DOCSLIB.ORG

Approximation of the Distance from a Point to an Algebraic Manifold

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Approximation of the Distance from a Point to an Algebraic Manifold Approximation of the Distance from a Point to an Algebraic Manifold Alexei Yu. Uteshev and Marina V. Goncharova Faculty of Applied Mathematics, St. Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg, 199034, Russia Keywords: Algebraic Manifold, Distance Approximation, Discriminant, Level Set. 2 Abstract: The problem of geometric distance d evaluation from a point X0 to an algebraic curve in R or manifold 3 G(X) = 0 in R is treated in the form of comparison of exact value with two its successive approximations d(1) and d(2). The geometric distance is evaluated from the univariate distance equation possessing the zero 2 set coinciding with that of critical values of the function d (X0), while d(1)(X0) and d(2)(X0) are obtained 2 via expansion of d (X0) into the power series of the algebraic distance G(X0). We estimate the quality of approximation comparing the relative positions of the level sets of d(X), d(1)(X) and d(2)(X). 1 INTRODUCTION Approximation (2) is known as Sampson’s dis- tance (Sampson, 1982). We aim at comparison of We treat the problem of Euclidean distance evaluation the qualities of approximations (2) and (3). We deal from a point X0 to manifold defined implicitly by the mostly with the case of algebraic manifolds (1), i.e. equation G(X) 2 R[X]. For this case, the tolerances of the ap- G(X) = 0 (1) proximations can be evaluated via comparison with n the true (geometric) distance value determined from in R ;n 2 f2;3g. Here G(X) is twice differentiable real valued function and it is assumed that the equa- the so-called distance equation (Uteshev and Gon- n charova, 2017; Uteshev and Goncharova, 2018). In tion (1) defines a nonempty set in R . This prob- lem arises in image processing, multi-object move- Section 2, we outline the background of this approach ment simulation, and in the scattered data approxima- for the case of a quadric manifold, while in Section 3 tion problems. Being the problem of nonlinear con- we extend it to the case of manifold of an arbitrary strained optimization, it can be solved with the aid order. In Section 4 we discuss an applicability of the of traditional Newton-like iteration methods. How- proposed approximations to the case of non algebraic ever for the applications connected with the parame- curve (1). ter synthesis such as, for instance, the manifold selec- tion best fitting to the given data set (Ahn et al., 2002; Aigner and Jutler, 2009; Cheng and Chiu, 2014), an 2 QUADRICS analytical representation is needed for the distance as a function of parameters of the problem (point coor- For the particular case of quadric polynomials G(X): dinates and coefficients of G(X) if the latter is a poly- G(X) := X>AX + 2B>X − 1 = 0; (4) nomial). > n×n n We concern here with the following approxima- where A = A 2 R and fX;Bg ⊂ R are the col- tions for the distance d umn vectors, formula (2) is represented as d = jGj=k∇Gk; (2) 1 jG(X0)j (1) d(1) = · : (5) 2 p(AX + B)>(AX + B) ∇G> · H (G) · ∇G 0 0 d = d 1 + G (3) (2) (1) 2k∇Gk4 The counterpart for the formula (3) can be originated from the following approximation Here ∇G stands for the gradient column vector, > — s 1 (AX + B)>A(AX + B) for transposition, H (G) is the Hessian of G(X), k · k d = d + 0 0 G(X ) e(2) (1) 1 > 2 0 ; is the Euclidean norm, and the right-hand sides in (2) 2 [(AX0 + B) (AX0 + B)] and (3) are calculated at X = X0. (6) 715 Uteshev, A. and Goncharova, M. Approximation of the Distance from a Point to an Algebraic Manifold. DOI: 10.5220/0007483007150720 In Proceedings of the 8th International Conference on Pattern Recognition Applications and Methods (ICPRAM 2019), pages 715-720 ISBN: 978-989-758-351-3 Copyright c 2019 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved ICPRAM 2019 - 8th International Conference on Pattern Recognition Applications and Methods suggested in (Uteshev and Goncharova, 2018). For a0 6= 0 , b0 6= 0; then this determinant equals this particular case, both approximations d and de (1) (2) a0 a1 a2 a3 a4 0 0 can be deduced via the following consideration. First 0 a0 a1 a2 a3 a4 0 compute the distance equation, i.e. an algebraic 0 0 a0 a1 a2 a3 a4 equation F (z) = 0;F (z) 2 [z] whose roots coin- R 0 0 0 b0 b1 b2 b3 cide with the critical values of the squared distance 0 0 b0 b1 b2 b3 0 function from X0 to (4), and, generically, the small- 0 b0 b1 b2 b3 0 0 est positive root of this equation equals d2. For the b0 b1 b2 b3 0 0 0 case of an ellipse G(x;y) := x2=a2 + y2=b2 − 1 = 0 and X0 = (x0;y0), this equation takes the form while for the general case it is composed similarly from degg rows of coefficients of f (x) and deg f rows 2 4 F (z;x0;y0) := L z (7) of coefficients of g(x). The resultant is a polynomial function of the coefficients of f (x) and g(x), and its vanishment yields the necessary and sufficient con- −2L L(a2 + b2 + x2 + y2) + a2y2 − b2x2 z3 0 0 0 0 dition for the existence of a common zero for these polynomials. 0 4 2 2 4 4 2 2 4 2 2 2 2 For the particular case g(x) ≡ f (x), the expression + 6L[a y0 + a y0 − b x0 − b x0 + L(a b + x0y0)] 0 Dx( f (x)) := Rx( f ; f )=a0 +[L2 − (a2x2 + b2y2)]2 z2 − 2a2b2 a2b2MG2 defines (up to a sign) the discriminant of the polyno- 0 0 0 mial f (x). Its vanishment yields the necessary and sufficient condition for the existence of a multiple 2 2 2 2 2 4 4 zero for f (x). − (a + b )M + 3a b M − 6a b S4 G0 Theorem 1. Let G(X0) 6= 0. Distance from X0 to the quadric (4) equals the square root from the minimal 2 2 2 4 4 2 2 2 2 +2a b M S4 z + a b G0 M + 4a b G0 = 0: positive zero of the distance equation (z) := (F(µ;z)) = 0; (8) 2 2 F Dµ Here L := a − b , G0 := G(x0;y0), where 2 2 2 2 2 4 2 4 M := x0 + y0 − a − b ; S4 := x0=a + y0=b : A B −I X F(µ;z) := det + µ 0 B> − X> z − X>X Represent the squared distance value as the formal se- 1 0 0 0 ries provided that this zero is not a multiple one. Here 2 2 3 2 n×n is the identity matrix. d (x0;y0) = `2G0 + `3G0 + ::: I R In (Uteshev and Goncharova, 2018) error esti- in powers of the algebraic distance G0 (which can be treated as a small parameter in a vicinity of the mations for the approximations (5) and (6) are de- considered ellipse) and substitute it into the distance duced in terms of maximal deviations of the mani- equation. Equate to zero the coefficients of G2 and folds d(1) = const and de(2) = const from the quadric 0 (4). G3. This results in the approximations (5) and (6). 0 2 2 2 2 Further expansion of the radical in (6) in power se- Example 1. For the ellipse x =18 + y =5 = 1, ries of G0, yields an analogue of approximation (3). approximate equidistant curves d(1)(x;y) = 3 and Similar distance equation of the degree 6 in z can be d(2)(x;y) = 3 in comparison with the true equidistant written down for the point-to-quadric problem in R3. d(x;y) = 3 are presented in Fig. 1 and 2 respectively. n The general expression (valid for a quadric in R for arbitrary n) can be represented via a special function of the coefficients of a polynomial known as the dis- 3 GENERAL ALGEBRAIC criminant. We first remind a more general notion. For MANIFOLDS the univariate polynomials f f (x);g(x)g ⊂ R[x] their For the general case of algebraic manifold, the con- resultant Rx( f ;g) can be defined in the form of Sylvester’s determinant (Uspensky, 1948). For in- struction of distance equation is also possible, at least, stance, if in principle. For the planar case, this construction is based on the following results (Uteshev and Gon- 4 3 f (x) := a0x + ··· + a4; g(x) := b0x + ··· + b3; charova, 2017): 716 Approximation of the Distance from a Point to an Algebraic Manifold Theorem 2. Let G(0;0) 6= 0 and G(x;y) be an even the minimal absolute value of real zeros of the equa- polynomial in y. Expand G in powers of y2 and de- tion G(x;0) = 0 or the square root from the minimal 2 note Ge(x;y ) ≡ G(x;y). Equation G(x;y) = 0 does positive zero of the equation F1(z) = 0 provided that not define a real curve if this zero is not a multiple one. (a) equation G(x;0) = 0 does not have real zeros Remark 1. The conditions (a) and (b) of Theo- and rems 2 and 3 (as well as their counterparts from the (b) equation undermentioned Theorem 5) can be verified without utilization of any numerical method.
Load more

Recommended publications
  • [Math.CV] 6 May 2005
    HOLOMORPHIC FLEXIBILITY PROPERTIES OF COMPLEX MANIFOLDS FRANC FORSTNERICˇ Abstract. We obtain results on approximation of holomorphic maps by al- gebraic maps, the jet transversality theorem for holomorphic and algebraic maps between certain classes of manifolds, and the homotopy principle for holomorphic submersions of Stein manifolds to certain algebraic manifolds. 1. Introduction In the present paper we use the term holomorphic flexibility property for any of several analytic properties of complex manifolds which are opposite to Koba- yashi-Eisenman-Brody hyperbolicity, the latter expressing holomorphic rigidity. A connected complex manifold Y is n-hyperbolic for some n ∈{1,..., dim Y } if every entire holomorphic map Cn → Y has rank less than n; for n = 1 this means that every holomorphic map C → Y is constant ([4], [16], [46], [47]). On the other hand, all flexibility properties of Y will require the existence of many such maps. We shall center our discussion around the following classical property which was studied by many authors (see the surveys [53] and [25]): Oka property: Every continuous map f0 : X → Y from a Stein manifold X is homotopic to a holomorphic map f1 : X → Y ; if f0 is holomorphic on (a neighbor- hood of) a compact H(X)-convex subset K of X then a homotopy {ft}t∈[0,1] from f0 to a holomorphic map f1 can be chosen such that ft is holomorphic and uniformly close to f0 on K for every t ∈ [0, 1]. Here, H(X)-convexity means convexity with respect to the algebra H(X) of all holomorphic functions on X (§2).
    [Show full text]
  • Moduli Spaces
    spaces. Moduli spaces such as the moduli of elliptic curves (which we discuss below) play a central role Moduli Spaces in a variety of areas that have no immediate link to the geometry being classified, in particular in alge- David D. Ben-Zvi braic number theory and algebraic topology. Moreover, the study of moduli spaces has benefited tremendously in recent years from interactions with Many of the most important problems in mathemat- physics (in particular with string theory). These inter- ics concern classification. One has a class of math- actions have led to a variety of new questions and new ematical objects and a notion of when two objects techniques. should count as equivalent. It may well be that two equivalent objects look superficially very different, so one wishes to describe them in such a way that equiv- 1 Warmup: The Moduli Space of Lines in alent objects have the same description and inequiva- the Plane lent objects have different descriptions. Moduli spaces can be thought of as geometric solu- Let us begin with a problem that looks rather simple, tions to geometric classification problems. In this arti- but that nevertheless illustrates many of the impor- cle we shall illustrate some of the key features of mod- tant ideas of moduli spaces. uli spaces, with an emphasis on the moduli spaces Problem. Describe the collection of all lines in the of Riemann surfaces. (Readers unfamiliar with Rie- real plane R2 that pass through the origin. mann surfaces may find it helpful to begin by reading about them in Part III.) In broad terms, a moduli To save writing, we are using the word “line” to mean problem consists of three ingredients.
    [Show full text]
  • Arxiv:Math/0608115V2 [Math.NA] 13 Aug 2006 H Osrcino Nt Lmn Cee N Oo.I H Re the the in Is On
    Multivariate Lagrange Interpolation and an Application of Cayley-Bacharach Theorem for It Xue-Zhang Lianga Jie-Lin Zhang Ming Zhang Li-Hong Cui Institute of Mathematics, Jilin University, Changchun, 130012, P. R. China [email protected] Abstract In this paper,we deeply research Lagrange interpolation of n-variables and give an application of Cayley-Bacharach theorem for it. We pose the concept of sufficient inter- section about s(1 ≤ s ≤ n) algebraic hypersurfaces in n-dimensional complex Euclidean space and discuss the Lagrange interpolation along the algebraic manifold of sufficient intersection. By means of some theorems ( such as Bezout theorem, Macaulay theorem (n) and so on ) we prove the dimension for the polynomial space Pm along the algebraic manifold S = s(f1, · · · ,fs)(where f1(X) = 0, · · · ,fs(X) = 0 denote s algebraic hyper- surfaces ) of sufficient intersection and give a convenient expression for dimension calcu- lation by using the backward difference operator. According to Mysovskikh theorem, we give a proof of the existence and a characterizing condition of properly posed set of nodes of arbitrary degree for interpolation along an algebraic manifold of sufficient intersec- tion. Further we point out that for s algebraic hypersurfaces f1(X) = 0, · · · ,fs(X) = 0 of sufficient intersection, the set of polynomials f1, · · · ,fs must constitute the H-base of ideal Is =<f1, · · · ,fs >. As a main result of this paper, we deduce a general method of constructing properly posed set of nodes for Lagrange interpolation along an algebraic manifold, namely the superposition interpolation process. At the end of the paper, we use the extended Cayley-Bacharach theorem to resolve some problems of Lagrange interpolation along the 0-dimensional and 1-dimensional algebraic manifold.
    [Show full text]
  • Mathematisches Forschungsinstitut Oberwolfach Singularities
    Mathematisches Forschungsinstitut Oberwolfach Report No. 43/2009 DOI: 10.4171/OWR/2009/43 Singularities Organised by Andr´as N´emethi, Budapest Duco van Straten, Mainz Victor A. Vassiliev, Moscow September 20th – September 26th, 2009 Abstract. Local/global Singularity Theory is concerned with the local/global structure of maps and spaces that occur in differential topology or theory of algebraic or analytic varieties. It uses methods from algebra, topology, alge- braic geometry and multi-variable complex analysis. Mathematics Subject Classification (2000): 14Bxx, 32Sxx, 58Kxx. Introduction by the Organisers The workshop Singularity Theory took place from September 20 to 26, 2009, and continued a long sequence of workshops Singularit¨aten that were organized regu- larly at Oberwolfach. It was attended by 46 participants with broad geographic representation. Funding from the Marie Curie Programme of EU provided com- plementary support for young researchers and PhD students. The schedule of the meeting comprised 23 lectures of one hour each, presenting recent progress and interesting directions in singularity theory. Some of the talks gave an overview of the state of the art, open problems and new efforts and results in certain areas of the field. For example, B. Teissier reported about the Kyoto meeting on ‘Resolution of Singularities’ and about recent developments in the geometry of local uniformization. J. Sch¨urmann presented the general picture of various generalizations of classical characteristic classes and the existence of functors connecting different geometrical levels. Strong applications of this for hypersurfaces was provided by L. Maxim. M. Kazarian reported on his new results and construction about the Thom polynomial of contact singularities; R.
    [Show full text]
  • §4 Grassmannians 73
    §4 Grassmannians 4.1 Plücker quadric and Gr(2,4). Let 푉 be 4-dimensional vector space. e set of all 2-di- mensional vector subspaces 푈 ⊂ 푉 is called the grassmannian Gr(2, 4) = Gr(2, 푉). More geomet- rically, the grassmannian Gr(2, 푉) is the set of all lines ℓ ⊂ ℙ = ℙ(푉). Sending 2-dimensional vector subspace 푈 ⊂ 푉 to 1-dimensional subspace 훬푈 ⊂ 훬푉 or, equivalently, sending a line (푎푏) ⊂ ℙ(푉) to 한 ⋅ 푎 ∧ 푏 ⊂ 훬푉 , we get the Plücker embedding 픲 ∶ Gr(2, 4) ↪ ℙ = ℙ(훬 푉). (4-1) Its image consists of all decomposable¹ grassmannian quadratic forms 휔 = 푎∧푏 , 푎, 푏 ∈ 푉. Clearly, any such a form has zero square: 휔 ∧ 휔 = 푎 ∧ 푏 ∧ 푎 ∧ 푏 = 0. Since an arbitrary form 휉 ∈ 훬푉 can be wrien¹ in appropriate basis of 푉 either as 푒 ∧ 푒 or as 푒 ∧ 푒 + 푒 ∧ 푒 and in the laer case 휉 is not decomposable, because of 휉 ∧ 휉 = 2 푒 ∧ 푒 ∧ 푒 ∧ 푒 ≠ 0, we conclude that 휔 ∈ 훬 푉 is decomposable if an only if 휔 ∧ 휔 = 0. us, the image of (4-1) is the Plücker quadric 푃 ≝ { 휔 ∈ 훬푉 | 휔 ∧ 휔 = 0 } (4-2) If we choose a basis 푒, 푒, 푒, 푒 ∈ 푉, the monomial basis 푒 = 푒 ∧ 푒 in 훬 푉, and write 푥 for the homogeneous coordinates along 푒, then straightforward computation 푥 ⋅ 푒 ∧ 푒 ∧ 푥 ⋅ 푒 ∧ 푒 = 2 푥 푥 − 푥 푥 + 푥 푥 ⋅ 푒 ∧ 푒 ∧ 푒 ∧ 푒 < < implies that 푃 is given by the non-degenerated quadratic equation 푥푥 = 푥푥 + 푥푥 .
    [Show full text]
  • Arxiv:Math/0005288V1 [Math.QA] 31 May 2000 Ilb Setal H Rjciecodnt Igo H Variety
    Mannheimer Manuskripte 254 math/0005288 SINGULAR PROJECTIVE VARIETIES AND QUANTIZATION MARTIN SCHLICHENMAIER Abstract. By the quantization condition compact quantizable K¨ahler mani- folds can be embedded into projective space. In this way they become projec- tive varieties. The quantum Hilbert space of the Berezin-Toeplitz quantization (and of the geometric quantization) is the projective coordinate ring of the embedded manifold. This allows for generalization to the case of singular vari- eties. The set-up is explained in the first part of the contribution. The second part of the contribution is of tutorial nature. Necessary notions, concepts, and results of algebraic geometry appearing in this approach to quantization are explained. In particular, the notions of projective varieties, embeddings, sin- gularities, and quotients appearing in geometric invariant theory are recalled. Contents Introduction 1 1. From quantizable compact K¨ahler manifolds to projective varieties 3 2. Projective varieties 6 2.1. The definition of a projective variety 6 2.2. Embeddings into Projective Space 9 2.3. The projective coordinate ring 12 3. Singularities 14 4. Quotients 17 4.1. Quotients in algebraic geometry 17 4.2. The relation with the symplectic quotient 19 References 20 Introduction arXiv:math/0005288v1 [math.QA] 31 May 2000 Compact K¨ahler manifolds which are quantizable, i.e. which admit a holomor- phic line bundle with curvature form equal to the K¨ahler form (a so called quantum line bundle) are projective algebraic manifolds. This means that with the help of the global holomorphic sections of a suitable tensor power of the quantum line bundle they can be embedded into a projective space of certain dimension.
    [Show full text]
  • The Symplectic Topology of Projective Manifolds with Small Dual 3
    THE SYMPLECTIC TOPOLOGY OF PROJECTIVE MANIFOLDS WITH SMALL DUAL PAUL BIRAN AND YOCHAY JERBY Abstract. We study smooth projective varieties with small dual variety using methods from symplectic topology. For such varieties we prove that the hyperplane class is an invertible element in the quantum cohomology of their hyperplane sections. We also prove that the affine part of such varieties are subcritical. We derive several topological and algebraic geometric consequences from that. The main tool in our work is the Seidel representation associated to Hamiltonian fibrations. 1. introduction and summary of the main results In this paper we study a special class of complex algebraic manifolds called projective manifolds with small dual. A projectively embedded algebraic manifold X ⊂ CP N is said to have small dual if the dual variety X∗ ⊂ (CP N )∗ has (complex) codimension ≥ 2. Recall that the dual variety X∗ of a projectively embedded algebraic manifold X ⊂ CP N is by definition the space of all hyperplanes H ⊂ CP N that are not transverse to X, i.e. X∗ = {H ∈ (CP N )∗ | H is somewhere tangent to X}. Let us mention that for “most” manifolds the codimension of X∗ is 1, however in special situations the codimension might be larger. To measure to which extent X deviates from the typical case one defines the defect 1 of an algebraic manifold X ⊂ CP N by ∗ def(X) = codimC(X ) − 1. arXiv:1107.0174v2 [math.AG] 27 Jun 2012 Thus we will call manifolds with small dual also manifolds with positive defect. Note that this is not an intrinsic property of X, but rather of a given projective embedding of X.
    [Show full text]
  • The Concept of a Simple Point of an Abstract Algebraic Variety
    THE CONCEPT OF A SIMPLE POINT OF AN ABSTRACT ALGEBRAIC VARIETY BY OSCAR ZARISKI Contents Introduction. 2 Part I. The local theory 1. Notation and terminology. 5 2. The local vector space?á{W/V). 6 2.1. The mapping m—»m/tri2. 6 2.2. Reduction to dimension zero. 7 2.3. The linear transformation 'M(W/V)->'M(W/V'). 8 3. Simple points. 9 3.1. Definition of simple loci. 9 3.2. Geometric aspects of the definition. 10 3.3. Local ideal bases at a simple point. 12 3.4. Simple zeros of ideals. 14 4. Simple subvarieties. 15 4.1. Generalization of the preceding results. 15 4.2. Singular subvarieties and singular points. 16 4.3. Proof of Theorem 3. 17 4.4. Continuation of the proof. 17 4.5. The case of a reducible variety. 19 5. Simple loci and regular rings. 19 5.1. The identity of the two concepts. 19 5.2. Unique local factorization at simple loci. 22 5.3. The abstract analogue of Theorem 3 and an example of F. K. Schmidt.... 23 Part II. Jacobian criteria for simple loci 6. The vector spaceD(W0 of local differentials. 25 6.1. The local ^-differentials in S„. 25 6.2. The linear transformationM(W/S„)-+<D(W). 25 7. Jacobian criterion for simple points: the separable case. 26 7.1. Criterion for uniformizing parameters. 26 7.2. Criterion for simple points. 28 8. Continuation of the separable case: generalization to higher varieties in Sn. 28 8.1. Extension of the preceding results.
    [Show full text]
  • Complex Algebraic Geometry
    Complex Algebraic Geometry Jean Gallier∗ and Stephen S. Shatz∗∗ ∗Department of Computer and Information Science University of Pennsylvania Philadelphia, PA 19104, USA e-mail: [email protected] ∗∗Department of Mathematics University of Pennsylvania Philadelphia, PA 19104, USA e-mail: [email protected] February 25, 2011 2 Contents 1 Complex Algebraic Varieties; Elementary Theory 7 1.1 What is Geometry & What is Complex Algebraic Geometry? . .......... 7 1.2 LocalStructureofComplexVarieties. ............ 14 1.3 LocalStructureofComplexVarieties,II . ............. 28 1.4 Elementary Global Theory of Varieties . ........... 42 2 Cohomologyof(Mostly)ConstantSheavesandHodgeTheory 73 2.1 RealandComplex .................................... ...... 73 2.2 Cohomology,deRham,Dolbeault. ......... 78 2.3 Hodge I, Analytic Preliminaries . ........ 89 2.4 Hodge II, Globalization & Proof of Hodge’s Theorem . ............ 107 2.5 HodgeIII,TheK¨ahlerCase . .......... 131 2.6 Hodge IV: Lefschetz Decomposition & the Hard Lefschetz Theorem............... 147 2.7 ExtensionsofResultstoVectorBundles . ............ 162 3 The Hirzebruch-Riemann-Roch Theorem 165 3.1 Line Bundles, Vector Bundles, Divisors . ........... 165 3.2 ChernClassesandSegreClasses . .......... 179 3.3 The L-GenusandtheToddGenus .............................. 215 3.4 CobordismandtheSignatureTheorem. ........... 227 3.5 The Hirzebruch–Riemann–Roch Theorem (HRR) . ............ 232 3 4 CONTENTS Preface This manuscript is based on lectures given by Steve Shatz for the course Math 622/623–Complex Algebraic Geometry, during Fall 2003 and Spring 2004. The process for producing this manuscript was the following: I (Jean Gallier) took notes and transcribed them in LATEX at the end of every week. A week later or so, Steve reviewed these notes and made changes and corrections. After the course was over, Steve wrote up additional material that I transcribed into LATEX. The following manuscript is thus unfinished and should be considered as work in progress.
    [Show full text]
  • A Model for Configuration Spaces of Points
    A MODEL FOR CONFIGURATION SPACES OF POINTS RICARDO CAMPOS AND THOMAS WILLWACHER Abstract. The configuration space of points on a D-dimensional smooth framed manifold may be compactified so as to admit a right action over the framed little D-disks operad. We construct a real combinatorial model for these modules, for compact smooth manifolds without boundary. Contents 1. Introduction 1 2. Compactified configuration spaces 5 3. TheCattaneo-Felder-Mnevgraphcomplexandoperad 7 ∗ 4. Twisting GraM and the co-module GraphsM 10 5. Cohomology of the CFM (co)operad 14 6. The non-parallelizable case 21 ∗ 7. A simplification of GraphsM andrelationstotheliterature 23 8. The real homotopytype of M and FMM 29 9. The framed case in dimension D = 2 33 Appendix A. Remark: Comparison to the Lambrechts-Stanley model through cyclic C∞ algebras 36 Appendix B. Example computation: The partition function of the2-sphere 38 Appendix C. Pushforward of PA forms 41 References 43 1. Introduction Given a smooth manifold M, we study the configuration space of n non-overlapping points on M n Confn(M) = {(m1,..., mn) ∈ M | mi , m j for i , j}. These spaces are classical objects in topology, which have been subject to intensive study over the decades. Still, even the rational homotopy type of the spaces Confn(M) is not arXiv:1604.02043v3 [math.QA] 20 Dec 2016 understood in general, though some models exist [LS, I]. The first main result of this paper is the construction of a real dg commutative algebra ∗ model GraphsM for Confn(M), in the case when M is a D-dimensional compact smooth manifold without boundary.
    [Show full text]
  • Introduction
    Introduction There will be no specific text in mind. Sources include Griffiths and Harris as well as Hartshorne. Others include Mumford's "Red Book" and "AG I- Complex Projective Varieties", Shafarevich's Basic AG, among other possibilities. We will be taught to the test: that is, we will be being prepared to take orals on AG. This means we will de-emphasize proofs and do lots of examples and applications. So the focus will be techniques for the first semester, and the second will be topics from modern research. 1. Techniques of Algebraic Geometry (a) Varieties and Sheaves on Varieties i. Cohomology, the basic tool for studying these (we will start this early) ii. Direct Images (and higher direct images) iii. Base Change iv. Derived Categories (b) Topology of Complex Algebraic Varieties i. DeRham Theorem ii. Hodge Theorem iii. K¨ahlerPackage iv. Spectral Sequences (Specifically Leray) v. Grothendieck-Riemann-Roch (c) Deformation Theory and Moduli Spaces (d) Toric Geometry (e) Curves, Jacobians, Abelian Varieties and Analytic Theory of Theta Functions (f) Elliptic Curves and Elliptic fibrations (g) Classification of Surfaces (h) Singularities, Blow-Up, Resolution of Singularities 2. Problems in AG (mostly second semester) (a) Torelli and Schottky Problems: The Torelli question is "Is the map that takes a curve to its Jacobian injective?" and the Schottky Prob- lem is "Which abelian varieties come from curves?" (b) Hodge Conjecture: "Given an algebraic variety, describe the subva- rieties via cohomology." (c) Class Field Theory/Geometric Langlands Program: (Curves, Vector Bundles, moduli, etc) Complicated to state the conjectures. (d) Classification in Dim ≥ 3/Mori Program (We will not be talking much about this) 1 (e) Classification and Study of Calabi-Yau Manifolds (f) Lots of problems on moduli spaces i.
    [Show full text]
  • Solomon Lefschetz
    NATIONAL ACADEMY OF SCIENCES S O L O M O N L EFSCHETZ 1884—1972 A Biographical Memoir by PHILLIP GRIFFITHS, DONALD SPENCER, AND GEORGE W HITEHEAD Any opinions expressed in this memoir are those of the author(s) and do not necessarily reflect the views of the National Academy of Sciences. Biographical Memoir COPYRIGHT 1992 NATIONAL ACADEMY OF SCIENCES WASHINGTON D.C. SOLOMON LEFSCHETZ September 3, 1884-October 5, 1972 BY PHILLIP GRIFFITHS, DONALD SPENCER, AND GEORGE WHITEHEAD1 OLOMON LEFSCHETZ was a towering figure in the math- Sematical world owing not only to his original contribu- tions but also to his personal influence. He contributed to at least three mathematical fields, and his work reflects throughout deep geometrical intuition and insight. As man and mathematician, his approach to problems, both in life and in mathematics, was often breathtakingly original and creative. PERSONAL AND PROFESSIONAL HISTORY Solomon Lefschetz was born in Moscow on September 3, 1884. He was a son of Alexander Lefschetz, an importer, and his wife, Vera, Turkish citizens. Soon after his birth, his parents left Russia and took him to Paris, where he grew up with five brothers and one sister and received all of his schooling. French was his native language, but he learned Russian and other languages with remarkable fa- cility. From 1902 to 1905, he studied at the Ecole Centrale des Arts et Manufactures, graduating in 1905 with the de- gree of mechanical engineer, the third youngest in a class of 220. His reasons for entering that institution were com- plicated, for as he said, he had been "mathematics mad" since he had his first contact with geometry at thirteen.
    [Show full text]