This is where I will put what happens during my journey into mathematics

1. Group Theory

   A. Introduction to group theory

   i. What is a group? Some standard groups like $(S(F), \circ)$, $(\mathbb{Z_n}, +_n)$, etc. Link between groups and fields.

   ii. What are the group axioms? How to test if something is a group? How to draw up a Cayley table of a small, finite group?

   iii. What is a subgroup? What are cyclic groups and subgroups? How to test if something is a subgroup of a group? How to find subgroups.

   B. Permutations in group theory

   i. How to write cycle forms? How to find the inverse of a cycle form? How to find the parity and order of a permutation?

   ii. The symmetric group of degree $n$, $S_n$ and the alternating group $A_n$.

   iii. How to find permutations of symmetries in both 2D and 3D shapes? How to compose permutations?

   iv. What is an isomorphism? What are some standard isomorphic groups? What is a homomorphism? Finding the image and kernel of a homomorphism.

   C. Lagrange's Theorem and finding all subgroups of groups up to order 8

   D. Other types of subgroups

   i. What are left and right cosets? What are normal subgroups? Identifying normal subgroups.

   ii. What is a quotient group? Group tables of quotient groups.

   iii. What is a conjugate subgroup? How do you find conjugates of elements?

   iv. How do you find conjugates in a 3D solid?

   E. Group actions

   i. What is a group action? How to test that something is a group action?

   ii. Group actions in symmetry groups.

   iii. The Orbit-Stabilizer Theorem

   iv. Fixed sets and the Counting Problem

2. Linear Algebra

   A. Introduction to matrices

   i. What is a matrix. Matrix properties, and arithmetic. Is the set of $2 \times 2$ matrices a field? Symmetric matrices, elementary matrices, diagonal matrices and transposition of matrices.

   ii. Finding the inverse of a matrix. Finding the determinant of a matrix including expansion of large matrices using cofactors.

   B. Finding solutions to systems of linear equations

   i. Using row-reduction, determining number of solutions, consistency of the system and so on.

   ii. Using Gauss-Jordan elimination.

   C. Introduction to vector spaces

   i. What is a vector space? What are the properties of a vector space and the relationship between a vector space and a group?

   ii. What is a linear combination? How to determine linear independence? What is a basis of a vector space? How to check if a set is a basis of a vector space?

   iii. Examples of vector spaces like $\mathbb{R^n}$, $P_n$, etc.

   D. More on vector spaces

   i. What is the dimension of a vector space?

   ii. What is a subspace?

   iii. How to find the orthogonal and orthonormal basis of a space? Gram-Schmidt orthogonalisation.

   E. Linear transformations

   i. What is a linear transformation? Examples of linear transformations.

   ii. How to recognize if a function is a linear transformation? How to represent it in matrix form? How to compose linear transformations or find their inverse?

   iii. What are the image and kernel of a linear transformation and how do you find them? The Dimension Theorem. How do you find a basis for the image of the linear transformation?

   iv. How to determine the number of solutions from the number of equations and unknowns of a linear transformation thanks to the image and kernel?

   F. Eigenvectors and eigenvalues

   i. What is an eigenvector and what is an eigenvalue? How to find the characteristic equation and the eigenvector equations.

   ii. How to find the eigenspaces of a linear equation? How to find an eigenvector basis?

   iii. How to diagonalise a matrix? What is a transition matrix? How to orthogonally diagonalise a linear transformation?

   G. How to classify conics and quadrics

3. Analysis

   A. Inequalities

   i. The triangle inequality. Several properties of real numbers. What is an irrational number? What is a rational number?

   ii. Rules for manipulating inequalities.

   iii. Solving inequalities vs. proving inequalities.

   B. Sequences, series and limits

   i. Convergence vs. divergence. Null sequences. Finding the limits of sequences and series.

   ii. Looking at which test to use given certain conditions for both sequences and series.

   iii. The Combination Rules for sequences and series.

   C. Continuity in functions and more on limits

   i. How to determine if a function is continuous? Classical definition of continuity.

   ii. Looking at the exponential and logarithmic functions.

   iii. Looking at properties of continuous functions like the Intermediate Value Theorem.

   D. Limits of functions

   i. What is a punctured neighbourhood? Showing whether a function has a limit or not.

   ii. Combination Rules for functions and other rules to find limits. One-sided limits.

   iii. Asymptotic behaviour of functions.

   E. Differentiation and integration of functions

   i. How to determine if a function is differentiable? How to determine if a function is integrable?

   ii. How to determine the upper and lower Riemann sums of a function?

   iii. Several functions: Blancmange, sawtooth and Riemann functions.

   iv. Rolle's Theorem and l'Hôpital's Rule.

   v. The Mean Value Theorem, Increasing-Decreasing Theorem, the First and Second Derivative Tests.

   vi. Combination Rules, Composition Rules, and so on.

   F. Power Series

   i. The generalised Binomial Theorem

   ii. Taylor series and other power series

   iii. The radius and interval of convergence of power series.

   iv. Using power series for estimates of $\pi$ and others.

This should be seen as if we're trying to run a marathon! As you can see from the list above, we learned a lot of different subjects. It was ten months of pretty intensive mathematical content, much of it proofs. Each week we had to read over a hundred pages of material and do exercises, so how do you study for an exam in a month and a half over material that is very ample?

It took a lot of planning first. I developed a means of preparing for the exam that did not include going over the text books again. While I studied the material, I took notes. Speaking to various other students, I realized that I had to take these notes a special way. Some students did not take notes at all because they felt that a Handbook that was provided by the course was enough. However, I felt that it was not enough to rely on this Handbook because it was just a summary without some of the links and context needed to fully integrate its meaning in my mind.

Instead, I developed a system of note taking that relied on the Zettelkasten method and on free tools like Zettlr and Logseq. I summarized, in my own words, each section of the material and used extensive backlinks to go to and from the subjects. Since I had linked and curated my notes while reading the material, the work was already done by the time I got to exam revision. I just went over my notes hand-in-hand with the Handbook in order to highlight the most important parts and annotate ideas that needed further explanation.

This was enough for me to revise the material and feel fairly confident about the subjects.

The rest was practice using old exercises and mainly old exam papers with solutions. I let myself do three terrible exam papers to familiarize myself with the questions they ask and also how to answer them (format, level of mathematical communication, etc). I then started to buckle down on the quality of my answers and finally ended up timining myself on the exams as if they were the real thing.

By the time the real exam had rolled around, I had already practiced 7 exams! By the time I embarked on the real exam, most of my nervousness was gone and I had developed the mental fortitude to do a 3 hour exam without panicking. While performing the exam, I made sure to have ample drink and a snack, but also to be sure to reread the exam questions first, leave those questions I felt a bit stuck on and do others then come back, and also to make sure that I did not feel tired. Focusing for 3 hours on a complex subject is still pretty hard, even for a person who is used to focusing on coding for hours!

I found it very challenging since my pure math knowledge was not great. I feel much better and indeed very attracted to the subjects we covered. I noticed links between making proofs and designing code. I hope to continue on my mathematical journey and run on this path alongside programming. They go hand-in-hand!

I will be embarking on two new courses in October: applied mathematics and graphs, networks and game theory. I am looking forward to both courses. I will also practice the knowledge I gained in the course described above by doing some of the harder exercises I left on the side but also to practice the proofs in the books and reread my notes.