Topology/Geometry Jan 2012

... (b) Describe the one-point compactification of T2 minus two distinct points. What is the fundamental group of the one-point compactification of T2 minus two distinct points? Q.3 Prove that the singular homology Ht (X) of the space X = pt consisting of a single point is equal to ...

... (b) Describe the one-point compactification of T2 minus two distinct points. What is the fundamental group of the one-point compactification of T2 minus two distinct points? Q.3 Prove that the singular homology Ht (X) of the space X = pt consisting of a single point is equal to ...

Final Exam

... two-planes (two-dimensional distribution)? Is this field of two-planes integrable on M ? b) Consider a third vector field Z = −x3 ...

... two-planes (two-dimensional distribution)? Is this field of two-planes integrable on M ? b) Consider a third vector field Z = −x3 ...

Assignment 2

... a subset I = {i1 , . . . , im } ⊂ {1, . . . , n} and an open neighborhood U of p in M , such that φ : U → Rm given by (x1 , . . . , xn ) 7→ (xi1 , . . . , xim ) maps U homeomophically to an open subset Ω ⊂ Rm , and U is the graph of a C ∞ map F : Ω → RJ , where J = {1, . . . , n} \ I. In other words ...

... a subset I = {i1 , . . . , im } ⊂ {1, . . . , n} and an open neighborhood U of p in M , such that φ : U → Rm given by (x1 , . . . , xn ) 7→ (xi1 , . . . , xim ) maps U homeomophically to an open subset Ω ⊂ Rm , and U is the graph of a C ∞ map F : Ω → RJ , where J = {1, . . . , n} \ I. In other words ...

Applied Math Seminar The Geometry of Data Spring 2015

... that as data is spread into high dimensions, the distance between points becomes large and the corresponding density very low and difficult to estimate. In order to avoid this issue, one imagines that only the data representation is high dimensional but the data actually lies along curved low-dimens ...

... that as data is spread into high dimensions, the distance between points becomes large and the corresponding density very low and difficult to estimate. In order to avoid this issue, one imagines that only the data representation is high dimensional but the data actually lies along curved low-dimens ...

1.6 Smooth functions and partitions of unity

... Given a smooth map ϕ : M −→ N of manifolds, we obtain a natural operation ϕ∗ : C ∞ (N, R) −→ C ∞ (M, R), given by f 7→ f ◦ ϕ. This is called the pullback of functions, and defines a homomorphism of rings since ∆ ◦ ϕ = (ϕ × ϕ) ◦ ∆. The association M 7→ C ∞ (M, R) and ϕ 7→ ϕ∗ takes objects and arrows ...

... Given a smooth map ϕ : M −→ N of manifolds, we obtain a natural operation ϕ∗ : C ∞ (N, R) −→ C ∞ (M, R), given by f 7→ f ◦ ϕ. This is called the pullback of functions, and defines a homomorphism of rings since ∆ ◦ ϕ = (ϕ × ϕ) ◦ ∆. The association M 7→ C ∞ (M, R) and ϕ 7→ ϕ∗ takes objects and arrows ...

BOOK REVIEW

... tangent bundle T M carries the structure of a smooth manifold such that the projection π : T M → M is a smooth map and every point x ∈ M has a neighbourhood U such that the inverse image π −1 (U ) of U is diffeomorphic to the direct product U × Rn , n = dim M, is proved. The theorem that every close ...

... tangent bundle T M carries the structure of a smooth manifold such that the projection π : T M → M is a smooth map and every point x ∈ M has a neighbourhood U such that the inverse image π −1 (U ) of U is diffeomorphic to the direct product U × Rn , n = dim M, is proved. The theorem that every close ...

CIS 736 (Computer Graphics) Lecture 1 of 30 - KDD

... – Affine subspace – nonempty subset S of VS (V, +, ·) such that closure S’ of S under point subtraction is a linear subspace of V – Span – set of all linear combinations of set of vectors – Linear independence – property of set of vectors that none lies in span of others – Basis – minimal spanning s ...

... – Affine subspace – nonempty subset S of VS (V, +, ·) such that closure S’ of S under point subtraction is a linear subspace of V – Span – set of all linear combinations of set of vectors – Linear independence – property of set of vectors that none lies in span of others – Basis – minimal spanning s ...

characterization of curves that lie on a surface in euclidean space

... Bishop’s proof. However, in order to achieve this goal, one is naturally led to the study of the geometry of Lorentz-Minkowski spaces, Eν3 [2], since hHessF ·, ·i may have a non-zero index ν. This study present some difficulties due to the many possibilities for the casual character of a curve β : ...

... Bishop’s proof. However, in order to achieve this goal, one is naturally led to the study of the geometry of Lorentz-Minkowski spaces, Eν3 [2], since hHessF ·, ·i may have a non-zero index ν. This study present some difficulties due to the many possibilities for the casual character of a curve β : ...

Algebra II — exercise sheet 9

... isomorphism of varieties. Extend this to an example of a morphism of projective varieties with the same properties. Solution: The map is given by polynomials, so it is regular. It is bijective, because for any point (x, y) ∈ X, we have y 2 = x3 and (with K algebraically closed), there are two square ...

... isomorphism of varieties. Extend this to an example of a morphism of projective varieties with the same properties. Solution: The map is given by polynomials, so it is regular. It is bijective, because for any point (x, y) ∈ X, we have y 2 = x3 and (with K algebraically closed), there are two square ...

Math 8306, Algebraic Topology Homework 12 Due in-class on Wednesday, December 3

... 1. Show that if a principal bundle P → B has a section, then there is a homeomorphism to the trivial principal bundle: P ∼ = B × G as right G-spaces. 2. Let G and H be topological groups. Suppose P1 → B is a principal Gbundle and P2 → B is a principal H-bundle. Show P1 ×B P2 (pullback!) is a princip ...

... 1. Show that if a principal bundle P → B has a section, then there is a homeomorphism to the trivial principal bundle: P ∼ = B × G as right G-spaces. 2. Let G and H be topological groups. Suppose P1 → B is a principal Gbundle and P2 → B is a principal H-bundle. Show P1 ×B P2 (pullback!) is a princip ...

Lesson 34 – Coordinate Ring of an Affine Variety

... understand groups, for instance, we study homomorphisms; to understand topological spaces, we study continuous functions; to understand manifolds in differential geometry, we study smooth functions. In algebraic geometry, our objects are varieties, and because varieties are defined by polynomials, i ...

... understand groups, for instance, we study homomorphisms; to understand topological spaces, we study continuous functions; to understand manifolds in differential geometry, we study smooth functions. In algebraic geometry, our objects are varieties, and because varieties are defined by polynomials, i ...

ALGEBRAIC GEOMETRY - University of Chicago Math

... (b) Prove that C is smooth at every one of its points if and only if the equation for C is irreducible if and only if its equation can be put in the first form of (a). (c) Prove that C is singular at every one of its points if and only if the equation for C can be put in the third form of (a). (d) P ...

... (b) Prove that C is smooth at every one of its points if and only if the equation for C is irreducible if and only if its equation can be put in the first form of (a). (c) Prove that C is singular at every one of its points if and only if the equation for C can be put in the third form of (a). (d) P ...

GR in a Nutshell

... The first equation has reproduced Einstein’s equations. The second equation involves a spin density tensor S and a modified torsion tensor T, and therefore couples spin with torsion. ...

... The first equation has reproduced Einstein’s equations. The second equation involves a spin density tensor S and a modified torsion tensor T, and therefore couples spin with torsion. ...

Unit 6 Lesson 7 Outline

... Lesson Plan Outline Geometry in Construction Title: Tangent Lines to Circles ...

... Lesson Plan Outline Geometry in Construction Title: Tangent Lines to Circles ...

0 1 0 0 0 0 1 0 0 0 0 1

... • A transformation is a function that maps a point (or vector) into another point (or vector). • An affine transformation is a transformation that maps lines to lines. Why are affine transformations "nice"? We can define a polygon using only points and the line segments joining the points. To move t ...

... • A transformation is a function that maps a point (or vector) into another point (or vector). • An affine transformation is a transformation that maps lines to lines. Why are affine transformations "nice"? We can define a polygon using only points and the line segments joining the points. To move t ...

Appendix B Topological transformation groups

... The following affine subgroup has no official name, therefore it is named here: the stretch transformations, denoted by Stretk , are affine transformations for which the matrix L in Equation B.2 is restricted to be a diagonal matrix, that is, a matrix in which elements off the diagonal are zero. The ...

... The following affine subgroup has no official name, therefore it is named here: the stretch transformations, denoted by Stretk , are affine transformations for which the matrix L in Equation B.2 is restricted to be a diagonal matrix, that is, a matrix in which elements off the diagonal are zero. The ...

Differential geometry of surfaces in Euclidean space

... Differential geometry of surfaces in Euclidean space In this short note I would like to illustrate how the general concepts used in Riemannian geometry arise naturally in the context of curved surfaces in ordinary (Euclidean) space. Let us assume the n-dimensional space Rn whose coordinates will be ...

... Differential geometry of surfaces in Euclidean space In this short note I would like to illustrate how the general concepts used in Riemannian geometry arise naturally in the context of curved surfaces in ordinary (Euclidean) space. Let us assume the n-dimensional space Rn whose coordinates will be ...

Homework sheet 6

... in Ui (when Ui is given the induced topology) for all i. [This verifies a claim made in class.] 5. Let C be a smooth projective plane curve. Let ` be a fixed line in the projective plane, and assume that ` is not tangent to C at any of the points where it intersects C. If P ∈ C, let tP denote the ta ...

... in Ui (when Ui is given the induced topology) for all i. [This verifies a claim made in class.] 5. Let C be a smooth projective plane curve. Let ` be a fixed line in the projective plane, and assume that ` is not tangent to C at any of the points where it intersects C. If P ∈ C, let tP denote the ta ...

Fractals

... 2. To produce a finite straight line continuously in a straight line. 3. To describe a circle with any centre and distance. 4. That all right angles are equal to each other. 5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles ...

... 2. To produce a finite straight line continuously in a straight line. 3. To describe a circle with any centre and distance. 4. That all right angles are equal to each other. 5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles ...

geometric congruence

... • AAS. (Also known as SAA.) If two triangles have 2 pairs of corresponding angles equal, as well as a pair of corresponding sides which is not in between them, the triangles are congruent. ∗ hGeometricCongruencei created: h2013-03-21i by: hCWooi version: h31467i Privacy setting: h1i hDefinitioni h51 ...

... • AAS. (Also known as SAA.) If two triangles have 2 pairs of corresponding angles equal, as well as a pair of corresponding sides which is not in between them, the triangles are congruent. ∗ hGeometricCongruencei created: h2013-03-21i by: hCWooi version: h31467i Privacy setting: h1i hDefinitioni h51 ...

Some comments on Heisenberg-picture QFT, Theo Johnson

... • C ∈ ALG1 (PRESK ) is moreover SO(2)-invariant (i.e. which is local for the Weiss topology. defines 2-dim oriented TQFT) if it is moreover equipped Example (Dwyer–Stolz–Teichner, as seen through my with a “pivotal” structure. • C ∈ ALG2 (PRESK ) is 3-dualizable if 1C ∈ C is compact glasses): Given ...

... • C ∈ ALG1 (PRESK ) is moreover SO(2)-invariant (i.e. which is local for the Weiss topology. defines 2-dim oriented TQFT) if it is moreover equipped Example (Dwyer–Stolz–Teichner, as seen through my with a “pivotal” structure. • C ∈ ALG2 (PRESK ) is 3-dualizable if 1C ∈ C is compact glasses): Given ...

In the branch of mathematics called differential geometry, an affine connection is a geometric object on a smooth manifold which connects nearby tangent spaces, and so permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space. The notion of an affine connection has its roots in 19th-century geometry and tensor calculus, but was not fully developed until the early 1920s, by Élie Cartan (as part of his general theory of connections) and Hermann Weyl (who used the notion as a part of his foundations for general relativity). The terminology is due to Cartan and has its origins in the identification of tangent spaces in Euclidean space Rn by translation: the idea is that a choice of affine connection makes a manifold look infinitesimally like Euclidean space not just smoothly, but as an affine space.On any manifold of positive dimension there are infinitely many affine connections. If the manifold is further endowed with a Riemannian metric then there is a natural choice of affine connection, called the Levi-Civita connection. The choice of an affine connection is equivalent to prescribing a way of differentiating vector fields which satisfies several reasonable properties (linearity and the Leibniz rule). This yields a possible definition of an affine connection as a covariant derivative or (linear) connection on the tangent bundle. A choice of affine connection is also equivalent to a notion of parallel transport, which is a method for transporting tangent vectors along curves. This also defines a parallel transport on the frame bundle. Infinitesimal parallel transport in the frame bundle yields another description of an affine connection, either as a Cartan connection for the affine group or as a principal connection on the frame bundle.The main invariants of an affine connection are its torsion and its curvature. The torsion measures how closely the Lie bracket of vector fields can be recovered from the affine connection. Affine connections may also be used to define (affine) geodesics on a manifold, generalizing the straight lines of Euclidean space, although the geometry of those straight lines can be very different from usual Euclidean geometry; the main differences are encapsulated in the curvature of the connection.