly the solution of the combinatorial problem which, provided the existence of certain subdivisions of a hypercube, give rise to the resolution of the geometric Abel map. [ABSTRACT FROM AUTHOR]

This paper is concerned with configurations of points in a plane lattice which determine angles that are rational multiples of \pi. We shall study how many such angles may appear in a given lattice and in which positions, allowing the lattice to vary arbitrarily. This classification turns out to be much less simple than could be expected, leading even to parametrizations involving rational points on certain algebraic curves of positive genus. [ABSTRACT FROM AUTHOR]

The problem of algebraic dependence of solutions to (non-linear) first order autonomous equations over an algebraically closed field of characteristic zero is given a 'complete' answer, obtained independently of model theoretic results on differentially closed fields. Instead, the geometry of curves and generalized Jacobians provides the key ingredient. Classification and formal solutions of autonomous equations are treated. The results are applied to answer a question on D^n-finiteness of solutions of first order differential equations. [ABSTRACT FROM AUTHOR]

Using the arithmetic Schottky uniformization theory, we show the arithmeticity of PSL2(C) Chern-Simons invariant. In terms of this invariant, we give an explicit formula of the Deligne-Riemann-Roch isomorphism as the Zograf-McIntyre-Takhtajan infinite product for families of algebraic curves. Applying this formula to the Liouville theory, we determine the unknown constant which appears in the holomorphic factorization formula of determinants of Laplacians on Riemann surfaces. [ABSTRACT FROM AUTHOR]

Gasull and Sabatini in [Ann. Mat. Pura Appl. 198 (2019), pp. 1985-2006] studied limit cycles of a Liénard system which has a fixed invariant curve, i.e., a Wilson polynomial Liénard system. The Liénard system can be changed into x = y − (x2 − 1)(x3 − bx), y = −x(1+y(x3 −bx)). For b ≤ 0.7 and b ≥ 0.76, limit cycles of the system are studied completely. But for 0.7 < b < 0 .76, the exact number of limit cycles is still unknown, and Gasull and Sabatini conjectured that the exact number of limit cycles is two (including multiplicities). In this paper, we give a positive answer to this conjecture and study all bifurcations of the system. Finally, we show the expanding of the moving limit cycle as b > 0 increases and give all global phase portraits on the Poincaré disk of the system completely. [ABSTRACT FROM AUTHOR]

We consider the geometry of regular holomorphic curves in C2 viewed as surfaces in the affine space R4. We study the A-singularities of parallel projections of generic such surfaces along planes to transverse planes. We show that at any point on the surface which is not an inflection point of the curve there are two tangent directions determining two planes along which the projection has singularities of type butterfly or worse. The integral curves of these directions form a pair of foliations on the surface defined by a binary differential equation (BDE). The singularities of this BDE are the inflection points of the curve together with other points that we call butterfly umbilic points. We determine the configurations of the solution curves of the BDE at its singularities. Finally, we prove that an affine view of an algebraic curve of degree d ≥ 2 in CP2 has 8d(d − 2) butterfly umbilic points. [ABSTRACT FROM AUTHOR]

This is an introduction, aimed at a general mathematical audience, to recent work of Aprodu, Farkas, Papadima, Raicu, andWeyman. These authors established a long-standing folk conjecture concerning the equations defining the tangent developable surface of the rational normal curve. This in turn led to a new proof of a fundamental theorem of Voisin on the syzygies of generic canonical curves. The present note, which is the write-up of a talk given by the second author at the Current Events seminar at the 2019 JMM, surveys this circle of ideas. [ABSTRACT FROM AUTHOR]

Given a real hyperelliptic algebraic curve X with non-empty real part and a real effective divisor D arising via pullback from P1 under the hyperelliptic structure map, we study the real inflection points of the associated complete real linear series |D| on X. To do so we use Viro's patchworking of real plane curves, recast in the context of some Berkovich spaces studied by M. Jonsson. Our method gives a simpler and more explicit alternative to limit linear series on metrized complexes of curves, as developed by O. Amini and M. Baker, for curves embedded in toric surfaces. [ABSTRACT FROM AUTHOR]

We consider the family of hyperelliptic curves over Q of fixed genus along with a marked rational Weierstrass point and a marked rational non-Weierstrass point. When these curves are ordered by height, we prove that the average Mordell-Weil rank of their Jacobians is bounded above by 5/2, and that most such curves have only three rational points. We prove this by showing that the average rank of the 2-Selmer groups is bounded above by 6. We also consider another related family of curves and study the interplay between these two families. There is a family φ of isogenies between these two families, and we prove that the average size of the φ-Selmer groups is exactly 2. [ABSTRACT FROM AUTHOR]

In this paper, we consider representations of the absolute Galois group Gal(Q/Q) attached to modular forms for noncongruence subgroups of SL2(Z). When the underlying modular curves have a model over Q, these representations are constructed by Scholl in [Invent. Math. 99 (1985), pp. 49-77] and are referred to as Scholl representations, which form a large class of motivic Galois representations. In particular, by a result of Belyi, Scholl representations include the Galois actions on the Jacobian varieties of algebraic curves defined over Q. As Scholl representations are motivic, they are expected to correspond to automorphic representations according to the Langlands philosophy. Using recent developments on automorphy lifting theorem, we obtain various automorphy and potential automorphy results for potentially GL2-type Galois representations associated to noncongruence modular forms. Our results are applied to various kinds of examples. In particular, we obtain potential automorphy results for Galois representations attached to an infinite family of spaces of weight 3 noncongruence cusp forms of arbitrarily large dimensions. [ABSTRACT FROM AUTHOR]

We describe an algorithm, based on the properties of the characteristic polynomials of Frobenius, to compute EndK(A) when A is the Jacobian of a nice genus-2 curve over a number field K. We use this algorithm to confirm that the description of the structure of the geometric endomorphism ring of Jac(C) given in the LMFDB (L-functions and modular forms database) is correct for all the genus-2 curves C currently listed in it. We also discuss the determination of the field of definition of the endomorphisms in some special cases. [ABSTRACT FROM AUTHOR]

Given an algebraic variety X defined over an algebraically closed field, we study the space RZ(X, x) consisting of all the valuations of the function field of X which are centered in a closed point x of X. We concentrate on its homeomorphism type. We prove that, when x is a regular point, this homeomorphism type only depends on the dimension of X. If x is a singular point of a normal surface, we show that it only depends on the dual graph of a good resolution of (X, x) up to some precise equivalence. This is done by studying the relation between RZ(X, x) and the normalized non-Archimedean link of x in X coming from the point of view of Berkovich geometry. We prove that their behavior is the same. [ABSTRACT FROM AUTHOR]

We give an explicit description of fundamental domains associated with the p-adic uniformisation of families of Shimura curves of discriminant Dp and level N ≥ 1, for which the one-sided ideal class number h(D, N) is 1. The results obtained generalise those in Schottky groups and Mumford curves, Springer, Berlin, 1980 for Shimura curves of discriminant 2p and level N = 1. The method we present here enables us to find Mumford curves covering Shimura curves, together with a free system of generators for the associated Schottky groups, p-adic good fundamental domains, and their stable reduction-graphs. The method is based on a detailed study of the modular arithmetic of an Eichler order of level N inside the definite quaternion algebra of discriminant D, for which we generalise the classical results of Hurwitz. As an application, we prove general formulas for the reduction-graphs with lengths at p of the families of Shimura curves considered. [ABSTRACT FROM AUTHOR]

Rational curves on Hilbert schemes of points on K3 surfaces and generalised Kummer manifolds are constructed by using Brill-Noether theory on nodal curves on the underlying surface. It turns out that all wall divisors can be obtained, up to isometry, as dual divisors to such rational curves. The locus covered by the rational curves is then described, thus exhibiting algebraically coisotropic subvarieties. This provides strong evidence for a conjecture by Voisin concerning the Chow ring of irreducible holomorphic symplectic manifolds. Some general results concerning the birational geometry of irreducible holomorphic symplectic manifolds are also proved, such as a non-projective contractibility criterion for wall divisors. [ABSTRACT FROM AUTHOR]

We analyze when integral points on the complement of a finite union of curves in P² are potentially dense. When the logarithmic Kodaira dimension ... is -∞, we completely characterize the potential density of integral points in terms of the number of irreducible components at infinity and the number of multiple members in a pencil naturally associated to the surface. When ... = 0, we prove that integral points are always potentially dense. The bulk of our analysis concerns the subtle case of ... = 1. We determine the potential density of integral points in a number of cases by incorporating the structure theory of affine surfaces and developing an arithmetic framework for studying integral points on surfaces fibered over curves. We also prove, assuming Lang-Vojta's conjecture, that an orbit under an endomorphism φ of P² can contain a Zariski-dense set of integral points only if there is a nontrivial completely invariant proper Zariski-closed subset of P² under φ. This may be viewed as a generalization of a result of Silverman on P¹. [ABSTRACT FROM AUTHOR]

The cap set problem asks how large can a subset of (Z/3Z)n be and contain no lines or, more generally, how can large a subset of (Z/pZ)n be and contain no arithmetic progressions. This problem was motivated by deep questions about structure in the prime numbers, the geometry of lattice points, and the design of statistical experiments. In 2016, Croot, Lev, and Pach solved the analogous problem in (Z/4Z)n, showing that the largest set without arithmetic progressions had size at most cn for some c < 4. Their proof was as elegant as it was unexpected, being a departure from the tried and true methods of Fourier analysis that had dominated the field for half a century. Shortly thereafter, Ellenberg and Gijswijt leveraged their method to resolve the original cap set problem. This expository article covers the history and motivation for the cap set problem and some of the many applications of the technique: from removing triangles from graphs, to rigidity of matrices, and to algorithms for matrix multiplication. The latter application turns out to give back to the original problem, sharpening our understanding of the techniques involved and of what is needed for showing that the current bounds are tight. Most of our exposition assumes only familiarity with basic linear algebra, polynomials, and the integers modulo N. [ABSTRACT FROM AUTHOR]

We construct a well-behaved compactification of the space of finite covers of a stacky curve using admissible cover degenerations. Using our construction, we compactify the space of tetragonal curves on Hirzebruch surfaces. As an application, we explicitly describe the boundary divisors of the closure in M6 of the locus of smooth plane quintic curves. [ABSTRACT FROM AUTHOR]

Let E be an elliptic curve over ℚ, such that End... (E) = ℤ and admitting a non-trivial cyclic ℚ,-isogeny. We prove that, for p > 37, the residual mod p Galois representation ...E,p : Gℚ → GL2( 픽p) is surjective. [ABSTRACT FROM AUTHOR]

We construct certain subgroups of hyperbolic triangle groups which we call "congruence" subgroups. These groups include the classical congruence subgroups of SL2(ℤ.), Hecke triangle groups, and 19 families of arithmetic triangle groups associated to Shimura curves. We determine the field of moduli of the curves associated to these groups and thereby realize the groups PSL2(Fq) and PGL2(픽q) regularly as Galois groups. [ABSTRACT FROM AUTHOR]

Let R be a reduced affine . . .-algebra with corresponding affine algebraic set X. Let C(X) be the ring of continuous (Euclidean topology) . . .-valued functions on X. Brenner defined the continuous closure Icont of an ideal I as IC(X)∩R. He also introduced an algebraic notion of axes closure Iax that always contains Icont, and asked whether they coincide. We extend the notion of axes closure to general Noetherian rings, defining f ∈ Iax if its image is in IS for every homomorphism R → S, where S is a one-dimensional complete seminormal local ring. We also introduce the natural closure I. . . of I. One of many characterizations is I. . . = I+{f |∈ R : ∃n > 0 with fn ∈ In+1}. We show that I. . . ⊆ Iax and that when continuous closure is defined, I. . . ⊆ Icont ⊆ Iax. Under mild hypotheses on the ring, we show that I. . . = Iax when I is primary to a maximal ideal and that if I has no embedded primes, then I = I. . . if and only if I = Iax, so that Icont agrees as well. We deduce that in the polynomial ring C[x1, . . . , xn], if f = 0 at all points where all of the ∂f/∂xi are 0, then f ∈ ( ∂f/∂x1, . . . , ∂f/∂xn)R. We characterize Icont for monomial ideals in polynomial rings over . . . , but we show that the inequalities I. . . ⊆ Icont and Icont ⊆ Iax can be strict for monomial ideals even in dimension 3. Thus, Icont and Iax need not agree, although we prove they are equal in . . .[x1, x2]. [ABSTRACT FROM AUTHOR]

A new method, in the interface of stochastic differential equations with boundary value problems, is developed in this work, aiming at representing solutions of exterior boundary value problems in terms of stochastic processes. The main effort concerns exterior harmonic problems but furthermore special attention has been paid to the investigation of time-reduced scattering processes (involving the Helmholtz operator) in the realm of low frequencies. The method, in principle, faces the construction of the solution of the direct versions of the aforementioned boundary value problems but the special features of the method assure definitely the usefulness of the approach to the solution of the corresponding inverse problems as clearly indicated herein. [ABSTRACT FROM AUTHOR]

From a topological viewpoint, a rational curve in the real projective plane is generically a smoothly immersed circle and a finite collection of isolated points. We give an isotopy classification of generic rational quintics in ℝℙ2 in the spirit of Hilbert’s 16th problem. [ABSTRACT FROM AUTHOR]

The Serre-Tate correspondence contains a lot of Tate's work in a casual form. We present some excerpts that show how some of Tate's best known contributions came into being. [ABSTRACT FROM AUTHOR]

Let Φ∞(d) denote the set of finite abelian groups that occur infinitely often as the torsion subgroup of an elliptic curve over a number field of degree d. The sets Φ∞(d) are known for d ≤ 4. In this article we determine Φ∞(5) and Φ∞(6). [ABSTRACT FROM AUTHOR]

We introduce the class of everywhere like functions, which helps us to recover some known classes (such as that of everywhere surjective ones). We also study the algebraic genericity of this new class together with the class of fractal functions. [ABSTRACT FROM AUTHOR]

A recent paper of Shekhar compares the ranks of elliptic curves E1 and E2 for which there is an isomorphism E1[p] ∼ E2[p]asGal(.../Q)- modules, where p is a prime of good ordinary reduction for both curves. In this paper we prove an analogous result in the case where p is a prime of good supersingular reduction. [ABSTRACT FROM AUTHOR]

For n ⩾ 2, let K = Q(Pn) = Q(T1,. . ., Tn). Let E/K be the elliptic curve defined by a minimal Weierstrass equation y2 = x3+Ax+B, with A,B Q[T1,..., Tn]. There's a canonical height ĥE on E(K) induced by the divisor (O), where O is the zero element of E(K). On the other hand, for each smooth hypersurface Γ in Pn such that the reduction mod Γ of E, EΓ/Q(Γ) is an elliptic curve with the zero element OΓ, there is also a canonical height ĥEΓ on EΓ(Q(Γ)) that is induced by (OΓ). We prove that for any P ∊(K), the equality ĥEΓ(PΓ)/ deg Γ = ĥE(P) holds for almost all hypersurfaces in Pn) . As a consequence, we show that for infinitely many t ∊ Pn) (Q), the specialization map σt : ∊(K) → Et(Q) is injective. [ABSTRACT FROM AUTHOR]

This paper investigates which integers can appear as 2-Selmer ranks within the quadratic twist family of an elliptic curve E defined over a number field K with E(K)[2] ≃ Z/2Z. We show that if E does not have a cyclic 4-isogeny defined over K(E[2]) with kernel containing E(K)[2], then subject only to constant 2-Selmer parity, each non-negative integer appears infinitely often as the 2-Selmer rank of a quadratic twist of E. If E has a cyclic 4-isogeny with kernel containing E(K)[2] defined over K(E[2]) but not over K, then we prove the same result for 2-Selmer ranks greater than or equal to r2, the number of complex places of K. We also obtain results about the minimum number of twists of E with rank 0 and, subject to standard conjectures, the number of twists with rank 1, provided E does not have a cyclic 4-isogeny defined over K. [ABSTRACT FROM AUTHOR]

For n ⩾ 2, let K = Q(Pn) = Q(T1,. . ., Tn). Let E/K be the elliptic curve defined by a minimal Weierstrass equation y2 = x3+Ax+B, with A,B Q[T1,..., Tn]. There's a canonical height ĥE on E(K) induced by the divisor (O), where O is the zero element of E(K). On the other hand, for each smooth hypersurface Γ in Pn such that the reduction mod Γ of E, EΓ/Q(Γ) is an elliptic curve with the zero element OΓ, there is also a canonical height ĥEΓ on EΓ(Q(Γ)) that is induced by (OΓ). We prove that for any P ∊(K), the equality ĥEΓ(PΓ)/ deg Γ = ĥE(P) holds for almost all hypersurfaces in Pn) . As a consequence, we show that for infinitely many t ∊ Pn) (Q), the specialization map σt : ∊(K) → Et(Q) is injective. [ABSTRACT FROM AUTHOR]

We calculate the intersection ring of 3-dimensional graph manifolds with rational coefficients and give an algebraic characterization of these rings when the manifold's underlying graph is a tree. We are able to use this characterization to show that the intersection ring obstructs arbitrary 3-manifolds from being homology cobordant to certain graph manifolds. [ABSTRACT FROM AUTHOR]

The genus 4 modular curve Xns(11) attached to a non-split Cartan group of level 11 admits a model defined over Q. We compute generators for its function field in terms of Siegel modular functions. We also show that its Jacobian is isomorphic over Q to the new part of the Jacobian of the classical modular curve X0(121). [ABSTRACT FROM AUTHOR]

Let S be a Dedekind scheme with field of functions K. We show that if XK is a smooth connected proper curve of positive genus over K, then it admits a Néron model over S, i.e., a smooth separated model of finite type satisfying the usual Néron mapping property. It is given by the smooth locus of the minimal proper regular model of XK over S, as in the case of elliptic curves. When S is excellent, a similar result holds for connected smooth affine curves different from the affine line, with locally finite type Néron models. [ABSTRACT FROM AUTHOR]

Motivated by the Coulomb potential of an equidistributed charge on a curve, Jun OfHara introduced and investigated the first geometric knot energy, the MNobius energy. We prove that every critical curve of this MNobius energy is of class C∞ and thus extend the corresponding result due to Freedman, He, and Wang for minimizers of the MNobius energy. In contrast to the techniques used by Freedman, He, and Wang, our methods do to not use the Mobius invariance of the energy, but rely on purely analytic methods motivated from a formal similarity of the Euler-Langrange equation to the half harmonic map equation for the unit tangent. [ABSTRACT FROM AUTHOR]

Isogenies are the morphisms between elliptic curves and are, accordingly, a topic of interest in the subject. As such, they have been well studied, and have been used in several cryptographic applications. V'elu's formulas show how to explicitly evaluate an isogeny, given a specification of the kernel as a list of points. However, V'elu's formulas only work for elliptic curves specified by a Weierstrass equation. This paper presents formulas similar to V'elu's that can be used to evaluate isogenies on Edwards curves and Huff curves, which are normal forms of elliptic curves that provide an alternative to the traditional Weierstrass form. Our formulas are not simply compositions of V'elu's formulas with mappings to and from Weierstrass form. Our alternate derivation yields efficient formulas for isogenies with lower algebraic complexity than such compositions. In fact, these formulas have lower algebraic complexity than V'elu's formulas on Weierstrass curves. [ABSTRACT FROM AUTHOR]

In this article we consider outer Galois actions on a free profinite group of rank two, induced by the étale fundamental group of a projective line minus three points or of a pointed elliptic curve over a number field. Under mild technical assumptions their respective images uniquely determine the curves and the number fields. [ABSTRACT FROM AUTHOR]

In this article we consider outer Galois actions on a free profinite group of rank two, induced by the étale fundamental group of a projective line minus three points or of a pointed elliptic curve over a number field. Under mild technical assumptions their respective images uniquely determine the curves and the number fields. [ABSTRACT FROM AUTHOR]

In this paper we construct infinite families of elliptic curves with given torsion group structures over cyclic cubic number fields. [ABSTRACT FROM AUTHOR]

Mazur, Tate, and Teitelbaum gave a p-adic analogue of the Birch and Swinnerton-Dyer conjecture for elliptic curves. We provide a generalization of their conjecture in the good ordinary case to higher dimensional modular abelian varieties over the rationals by constructing the p-adic L-function of a modular abelian variety and showing that it satisfies the appropriate interpolation property. This relies on a careful normalization of the p-adic L-function, which we achieve by a comparison of periods. Our generalization agrees with the conjecture of Mazur, Tate, and Teitelbaum in dimension 1 and the classical Birch and Swinnerton-Dyer conjecture formulated by Tate in rank 0. We describe the theoretical techniques used to formulate the conjecture and give numerical evidence supporting the conjecture in the case when the modular abelian variety is of dimension 2. [ABSTRACT FROM AUTHOR]

We classify the endomorphism algebras of factors of the Jacobians of certain hypergeometric curves over a field of characteristic zero. Other than a few exceptional cases, the endomorphism algebras turn out to be either a cyclotomic field E = Q(ζq), or a quadratic extension of E, or E⊗E. This result may be viewed as a generalization of the well known results of the classification of endomorphism algebras of elliptic curves over C. [ABSTRACT FROM AUTHOR]

In this paper, we study a complete noncompact nonnegatively curved Alexandrov space A with a soul S of codimension two. We establish some structural results under additional regularity assumptions. As an application, we conclude that in this case Sharafutdinov retraction, π : A → S, is a submetry. [ABSTRACT FROM AUTHOR]

In this article, we consider Leray solutions of the Navier-Stokes equations in the exterior of one obstacle in 3D and we study the asymptotic behavior of these solutions when the obstacle shrinks to a curve or to a surface. In particular, we will prove that a solid curve has no effect on the motion of a viscous fluid, so it is a removable singularity for these equations. [ABSTRACT FROM AUTHOR]

We prove a general form of Green's Formula and the Cauchy Integral Theorem for arbitrary closed rectifiable curves in the plane. We use VituSkin's localization of singularities method and a decomposition of a rectifiable curve in terms of a sequence of Jordan rectifiable sub-curves due to Carmona and Cufi. [ABSTRACT FROM AUTHOR]

This paper explores the relationship between the existence of an exact embedded Lagrangian filling for a Legendrian knot in the standard contact R3 and the hierarchy of positive, strongly quasi-positive, and quasipositive knots. On one hand, results of Eliashberg and especially Boileau and Orevkov show that every Legendrian knot with an exact, embedded Lagrangian filling is quasi-positive. On the other hand, we show that if a knot type is positive, then it has a Legendrian representative with an exact embedded Lagrangian filling. Further, we produce examples that show that strong quasi-positivity and fillability are independent conditions. [ABSTRACT FROM AUTHOR]

Hurwitz spaces are moduli spaces of curve covers. The isomorphism classes of covers of P¹ℂ with given ramification data are parameterized combinatorially by Nielsen tuples in the monodromy group G. The Artin braid group acts on Nielsen tuples, and the orbits of this action correspond to the connected components of the corresponding Hurwitz space. In this article we consider the case G = A5. We give a complete classification of the braid orbits for all ramification types, showing that the components are always distinguishable by the Fried-Serre lift invariant. [ABSTRACT FROM AUTHOR]