This post is dedicated to the mathematics students of Yachay Tech that are about to start their third year, the year at which they start to see real mathematics.

Updated: August 10, 2024

Before the begining of a new semester, it is common ammong students to ask for study material to other students that have already passed certain classes. With the newcomers in mind, I have collected below study resources of the most important courses I have taken as an undergradute mathematics student. Based on discussions with classmates and professors, I also share specific insights and general recommendations on certain subjects of the mathematics program of Yachay Tech.

Among other things, you will find:

  • past exams and quizzes
  • assignments
  • problem sets & solutions to exercises
  • book recommendations

The list belowed is arranged in the same order in which I took the subjects.

Real analysis

Professor Eusebio Ariza taught this course the second semester of 2022. This course was an introduction to classical real analysis. You may want to take a look at the syllabus or study plan.

Some classmates recommend to take this course before topology. In a way, topology generalizes some of the concepts covered in this course. Taking this course first may have the advantage of fostering a bit of mathematical maturity (which would be helpful for topology).

Here are some resources:

  • Past exams & quices: here.
  • Assignments, problem sets, lecture notes: here.
  • Book: Understanding Analysis by Stephen Abbott (second edition). I strongly recommend this book; it goes straight to the point, it is very well written and it has a solution manual. Other useful references are:
    • Analysis I by Terence Tao. This book is awesome but takes some more time to read. The intuition gained is invaluable, however. Terence Tao builds the real numbers from the ground. Stephen Abbott uses the axiom of completeness, i.e. assumes the real numbers exist.
    • Mathematical Analysis by Tom Apostol. This is a classic.
    • Introduction to Classical Real Analysis by Karl Stromberg. This is the reference we used at the begining of our course.
    • Introducción al Análisis Real by Tineo & Uzcátegui.
    • Supplement for Measure, Integration & Real Analysis by Sheldon Axler, freely available here. This is a concise reference that one may use to review key concepts of the course.
    • Principles of Mathematical Analysis by Walter Rudin. This book is not meant to be an introductory textbook. Use it as reference only.

Mastering the contents of this course is recommended.

Topology

It is given by Professor Wilman Brito since the second semester 2022. I took it the same semester when I took Real Analysis, although it may be advisable to take Real Analysis first. Both subjects demand a lot of time. A good understanding of topology is essential for future courses. Wilman has always put emphasis on how important this subject is, and now that I’m in my last year, I can confirm that it is indeed the case.

This is an introductory course to topology or, better said, to what is called point set topology or general topology. A deeper study of topology is Algebraic Topology. Munkres does cover this subject from chapter 9 onwards. Last time Wilman took a metric spaces first approach to the course. He will probably do the same next semester, so keep this in mind if you want to study ahead.

I learned topology mainly using this lecture notes from the University of Toronto for the course MAT327 • Topology, which has a lot of material already. I recommend to check the lecture summaries section and the big list of problems. This lecture notes cover most of the material that Wilman covers.

Here are some resources:

  • Past exams & quices: here.
  • Assignments, problem sets, lecture notes: here. Take a look at the lectures folder to see the topics covered at every class when I took the course.
  • Book: Topology by James Munkres (second edition), but other good ones are:
    • Elementary Topology Problem Textbook by O. Ya. Viro et al. I liked it because it has a nice introduction to algebraic topology.
    • Topology without tears by Sidney A. Morris. A lot of classmates recommended it. It lacks some topics though.
    • Topology by Fairchild and Ionescu. As Professor Mayorga says, it seems this book was made so that you learn functional analysis later on.
    • General topology by Engelking. I liked it because it has a good discussion on the separation axioms.
    • Introduction to Topology by Tej Bahadur
    • Introduction to Topology by Gamelin

Abstract Algebra

I took this course the first semester of 2023. My professor was Pablo Rosero. Here is the syllabus. And here is the study plan for the upcoming semester that Pablo kindly prepared. Studying ahead is a very good idea.

This is an introductory course to Abstract Algebra also called Modern Algebra. (When a mathematician talks about algebra it refers to this subject, not to linear algebra or the algebra you learn in high school.) This subject is not like the other ones of this list. Let’s say the modus operandi is different: you develop a new way of thinking. In a way, you cannot apply the heuristics used for Analysis. As Cesc would say, think a lot and write very little. The course is not difficult, but one has to solve as many exercises as possible. I enjoyed it very much. On the other hand, I’d say that skills on proofs and writing are necessary to take this course. Be open minded to learn a new and beautiful part of mathematics.

Some resources you may want to review are:

  • Past quices & exams: here.
  • Assignments, problem sets, lecture notes: here.
  • Book: Abstract Algebra by Dummit & Foote. This is a very popular book and has tons of exercises. Some people even consider it to be an encyclopedia of algebra. Some other references are:
    • An unofficial solution manual to Dummit & Foote’s Abstract Algebra.
    • A Book of Abstract Algebra by Charles Pinter. It has very nice explanations and exercises. The author takes his time to explain in detail what you need to learn.
    • Contemporary Abstract Algebra by Joseph Gallian. I liked very much the way it is written. It is easy to read and the author makes it clear what are the main results one needs to learn and which others one may skip. Dummit and Foote do not do this. You can find it in Yachay’s library.
    • Algebra by Hungerford. It was a very good complement to our main book, although it is oriented to the graduate level. Hungerford wrote another book but oriented to the undergraduate level: Abstract Algebra, an Introduction.
    • The study guide used for previous courses on the subject.
    • Abstract Algebra by Herstein. A classic. Do not use it as an introductory textbook. Take a loot at this review.
    • This lecture notes on Galois Theory of the University of Edinburgh. Very accessible. Check the section where the general formula for the cubic equation is written. Neat.
    • This playlist by Socratica. Watch the videos for each topic before you go to class. Very recommended.
    • This playlist on a course of Abstract Algebra at Harvard by Professor Benedict Gross.
    • Visual group theory, a book with a visual approach to abstract algebra.

There were two presentation we had to prepare for this course. I chose to make the first one on cryptography and the latter on category theory.

Probability and stochatic processes

I took this course last semester (2023, second semester) with Professor Saba Infante. We focused on the fundamentals of the theory.

  • Quices & exams: here
  • (lecture) notes: here
  • Book: We used several references. Take a look at the syllabus.

Dynamical systems

I took this course last semester (2023, second semester) with Professor Bladismir Ruiz.

  • Quices & exams: here
  • Assignments: here
  • (lecture) notes: here
  • Book: We used mainly Equações diferenciais ordinárias by Jorge Sotomayor, but also some other textbooks like Dynamics and Bifurcations by Hale and Koçak, and Dynamical systems by Robinson.

Functional analysis

I took this course last semester (2023, second semester) with Professor Juan Mayorga. A good understanding of topology was crucial for taking this class, but also knowledge of Real Analysis and Linear Algebra. We covered all the material the professor had devised and more. The course was very well organized. Here you can find the syllabus.

This course is supposed to be an introductory class on Functional Analysis. Basically, here we study vector spaces (and the maps between them) but in infinite dimensions. The main part of the course consist of the Fundamental Theorems of Functional Analysis. (Yes, they have a special name.) It turns out these theorems are consequences of Baire’s Theorem, a result of general topology. (Topology again!) Except the Hahn Banach Theorem, which is an algebraic result in its more general form. Anyway, they are all consequences of the axiom of choice.

Some resources you may find useful:

  • Past quices & exams: here
  • Some notes and other stuff: here. For this course, I decided (was required to) take notes on paper. Look at this.
  • Book: A Course of Functional Analysis with Calculus of Variations by Juan Mayorga. This was the main reference for our class, especially chapter 5. A preliminary version is available here, kindly provided by the author. Other books I used are listed below.
    • Introductory Functional Analysis with Applications by Erwin Kreyszig. Basically this was our second main reference. It is an excellent book. However, I didn’t like some notation he uses throughout (e.g., ‘sum from 1 to n’).
    • A First Course in Functional Analysis by David Promislow. I liked very much this book. I used it for studying the fundamental theorems. The author outlines the logic behind the steps needed to prove the results so one can get the general idea at the begining. It is available in Yachay’s library.
    • Functional Analysis, Sobolev Spaces and Partial Differential Equations by Haim Brezis. This is a classic and very popular book, but it is not an introductory textbook. Use it as a reference. It starts straight with the Hahn Banach Theorem.
    • Elements of the Theory of Functions and Functional Analysis by Kolmogorov & Fomin. It goes very fast. I only used to compare results and proofs.
    • Handbook of Analysis and its Foundations by Eric Schechter. This book is not on Functional Analysis but I founded when I searched for a related topic. Actually it is a handbook on analysis, a must have reference. It has a lot of content. A ‘bible’ of analysis.

Some thoughts: Linear Algebra Done Right by Sheldon Axler must be the go to reference to use for the course of Advanced Linear Algebra (or the newer course: Linear and Multilinear Algebra). It is freely available here, provided by the author. A glance at the table of contents reveils it would be a very good preparation for (but not limited to) Functional Analysis.

Differential Geometry

Differential Geometry Diagram

I took this course in the first semester of 2024 with Professor Bladismir Ruiz. It was one of the best courses I’ve ever taken. The prerequisites include multivariable calculus and some real analysis. The focus of the course is on the geometry of curves and surfaces, rather than the differential geometry of smooth manifolds. The reason for this point of view is that the plane and space curves and surfaces in ℝ³ are the simplest examples of smooth spaces. Thus, their study serves as a good introduction for advanced differential geometry. One of the most important results we studied was the Gauss-Bonnet theorem, which we covered towards the end of the course.

  • Syllabus
  • Some notes and other stuff: here
  • Assignments: here
  • Book(s): Differential geometry of curves and surfaces by Manfredo P. do Carmo. This is a very popular book, and the main reference for our class.
    • Elementary differential geometry by Andrew Pressley. This is a very rigourous book and very well written.
    • Apontamentos de geometria diferencial by Jorge Picado. This was the main reference used by our professor at the begining of the course, especially for the part of curves.
    • Elementary differential geometry by Barrett O’Neill. The viewpoint is similar to that of do Carmo’s, but it is aimed at a less advanced level.

    Some complementary references:

    • A comprehensive introduction to differential geometry by Michael Spivak.
    • Elements of differential geometry by Richard Millman and George Packer.
    • Visual differential geometry and forms by Tristan Needham.
    • The geometry of curves and surfaces (lecture notes) by Weiyi Zhang, available here.

Close

This is just my opinion. Feel free to look for your own resources. Also, feel free to contact me if you have any question or recommendation, or if you need any specific material.