In [1], Frenkel, Teleman, and Tolland consider the compactification of the **moduli** space of **maps** **from** **curves** to a space with automorphisms. They define the **moduli** stack of Gieseker bundles on stable **curves**, M f g,n , and showed that there exist well- defined K-theoretic invariants in the case where the target is [ pt / C × ] . Proving well-definedness of these invariants is difficult because the resulting **moduli** space is complete but not finite type. Their proof of well-definedness of invariants relies on their description of local charts on the **moduli** stack. While the use of charts allows them to conclude that the invariants are indeed finite, it does not tell us how the invariants can be computed and does not easily generalize to [ X / C × ] for arbitrary scheme X .

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major surprise, [HM82], [Har84] and [EH87] proved it wrong showing that for g ≥ 24 the **moduli** space M g is very far **from** being rational or unirational since it is of general type. After that, there has been a great deal of work trying to describe the **geometry** in the intermediate cases 11 ≤ g ≤ 23. For an exhaus- tive survey on the subject one can see [Far08] or [Ver13]. Summarizing, now it is known that **moduli** **spaces** of **curves** are characterized by the fact of being rational or unirational in the simplest and manageable cases, that is for small values of g, and of general type for large g. In particular, the transition of M g **from** the uniruledness case, that is g ≤ 16, to the case where M g is of general

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erated over its prime field. This is widely believed in the case of hyperbolic **curves** over number fields and is usually referred as the section conjecture. For a similar statement in the non-projective case, one needs to consider the so-called cuspidal sections, see [48], Section 18. Although we will discuss non-projective varieties in what follows, we will not need to specify the no- tion of cuspidal sections. The reason for this is that we will be considering sections that locally come **from** points (the Selmer set defined below) and these will not be cuspidal.

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Example 5. 2. Let X 0 : Ring → Gpd, be the classical **moduli** problem of example 2. 3, which assigns to each commutative ring R , the grupoid Hom ( Spec R , M 1, 1 ), of elliptic **curves** over R . It is possible to make sense of the notion of an elliptic curve over R , when R , is an arbitrary E ∞ − ring, and thereby obtain an enhancement X : CAlg(Sp) → S , of

The aim of this paper is to determine the topological structure of the space of m-spin bundles on hyperbolic Klein surfaces. A Klein surface is a non-orientable topological surface with a maximal atlas whose transition **maps** are dianalytic, i.e. either holomorphic or anti-holomorphic, see [AG]. Klein surfaces can be described as quotients P/ h τ i , where P is a compact Riemann surface and τ : P → P is an anti-holomorphic involution on P . The category of such pairs is isomorphic to the category of Klein surfaces via (P, τ ) 7→ P/ h τ i . Under this correspondence the fixed points of τ correspond to the boundary points of the Klein surface. In this paper a Klein surface will be understood as an isomorphy class of such pairs (P, τ ). We will only consider connected compact Klein surfaces. The category of connected compact Klein surfaces is isomorphic to the category of irreducible real algebraic **curves** (see [AG]).

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The birational **geometry** of **moduli** **spaces** of sheaves on surfaces has been studied a lot in recent years; see for instance Arcara, Bertram, Coskun and Huizenga [2], Bayer and Macrì [6; 7], Bertram, Martinez and Wang [9], Coskun and Huizenga [13; 14; 12], Coskun, Huizenga and Woolf [15], Li and Zhao [26] and Woolf [33]. The milestone work in [6; 7] completes the whole picture for K3 surfaces. In this paper, we give a complete description of the minimal model program of the **moduli** space of semistable sheaves on the projective plane via wall-crossings in the space of Bridgeland stability conditions. As a consequence, we deduce a description of their nef cone, movable cone and the chamber decomposition of their minimal models.

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It is proved by Mukai in [Mu84] that the **moduli** space of stable sheaves on an abelian or a projective K3 surface is smooth and has a natural symplectic structure. This construction has been generalized in two directions. On the one hand, the symplectic structure can be generalized to (holomorphic) Poisson structures. In the paper [Tyu88], the author showed that a Poisson structure on the surface will naturally determine an antisymmetric bivector field on the **moduli** space of stable sheaves. Bottacin [Bo95] then proved that such a bivector field satisfies the closure condition and endows the **moduli** space with a natural Poisson structure.

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Abstract In the primary visual cortex, the processing of information uses the distri- bution of orientations in the visual input: neurons react to some orientations in the stimulus more than to others. In many species, orientation preference is mapped in a remarkable way on the cortical surface, and this organization of the neural population seems to be important for visual processing. Now, existing models for the **geometry** and development of orientation preference **maps** in higher mammals make a crucial use of symmetry considerations. In this paper, we consider probabilistic models for V1 **maps** **from** the point of view of group theory; we focus on Gaussian random fields with symmetry properties and review the probabilistic arguments that allow one to es- timate pinwheel densities and predict the observed value of π . Then, in order to test the relevance of general symmetry arguments and to introduce methods which could be of use in modeling curved regions, we reconsider this model in the light of group representation theory, the canonical mathematics of symmetry. We show that through the Plancherel decomposition of the space of complex-valued **maps** on the Euclidean plane, each infinite-dimensional irreducible unitary representation of the special Eu- clidean group yields a unique V1-like map, and we use representation theory as a symmetry-based toolbox to build orientation **maps** adapted to the most famous non- Euclidean geometries, viz. spherical and hyperbolic **geometry**. We find that most of the dominant traits of V1 **maps** are preserved in these; we also study the link between symmetry and the statistics of singularities in orientation **maps**, and show what the striking quantitative characteristics observed in animals become in our curved mod- els.

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It is opportune to consider a generalization of the Radon- Schmid [1], on coherent D-modules [2], of the sheaf of holomorphic bundles into a cohomological context, [3] with the goal of establishing the equivalences between geometrical objects (vector bundles) and algebraic ob- jects as they are the coherent D-modules, these last with the goal of establishing the conformal classes useful to define adequately the differential operators that define the field equations of all the microscopic and macro- scopic phenomena of the space-time through the connec- tions or shape operators [4], of the complex holomorphic bundles in gauge theory. The class of these equivalences is precisely in our **moduli** space, which has in considera- tion the differential operator that defines the connection of the corresponding vector bundle which establish the relation among dimensions of cohomological conformal classes [5], and the vector bundles corresponding via the differential operators of the equations of shape (connec- tion) of the Riemannian manifold [5].

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For example, the space of sections of a smooth vector bundle over a compact connected ﬁnite-dimensional manifold is a nice Fréchet space and hence LM is locally modeled on nice Fréchet s[r]

| g i g i (8) It follows **from** Equation (7) that g takes both posi- tive and negative values, so **from** inequality (6), there is an index j such that g ( j ) and g ( j 1) have oppo- site sign; without loss of generality, g ( j ) > 0 and

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Now any compact Lie group can be embedded in U ( n ) , but this imposes conditions on the data so that it can be hard to find, cf. [34, Sect. 6]. Using the framework of [7], we solve this problem for O(n) and give an algorithm which is inductive on dimension, finding formulae for extended solutions for the group O(n) **from** those for O(n − 2) to end up with algebraic formulae for all harmonic **maps** of finite uniton number and their extended solutions **from** a surface to O ( n ) of finite uniton number in terms of free holomorphic data. Our method is to interpret the extended solution equations in the Lie group and replace the initial data of Burstall and Guest, which had to be integrated in [7], by data which gives the solution by differentiation and algebraic operations.

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to just look at the Bott and Puppe se quence s• out one of the differentials sequence... in the KO-theory.[r]

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First we show that an f-holomorphic map of a Hermitian cosymplectic manifold is harmonic provided that the f-structure on the co-domain satisfies (d^F)1,1 = 0, where V is the Levi-Civita connection. We then characterize those invariant f-structures and metrics on homogeneous **spaces** which satisfy this condition. On a homogeneous space whose tangent bundle splits as a direct sum of mutually distinct isotropy **spaces** (e.g. a flag manifold), we see that an f-structure which is horizontal (i.e. [F+, F J c h) satisfies (d^F)1*1 ■ 0 for any choice of invariant metric. Thus f-holomorphic **maps** are equi-harmonic (harmonic with respect to all invariant metrics). Equi-harmonic **maps** are seen to behave well in combination with homogeneous **geometry**.

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Let A be a subset of a topological space (X, τ). The closure and the interior of A are denoted by Cl (A) and Int(A), respectively. **Maps** and of course irresolute **maps** stand among the most important notions in the whole of pure and applied mathematical science. Various interesting problems arise when one considers openness. Its importance is significant in various areas of mathematics and related sciences. In 1972, Crossley and Hildebrand [1] introduced the notion of irresoluteness. . Many different forms of irresolute **maps** have been introduced over the years. Andrijevic [2] introduced a new class of generalized open sets in a topological space, the so-called b-open sets. This type of sets was discussed by Ekici and Caldas [3] under the name of γ-open sets. The class of γ-open sets is contained in the class of semi-preopen sets and contains all semi-open sets and preopen sets. The class of γ-open sets generates the same topology as the class of preopen sets. A subset A of a topological space X issaid to be γ-open [3] if

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The goal of this section is to calculate two neighbor graphs for the Williams tri- angle. Calculating more neighbor graphs will help us understand them better and two examples for the same type of generalised Sierpinski triangle will allow us to compare neighbor graphs for similar fractal attractors. The introduction of a second flip map to change a Steemson triangle to a Williams triangle does not affect the process used to determine the set of proper neighbor **maps** or how to construct the neighbor graph. One difference between 4F N N and 4F F N is that the Williams triangle has ‘more’ solutions to the algebraic condition. It is not necessary to make explicit what is meant by ‘more’; however, we can do a quick comparison to demonstrate the difference. Let the scaling factors for the triangles be (α, β, γ) = (λ i , λ j , λ k ) with i, j, k ∈ {1, 2, 3, 4} and not all three

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The comparison of numerical results of this paper with results of the numerical experiment **from** [21] shows that these results have analogous features: among them the lost of differentiability of CIC and appearance of the ramification points. But in contrast to [21] complication of topological structure of CIC in this paper is connected with appearance of loops. The more simple analytic formulas for the considered in this paper one-parameter family, than for Lorenz’s one-parameter family of quadratic **maps** in the plane introduced in [26] and considered in [21], makes it possible not only to give more detailed and transparent description of the rebuildings of CIC, but also to give the rigorous proofs of some observable phenomena.

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The intersection (also the self-intersection) problem is a fundamental process needed in model- ing complex shapes in CAD/CAM system. It is useful in the representation of the design of com- plex objects, in computer animation and in NC machining for trimming off the region bounded by the self-intersection **curves** of offset surfaces. It is also essential to Boolean operations nec- essary in the creation of boundary representation in solid modeling [18]. Self-intersections

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Students are now acquainted with the idea of “locus,” and how **Geometry** Expressions can be used to explore loci. Now, they will look more closely at the circle, defined as the locus of points on a plane equidistant **from** a fixed point. In particular, they will find the parametric and implicit equations of circles.