# Sergei Yakovenko's blog: on Math and Teaching

## 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!
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* 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
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## 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
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## 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.

## 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.

# Analgebraic Geometry

It so happened that at the beginning of 2016 I gave a talk on the conference “Geometric aspects of modern dynamics” in Porto, delivered a minicourse at Journées Louis Antoine in Rennes and wrote an expository paper for the European Mathematical Society Newsletter, all devoted to the same subject. The subject, provisionally dubbed as “Analgebraic geometry”, deals with algebraic-like properties (especially from the point of view of intersection theory) of real and complex analytic varieties defined by ordinary and Pfaffian differential equations with polynomial right hand sides. Thus

analgebraic = un-algebraic + analytic + algebraic (background) + weak algebraicity-like properties.

It turns out that this analgebraic geometry has very intimate connections with classical problems like Hilbert 16th problem, properties of periods of algebraic varieties, analytic number theory and arithmetic geometry.

For more details see the presentation prepared for the minicourse (or the shorter version of the talk) and the draft of the paper.

Any remarks and comments will be highly appreciated.

## Monday, February 1, 2016

### Finally, exam!

Filed under: lecture,Rothschild course "Analysis for high school teachers" — Sergei Yakovenko @ 3:41
Tags:

# Exam

The exam is posted online on Feb 1, 2016, and must be submitted on the last day of the exams’ period, February 26. Its goals are, besides testing your acquired skills in the Analysis, to teach you a few extra things and see your ability for logical reasoning, not your proficiency in performing long computations. If you find yourself involved in heavy computations, better double check whether you understand the formulation of the problem correctly. Remember, small details sometimes matter!

Please provide argumentation, better in the form of logical formulas, not forgetting explicit or implicit quantifiers $\forall$ and $\exists$. They really may change the meaning of what you write!

Problems are often subdivided into items. The order of these items is not accidental, try to solve them from the first till the last, and not in a random order (solution of one item may be a building block for the next one).

To get the maximal grade, it is not necessary to solve all problems, but it is imperative not to write stupid things. Please don’t try to shoot in the air.

The English version is the authoritative source, but if somebody translates it into Hebrew (for the sake of the rest of you) and send me the translation, I will post it for your convenience, but responsibility will be largely with the translator.

If you believe you found an error or crucial omission in the formulation of a problem, please write me. If this will be indeed the case (errare humanum est), the problem will be either edited (in case of minor omissions) or cancelled (on my account).

That’s all, folks!© Good luck to everybody!

Yes, and feel free to leave your questions/talkbacks here, whether addressed to Michal/Boaz/me or to yourself, if you feel you want to ask a relevant question.

# Corrections

## Correction 1

The formulation of Problem 1 was indeed incorrect. The set $A'$ was intended to be the set of accumulation points for a set $A\subseteq [0,1]$. The formal definition is as follows.

Definition. A point $p\in [0,1]$ belongs to to the set of limit points $A'$ if and only if $\forall\varepsilon$>0 the intersection $(p-\varepsilon,p+\varepsilon)\cap A$ is infinite. The point $p$ itself may be or may be not in $A$.

Isolated points of $A$ are never in $A'$, but $A'$ may contain points $p\notin A$.

Apologies for the hasty formulation.

## Correction 2: Problem 3(b) cancelled!

The statement requested to prove in Problem 3(b) is wrong, and I am impressed how fast did you discover that. Actually, the problem was taken from the textbook by Zorich, vol. 1, where it appears on p. 169, sec. 4.2.3, as Problem 4.

The assertion about existence of the common fixed point of two commuting continuous functions $f,g\colon [0,1]\to[0,1]$ becomes true if we require these functions to be continuously differentiable on $[0,1]$ (in particular, for polynomials), but the proof of this fact is too difficult to be suggested as a problem for the exam.

Thus Problem 3(b) is cancelled.

## Tuesday, January 26, 2016

### Lecture 13, Jan 26, 2016

Questions concerned integrability of discontinuous functions, notions of improper integrals (how and when they can be defined), topological properties (equivalent definitions of compactness, connectedness etc.)

Here are some textbooks that I recommend for preparing when working on the exam. Keep them on your virtual bookshelf: they cover much more that I explained in the course, but who knows what questions related to analysis you might have.

1. V. Zorich, vol. 1: Chapters 1-6, pp.1-371.
2. V. Zorich, vol. 2: Parts of Chapter 9 (continuous maps) and the first part of Chapter 18 on Fourier series.
3. W. Rudin: Chapters 1-6, pp.1-165.

The problems for exam will be posted on February 1st (at least the English version).

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