"I write not because I know something but to learn something."
"The most important book to me is my own notebook because it is written in the way I understood."
This pinned post contains the list of my lecture notes on mathematics, physics, and related areas (theoretical computer science, mathematical biology, and mathematical finance). They are categorized by subjects in each area for readers' convenience. This post is subject to updates as I continue to add new lecture notes.
Mathematics
Abstract Algebra I: Groups
Lecture Notes
Problem Sets
References and Further Reading
I. N. Herstein, Abstract Algebra, 3rd Edition, Wiley, 1996
I. N. Herstein, Topics in Algebra, 2nd Edition, John Wiley & Sons, 1975
Ramji Lal, Algebra 1, Groups, Rings, Fields and Arithmetic, Springer, 2017
Abstract Algebra II: Rings, Fields, Galois Theory, Modules, Representation Theory
Lecture Notes
Problem Sets
References and Further Reading
I. N. Herstein, Abstract Algebra, 3rd Edition, Wiley, 1996
I. N. Herstein, Topics in Algebra, 2nd Edition, John Wiley & Sons, 1975
Ramji Lal, Algebra 1, Groups, Rings, Fields and Arithmetic, Springer, 2017
Ramji Lal, Algebra 2, Linear Algebra, Galois Theory, Representation Theory, Group Extensions and Schur Multiplier, Springer, 2017
Algebraic Topology: Homotopy, Homology, Cohomology
Lecture Notes
Homotopy
Homology
Cohomology
Problem Sets
Homotopy
Homology
Cohomology
References and Further Reading
Calculus
Differential Calculus
Lecture Notes
Problem Sets
Integral Calculus
Lecture Notes
Problem Sets
Sequences, Series, and Vector-Valued Functions
Lecture Notes
Problem Sets
Functional Analysis
Functional Analysis
Lecture Notes
Problem Sets
References and Further Reading
H. L. Royden, Real Analysis, Second Edition, The Macmillan Company
Peter Szekeres, A Course in Modern Mathematical Physics, Groups, Hilbert Space and Differential Geometry, Cambridge University Press, 2004
Functions of a Complex Variable
Lecture Notes
Problem Sets
References and Further Reading
Linear Algebra
Lecture Notes
Problem Sets
References and Further Reading
Ramji Lal, Algebra 2, Linear Algebra, Galois Theory, Representation Theory, Group Extensions and Schur Multiplier, Springer, 2017
Serge Lang, Introduction to Linear Algebra, Second Edition, Springer, 1986
Serge Lang, Linear Algebra, Third Edition, Springer, 2004
Number Theory and Cryptography
Computer Software for Computation
Study of number theory is often accompanied by heavy computations for which computers can be effectively used. From time to time, I will be using Maxima, an open source computer algebra system (abbreviated as CAS, a software package for symbolic computation) and Python. Sage (CoCalc) is also a good and powerful open source CAS but I will not be using it here. Main reason is that I find it restrictive as it requires a web-based interface (a browser) and an internet connection. Unless you run your own CoCalc server (if you are going to use it just for yourself, why would you run such a server?), you need to have an access to a CoCalc server that is run by someone else. I personally prefer a crude and low tech computing environment that does not require a particular interface and a lot of resources. Maxima and Python fit into such preference of mine. Maxima and Python do have interfaces but they also can be run interface-free in a command shell.
MAXIMA
What is Maxima?: MIT has developed a computer algebra system, called Macsyma, from 1968 to 1982 as part of Project MAC. They turned over a copy of the Macsyma source code to the Department of Energy (DOE). That version is known as DOE Macsyma. It had been maintained by William Schelter at the University of Texas from 1982 until his death in 2001. In 1998, Schelter obtained permission from the DOE to release the DOE Macsyma source code under the GNU Public License, and in 2000 he initiated the Maxima project at SourceForge to maintain and continue developing DOE Macsyma, now called Maxima.
2. For those who are using MacOS, Maxima can be download from here and an installation instruction can be found here.
3. If you are using Ubuntu Linux OS, simply run the following command in command shell:
sudo apt install gnuplot
4. If you are using FreeBSD Unix, as root run:# pkg install maxima && pkg install wxmaxima
and also install Gnuplot by running# pkg install gnuplot
5. For all other operating systems, refer to the instruction at Maxima Downloads page here.Maxima documentation page is an important source on how to use Maxima for your computational needs and purposes. Another important source on using Maxima, especially for doing mathematics is The MaximaList. The html version of Maxima 5.29.0 manual on Number Theory is available online here.
PYTHON
You can find download/installtion instruction for your OS at Python.org. Make sure that you install Python 3.
1. For Windows system, select and download Python Windows installer, for example python-3.3.0.msi and run it.
2. For MacOS, select and download .dmg Mac Installer Disk Image and run it.
3. For Ubuntu Linux OS, run in command shell, for example
sudo apt install python3.3 && sudo apt install idle3
IDLE is a covenient interface for Python. I love iPython (Interactive Python Shell). It allows you to run .py files in command shell. To install ipython runsudo apt install ipython3
4. For FreeBSD Unix, as root run:# pkg install python
This command will install Python 3 and IDLE 3 as well. To install ipython, as root run:# pkg install py37-ipython
Lecture Notes
Problem Sets
References and Further Reading
Elementary Number Theory
Cryptography
Steven Galbraith, Mathematics of Public Key Cryptography
Neal Koblitz, A Course in Number Theory and Cryptogtraphy, Graduate Texts in Mathematics 114, Springer-Verlag, 1994
Gabor Wiese, Théorie des nombres et applications à la cryptographie
Elliptic Curves
Probability
Lecture Notes
Problem Sets
References and Further Reading
Stochastic Differential Equations
Lecture Notes
References and Further Reading
Don S. Lemons, An Introduction to Stochastic Processes in Physics, Containing "On the Theory of Brownian Motion" by Paul Langevin, translated by Anthony Gythiel, The Johns Hopkins University Press, 2002
Bernt Øksendal, Stochastic Differential Equations, An Introduction with Applications, Fifth Edition, Corrected Printing, Springer-Verlag, 2000
Philip E. Protter, Stochastic Integration and Differential Equations, Second Edition, Springer, 2004
Sheldon Ross, A First Course in Probability, Fifth Edition, Prentice Hall, 1997
Physics
Fields
Lecture Notes
References and Further Reaing
Quantum Mechanics
Lecture Notes
References and Fuerther Reading
Wormhole Physics
Lecture Notes
References and Further Reading
A pretty big book with more than 1300 pages. This book is dubbed the bible of general relativity.
Michael S. Morris and Kip S. Thorne, Wormholes in spacetime and their use for interstellar travel: A tool for teaching geneal relativity, Am. J. Phys. 56 (5), 395-412, May 1988Michael S. Morris, Kip S. Thorne, and Ulvi Yurtsever, Wormholes, Time Machines, and the Weak Energy Condition, Physical Review Letters, Volume 61, Number 13, pp. 1446-1449, 26 September 1988
Hans Stephani, Relativity, An Introduction to Special and General Relativity, Third Edition, Cambridge University Press, 2004
Matt Visser, Lorentzian Wormholes, From Einstein to Hawking, AIP Series in Computational and Applied Mathematical Physics, 2008
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