Book
List of Math and Computer Science books and courses I have taken over the yearsPoint-Set Topology (from Category Theory point of view )
Tai-Danae Bradley, Tyler Bryson, John Terilla - Topology: A Categorical Approach
The book is a refresh in point-set topology and homotopy, hence, readers are expected to know what's happening.
Explanations might not be the easiest to follow but sufficient for determined people.
Overall, must read book.
Self-taught (2024)
Algebraic Topology I
Supplements:
Allen Hatcher - Algebraic Topology
MIT OpenCourseWare - Algebraic Topology I
Roman Sauer - Algebraic Topology youtube series
This is the most difficult as well as the most beautiful mathematical course I have ever taken.
Algebraic Topology (AT) uses tools from Category Theory (CT), Homological Algbera (HA) to study topological objects.
While the Hatcher's book focuses more on geometric approach to AT, Dr. Charmaine Sia's approach to AT is more abstract, explaining everything by abstraction from CT and HA.
The approach is wellrounded, covers many aspects of the subject, without her introduction, I would never be able to grasp any idea from AT.
The course includes Homology, Cohomology, Cup Product, Fundamental Group, van Kampen theorem, Covering Spaces
For further references, one should benefit from reading MIT OpenCourseWare AT I notes (for exercises), Roman Sauer AT youtube series (for HA approach to AT)
NUS MA5209 by Dr. Charmaine Sia (2024)
Analysis I
Richard L. Wheeden, Antoni Zygmund - Measure and Integral: An Introduction to Real Analysis
Walter Rudin - Real and Complex Analysis
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NUS MA5205 by Dr. Seng Kee Chua (2023)
Differential Geometry of Curves and Surfaces
Mantredo P. do Carmo - Differential Geometry of Curves and Surfaces
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NUS MA4271 by Dr. Daren Wei (2023)
Point-Set Topology
Sidney A. Morris - Topology without tears
James Munkres - Topology
My first experience with topology.
Coming up with the notion and defintion in topology is unintuitive and requires years of refinement.
It often requires a significant amount of time doing exercise to have the sense of what happening.
A solid basic set theory is required.
The book consists of several parts: (1) notion of continuous mapping, open mapping, homeomorphism, subspace, metric space (2) topological properties: open, closed, connected, compact (3) embedding theorem (4) product topology abd Tychonoff theorem (5) quotient space
Self Taught (since 2023)
Abstract Algebra
Dan Saracino - Abstract Algebra: A First Course
Dummit, David Steven, and Richard M. Foote - Abstract Algebra (Chapter 1 to Chapter 5)
A book on discrete structure: groups and rings. Some ideas about number theory is required.
Algebra is highly abstract and complex, however, it will be intuitive if one dedicates time on it
Self Taught (2023)
Complex Analysis
Dr. Michael Penn's Youtube Series on Complex Analysis here
Complex numbers,
Sequence - Limit - Derivative,
Cauchy-Riemann Equation,
Conformal Mapping - Mobius Transformation,
Gauss's Mean Value Property - Maximum Modulus,
Riemann Integeration for Complex numbers,
Cauchy's Integral Formula - Lioville's Theorem,
Morena's Theorem,
Complex Green's Theorem,
Sequence of Complex functions,
Taylor Series - Zeroes of Analytic functions,
Laurent Decomposition/Series,
Singularities of Analytic functions,
Residues - Residue Therem
Feels like a vector calculus courses on the multiplication of complex numbers. Misterious
Self Taught (2023)
Nonlinear Dynamics and Chaos
Steven Strogatz - Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering
A pleasant introduction to nonlinear dynamics and chaos. The book focuses on graphical methods and high level ideas.
Downside: Too much non-mathematical details. Most students don't need to understand the mechanisms of reproduction in biology, mass preservation in physics, or chemical equations in chemistry.
Self Taught (2023)
Introduction to Functional Analysis
18.102 | Spring 2021 | Undergraduate
Introduction to Functional Analysis
Analysis is the study of limit and its corollaries. Functional Analysis is the course on limits of sequence of functions. Once in a library, I wanted to refresh my skills, I tried to derive Fourier Transform from the Hibert space L^2 with sinusoids basis but realized that I never taken any function analysis course. The course introduces concept of Banach space, σ-algebra, Borel set, Algbemeasurable set on reals, measurable functions on reals, Hibert Space.
MIT 18.102 by Dr. Casey Rodriguez. Self Taught (2023)
Real Analysis
Stephen Abbott - Understanding Analysis
Walter Rudin - Principles of Mathematical Analysis
Analysis is the study of limit and its corollaries. Real Analysis is the course on limits of sequence of reals. What I was impressed the most from this course is the concept and limit and infinity on reals. The book is like a journey explaining many concepts for real numbers. And at the end, Abbott gave readers a little gift on the construction real numbers. It's like the twist in a movie, after that they all make sense.
Self Taught (2021)
Linear Algebra
Sheldon Axler - Linear Algbera Done Right
Beside Sipser's book, this is my second most favorite textbook and also my first proper Mathematics textbook. The book is a sequence of true statements with explainations. They're just too beautiful of a construction. The pure math textbook for linear algebra, built up from the concepts of vector space and invariant subspace to trace and determinant. the must read 2nd book in linear algebra. Different from Computer Science, Mathematics has a kind of existential proof that does not require to construct an example. (Similar to the concept of certificate in Computer Science)
Self Taught (2021)
Theory of Computation, Complexity Theory
Introduction to the theory of computation - Michael Sipser
Recommended by Dr. Zinovi Rabinovich. I studied this book at the same time with Advanced Topics in Algorithms. The Sipser's book really openned my mind about theoretical computer science. Really satisfied with the rigorousness of concepts in computer science, this was the first time I really know what mathematicians do. The book covers most basic concepts about automata theory, computability theory and complexity theory with rigorous proofs
Self Taught (2021)
Advanced Topics in Algorithms
Cormen, Thomas H and Leiserson, Charles E and Rivest, Ronald L and Stein, Clifford - Introduction to algorithms
Dr. Zinovi Rabinovich inspired me a lot in this course. The course is about construction of all concepts and theoretical guarantee of all algorithms in Computer Science. I was a Physics student in high school, Mathematics for me is just a bunch of unneccessary non-sense mostly due to the way they teached Mathematics in my secondary school. At the time, I thought Physics is the way of describing nature, describing everything. But after this course, I really surpised on how perfect everything is in Theoretical Computer Science is. They are just beautiful.
Analysis techniques: solving recurrence, master theorem, lower bound:, np-completeness: turing machine, hardness classes, P and NP, decision and optimisation, reduction and reducibility, NP-hardness, NP-completeness, Cook-Levin theorem, dynamic programming: optimal sub-structure and momorization, sub-problem dependency graph, search techniques: backtracking, branch and bound, computation geometry: polygon triangulation, convex hull, mathematical primer for convexity, min-cut/max-flow: Fold-Fulkerson, Edmonds-Karp, maximum bipartite matching, approximation algorithm and heuristic: heuristic and approximation, randomized algorithm: Monte Carlo and Las Vegas,
NTU CZ4016 by Dr. Zinovi Rabinovich (2020)
Simulation and Modelling
Banks, Jerry and Carson, John S and Nelson, Barry L and Nicol, David M - Discrete-event system simulation: Pearson new international edition
Types of simulation: Monte Carlo, continuous system, discrete event simulation
Simulation world view: event-scheduling, process-interaction, random numbers and random variate generation: middle-square, linear congruential (lcg), inverse transform, convolution, composition, acceptance-rejection, input modelling: maximum likelihood estimation, goodness-of-fit tests (chi-square, kolmogorov), arrival process, verification and validation:, output analysis: variance reduction techniques, antithetic variates, comparison: pair-t, queueing model: Kendall notation, Little's Law, analysis of M/M/1
NTU CZ4015 by Dr. Huang Shell Ying and ___ (2019)
Compiler Techniques
Lexical analysis: regular language, regular expression, finite automata, deterministic and non-deterministic finite automata (dfa, nfa)
Syntax analysis: context-free language and grammar, recursive descent parsing, LR parsing, LR(0), LALR(1), error recovery, sematic analysis: abstract syntax tree, abstract grammar, attribute grammar, scope checking, type checking, code generation: intermediate language, run-time storage organisation, stack vs heap, stack machine, code generation for physical machine, register allocation, optimisation: intra-procedural, data flow analysis, simple local optimisation, liveness analysis, common subexpression elimination, overview of inter-procedural optimisation,
NTU CZ3007 Dr. Huang Shell Ying (2019)
Artificial Intelligence
Stuart J. Russell, Peter Norvig - Artificial Intelligence: A Modern Approach
search: uninformed search (breath first search, depth first search, iterative deepening depth first search, uniform cost search), informed search (best first, A*), heuristic function, constraint satisfaction problem, adversarial search (minimax), first order logic,
NTU CZ3005 (2019)
Algorithms
Analysis of algorithms, search: sequential search, binary search, hashing, sort: insertion sort, heap sort, quick sort, merge sort, graph: breath first search, depth first search, backtracking, dijkstra, minimum spanning tree
NTU CZ2001 (2018)
Data Structures
Basic C programming, recursion, memory management, array, linked list, stack and queue, binary tree, binary tree traversal, graph
NTU CE1007 (2017)