Abstract. In this paper, using the formula for the integrals of the ψ-classes over the double ramification cycles found by S. Shadrin, L. Spitz, D. Zvonkine and the author, we derive a new explicit formula for the n -point function of the intersection numbers on the **moduli** **space** of **curves**.

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In concise the proposed method begins from Witten’s (1991) conjecture and Kontsevich’s (1992) model, which we hipothesise can be used to formulate the interesting connection between the two entities, the ‘elusive’ Feynman integral and the ‘predictable’ completely integrable systems, that originally seems disparate in nature. Technically, this refers to a generator function of intersection numbers on **moduli** **space** for stable **curves** and the τ-function of the KdV (Korteweg-de Vries) hierarchy. This conjecture is derived fundamentally from two approaches with respect to two- dimensional quantum gravity. Essentially, the correlator of the two-dimensional quantum gravity is the Feynman integral with respect to the ‘metric **space**’ of the two-dimensional topological real **space**. The key point of the connection between topology of the **moduli** spaces of algebraic **curves** and the ‘matrix’ integrals (of the form of Feynman integral) is the asymptotic expansion of these integrals in terms of the Feynman diagrams. The technique of Feynman diagram expansion was invented by Feynman (see Ramond 1981) for reducing the infinite-dimensional Feynman integral appeared in quantum electrodynamics (QED) to an infinite-series of finite-dimensional integrals. The infinite- series is a summation over all graphs with certain properties representing the physical process, such as the collision pattern of elementary particles.

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Abstract. We prove that a very general hypersurface of bidegree (2, n) in P 2 × P 2 for n bigger than or equal to 2 is not stably rational, using Voisin’s method of integral Chow-theoretic decompositions of the di- agonal and their preservation under mild degenerations. At the same time, we also analyse possible ways to degenerate Prym **curves**, and the way how various loci inside the **moduli** **space** of stable Prym **curves** are nested. No deformation theory of stacks or sheaves of Azumaya alge- bras like in recent work of Hassett-Kresch-Tschinkel is used, rather we employ a more elementary and explicit approach via Koszul complexes, which is enough to treat this special case.

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**space** of stable **curves** can be described by integrable hierarchies in two different ways ([Kaz09, Bur13]). Witten’s original conjecture can be generalized to the **moduli** **space** of r-spin **curves** and its generalizations ([Wit93, FSZ10, FJR13]). An integrable hierarchy corresponding to the Gromov-Witten theory of the resolved conifold is discussed in [BCR12, BCRR14]. A hierarchy that governs the degree zero Gromov-Witten invariants was constructed in [Dub13]. KdV type equations for the intersection numbers on the **moduli** **space** of Riemann surfaces with boundary were introduced in [PST14] and further studied in [Bur14a, Bur14b].

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Using a birational correspondence between the twistor **space** of S 2n and projective **space**, we describe, up to birational equivalence, the **moduli** **space** of superminimal surfaces in S 2n of degree d as **curves** of degree d in projective **space** satisfying a certain diﬀerential system. Using this approach, we show that the **moduli** **space** of linearly full maps is nonempty for suﬃciently large degree and we show that the dimension of this **moduli** **space** for n = 3 and genus 0 is greater than or equal to 2d + 9. We also give a direct, simple proof of the connectedness of the **moduli** **space** of superminimal surfaces in S 2n of degree d.

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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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By a conformal string in Euclidean **space** is meant a closed critical curve with non constant conformal curvature of the conformal arclength functional. The conformal geometry of **space** **curves** was mainly developed in the first half of the past century and later taken up starting from the early 1980's. This subject has got much attention for its many fields of application, including the theory of integrable systems [3], [12], [8], topology and M • 𝑜 bius energy of knots [2], [6], [9], and the geometric approach to shape analysis and medical imaging [13].

We apply Lie group based similarity methods to the study of a new, and widely relevant, class of objects, namely motions of a **space** curve. In particular, we consider the motion of a curve evolving with a curvature κ and torsion τ dependent velocity law. We systematically derive the Lie point symmetries of all such laws of motion and use these to catalogue all their possible similarity reductions. This calculation reveals special classes of law with high degrees of symmetry (and a correspondingly large number of similarity reductions). Of particular note is one class which is invariant under general linear transformations in **space**. This has potential applications in pattern and signal recognition.

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Quintic threefolds are a class of projective varieties which occupy a special place in algebraic geometry. They are some of the simplest examples of Calabi-Yau varieties. Calabi-Yau varieties have received a great deal of attention in the last 30 years because they give the right geometric conditions for some superstring compactifications [4]. A surprising geometric relationship found for Calabi-Yau varieties, due to string theory, is a phenomenon called mirror symmetry. Mirror symmetry, in its original form, states that a Calabi-Yau variety Y has a mirror variety Y where the complex structure **moduli** **space** of Y is the same as the complexified K¨ahler **moduli** **space** of Y . This was explicitly calculated in [5] for the case of quintic threefolds. The purpose of this dissertation is to give an explicit description of the **space** of complex structure **moduli** **space** for Calabi-Yau quintic threefolds using geometric invariant theory (GIT).

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results previously known for convolution estimates related to **space** **curves**, namely [1-6]. This article is organized as follows: in the following section, a uniform estimate for convolution operators with measures supported on plane **curves**. The proof of Theo- rem 1.1 based on a T*T method is given in Section 3.

These results just described also have signiﬁcant topological implications. In fact, by combining them with a theorem of Atiyah [1], the author [14] was able to com- pletely classify spaces in the genus of BSU(2) which admit nonnullhomotopic maps from inﬁnite complex projective **space** (“the maximal torus”) and to compute the maps explicitly in these cases. This extends the classical result of McGibbon [8] and Rector [11] on maximal torus admissibility of spaces in the genus of BSU(2).

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Once the symbol is completed (this is possible because, in our case, the topological obstruction vanishes, see 3.2.3) we obtain a Fredholm operator in the edge algebra with a parametrix (with asymptotics) of inverse order. At this point we want to use the Implicit Function theorem for Banach spaces to obtain finite dimensionality and smoothness of the **moduli** **space** of deformations. However the possible non-surjectivity of the linearised deformation map produces an obstruction **space**. The presence of an obstruction **space** is not unexpected because even in the case of a compact manifold with isolated conical singularities each of the singular cones contributes to the obstruction **space**. This was studied in detail by Joyce [Joy04b].

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between the curvatures for any unit speed curve to be congruent to a normal curve in the n-dimensional Euclidean **space**. Moreover, the diﬀerentiable function f (s) is introduced by using the relationship between the curvatures of any unit speed curve in E n . Finally, the diﬀerential equation characterizing a normal curve can be solved explicitly to determine when the curve is congruent to a normal curve.

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Further work would focus on the generalisation of the class of regression partition models searched by MAP techniques to include models that exhibit the type of depen- dence between clusters that might be expected were one cluster to regulate another. The statistical models added to the search **space** need not only to be parsimonious so that they can be estimated and scored at speed, utilising properties of conjugacy, but will also need to mirror closely the types of regulatory dependence expected within this context. The fact that each dependency model within the class can be associated with a particular network of relationships means that it will be possible to depict the MAP model graphically using the semantics of excitation and inhibition familiar to the sci- entist rather than through conditional independence graphs whose meanings are easily misunderstood. In the context of our main example each node in these new graphs will be indexed by a set of clusters and their profiles that are potentially regulatory. Each edge will be either directed - representing excitation between these sets - or blocked - representing inhibition. The genes within the sets of clusters will of course be annotated allowing the scientist to explore their signatures and how these might relate to the actual postulated message passing mechanisms. This search methodology, if employed on a new microarray experiment, would be able to provide possible explanatory regulatory graphs together and we could provide such a schemata back to the biologists.

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So we have defined the deformed phase **space** and Gauss-law constraints and now one should just remember that the same phase **space** and constraints recently appeared in [12, 13] where the relation between the Heisenberg double and the symplectic structure of the **moduli** **space** of flat connections on a Riemann surface was studied. Namely it was shown in [12, 13] that there is such a change of variables that the Poisson structure in terms of the new variables coincides with the Poisson structure which was introduced by Fock and Rosly [14] to describe the **moduli** **space**. For reader’s convinience we present here the corresponding formulas in our notations.

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The classical work on the scattering of acoustic waves by semi-infinite gratings is a pa- per by Hills and Karp [20], which addressed the case of small sound-soft circular cylinders. They followed the technique of [21], which assumed isotropic point scatterers so that in the neighbourhood of an individual defect, the scattered field may be represented by a term incorporating an unknown coefficient and the free-**space** Green’s function for the Helmholtz operator centred at that scatterer. This method is particularly well adapted to the problem presented here, since the assumption that the grating’s elements are small, and therefore scatter isotropically, is not formally required for the pinned biharmonic plate. The additional attribute of the non-singular Green’s function (4) is also advantageous for this wave scattering method.

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We presented new geometrical models for a curve evolving in **space** by a method different those in [7]. In addition to, the evolution equation for spherical images, Smarandache **curves** are obtained. Finally we displayed geometric visualization of the evolving curve, spherical images and Smarandache **curves**.

coassociative ALE-fibrations over a compact flat Riemannian 3-manifold Q. The flatness condition allows an explicit description of the deformation **space** of closed G2-structures, and hence also the **moduli** **space** of supersymmetric vacua: it is modeled by flat sections of a bundle of Brieskorn-Grothendieck resolutions over Q. Moreover, when instanton corrections are neglected, we obtain an explicit description of the **moduli** **space** for the dual type IIA string compactification. The two **moduli** spaces are shown to be isomorphic for an important example involving A1-singularities, and the result is conjectured to hold in generality. We also discuss an interpretation of the IIA **moduli** **space** in terms of "flat Higgs bundles" on Q and explain how it suggests a new approach to SYZ mirror symmetry, while also providing a description of G2-structures in terms of B-branes. The net result is two algebro-geometric descriptions of the **moduli** **space** of complexified G2-structures: one as a character variety and a mirror description in terms of a Hilbert scheme of points. Usual G2-deformations are parametrized by spectral covers of flat Higgs bundles.

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Let 𝛼 = 𝛼(𝑠) and 𝛽 = 𝛽(𝑠 ∗ ) be two regular unit speed **curves** lying fully in pseudohyperbolic **space** 𝐻 0 2 in R 3 1 and let {𝛼, 𝑇, 𝜉} and {𝛽, 𝑇 𝛽 , 𝜉 𝛽 } be the moving Sabban frames of these **curves**, respectively. Then we have the following definitions of pseudohyperbolical Smarandache **curves**.

It is a remarkable fact that much of the geometry of a “geometric background” (algebraic **space**, symplectic manifold,...) is encoded in the noncommutative shadow attached to it, and the associated 2D-TFT. For instance, the smoothness and properness of a scheme X is encoded in the (full) dualizability of QC(X) as an ob- ject of the symmetric monoidal (∞, 2)-category of k-linear presentable ∞-categories. Conjecturally, the Hodge structure of X can be recovered from the TFT associated to QC(X). There are similar statements for other geometric backgrounds, such as symplectic manifolds. We will briefly touch upon these ideas in Chapter 5. For a detailed treatment, the reader is referred to [KKP08, Lur09b].

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