Sergei Yakovenko's blog: on Math and Teaching

Saturday, February 11, 2017

Exam

Filed under: Calculus on manifolds course,lecture,problems & exercises — Sergei Yakovenko @ 5:22
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Problems for the take-home exam

Here is the file with the problems for the exam.

The rules are simple:

  1. The submission is due on the last day of the exam period (including the vacations) as per the FGS rules, that is, March 26, 2017.
  2. English is preferred over Hebrew, typeset solutions to the handwritten ones, although no punishment for the deviant behavior will ensue. Hardcopy solutions should be put into my mail box in the Zyskind building,  otherwise feel free.
  3. Nobody is perfect: if you believe you find an error and the problem as it is stated is wrong, don’t hesitate to write a talkback to this post. All bona fide errors will be corrected or the problem cancelled outright.
  4. I tried to make the exam as instructive as the course was. Most of the problems are things that I planned to include, but didn’t have time to. To simplify your life, they were split into what I believe are simple steps. Don’t hesitate to consult textbooks, but let me see that you indeed read and digested them. The presumptively harder items are marked by the asterisk.
  5. To get the perfect grade 100, you don’t have to submit solutions to all problems. The grade will be based on my purely subjective assessment of your exam and in any case will not be additive neither multiplicative.  Please be aware that writing patently stupid things may be more detrimental to the outcome than just skipping an item that you cannot cope with.
  6. I hope to post on this site the aggregated and slightly polished lecture notes in hope they might help you.
  7. I hope to be able to answer any questions you might have concerning the problems, better posted here than emailed to me. Moreover, I encourage open discussions here as long as they don’t result in posting complete solutions. Sometimes one stumbles over the most stupid things and needs to talk to other to overcome that. That’s fairly normal. To enter math formulas, you type the dollar sign $ immediately followed by the word “latex”, and after the blank space type in your formula. Don’t forget to close with another $.
  8. If you cannot meet the deadline for serious reasons, write me. Everything is negotiable.

Good luck and merry ט”ו בשבט!

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Monday, February 6, 2017

Lecture 14 (Feb 6, 2017)

Calculus on complex manifolds

If V is a complex vector space, then it is naturally also a real vector space (if you allow multiplication by complex numbers, then that by real numbers is automatically allowed). However, forgetting how to multiply by the imaginary unit results in the fact that the dimension \dim_{\mathbb R}V of the space over the real numbers is two times higher. If we regret our decision to forget the complex multiplication, we still can restore it by introducing the \mathbb R-linear operator J\colon V\to V such that J^2=-E, where E is the identity operator.

An even-dimensional real vector space with such an operator is called an almost complex space, and it obviously can be made into a complex vector space (over \mathbb C). However, if we consider an even-dimensional manifold M with the family of operators as above, it is somewhat less than a complex analytic manifold (a topological space equipped with an atlas of charts with biholomophic transition functions). For details, follow the lecture notes that will be available later.

Lecture 13 (Jan 30, 2017)

Symplectic manifolds

In parallel with the Riemannian manifolds equipped with a positive definite (symmetric) scalar product on each tangent space, it is interesting to consider manifolds equipped with an antisymmetric scalar product on each tangent space, i.e., with a differential 2-form \omega\in\Omega^2(M). This form is called a symplectic structure, if \mathrm d\omega=0 and an additional nondegeneracy condition is met.

It turns out that this structure naturally arises on the cotangent bundle M=T^*N of an arbitrary smooth manifold N. Moreover, this structure is intimately related with the mechanics of frictionless systems: the Hamiltonian differential equations can be naturally described by vector fields X which satisfy the Hamiltonian condition \mathrm i_X\omega=\mathrm d H, where H is a function (Hamiltonian, or full energy) on the symplectic manifold. Thus each Hamiltonian vector field is “encoded” by a single function, rather than by a tuple of functions. The commutator of Hamiltonian vector fields is again Hamiltonian: this is the invariant definition of the Poisson bracket.

There are two instant ramifications from this point. One can discuss integrability of the Hamiltonian vector fields. Another, less physically motivated direction is to study the symplectic geometry, first locally, then globally. It is a surprising twisted counterpart of the Riemannian geometry, which has no intrinsic curvature but nevertheless is very rich globally.

The lecture notes will be available later.

Lecture 12 (Jan 23, 2017)

Filed under: Calculus on manifolds course,lecture — Sergei Yakovenko @ 4:51
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Lie groups and Lie algebras

A Lie group is a smooth manifold with carries on it the structure of a group which is compatible with the smooth structure (i.e., the multiplication by an element of the group is a smooth self-map, necessarily a diffeomorphism, of the manifold).

This group structure means very high “homogeneity” of the manifold, in particular, existence of a flat connexion. On the other hand, there is a distinguished point on the manifold, corresponding to the group unit.

It turns out that the tangent space at the group unit is equipped with a natural operation, the antisymmetric bilinear bracket, closely related to the commutator of vector fields on the Lie group. This algebraic structure is called the Lie algebra, and it in a sense “encodes” the group structure.

The notes will be available later.

Monday, January 30, 2017

Lecture 11 (Jan 16, 2017)

Filed under: Calculus on manifolds course,lecture — Sergei Yakovenko @ 3:54
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Geodesics

Definitions of geodesic curves on a Riemmanian manifold. Differential equations of the second order. Local existence of solutions. Geodesic map. Geodesic spheres, orthogonality. Local minimality of geodesic curves. Metric and geodesic completeness of Riemannian manifolds.

Survey of adjacent areas. Behavior of the nearby geodesics. Jacobi field. On surfaces: conjugated points, the role of the Gauss curvature, global properties of manifolds with positive and negative curvature (comparison). Hyperbolic plane \mathbb H, geodesics on it. Realization of the hyperbolic geometry on a surface in \mathbb R^3_{++-}. Impossibility of global embedding of \mathbb H into \mathbb R^3_{+++}.

Wednesday, January 11, 2017

Lecture 10, Jan 9, 2017

Introduction to the Riemannian geometry

  1. Flat structure of the Euclidean space and coordinate-wise derivation of vector fields.
  2. Axiomatic definition of the covariant derivative and its role in defining the parallel transport along curves on manifolds. Connexion.
  3. Covariant derivative \overline\nabla\text{ on }\mathbb R^n and its properties (symmetry, flatness, compatibility with the scalar product).
  4. Smooth submanifolds of \mathbb R^n. The induced  Riemannian metric and connection. Gauss equation.
  5. Weingarten operator on hypersurfaces and its properties. Gauss map.
  6. Curvatures of normal 2-sections (the inverse radius of the osculating circles). Principal, Gauss and mean curvatures.
  7. Curvature tensor: a miracle of a 2-nd order differential operator that turned out to be a tensor (“0-th order” differential operator).
  8. Symmetries of the curvature and Ricci tensors.
  9. Uniqueness of the symmetric connexion compatible with a Riemannian metric. Intrinsic nature of the Gauss curvature.

The lecture notes are available here.

Lecture 9, Jan 2, 2017

Sundries

I briefly discussed the (simplicial) homology construction in application to smooth manifolds and described several pairings: de Rham pairing (integration) between homology and cohomology, intersection form between H_k(M^n,\mathbb Z) and H_{n-k}(M^n,\mathbb Z), the pairing H^k_\text{dR}(M^n,\mathbb R)\times H^{n-k}_\text{dR}(M^n,\mathbb R)\to\mathbb R, \quad (\alpha,\beta)\longmapsto \displaystyle \int_M \alpha\land\beta and the Poincare duality.

Then I mentioned without proofs several results stressing the role of smoothness, in particular, how different smooth structures can live on homeomorphic manifolds. The tale of planar curve eversion and sphere eversion was narrated. For the video of the sphere eversion go here.

In the second part of the lecture I discussed natural additional structures that can live on smooth manifolds, among them

  • Complex structure, almost complex structure,
  • Symplectic structure,
  • Parallel transport,
  • Riemannian metric,
  • Group structure.

Then we prepared the ground for the next lecture, discussing how examples of these structures naturally appear (e.g., on submanifolds of the Euclidean space, on quotient spaces, …)

There will be no notes for this lecture, because of its mostly belletristic style.

Wednesday, December 21, 2016

Lecture 7, Dec 19, 2016

Integration of differential forms and the general Stokes theorem

We defined integrals of differential k-forms over certain simple geometric objects (oriented cells, smooth images of an oriented cube [0,1]^k), and extended the notion of the integral to integer combinations of cells, finite sums \sigma=\sum c_i\sigma_i,\ c_i\in\mathbb Z, so that \langle \omega,\sigma\rangle=\displaystyle\int_\sigma \omega=\sum_i c_i\int_{\sigma_i}\omega=\sum c_i \langle \omega,\sigma_i\rangle. Such combinations are called k-chains and denoted C^k(M).

Then the notion of a boundary was introduced, first for the cube, then for cells and ultimately for all chains by linearity. The property \partial\partial\sigma=0 was derived from topological considerations.

The “alternative” external derivative D on the forms was introduced as the operation conjugate to \partial so that \langle D\omega,\sigma\rangle=\langle\omega, \partial \sigma\rangle for any chain \sigma with respect to the pairing \Omega^k(M)\times C^k(M)\to \mathbb R defined by the integration. A relatively simple straightforward computation shows that for a (k-1)-form \omega=f(x)\,\mathrm dx_2\land\cdots\land \mathrm dx_n we have
D\omega=\displaystyle\frac{\partial f}{\partial x_1}\,\mathrm d x_1\land \cdots\land \mathrm dx_n, that is, D\omega=\mathrm d\omega. It follows than that D=\mathrm d on all forms, and hence we have the Stokes theorem \langle \mathrm d \omega, \sigma\rangle=\langle \omega,\partial\sigma\rangle.

Physical illustration for the Stokes theorem was given in \mathbb R^3 for the differential 1-form which is the work of the force vector field and for the 2-form of the flow of this vector field.

The class concluded by discussion of the global difference between closed and exact forms on manifolds as dual to that between cycles (chains without boundary) and exact boundaries and the Poincare lemma was proved for chains in star-shaped subdomains of \mathbb R^n.

There will be no lecture notes for this lecture, since the ideal exposition (which I tried to follow as close as possible) is in the book by V. I. Arnold, Mathematical methods of classical mechanics (2nd edition), Chapter 7, sections 35 and 36.

Monday, December 12, 2016

Lecture 6, December 12, 2016

Exterior derivation

The differential \mathrm df of a smooth function f is in a sense container which conceals all directional derivatives L_Xf=\left\langle\mathrm df,X\right\rangle along all directions, and dependence on X is linear.

If we consider the directional Lie derivative L_X\omega for a form \omega\in\Omega^k(M) of degree k\ge 1, then simple computations show that L_{fX}\omega is no longer equal to f\cdot L_X\omega. However, one can “correct” the Lie derivative in such a way that the result will depend on X linearly. For instance, if \omega\in\Omega^1(M) and X is a vector field, we can define the form \eta_X\in\Omega^1(M) by the identity \eta_X=L_X\omega-\mathrm d\left\langle\omega,X\right\rangle and show that the 2-form \eta(X,Y)=\left\langle\eta_X,Y\right\rangle is indeed bilinear antisymmetric.

The 2-form \eta is called the exterior derivative of \omega and denoted \mathrm d\omega\in\Omega^2(M). The correspondence \mathrm d\colon\Omega^1(M)\to\Omega^2(M) is an \mathbb R-linear operator which satisfies the Leibniz rule \mathrm d(f\omega)=f\,\mathrm d\omega+(\mathrm df)\land \omega and \mathrm d^2 f=0 for any function f\in\Omega^0(M).

It turns out that this exterior derivation can be extended to all k-forms preserving the above properties and is a nice (algebraically) derivation of the graded exterior algebra \Omega^\bullet(M)=\bigoplus_{k=0}^n\Omega^k(M).

The lecture notes are available here.

Monday, December 5, 2016

Lecture 5, Dec 5, 2016

Multilinear antisymmetric forms and differential forms on manifolds

We discussed the module of differential 1-forms dual to the module of smooth vector fields on a manifold. Differential 1-forms are generated by differentials of smooth functions and as such can be pulled back by smooth maps.

The “raison d’être” of differential 1-forms is to be integrated over smooth curves in the manifold, the result being dependent only on the orientation of the curve and not on its specific parametrization.

At the second hour we discussed the notion of forms of higher degree, which required to introduce the Grassman algebra on the dual space T^* to an abstract finite-dimensional linear space T\simeq\mathbb R^n. The Grassmann (exterior) algebra is a mathematical miracle that was discovered by a quest for unusual and unknown, with only slight “motivations” from outside.

The day ended up with the definition of the differential k-forms and their functoriality (i.e., in what direction and how they are carried by smooth maps between manifolds).

The lecture notes are available here.

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