Sergei Yakovenko's blog: on Math and Teaching

De Rham and Cech cohomology of smooth manifolds

Using the exterior differential $d$ on smooth differential forms and the fact that $d^2=0$, we define the de Rham cohomology with real coefficients $H^k_{\mathrm dR}(M,\mathbb R)$ as the quotient space of closed $k$-forms by exact $k$-forms. This is a global invariant of a manifold $M$ (for non-compact manifolds we may also consider a version for compactly supported forms, which yields different results).

De Rham cohomology can be computed using the Poincare lemma. If $\mathfrak U=\{U_i\}$ is an open covering of $M$ such that all opens sets and all their non-empty finite intersections are topologically trivial (homeomorphic to open balls), then for any closed form $\omega\in\Omega^k(M)$ one can construct its primitives $\xi_i\in\Omega^{k-1}(U_i)$ such that $\mathrm d\xi_i=\omega$ in $U_i$. The $(k-1)$th forms $\xi_i$ may disagree on the intersections $U_{ij}=U_i\cap U_j$, but one can attempt to twist them by suitable closed forms $\mathrm d\phi_i$. The corresponding system of $(k-2)$-forms $\{\phi_i\}$ satisfies certain linear conditions on pairwise intersections; to satisfy these conditions one has to look for forms on triple intersections etc.

This construction gives rise to the notion of the Cech cohomology defined via systems of linear algebraic equations and reduces computation of the de Rham cohomology to a problem from linear algebra, determined by the combinatorics of the pairwise and multiple intersections of the sets $U_i$. In particular, one can conclude that the de Rham cohomology of compact manifolds is finite-dimensional.

An ultra-concise set of notes is available here (I hope to return and expand this text). The notion of Cech cohomology is further elaborated here.

NB. The class was shorter than usual because of the Hanukka lighting ceremony.

חנוכה שמח and Happy New Year, С наступающим Новым годом!

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.

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.

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.

Objects that live on manifolds: functions, curves, vector fields

We discussed how one may possibly define smooth functions on manifolds, smooth curves, tangent vectors, smooth vector fields. Next we discussed how these objects can be carried between manifolds if there exists a smooth map (or diffeomorphism) between these manifolds.

Flow of vector field. Lie derivatives.

Every vector field $X$ on a compact smooth manifold $M$ defines a family of automorphisms $F^t_X$ (diffeomorphic self-maps) of $M$ which form a one-parametric group, called the flow. Any object living on $M$ can be carried by the flow by the operators $\bigl(F^t_X\bigr)^*$, $t\in\mathbb R$. The Lie derivative along $X$ is the velocity of this action at $t=0$, namely, $L_X=\frac{\mathrm d}{\mathrm dt}\big|_{t=0}\bigl(F^t_X\bigr)^*$.

We show that the Lie derivative of functions coincides with the action of the corresponding derivations, and the Lie derivation of another vector field is the Lie bracket $L_XY=[X,Y]$.

At the end of the day we establish the identities $[L_X,L_Y]=L_{[X,Y]}$ and the Leibniz rule for $L_X$ with respect to the Lie bracket, $L_X[Y,Z]=[Y,L_XZ]+[L_XY,Z]$. Both turn out to be equivalent to the Jacobi identity $[X,[Y,Z]]+[Y,[X,Z]]+[Z,[X,Y]]=0$ for the Lie bracket.

The lecture notes are available here.

In addition to previously mentioned books, you may like the book I. Kolár, P. Michor, J. Slovák, Natural Operations in Differential Geometry, freely available from the Web.

Besides, I mentioned that the Jacobi identity has many different faces. One of them, discovered by V. Arnold, can be stated as follows: the three altitudes of a triangle intersect at one point because of the Jacobi identity*. You can find the explanations here and here. Enjoy!
________________________________________
* In fact, it is a slightly different Jacobi identity, not for the Lie bracket of vector fields, but for the vector product $\mathbb R^3\times\mathbb R^3\mapsto\mathbb R^3$, $u,v\mapsto [u,v]=u\times v$. But later we will see that this vector product is the commutator in the Lie algebra of vector fields on the group of orthogonal transformations of $\mathbb R^3$, thus the difference is purely technical.

Monday, November 21, 2016

Lecture 3, Nov 21, 2016

Filed under: Calculus on manifolds course,links — Sergei Yakovenko @ 4:51
Tags: ,

Concept of Manifold

The entire lecture was devoted to motivation and examples of $C^\infty$-smooth manifolds (submanifolds of $\mathbb R^n$, spheres, tori, projective spaces, matrix groups etc).

Slightly more detailed plan of the lecture is here.

If you want to read more (which is most highly welcome), here are a few recommendations:

A few thoughts on how to use these books. The subject (calculus on manifolds) is difficult because it involves both complicated concepts and the new language describing these concepts, and there is no way to learn these things but in parallel. One possibility to practice in the new language is to read as many texts about familiar subjects, as possible. This is what I suggest: if you believe you understand certain things, try to read about them in different books and make sure that different notation adopted by different authors does not detract you from the core.

A little bit more specific note. A closely related beautiful subject, Algebraic Geometry, was born from studies of how subsets of real or complex Euclidean space may look like. For some time it developed using mostly geometric/analytic tools, but eventually it was realized that to avoid problems with singularities, “double points”, “points at infinity” etc., one should start with the algebra of polynomials in one and several variables, its ideals, the quotient algebras and schemes in general. This approach brought tremendous achievements.

In my attempt to present the basic constructions of Calculus on Manifolds and, more generally, Differential Geometry, I decided to make the first several steps in a similar spirit and build objects from the algebra of $C^\infty$-smooth functions on a manifold. Of course, these algebras are very different from the algebras of polynomials (in particular, they are not Noetherian), which makes life some times easier, some times more difficult.

See you in a week.

Tuesday, November 15, 2016

Lecture 2 (Nov. 14, 2016).

Filed under: Calculus on manifolds course — Sergei Yakovenko @ 5:07
Tags: , ,

Tangent vectors, vector fields, integration and derivations

Continued discussion of calculus in domains lf $\mathbb R^n$.

• Tangent vector: vector attached to a point, formally a pair $(a,v):\ a\in U\subseteq\mathbb R^n, \ v\in\mathbb R^n$. Tangent space $T_a U=\ \{a\}\times\mathbb R^n$.
• Differential of a smooth map $F: U\to V$ at a point $a\in U$: the linear map from $T_a U$ to $T_b V,\ b=F(a)$.
• Vector field: a smooth map $v(\cdot): a\mapsto v(a)\in T_a U$.  Vector fields as a module $\mathscr X(U)$ over $C^\infty(U)$.
• Special features of $\mathbb R^1\simeq\mathbb R_{\text{field}}$. Special role of functions as maps $f:\ U\to \mathbb R_{\text{field}}$ and curves as maps $\gamma: \mathbb R_{\text{field}}\to U$.
• Integral curves and derivations.
• Algebra of smooth functions $C^\infty(U)$. Contravariant functor $F \mapsto F^*$ which associates with each smooth map $F:U\to V$ a homomorphism of algebras $F^*:C^\infty(V)\to C^\infty(V)$. Composition of maps vs. composition of morphisms.
• Derivation: a $\mathbb R$-linear map $L:C^\infty(U)\to C^\infty(U)$ which satisfies the Leibniz rule $L(fg)=f\cdot Lg+g\cdot Lf$.
• Vector fields as derivations, $v\simeq L_v$. Action of diffeomorphisms on vector fields (push-forward $F_*$).
• Flow map of a vector field: a smooth map $F: \mathbb R\times U\to U$ (caveat: may be undefined for some combinations unless certain precautions are met) such that each curve
$\gamma_a=F|_{\mathbb R\times \{a\}}$ is an integral curve of $v$ at each point $a$. The “deterministic law” $F^t\circ F^s=F^{t+s}\ \forall t,s\in\mathbb R$.
•  One-parametric (commutative) group of self-homomorphisms $A^t=(F^t)^*: C^\infty(U)\to C^\infty(U)$. Consistency: $L=\left.\frac{\mathrm d}{\mathrm dt}\right|_{t=0}A^t=\lim_{t\to 0}\frac{A^t-\mathrm{id}}t$ is a derivation (satisfies the Leibniz rule). If $A^t=(F^t)^*$ is associated with the flow map of a vector field $v$, then $L=L_v$.

Update The corrected and amended notes for the first two lectures can be found here. This file replaces the previous version.

Crash course on linear algebra and multivariate calculus

Real numbers as complete ordered field. Finite dimensional linear spaces over $\mathbb R$. Linear maps. Linear functionals, the dual space. Linear operators (self-maps of linear space), invertibility via determinant. Affine maps, affine spaces.

Polynomial nonlinear maps and functions, re-expansion as a tool to construct linear (affine) approximation. Differential. Differentiability of maps, smoothness of functions.

Inverse function theorem.

Vector fields, parameterized curves, differential equations.

The first set of notes is available here here.

Calculus on manifolds: new course announcement

Filed under: Calculus on manifolds course — Sergei Yakovenko @ 4:19

‘שלום כיתה א

This academic year (תשע”ז) after a long pause I will “again” teach “this” course (which previously was delivered under the name of “Differential Geometry”). This time I decided to give it the name more appropriate for the content.

Each week some 90+ minutes of lecture will take place in Room 155, Zyskind Building every Monday 10:15–12:00, starting from November 7, 2016 and the last lecture scheduled for February 6, 2017. Then there will be a take-home exam with about a month to submit solutions.

I will post the lecture notes on this blog: to simplify search, look at the respective category. Usually notes will be prepared before the talk, but to keep records straight I will also briefly summarize which parts I succeeded to cover in the real 90+ minutes of time.

Previous courses’ notes, some reading recommendations and other relevant information are available from the stationary repository.

You are most welcome to ask questions, answer them, report grave errors in the notes and interact with this blog in any other way. Please, no likes! 😉

Monday, March 7, 2016

Wonderful interview with Sir Michael Atiyah

Filed under: links — Sergei Yakovenko @ 4:48
Tags: ,

One of the most exquisite minds of our times shares his insights of how a Mathematician perceives the worlds of Mathematics and Physics.

A must read for all ages and all specializations, from students to retirees.

« Previous PageNext Page »

Create a free website or blog at WordPress.com.