DOCSLIB.ORG

DIFFERENTIAL GEOMETRY, D COURSE Gabriel P. Paternain

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
DIFFERENTIAL GEOMETRY, D COURSE Gabriel P. Paternain DIFFERENTIAL GEOMETRY, D COURSE Gabriel P. Paternain Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 0WB, England E-mail address: [email protected] Contents Chapter 1. Smooth manifolds and smooth maps 1 1.1. Definitions 1 1.2. Tangent spaces and derivatives 2 1.3. Regular values and Sard’s theorem 3 1.4. Transversality 5 1.5. Manifolds with boundary 6 1.6. Degree modulo 2 7 1.7. Abstract manifolds and Whitney’s embedding theorem 11 Chapter 2. Length, area and curvature 13 2.1. Arc-length, curvature and torsion of curves 13 2.2. The isoperimetric inequality in the plane 14 2.3. First fundamental form and Area 16 2.4. The Gauss map 19 2.5. The second fundamental form in local coordinates 20 2.6. Theorema Egregium 21 Chapter 3. Critical points of length and area 23 3.1. Geodesics 23 3.2. Covariant derivative and parallel transport 24 3.3. Minimal surfaces 25 3.4. The Weierstrass representation 26 3.5. The meaning of g in the Weierstrass representation 29 Chapter 4. Global Riemannian geometry 31 4.1. The exponential map and geodesic polar coordinates 31 4.2. The Gauss-Bonnet theorem 33 4.3. Applications 38 4.4. Fenchel’s theorem 39 4.5. The F´ary-Milnor theorem 40 3 CHAPTER 1 Smooth manifolds and smooth maps 1.1. Definitions Definition 1.1. Let U be an open set of Rn. A map f : U → Rm is called smooth if it has continuous partial derivatives of all orders. A map f : X → Rm defined on an arbitrary subset X of Rn is called smooth if for each x ∈ X there is an open set U ⊂ Rn containing x and a smooth map F : U → Rm such that F equals f on U ∩ X. We will use the shorthand term local when we wish to refer to behaviour only in a neighbourhood of a point. Smoothness is a local property. (The term global refers to the whole space X.) Definition 1.2. A smooth map f : X → Y between subsets of Euclidean space is a diffeomorphism if it is a bijection, and if the inverse f −1 : Y → X is also smooth. X and Y are diffeomorphic if such an f exists. Differential Topology is about properties of a set X ⊂ Rn which are invariant under diffeomorphisms. Arbitrary sets of Rn may be too wild. We would like to have a class of sets on which can do locally the same as in Euclidean space. Definition 1.3. Let X be a subset of RN . We say that X is a k-dimensional manifold if each point possesses a neighbourhood V in X which is diffeomorphic to an open set of Rk. A diffeomorphism φ : U → V , where U is an open set of Rk, is called a parametrization of the neighbourhood V . The inverse diffeomorphism φ−1 : V → U is called a coordinate system or a chart on V . −1 If we write φ = (x1, . , xk) the k smooth functions x1, . , xk on V are called coordinate functions. The dimension k of X is written as dim X. Example 1.4. The unit sphere S2 given by all (x, y, z) ∈ R3 with x2 +y2 +z2 = 1 is a smooth manifold of dimension 2. The diffeomorphism p (x, y) 7→ (x, y, 1 − x2 − y2), for x2 +y2 < 1, parametrizes the region z > 0 of S2. By interchanging the roles of x, y and z and changing the sign of the variables, we obtain similar parametrizations of the regions x > 0, y > 0, x < 0, y < 0 and z < 0. Since these cover S2, it follows that S2 is a smooth manifold. Similarly Sn := {x ∈ Rn+1 : |x| = 1} is a smooth manifold of dimension n. Exercise 1.5. If X and Y are manifolds, so is X × Y , and dim X × Y = dim X + dim Y . 1 2 1. SMOOTH MANIFOLDS AND SMOOTH MAPS Definition 1.6. If X and Z are both manifolds in RN and Z ⊂ X, then Z is a submanifold of X. In particular, X itself is a submanifold of RN . Any open set of X is a submanifold of X. The codimension of Z in X is dim X − dim Z. 1.2. Tangent spaces and derivatives n For any open set U ⊂ R and x ∈ U, the tangent space to U at x (denoted TxU) is defined to be Rn. Recall from Analysis II that for any smooth map f : U → Rm, the derivative of f at x ∈ U is the linear map n m dfx : R → R given by f(x + th) − f(x) dfx(h) = lim t→0 t for h ∈ Rn. Now let us define the tangent space TxX of an arbitrary smooth manifold X ⊂ RN . Choose a parametrization φ : U → X around x where U is an open set of Rk. Without loss of generality assume that φ(0) = x. Think of φ as a map from U to N k N R , so dφ0 : R → R is defined. Set k TxX := dφ0(R ). Lemma 1.7. TxX does not depend on φ and dim TxX = k. Proof. Suppose ψ : V → X is another choice, with ψ(0) = x. By shrinking both U and V , we may assume that φ(U) = ψ(V ). Then the map h = ψ−1 ◦ φ : U → V is a diffeomorphism. Now we write φ = ψ ◦ h and we differentiate. Using the chain rule we have: dφ0 = dψ0 ◦ dh0. Since dh0 is an invertible linear map, it follows at once that dφ0 and dψ0 have the same image. Since φ−1 : φ(U) → U is a smooth map, we can choose an open set W in RN containing x and a smooth map Φ : W → Rk that extends φ−1. Then Φ ◦ φ is the identity map of U, so the chain rule implies that the composition of linear maps k dφ0 dΦx k R → TxX → R k k is the identity map of R . Thus dφ0 : R → TxX is an isomorphism and dim TxX = k. We can now define the derivative of a smooth map f : X → Y between arbitrary manifolds. Let x be a point in X and set y = f(x). The derivative must be a linear map dfx : TxX → TyY which gives us back the usual derivative when X and Y are open sets in Euclidean space and it must also satisfy the chain rule. Keeping this in mind, let φ : U → X and ψ : V → Y be parametrizations around x and y respectively, where U is an open set of Rk and V is an open set of Rl. As before suppose φ(0) = x and ψ(0) = y. If U is small enough we have the diagram: f X −−−−→ Y x x φ ψ h=ψ−1◦f◦φ U −−−−−−−−→ V 1.3. REGULAR VALUES AND SARD’S THEOREM 3 Hence our definition of dfx must be so that the following diagram commutes: dfx TxX −−−−→ TyY x x dφ0 dψ0 dh0 Rk −−−−→ Rl Since dφ0 is an isomorphism, we must have −1 dfx := dψ0 ◦ dh0 ◦ dφ0 . We must verify that this definition does not depend on the choices of φ and ψ. Exercise 1.8. Prove that this is indeed the case. f g Chain Rule. If X → Y → Z are smooth maps of manifolds, then d(g ◦ f)x = dgf(x) ◦ dfx. 1.2.1. The inverse function theorem. Let f : X → Y be a smooth map between manifolds. We say that f is a local diffeomorphism at x if f maps a neighbourhood of x diffeomorphically onto a neighbourhood of f(x). Theorem 1.9 (The inverse function theorem). Suppose that f : X → Y is a smooth map whose derivative dfx at the point x is an isomorphism. Then f is a local diffeomorphism at x. Exercise 1.10. Prove the theorem assuming the inverse function theorem for smooth functions between open sets of Euclidean space, as you have seen it in Analysis II. 1.3. Regular values and Sard’s theorem Let f : X → Y be a smooth map between manifolds. Let C be the set of all points x ∈ X such that dfx : TxX → Tf(x)Y is not surjective. Definition 1.11. A point in C will be called a critical point. A point in f(C) will be called a critical value. A point in the complement of f(C) will be called a regular value. Remark 1.12. Note that if dim X < dim Y , then C = X and the preimage of a regular value is the empty set. Theorem 1.13 (Preimage theorem). Let y be a regular value of f : X → Y with dim X ≥ dim Y . Then the set f −1(y) is a submanifold of X with dim f −1(y) = dim X − dim Y . −1 Proof. Let x ∈ f (y). Since y is a regular value, the derivative dfx maps TxX onto TyY . The kernel of dfx is a subspace K of TxX of dimension p := dim X − dim Y . Suppose X ⊂ RN and let T : RN → Rp be any linear map such that Ker(T ) ∩ K = {0}. Consider the map F : X → Y × Rp given by F (z) = (f(z),T (z)). The derivative of F is given by dFx(v) = (dfx(v),T (v)) 4 1. SMOOTH MANIFOLDS AND SMOOTH MAPS which is clearly nonsingular by our choice of T . By the inverse function theorem, F is a local diffeomorphism at x, i.e.
Load more

Recommended publications
  • LECTURE 6: FIBER BUNDLES in This Section We Will Introduce The
    LECTURE 6: FIBER BUNDLES In this section we will introduce the interesting class of fibrations given by fiber bundles. Fiber bundles play an important role in many geometric contexts. For example, the Grassmaniann varieties and certain fiber bundles associated to Stiefel varieties are central in the classification of vector bundles over (nice) spaces. The fact that fiber bundles are examples of Serre fibrations follows from Theorem ?? which states that being a Serre fibration is a local property. 1. Fiber bundles and principal bundles Definition 6.1. A fiber bundle with fiber F is a map p: E ! X with the following property: every ∼ −1 point x 2 X has a neighborhood U ⊆ X for which there is a homeomorphism φU : U × F = p (U) such that the following diagram commutes in which π1 : U × F ! U is the projection on the first factor: φ U × F U / p−1(U) ∼= π1 p * U t Remark 6.2. The projection X × F ! X is an example of a fiber bundle: it is called the trivial bundle over X with fiber F . By definition, a fiber bundle is a map which is `locally' homeomorphic to a trivial bundle. The homeomorphism φU in the definition is a local trivialization of the bundle, or a trivialization over U. Let us begin with an interesting subclass. A fiber bundle whose fiber F is a discrete space is (by definition) a covering projection (with fiber F ). For example, the exponential map R ! S1 is a covering projection with fiber Z. Suppose X is a space which is path-connected and locally simply connected (in fact, the weaker condition of being semi-locally simply connected would be enough for the following construction).
    [Show full text]
  • MTH 304: General Topology Semester 2, 2017-2018
    MTH 304: General Topology Semester 2, 2017-2018 Dr. Prahlad Vaidyanathan Contents I. Continuous Functions3 1. First Definitions................................3 2. Open Sets...................................4 3. Continuity by Open Sets...........................6 II. Topological Spaces8 1. Definition and Examples...........................8 2. Metric Spaces................................. 11 3. Basis for a topology.............................. 16 4. The Product Topology on X × Y ...................... 18 Q 5. The Product Topology on Xα ....................... 20 6. Closed Sets.................................. 22 7. Continuous Functions............................. 27 8. The Quotient Topology............................ 30 III.Properties of Topological Spaces 36 1. The Hausdorff property............................ 36 2. Connectedness................................. 37 3. Path Connectedness............................. 41 4. Local Connectedness............................. 44 5. Compactness................................. 46 6. Compact Subsets of Rn ............................ 50 7. Continuous Functions on Compact Sets................... 52 8. Compactness in Metric Spaces........................ 56 9. Local Compactness.............................. 59 IV.Separation Axioms 62 1. Regular Spaces................................ 62 2. Normal Spaces................................ 64 3. Tietze's extension Theorem......................... 67 4. Urysohn Metrization Theorem........................ 71 5. Imbedding of Manifolds..........................
    [Show full text]
  • Basics of the Differential Geometry of Surfaces
    Chapter 20 Basics of the Differential Geometry of Surfaces 20.1 Introduction The purpose of this chapter is to introduce the reader to some elementary concepts of the differential geometry of surfaces. Our goal is rather modest: We simply want to introduce the concepts needed to understand the notion of Gaussian curvature, mean curvature, principal curvatures, and geodesic lines. Almost all of the material presented in this chapter is based on lectures given by Eugenio Calabi in an upper undergraduate differential geometry course offered in the fall of 1994. Most of the topics covered in this course have been included, except a presentation of the global Gauss–Bonnet–Hopf theorem, some material on special coordinate systems, and Hilbert’s theorem on surfaces of constant negative curvature. What is a surface? A precise answer cannot really be given without introducing the concept of a manifold. An informal answer is to say that a surface is a set of points in R3 such that for every point p on the surface there is a small (perhaps very small) neighborhood U of p that is continuously deformable into a little flat open disk. Thus, a surface should really have some topology. Also,locally,unlessthe point p is “singular,” the surface looks like a plane. Properties of surfaces can be classified into local properties and global prop- erties.Intheolderliterature,thestudyoflocalpropertieswascalled geometry in the small,andthestudyofglobalpropertieswascalledgeometry in the large.Lo- cal properties are the properties that hold in a small neighborhood of a point on a surface. Curvature is a local property. Local properties canbestudiedmoreconve- niently by assuming that the surface is parametrized locally.
    [Show full text]
  • Chapter 1 I. Fibre Bundles
    Chapter 1 I. Fibre Bundles 1.1 Definitions Definition 1.1.1 Let X be a topological space and let U be an open cover of X.A { j}j∈J partition of unity relative to the cover Uj j∈J consists of a set of functions fj : X [0, 1] such that: { } → 1) f −1 (0, 1] U for all j J; j ⊂ j ∈ 2) f −1 (0, 1] is locally finite; { j }j∈J 3) f (x)=1 for all x X. j∈J j ∈ A Pnumerable cover of a topological space X is one which possesses a partition of unity. Theorem 1.1.2 Let X be Hausdorff. Then X is paracompact iff for every open cover of X there exists a partition of unity relative to . U U See MAT1300 notes for a proof. Definition 1.1.3 Let B be a topological space with chosen basepoint . A(locally trivial) fibre bundle over B consists of a map p : E B such that for all b B∗there exists an open neighbourhood U of b for which there is a homeomorphism→ φ : p−1(U)∈ p−1( ) U satisfying ′′ ′′ → ∗ × π φ = p U , where π denotes projection onto the second factor. If there is a numerable open cover◦ of B|by open sets with homeomorphisms as above then the bundle is said to be numerable. 1 If ξ is the bundle p : E B, then E and B are called respectively the total space, sometimes written E(ξ), and base space→ , sometimes written B(ξ), of ξ and F := p−1( ) is called the fibre of ξ.
    [Show full text]
  • Introduction to Commutative Algebra
    MATH 603: INTRODUCTION TO COMMUTATIVE ALGEBRA THOMAS J. HAINES 1. Lecture 1 1.1. What is this course about? The foundations of differential geometry (= study of manifolds) rely on analysis in several variables as \local machinery": many global theorems about manifolds are reduced down to statements about what hap- pens in a local neighborhood, and then anaylsis is brought in to solve the local problem. Analogously, algebraic geometry uses commutative algebraic as its \local ma- chinery". Our goal is to study commutative algebra and some topics in algebraic geometry in a parallel manner. For a (somewhat) complete list of topics we plan to cover, see the course syllabus on the course web-page. 1.2. References. See the course syllabus for a list of books you might want to consult. There is no required text, as these lecture notes should serve as a text. They will be written up \in real time" as the course progresses. Of course, I will be grateful if you point out any typos you find. In addition, each time you are the first person to point out any real mathematical inaccuracy (not a typo), I will pay you 10 dollars!! 1.3. Conventions. Unless otherwise indicated in specific instances, all rings in this course are commutative with identity element, denoted by 1 or sometimes by e. We will assume familiarity with the notions of homomorphism, ideal, kernels, quotients, modules, etc. (at least). We will use Zorn's lemma (which is equivalent to the axiom of choice): Let S; ≤ be any non-empty partially ordered set.
    [Show full text]
  • Local Structure of Principally Polarized Stable Lagrangian Fibrations 3
    LOCAL STRUCTURE OF PRINCIPALLY POLARIZED STABLE LAGRANGIAN FIBRATIONS JUN-MUK HWANG, KEIJI OGUISO Abstract. A holomorphic Lagrangian fibration is stable if the characteristic cycles of the singular fibers are of type Im, 1 ≤ m< ∞, or A∞. We will give a complete description of the local structure of a stable Lagrangian fibration when it is principally polarized. In particular, we give an explicit form of the period map of such a fibration and conversely, for a period map of the described type, we construct a principally polarized stable Lagrangian fibration with the given period map. This enables us to give a number of examples exhibiting interesting behavior of the characteristic cycles. 1. Introduction For a holomorphic symplectic manifold (M,ω), i.e., a 2n-dimensional complex manifold 0 2 with a holomorphic symplectic form ω H (M, ΩM ), a proper flat morphism f : M B over an n-dimensional complex manifold∈ B is called a (holomorphic) Lagrangian fibration→ if all smooth fibers are Lagrangian submanifolds of M. The discriminant D B, i.e., the set of critical values of f, is a hypersurface if it is non-empty. In [HO1], the⊂ structure of the singular fiber of f at a general point b D was studied. By introducing the notion of characteristic cycles, [HO1] shows that the∈ structure of such a singular fiber can be described in a manner completely parallel to Kodaira’s classification ([Kd], see also V. 7 in [BHPV]) of singular fibers of elliptic fibrations. Furthermore, to study the multiplicity of the singular fibers, [HO2] generalized the stable reduction theory of elliptic fibrations (cf.
    [Show full text]
  • In This Section We Define Precisely Matrix Lie Groups And
    2.8. LIE GROUPS AND LIE ALGEBRAS, THE “BABY CASE” 65 2.8 Lie Groups and Lie Algebras, the “Baby Case” In this section we define precisely matrix Lie groups and their Lie algebras One of the reasons that Lie groups are nice is that they have a differential structure, which means that the no- tion of tangent space makes sense at any point of the group. Furthermore, the tangent space at the identity happens to have some algebraic structure, that of a Lie algebra. Roughly, the tangent space at the identity provides a “lin- earization” of the Lie group, and it turns out that many properties of a Lie group are reflected in its Lie algebra. The challenge that we are facing is that a certain amount of basic differential geometry is required to define Lie groups and Lie algebras in full generality. 66 CHAPTER 2. BASICS OF CLASSICAL LIE GROUPS Fortunately, most of the Lie groups that we need to con- sider are subspaces of RN for some sufficiently large N. In fact, they are all isomorphic to subgroups of GL(N,R) for some suitable N,evenSE(n), which is isomorphic to a subgroup of SL(n +1). Such groups are called linear Lie groups (or matrix groups). Since the groups under consideration are subspaces of RN , we do not need immediately the definition of an abstract manifold. We just have to define embedded submanifolds (also called submanifolds) of RN (in the case of GL(n, R), N = n2). In general, the difficult part in proving that a subgroup of GL(n, R) is a Lie group is to prove that it is a manifold.
    [Show full text]
  • Topology: Notes and Problems
    TOPOLOGY: NOTES AND PROBLEMS Abstract. These are the notes prepared for the course MTH 304 to be offered to undergraduate students at IIT Kanpur. Contents 1. Topology of Metric Spaces 1 2. Topological Spaces 3 3. Basis for a Topology 4 4. Topology Generated by a Basis 4 4.1. Infinitude of Prime Numbers 6 5. Product Topology 6 6. Subspace Topology 7 7. Closed Sets, Hausdorff Spaces, and Closure of a Set 9 8. Continuous Functions 12 8.1. A Theorem of Volterra Vito 15 9. Homeomorphisms 16 10. Product, Box, and Uniform Topologies 18 11. Compact Spaces 21 12. Quotient Topology 23 13. Connected and Path-connected Spaces 27 14. Compactness Revisited 30 15. Countability Axioms 31 16. Separation Axioms 33 17. Tychonoff's Theorem 36 References 37 1. Topology of Metric Spaces A function d : X × X ! R+ is a metric if for any x; y; z 2 X; (1) d(x; y) = 0 iff x = y. (2) d(x; y) = d(y; x). (3) d(x; y) ≤ d(x; z) + d(z; y): We refer to (X; d) as a metric space. 2 Exercise 1.1 : Give five of your favourite metrics on R : Exercise 1.2 : Show that C[0; 1] is a metric space with metric d1(f; g) := kf − gk1: 1 2 TOPOLOGY: NOTES AND PROBLEMS An open ball in a metric space (X; d) is given by Bd(x; R) := fy 2 X : d(y; x) < Rg: Exercise 1.3 : Let (X; d) be your favourite metric (X; d). How does open ball in (X; d) look like ? Exercise 1.4 : Visualize the open ball B(f; R) in (C[0; 1]; d1); where f is the identity function.
    [Show full text]
  • Differential Geometry
    University of Cambridge Mathematics Tripos Part II Differential Geometry Lent, 2018 Lectures by M. Dafermos Notes by Qiangru Kuang Contents Contents 1 1 Smooth Manifolds 1 Smooth Manifolds 1.1 Definitions Definition (Smooth). Let 푈 ⊆ R푛 be an open set and 푓 ∶ 푈 → R푚. 푓 is said to be smooth if 휕|훼|푓 휕푥훼 exists for all multi-indices 훼. Notation. A multi-index 훼 = (훼1, … , 훼푛) is a tuple of non-negative integers |훼| = ∑푛 훼 and 푖=1 푖. Definition (Smooth). Let 푋 ⊆ R푛 be a subset. 푓 ∶ 푋 → R푚 is smooth if for all 푥 ∈ 푋, there exists 푈 ⊆ R푛, 푥 ∈ 푈 such that there exists smooth ̃ 푚 푓 ∶ 푈 → R extending 푓|푋∩푈. i.e. the following diagram commutes 푓 ̃ 푈 R푚 휄 푓 푈 ∩ 푋 Remark. 1. This is a local property. 2. 푋 is a topological space with subspace topology induced from R푛. 푓 is smooth implies that 푓 is continuous. Definition (Diffeomorphism). Let 푋 ⊆ R푛 and 푌 ⊆ R푚. A map 푓 ∶ 푋 → 푌 is a diffeomorphism if 푓 is smooth and bijective and its inverse 푓−1 ∶ 푌 → 푋 is also smooth. Remark. 1. Diffeomorphism implies homeomorphism (with repsect to the subspace topology). 2. Diffeomorphism is an equivalence relation. Definition (Manifold). A 푘-dimensional manifold is a set 푋 ⊆ R푁 such that for all 푥 ∈ 푋 there exists 푉 ⊆ 푋 open, 푥 ∈ 푉 such that 푉 is diffeomorphic to an open subset 푈 ⊆ R푘, i.e. 푋 is locally diffeomorphic to R푘. Remark. The diffeomorphism 휑 ∶ 푈 → 푉 is called a local parameterisation and its inverse 휑−1 ∶ 푉 → 푈 is called coordinate charts.
    [Show full text]
  • 1.4 Group Theory 2: Lie Algebras and Representations 1.4.1 Lie Algebras
    1.4 Group theory 2: Lie algebras and representations 1.4.1 Lie algebras: the compact case Finite groups, like the symmetry group of a cube, have a finite number of group elements and are characterised by the multiplication table of the group elements. For continuous groups, i.e., Lie groups, such a table is not meaningful. Instead one turns to the Lie algebra associated to the Lie group, which is a local property of the Lie group. Adding information about the global, or topological, properties of the Lie group manifold one has a full description of the group. The standard example is the Lie algebras so(3) and su(2) which are the same (i.e., isomorphic) while their groups SO(3) and SU(2) are locally the same but topologically di↵erent: SU(2) = S3 and SO(3) = RP 3 which is the set of all lines 3 through the center of the three-sphere S . This relation is written SO(3) = SU(2)/Z2. We will in general denote groups with upper case and Lie algebras with lower case letters. The Lie algebra can be extracted from a Lie group G by writing its elements g as the exponential of a linear combination of a new set of matrices, the so called generators T i. That is i i g G : g := ei↵ T ,i=1, 2,...,d, (1.13) 2 with the exponential of a matrix defined by its power series expansion. Here d is the dimension of the group or the Lie algebra, and one should note that the parameters, or ”angles”, ↵i are real numbers.
    [Show full text]
  • Metric Geometry and Geometric Group Theory Lior Silberman
    Metric Geometry and Geometric Group Theory Lior Silberman Contents Chapter 1. Introduction 5 1.1. Introduction – Geometric group theory 5 1.2. Additional examples 5 Part 1. Basic constructions 8 1.3. Quasi-isometries 9 1.4. Geodesics & Lengths of curves 10 Problem Set 1 13 1.5. Vector spaces 15 1.6. Manifolds 16 1.7. Groups and Cayley graphs 16 1.8. Ultralimits 17 Problem set 2 20 Part 2. Groups of polynomial growth 21 1.9. Group Theory 22 Problem Set 3 26 1.10. Volume growth in groups 28 1.11. Groups of polynomial growth: Algebra 28 1.12. Solvability of amenable linear groups in characteristic zero (following Shalom [11]) 30 1.13. Facts about Lie groups 30 1.14. Metric geometry – proof of theorem 94 30 Problem Set 4 34 Part 3. Hyperbolic groups 36 1.15. The hyperbolic plane 37 1.16. d-hyperbolic spaces 37 1.17. Problem set 5 40 Part 4. Random Groups 41 1.18. Models for random groups 42 1.19. Local-to-Global 42 1.20. Random Reduced Relators 47 Part 5. Fixed Point Properties 48 1.21. Introduction: Lipschitz Involutions and averaging 49 1.22. Expander graphs 51 Problem Set 54 3 Bibliography 55 Bibliography 56 4 CHAPTER 1 Introduction Lior Silberman, [email protected], http://www.math.ubc.ca/~lior Course website: http://www.math.ubc.ca/~lior/teaching/602D_F08/ Office: Math Building 229B Phone: 604-827-3031 Office Hours: By appointment 1.1. Introduction – Geometric group theory We study the connection between geometric and algebraic properties of groups and the spaces they act on.
    [Show full text]
  • Introduction to Commutative Algebra
    INTRODUCTION TO COMMUTATIVE ALGEBRA Mel Hochster Department of Mathematics, University of Michigan, Ann Arbor, MI 48109{1043, USA E-mail address: [email protected] INTRODUCTION TO COMMUTATIVE ALGEBRA Prof. Mel Hochster, instructor [email protected] (734)764{4924 (office1) These are lecture notes for Math 614, Fall 2020. The individual lectures are organized into sections. c Mel Hochster 2020 1However, I will not be using my office in Fall, 2020. Contents Chapter 1. Overview, notation, review of topology, categories and Spec 5 1. Lecture of August 31 5 1.1. Algebraic sets 8 2. Lecture of September 2 10 2.1. Filters and ultrafilters 11 2.2. Categories 11 2.3. Metric spaces. 14 3. Lecture of September 4 15 Chapter 2. Equivalence of categories, products and coproducts, free modules, semigroup rings, and localization 19 1. Lecture of September 9 19 2. Lecture of September 11 23 3. Lecture of September 14 26 4. Lecture of September 16 30 5. Lecture of September 18 32 Chapter 3. Embedding in products of domains, integral and module-finite extensions, height, and Krull dimension 37 1. Lecture of September 21 37 2. Lecture of September 23 43 3. Lecture of September 25 46 4. Lecture of September 28 50 5. Lecture of September 30 53 Chapter 4. Noether normalization, Hilbert's Nullstellensatz, and dimension in finitely generated algebras over a field 57 1. Lecture of October 2 57 1.1. Algebraic independence and transcendence bases 58 2. Lecture of October 5 62 3. Lecture of October 7 67 Chapter 5. Chain conditions on rings and modules 71 1.
    [Show full text]