This December I unofficially [just knowledge, no certificate] attended Annual Foundation School – 1, held at National Institute of Science Education and Research. This was possible due to Sagar Shrivastava.

This winter I just wanted to explore maximum number of mathematics topics unknown to me. Also this gave me an excellent opportunity to meet aspiring as well as established mathematicians from all over India. It was wonderful experience to have mathematics being talked all around me.

In this post I will tell about “*what new things I learned this December*”

**Week 1**

* Real Analysis 1* by Pradipta Bandyopadhyay: First topic to be covered was Measure theory and integration. I could understand just some basic definitions and 2.5 out of 4 classes were fruitful for me.

* Topology *by Parameswaran Sankaran: He spent one class on discussing basics and then jumped to problem solving, so that all students who have done the course can also benefit from his classes. From the time I saw proof of Euler’s polyhedron formula, I was very much attracted to Topology and thus enjoyed these classes. I can illustrate this with following incident:

On 7th December 2014 (first Sunday of AFS 1) I was sitting outside barber shop (as inside it was already very much crowded), I started thinking about Klein Bottle and how to transform it into a sphere with 2 discs removed and replaced by Möbius strips. Then the barber came out (after half an hour) and called me in, but I didn’t want to be disturbed and rather sent a man who had just come, inside. Then after I was done with my thinking I realized my mistake and as a result of which I had to wait for another 15 minutes.

* Group theory *by Brundaban Sahu : All basics were discussed, thus this was most fruitful course for me (also I had already done some homework.)

**Week 2**

* Group theory* by Brundaban Sahu : Learnt awesome awesome new proof for “there exist only 5 platonic solids” (earlier I knew proof by topology only)

* Real Analysis 1* by Saugata Bandyopadhyay :Saw the most dreadful proof of real analysis [Resiz Representation Theorem] (around 13 different steps, took 3 classes to complete the proof)

* Real Analysis 2* by Varadharajan Muruganandam: Discussed, the topic “Functions of several real variables” – This was most weird course for me. I just made myself familiar with some elementary theorem (could not understand their proofs) but this course forced me to kick start a systematic study of Real Analysis and Linear Algebra.

**Week 3**

* Group theory* by Binod Sahoo : Learnt awesome awesome new proof for ” Fundamental Theorem of Arithmetic” using Jordan-Hölder Theorem. Also a bit of Linear Algebra was discussed in last class.

* Real Analysis 1* by Sanjay Parui: Application of vitalli cover and Lebesgue integration were discussed.

* Real Analysis 2* Varadharajan Muruganandam: Got a flavor of differential geometry.

**Week 4**

This week started with celebration of Ramanujan’s birthday. This week was rather* Mathemusical*., as every evening I attended Spic Macay, Rural School Intensive programe..

* Group theory *by Varadharajan Muruganandam: He, taught topology groups, so as to prove that SO(n) groups are simple.

* Real analysis 1* by S. Thangavelu : Measure theory was brought to it’s end by discussing convolutions etc.

* Topology* by Samik Basu [Dept. of Mathematics, Ramakrishna Mission Vivekananda University, Belur Math, W. B.] : Main focus was on quotient topology and its applications.