Annual Foundation School - II (2017) - Kozhikode

Speakers and Syllabus 

I gave five lectures covering the tenth chapter of Artin's Algebra (first edition). I didn't have time to start the next chapter. Following is a brief description of the material covered:

Rings, homomorphisms, ideals, quotient rings by ideals, adjunction of elements, integral domains, prime
and maximal ideals, Hilbert's Nullstellensatz (two versions below and their equivalence)

(i) maximal ideals in C[x_1,...,x_n] are in bijective correspondence with C^n; and
(ii) for an ideal I in C[x_1,...,x_n], I(V(I)) is equal to the radical of I.

Finally, I gave a brief introduction to algebraic geometry by defining algebraic varieties in C^n, Zariski topology, and giving some examples of curves in C^2.

I focused on giving many examples and describing the results in concrete situations. I think most students have become comfortable working with subrings of C and polynomial rings, which were the main focus of my lectures. During the tutorials, we solved several exercises from Artin and other places.

I was quite happy with the enthusiasm and hard work of the students. They worked diligently even outside the class and tutorial times and I enjoyed sitting with them to discuss, mainly after dinners.

Here is a brief account of what was taught in my course. Motivation from number theory, Fundamental Theorem of Arithmetic (Division algorithm) in the ring of integers and polynomial rings over a field and the Gaussian integers, pointing out, division algorithm by a monic polynomial holds for polynomials over any commutative ring with identity. Euclidean domains are PID’s and PID’s are UFD’s. Gauss Lemma. A necessary condition for the existence of a Euclidean function in terms of universal side divisors, the Dedekind Hasse function, existence of Dedekind-Hasse function as a necessary and sufficient condition for an integral domain to be a PID. Historical introduction to algebraic number theory, the theorem of Diophantus and Fermat’s last theorem. Definition of algebraic numbers and integers, the ring of integers of a quadratic field. Unique factorization of ideals in the ring of integers of a number field, the definition of Dedekind domains, statement that for a Dedekind domain, UFD is equivalent to PID. Proof of this last statement was given for the ring of integers of an imaginary quadratic field. Definition of Ideal Class group and its significance, Class number one is equivalent to PID and hence equivalent to UFD. An outline of the proof through guided exercises, that the norm function is Dedekind-Hasse function on the ring of integers of Q( sqrt ( − 19)) and hence it is a PID. But there does not exist any Euclidean function on this ring, thus providing an example that PID does not imply Euclidean. List of nine values of d for which the ring of integers of Q( √ d) is PID and out of which the first five are Euclidean and there can be no Euclidean function on the remaining four. Units in imaginary quadratic fields contrast with real quadratic fields, by examples; relation and history of Brahma Gupta- Pell equation. Eisenstein’s irreducibility criterion and application to cyclotomic polynomials were also done. Tutorial sheets were distributed and some exercises were solved completely some were partially done with enough hints. Students participation was good. Except one of the tutorial sessions, attendance was nearly hundred percent. A few were very good, some average and some not up to the expectation. Overall, most of them made efforts to learn, but mostly in the classroom. Some tutorial sessions were extended, since out of class, study was lacking.

During my five lectures and tutorials, Chapter 14 of M.Artin second edition has been covered. The concept of modules, sub-modules, free modules finitely generated modules and noetherian rings were introduced and developed the theory in order to solve a linear equation over the ring of integers and then derived structure theorem of abelian groups and the same for modules over polynomial rings. Then as an application the linear space of finite dimension over F is equivalent to the F[t] module has been showed and finally rational canonical form associated with one such linear operator has been derived.

I gave the initial 5 lectures and conducted 5 tutorials in the geometry section of the school. The focus was on differential geometry of plane curves; more specifically the first 3 chapters from Pressley's book on differential geometry. In the first lecture, reviewed some of the basic aspects of multivariable calculus. Good part of the focus was on the Implicit and the Inverse function theorems. In the next lecture, I began discussing parametrized curves. I then discussed length of a parametrized curve and reparametrization of a parametrized curve, especially the unit speed reparametrizations of regular parametrized curves. I then introduced the notion of curvature of plane curves and derived an expression for the same. A related notion of signed curvature was also introduced. I then proved the fundamental theorem on existence and uniqueness of plane curves with a given signed curvature. In the later lectures I began talking about spaces curves. I introduced torsion of space curves and derived an expression for the same. I then discussed the Serret-Frenet equations. I then stated (without proof) the analogues theorem for existence and uniqueness of space curves with given curvature and torsion. In the last lecture I proved two global theorems on plane curves; namely the isoperimetric inequality and the 4 vertex theorem. The tutorials were devoted to discussing how to prove Inverse function theorem from Implicit function theorem and vice versa, comparison between level curves and parametrized curves, exercises discussing properties of curvature and fixing some loose ends from the lectures.

The students were alert and enthusiastic. The teaching experience was rewarding. The facilities and arrangements at KSOM were good.

The partcipants were taught through first three chapters from Pressley's book during the first week which is about curves. So I began with the question: What is curve? I got some very interesting answers which helped me to take the course that I should for these audience.

I began with recalling inverse and implicit function theorems smooth version of invariance of domain, and lead them the concept of manifolds with boundary in euclidean spaces. Most of the examples were taken from the book Pressley and therefore were surfaces. In particular, the of all the classes of quadratics was discussed. I also gave them some glimpses of what is Topology, Algebraic Topology, Differential Topology and Differential Geometry. Then I proved the regular inverse image theorem, Jacobian Criterion for a subset of euclidean space to be a manifold. Then tangent spaces derivative of smooth maps on tangent spaces were introduced.

In lecture three, I introduced the first fundamental form of any manifold in a euclidean space and and indicated how this leads to the concept of Riemannian manifolds in general. We then discussed local isometry, conformal mappings and equi-areal mappings. As a consequence, we obtained Archimedes theorem and computed the area of a triangle on the sphere. The last lecture was devoted to the study of the geometry of the sphere, and the unit 2-disc with the Poincare metric as examples of non euclidean geometry.

The participants were on the whole quite enthusiastic though with diverse background. The distinction between those who had attended AFS-I earlier and those who had not was very clear. Often the discussion hours went beyond the stipulated time. I also engaged a session of making Platonic solids, which the participants enjoyed.

I also gave two special talks in which I presented two proofs of Fundamental Theorem of Algebra, one using Linear algebra and the other using only real analysis.

We had only one tutor who was also not of much use.

The arrangements at KSOM were excellent.

Incidentally, I came to know on the very first day that there is a II edition of Pressely book and got a soft copy. This edition is far better than the first one and I recommend that we follow this in future.

After beginning with the definition of geodesics as curves on surfaces that obey Newton's First Law, with geodesic curvature as the tangential component of curvature, and Gaussian curvature as the product of minimum and maximum normal curvatures of curves passing through the given point of the surface, the Gauss-Bonnet Theorem was introduced as a statement of the form total curvature equals a constant. This was first stated for simple smooth closed curves, then for curvilinear polygons and then for compact orientable surfaces. The constant was motivated in the first two cases to be either a plane angle or a solid angle by considering the cases of a closed disc or a sphere. The proof for the compact surface case was deduced from the curvilinear polygon case after assuming the existence of triangulations and it turns out that the constant is the Euler characteristic, showing that the theorem amazingly relates the differential geometric curvature with a topological invariant. The implications of such a statement were explained.

The second fundamental form was introduced as arising naturally from an effort to compute the normal curvature of a curve on the surface. The Weingarten map was defined and the existence of real eigenvalues were shown. The eigenvalues were called principal curvatures and the corresponding eigenvectors principal eigenvectors. It was established that these vectors are orthogonal (when the eigenvalues are distinct) and Euler's Theorem was established relating the normal curvature of an arbitrary curve with these principal  curvatures. As a corollary to this, it followed that these principal curvatures were none other than the maximal and minimal normal curvatures. A formula for Gaussian curvature was deduced using the fact that it is the determinant of the Weingarten map, showing that it is a smooth function.

The above analysis allows approximation of the surface by an equation of degree two and the classification of its points as elliptic, hyperbolic, parabolic or planar was illustrated.

It was shown that the Wiengarten map is also the same as the Gauss map up to a sign,where the Gauss map records the variation of the unit normal to the surface. Gauss's Theorem on Gaussian Curvature as the limiting value of area spanned by the normal vector on the unit sphere per unit area of the surface was proved, generalising the statement that the curvature of a curve is the limiting value of change in the normal direction (or tangent direction) per unit arc length. The total curvature of the torus was shown to be zero easily by considering the Gauss map and the regions of elliptic and hyperbolic points on it.

Hopf's Umlaufsatz was motivated on the plane using the case of a circle and its proof for the surface case was explained using the ideas of isotopy. A calculation relative to a moving frame on the tangent plane and forming a right handed orthonormal system with the unit normal shows that the difference between the rate of angular change between the tangent and the first vector of the frame, and the geodesiccurvature, is measured on the bounding curve by the dot product of the first vector with the rate of change of the second (rates measured relative to arclength). This is essentially an infinitesimal version of the Gaussian curvature on the inner surface recorded by the bounding curve and integrating this on the curve leads to the proof of Gauss-Bonnet for the case of a smooth curve, from which the polygon case can be deduced. Green's Theorem and all basic formulas involving the second and first fundamental forms are put to use here.

It was pointed out that the same moving frames calculation applied to the Gram-Schmidt orthonormalisation of the frame of a patch (chart) leads to the proof of Gauss's Theorem Egregium, which showed that the notion of Gaussian curvature is intrinsic, is preserved by isometries, and showed that the determinant of the second fundamental form depends on the first fundamental form. It was thus explained why any map of the earth distorts distances, must distort either angles or areas and not preserve both.

Geodesics, their appearance in normal sections, and their determination on surfaces of revolution (Clairaut's Theorem) were explained. There was not enough time to cover the Geodesic Equations, Gauss Equations or Geodesic Coordinates. These the students were requested to study later on their own.

Several hours of extra classes were taken in order to complete the above proofs and most of the students attended these sessions from 5.30PM to 7.30PM with interest, enthusiasm and appreciable patience. The lecturer greatly enjoyed lecturing on these beautiful topics and wishes to do the same in future given the opportunity.

Lecture I:
Review of Riemann integration - The The Riemann characterisation of integrability -Fundamental theorem of integral calculus.

Lecture II:
(Abstract) Measurable spaces - sigma algebras-existence of sigma algebras - Measurable functions-Borel sigma algebras.

Lecture III:
(Abstract )Measure spaces - Measures- examples(Lebeque measure on R^n - no construction or detailed discussion here) - Sequences of measurable functions-properties and theorems

Lecture IV:
Complete measure spaces(this was done with complete detailed construction and important results) - the notion of almost everywhere and related results.

Lecture V:
Integration - simple functions-positive measurable functions - monotone convergence theorem- measurable functions - complex measurable functions-properties and examples.

Lecture VI:
Fatou's lemma-Dominated convergence theorem - absolute continuity of Lebesgue integral(an abstract measure was referred to as Lebesgue measure) - absolutely continuous measure and Radon Nykodm theorem were mentioned - Revisit of Riemann integration - the Lebesgue criterion of Riemann integrability.

In all the 6 tutorial hours, problems were worked out and left over verifications were done. As prescribed all topics in the first chapter of Rudin's book on Real and Complex Analysis were covered though I did not strictly follow the material there. Several other textbooks such as Real Analysis by Folland and Royden were made use of.

In the next two lecture hours and tutorials hours Spectral measure, spectral integrals and spectral theorems were done in detail. This was done as participants showed enthusiasm to see where measure theory methods can be used and applied.

Printed class notes were given after each class so that students could concentrate on attending and interacting rather than taking down notes.

The whole audience was interested in the subject and a few of them were really good. Quite a few of them need a push and some assistance to start on a problem. I have given them notes for all the classes. Some corrections are needed in the notes and there are typos as well. I will get back and give them the corrected notes in full.

Lecture 1:
Review of topological concepts- Locally Compact Hausdorff spaces, Urysohn's Lemma, Partition of Unity

Lecture 2:
Riesz Representation Theorem: Statement and preliminary steps in proof- Construction of sigma-algebra from a positive linear functional, Inner and Outer regularity, Uniqueness of measure thus obtained

Lecture 3:
Riesz Representation Theorem (continued): Approximation Lemma, Countable sub-additivity and additivity on pairwise disjoint coverings, inner regularity on Borel sets with finite measure.

Lecture 4:
Riesz Representation Theorem (continued): Completion of proof of RRT, Definition of sigma-compactness and sigma-finiteness, Regularity theorem for Riesz measures on sigma-compact, Locally Compact Hausdorff spaces

Lecture 5:
Equivalent formulation of Riemann integration on $\R^n$, Proof that continuous compactly supported functions are Riemann integrable, Definition of Lebesgue measure on $\R^n$ via the Riesz Representation Theorem, Caratheodory's approach using outer measures for constructing measures, equivalence of Lebesgue measure obtained via RRT and Caratheodory constructions.

Lecture 6:
$L^p$-spaces: Convex functions, Inequalities of Jensen, Holder and Minkowski

Lecture 7:
$L^p$-spaces: Definition of a Banach spaces, definition of $L^p$-spaces, Proof that $L^p$-spaces are Banach.

Lecture 8:
Hausdorff measures: definition of metric outer measure, definition, and properties of Hausdorff measures, Definition of Hausdorff dimension, Computation of Hausdorff dimension of the Cantor set

Haar measures: Definition of Topological groups, Definition of Radon and Haar measures, examples of Haar measures via Riesz Representation Theorem

In all tutorial classes, problems pertaining to relevant topics were given including some leftover exercises from Lecture classes.