Calendar for 2019 - 2020 Spring Semester MATH 3980T Section 103 (Class Number 14592), taught by Mark Barsamian

• Week 1: Mon Jan 13 - Fri Jan 17 Show/Hide Week 1

  Reading

  • Boothby Chapter I

  Background Terms:

  These come up in Chapter I and are not defined in Chapter I. If you know what these terms mean, explain them to me in a tutorial meeting. If you don�t know what some of them mean, flag them as important terms. You�ll learn about them in your Topology tutorial. (Explain them to me then.)

  • The natural topology on a metric space (page 1)
  • Vector Space (page 2)
  • Inner Product Space (page 2)
  • Normed Vector Space (page 3)
  • The sequence of inclusions:
  • Inner Product Spaces \( \subset \) Normed Vector Spaces \( \subset \) Metric Spaces \( \subset \) Topological Spaces
  • Can you give examples of spaces that are in each of these but not in the one to its left?
  • Hausdorff Space (page 6)
  • Basis for a topology (page 6)
  • Subspace topology (page 6)
  • Compact set (page 7)
  • Locally connected (page 8)
  • Locally compact (page 8)
  • Normal Space (page 8)
  • Metrizable Space (page 8)

  Discuss in Tutorial

  • Some of the Background terms from list above
  • Subspace topology (page 6)
  • Clarify Process of identifying sides of a rectangle, yielding cylinder, mobius band, torus, Klein bottle
  • Discuss issue of embedding these spaces in \( \mathbb{R}^3 \) (discussed on page 12). What goes wrong with Klein?
  • Discussion of Theorem 4.1 (page 12) mentions theorem classifying compact, orientable surfaces (2-manifolds), citing proof in Massey. (Mark B: remember to bring in clearer proof.)
  • Study of 3-manifolds, including classification of them, is a rich area of current research.

  Homework 1

  from Boothby (Think about all of these. Pick at least 5 to write up and turn in in Week 2.)

  • 1.1 # 1, 2, 4, 6
  • 1.3 # 1, 3, 4
  • 1.4 # 3, 4, 5
  • 1.5 # 1, 2, 3, 5, 6 (Ha Ha)
  Hide Week 1

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• Weeks 2 & 3: Mon Jan 20 - Fri Jan 31 Show/Hide Week 2

  Reading: Boothby Chapter II Sections 1 - 5

  Important Concepts:

  These come up in Ch 2. (If you have seen any of them before, explain what you know.)

  • The classes of functions \( C^k, C^\infty, C^\omega \)
  • The set \( C^\infty(a) \), the germs of functions at \( a \).
  • The distinction between linear maps from \( C^\infty(a) \) to \( C^\infty(a) \) and derivations from \( C^\infty(a) \)to \( C^\infty(a) \).

  Discuss in Tutorial

  • A vector space is a set and a field of scalars, with operation of addition of elements of the set and scalar multiplication of elements of the set by the scalars of the field.
  • For maps from one vector space to another, the only issue is whether the maps preserve the vector space structures of addtion and scalar multiplication. that is, does \( h(ax+by)=ah(x)+bh(y) \)? Maps that do preserve these structures are called linear .
  • An algebra is a vector space with the extra operation of multiplication \( * \) of elements of the set by each other such that
    • multiplication is associative: \( x*(y*z)=(x*y)*z \).
    • multiplication is distributive: \( x*(y+z)=x*y+x*z \).
    • scalar multiplication in the algebra is compatible with multiplication in the field in the following way: \( c(x*y)=(cx)*y=x*(cy) \).
  • For maps from one algebra to another, there is of course the issue of whether or not the algebra respects the vector space structure, that is, whether or not the map is linear. But there is also the issue of how the map behaves with respect to the multiplicative structure of the algebra.
    • maps that are multiplicative behave this way: \( h(x*y)=h(x)*h(y) \).
    • maps that are derivations behave this way: \( h(x*y)=h(x)*y+x*h(y) \)
  • Example: The set \( C^\infty(U) \) of differentiable functions has function addition and function multiplication. It is an algebra.
    • On this algebra, the evaluation map, \( E_a \), defined by \( E_a(f)=f(a) \), where \( f(a) \) denotes the constant function, is a linear map that is multiplicative. That is, \(E_a(f*g)=f(a)g(a)=E_a(f)*E_a (g)\).
    • But on this same algebra, the derivative map, \( D \), defined by \( D(f)=\frac{df(x)}{dx} \) is a linear map that is a derivation. That is, \( \frac{d}{dx}(fg)=\frac{df}{dx}*g+f*\frac{dg}{dx} \), the common product rule for derivatives.

  Homework 2 from Boothby Chapter 2 (Turn in on Wed Feb 5)

  • 2.1 # 4, 5, 6
  • 2.3 # 4
  • 2.4 # 3, 5
  Hide Week 2

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• Weeks 4 & 5: Mon Feb 3 - Fri Feb 14 Show/Hide Weeks 4 & 5

  Reading:

  • Boothby Chapter II Sections 6, 7 (don't get bogged down in these)
  • Boothby Chapter III Sections 1 - 5

  Important Concepts:

  • Revisit a couple of manifolds that you saw before ( \( S^2 \), for example) but with Boothby�s precision.
  • More sophisticated examples of manifolds not presented as subspaces of R^3.
  • Manifolds that can be presented as quotient spaces
    • The Projective Space \( P^n(\mathbb{R}) \)
    • The Grassman manifolds \( G(k,n) \)
  • Immersions, Embeddings, Submanifolds

  Discuss in Tutorial:

  • Tuesday, discuss quotient spaces (3.1 � 3.3 topics)
  • Thursday, discuss immersions, embeddings, submanifolds (3.4, 3.5 topics)

  Homework 3: from Boothby Chapter 3 (Turn in on Thu Feb 13)

  • 3.1 # 2
  • 3.2 # 1, 2, 3, 4, 5

  Exam 2 (Assigned Thu Feb 13Turn in on Fri Feb 14)

  • Boothby 3.1 # 5
  • (See definition (2.2) on page 60 of open equivalence relation .) Give an example of a topological space \( X \) and an equivalence relation \( ~ \) on \( X \) such that \( ~ \) is NOT open.
  • 3.3 # 5
  Hide Weeks 4 & 5

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• Week 6: Mon Feb 17 - Fri Feb 21 Show/Hide Week 6

  Reading: from Boothby Chapter III Differentiable Manifolds and Submanifolds

  • Section III.4 Rank of a mapping, Immersions
  • Section III.5 Submanifolds

  Discuss in Tutorial

  • immersions, embeddings, submanifolds (III.4, III.5 topics)

  Homework 4: from Boothby Chapter III (Turn in on Wed Feb 19)

  • III.4 # 3, 7
  • III.5 # 2, 6, 7
  Hide Week 6

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• Week 7: Mon Feb 24 - Fri Feb 28 Show/Hide Week 7

  Reading: from Boothby Chapter III Differentiable Manifolds and Submanifolds

  • Section III.6 Lie Groups
  • Section III.7 The Action of a Lie Group on a Manifold

  Homework 5 from Boothby Chapter III (Turn in on Wed Feb 26)

  • III.6 # 1, 6, 9
  • III.7 # 4, 5, 7, 9
  Hide Week 7

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• Week 8: Mon Mar 2 - Fri Mar 6 Show/Hide Week 8

  Reading: from Boothby Chapter IV Vector Fields on a Manifold

  • Section IV.1 The Tangent Space at a Point of a Manifold
  • Section IV.2 Vector Fields

  Homework 6: from Boothby Chapter IV (Turn in on Wed Mar 4)

  • IV.1 # 2, 5
  • IV.2 # 2, 5, 6, 7, 8, 10
  Hide Week 8

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• Weeks 9,10: Mon Mar 9 - Fri March 20: Spring Break; no meetings
• Weeks 11,12,13: Mon Mar 23 - Fri Apr 10 Show/Hide Weeks 11,12,13

  Reading: from Boothby Chapter IV Vector Fields on a Manifold

  • Section IV.3 One-Parameter and Local One-Parameter Groups Acting on a Manifold
  • Section IV.4 The Existence Theorem for Ordinary Differential Equations
  • Section IV.5 Some Examples of One-Parameter Groups

  Homework 7: from Boothby Chapter IV (Turn in on Thu Apr 9)

  • IV.3 # 2, 5
  • IV.4 # 3, 4, 5
  • IV.5
    • Show that the maps \(F_1, F_2\) in Example 5.10 and map \(F\) in Examples 5.11 are in fact homomorphisms.
    • IV.5 # 3, 4, 6
  Hide Week 11

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• Week 14: Mon Apr 13 - Fri Apr 17 Show/Hide Week 14

  Reading: from Boothby Chapter IV Vector Fields on a Manifold

  • Section IV.6 One Parameter Subgroups of Lie Groups
  • Section IV.7 The Lie Algebra of Vector Fields on a Manifold

  Homework 8

  • IV.6 # 1, 2, 6 (Only do these three if you have not previously done them in some other course.)
  • IV.6 # 3,4,7
  • IV.7 # 2, 3, 4, 6
  • IV.7 # 7 or {8,9}
  Show/Hide Week 14

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• Week 15: Mon Apr 20 - Fri Apr 24 Show/Hide Week 15

  Reading: from Boothby Chapter V Tensors and Vector Fields on Manifolds

  • V.1 Tangent Covectors
  • V.2 Bilinear Forms. The Riemannian Metric
  • V.3 Riemannian Manifolds as Metric Spaces

  Homework 9

  • V.1 # 2, 3, 5, 7
  • V.2 # 2, 4, 9
  • V.3 # 1, 2, 3, 4, 7
  Show/Hide Week 15

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• Summer: Mon May 25 meeting Show/Hide May 25 meeting

  Reading: from Boothby Chapter V, Tensors and Vector Fields on Manifolds , read

  • Section V.4 Partitions of Unity

  Homework

  • Old stuff:
    • Find the mistake in Remark (5.6) on page 77 in Section III.5.
    • What are the dimensions of \( Gl(n,\mathbb{R}),Sl(n,\mathbb{R}),O(n) \) as manifolds ? Explain why the numbers make sense.
    • Review Gauss Map from Pressley .
  • Boothby V.4
    • Exercises # 1, 2, 3, 4
    • Section V.4 starts with a remark about non-existence of a non-vanishing \( C^\infty \) vector field on \( S^2 \), and about the associated fact of non-existence of a non-vanishing covector field on \( S^2 \) as well. But by Theorem (4.5) , we see that there is a \( C^\infty \) Riemannian Metric on \( S^2 \). The proof involves the pullback \( \phi^*\psi \) of the usual inner product \( \psi \) on \( \mathbb{R}^2 \) using the coordinate maps \( \phi_i \) of a regular covering \( \{U_i,V_i,\phi_i\} \) and an associated partition of unity \( \{f_i\} \). What happens if an analogous sort of scheme is attempted to construct a non-vanishing vector field on \( S^2 \)? We know something must fail, but what? Think about such a scheme.
    • We have studied two surfaces that have immersions in \( \mathbb{R}^3 \):
      • 2-dimensional real projective space, \(P^2(\mathbb{R})\)
      • The Klein Bottle
      The Whitney imbedding Theorem ( Theorem (4.7) and remarks following) tells us that these can be imbedded in \( \mathbb{R}^4 \). Of course the theorem statement does not give any indication of how to find the imbedding. See if you can find imbeddings for these on the web (probably spelled embedding ) and present them.
  Show/Hide May 25 meeting

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• Summer: Thu May 28 meeting Show/Hide May 28 meeting

  Reading: from Boothby Chapter V, Tensors and Vector Fields on Manifolds , read

  • Section V.5 Tensor Fields
  • Section V.6 Multiplication of Tensors

  Homework

  • Boothby V.5 Exercises 1, 2, 3, 4, 5, 6
  • Boothby V.6 Exercises 1, 2, 3, 4, 5, 6, 7
  Show/Hide May 28 meeting

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