Having trouble following your lectures in university/school?
Can't make sense of what your professors/lecturers said?
Feeling that the subjects that you're studying are too hard?
Textbooks are too difficult/tedious to read?
Or simply want some guide for self-studying?
Get in touch with me and I'll be able to help! You will soon discover that my lessons are a lot more effective than what you might expect.
I am exceptionally good at giving introduction to new subjects/concepts in a way that is as intuitive as possible, and for most of the time, you just won't be able to find the same information elsewhere. Motivations and interpretations of new concepts are provided throughout. Moreover, I can perfectly link new concepts to the students' existing knowledge, which allows you to actually "own" the new concepts instead of merely knowing that they exist. If you are having trouble at school or studying by yourself, get in touch with me and I will solve your puzzles.
Lessons that I have given in the past include first courses in group theory, ring theory (together with basic algebraic geometry), and category theory. Please see the "Experience" section below for details.
All courses that I can teach include:
(1) Anything at the pre-university level;
(2) Calculus/analysis, up to the level of Spivak's Calculus, Rudin's Principles of Mathematical Analysis, and Hubbard&Hubbard's Vector Calculus, Linear Algebra, and Differential Forms;
(3) Linear algebra, up to the level of Axler's Linear Algebra Done Right;
(4) Abstract algebra, up to the level of Dummit&Foote's Abstract Algebra,
(5) Topology, up to the level of Munkres' Topology;
(6) Algebraic geometry and commutative algebra, up to the level of Cox, Little&O'Shea's Ideals, Varieties, and Algorithms (soon will be upgraded to Hartshorne's Algebraic Geometry),
(7) Manifold theory, up to the level of John Lee's three volumes of Introduction to Topological/Smooth/Riemannian Manifolds;
(8) Category theory (categories, functors, natural transformations, and adjoints; soon will be upgraded to a complete course on category theory);
(9) Classical mechanics, up to the level that is needed for quantum field theory, i.e., classical particle/field theory + special relativity + covariant formulation;
(10) Classical electromagnetism, up to the level that is needed for quantum field theory, i.e., covariant formulation of electromagnetism;
(11) Basic general relativity, either a mathematical approach or a physical approach;
(12) Quantum mechanics, up to the level of Sakurai's Modern Quantum Mechanics;
(13) Foundations/interpretations of quantum mechanics (with the standpoint that quantum mechanics is incomplete);
(14) Basic quantum field theory or gauge theories, either a mathematical or a physical approach, up to a level that leads you to string theory.
Please note that technically I belongs to the Department of Chemistry of the University of Oxford. This should mean that
(1) when contacting me for lessons, please be aware that you will be talking to a theoretical chemistry DPhil student rather than a maths or physics DPhil student, but you are very much welcome to believe that the lessons that I give could well be better than ones given by a true maths/physics DPhil student, and
(2) I will also be able to talk about chemistry/biochemistry/biology if you are interested, and talk about how these subjects may interact with maths/physics.
I speak both English and Chinese (Mandarin).
I currently have one student on Superprof, F, who did group theory, category theory, and ring theory with me (plus a bit of linear algebra). We are planning for modules, fields, more categories and algebraic geometry in the near future. We had some very good interactions and we really enjoyed talking to each other.
Thanks to F, I am now able to offer some more structured lessons on group theory, ring theory, and category theory.
1. Group Theory
Having trouble with your first course in group theory?
Group theory lectures at your university are poorly structured or are too hard to understand?
Having a hard time reading group theory textbooks?
Feeling confused what group theory is all about?
Losing confidence in studying group theory?
I'M HERE TO HELP!
In my lessons on group theory I will explain to you about all the intuitions and big pictures that you should have in mind when studying group theory. These "inner workings" of group theory will help you see why the proofs in the textbook/lectures are structured in that way, so that everything that you felt pointless or nonsense before will now make beautiful sense to you. If you are taking a course on group theory, you will be able to regain confidence in front of all people, interact with and challenge the lecturer much more often. If you are reading textbooks, you will find the textbook materials much easier to understand. In addition, these intuitions will become your guiding principles for solving problems, will allow you to understand why your lecturer/textbook has chosen those topics for a first course in group theory, will help you appreciate the entire subject, and will reveal the connections to other subjects that you are studying.
i. Symmetries of an object;
ii. The advertisement that the group of symmetries of an object is a bag of verbs;
iii. The advertisement that any group is a bag of verbs;
iv. The meaning of group actions (this will lead to Cayley's theorem);
v. Properties of group actions (this will lead to the discussion of orbits, class equation, cycle decomposition of a permutation, stabilizers, centralizers, normalizers, and finally the advertisement that the internal structure that controls how a group acts on sets is its subgroup structure);
vi. Sylow's theorem;
vii. Classification of finite abelian groups; and
viii. Stories about the classification of finite simple groups.
It may take up to 10 hours in total to cover all of the above topics. The main textbook that I follow is Dummit & Foote's Abstract Algebra, 3rd edition.
I would normally assume that you have already learned some basics about group theory. However, if you are not familiar with or are not comfortable with homomophisms/isomorphisms, quotients and normal subgroups, the isomorphism theorems, etc., these topics could be added to our discussion. If you would like to discuss homomorphisms/isomorphisms and related stuffs in a more general context, please see below for lessons on category theory.
2. Ring Theory and Algebraic Geometry
Students who are new to rings may have a strange feeling that although they know how to work with rings, they still don't have a satisfactory picture in mind about what rings really are. I believe that a deeper and more careful understanding of the two binary operations defined on a ring (which is usually not covered in textbooks or university lectures) will solve these puzzles, and hence my lessons on ring theory will start from here. Next, I will help you organize your thoughts on zero divisors and units, which are the new beasts that appear in rings in comparison with groups, so that you can acquire a more structured understanding of them. These will then make it easier to understand the properties of ideals, including prime/maximal ideals and the quotient by prime/maximal ideals, which will then be accompanied with a discussion of the Algebra-Geometry Dictionary based on Cox, Little & O'Shea.
The contents on ring theory may take about 5-6 hours to finish, while the contents on basic algebraic geometry can take longer, depending how deep you would like to go into it; if you merely want me to motivate the Algebra-Geometry Dictionary without filling out the details, I might be able to do it in 1-2 hours. In addition, if you would like to talk about the chain of inclusions concerning Euclidean domains, integral domains, principal ideal domains, unique factorization domains, etc., we could do that for another couple of hours.
I would generally assume that you are comfortable with ring homomorphisms, quotients, ideals, and the isomorphism theorems. If not, we may add these topics to our discussions as well.
3. Category Theory
Category theory is highly abstract, and you may find it hard to get started. If so, no matter whether you are taking a course on category theory or are learning category theory yourself by reading textbooks, I will be able to help you get onto the right track by explaining to you about categories, functors, natural transformations, universal properties, and adjoints, in a way that is very likely to be more intuitive and more acceptable than others. In particular, I have my unique way of introducing natural transformations and adjoints, which seems to work better than anywhere else. These discussions could take about 5 hours in total to finish. I am confident that after these lessons you will be happy with the language of category theory and you will be able to go on to study more category theory much more easily.
Acquaintance with groups and vector spaces is in general expected; knowledge of other subjects would also be very helpful. It will be hard to learn categories without knowing at least one or two examples. If you are having trouble with group theory, please see above for my lessons on group theory. If you are not happy with linear algebra, I could also go over some linear algebra for you before going into or during our discussions of category theory.
Previously, I worked as TA in the Department of Mathematics and Statistics at the University of Saskatchewan for 3 years. Courses that I have assisted with include
i. MATH 264 Linear Algebra (computation-based, for students in physics, engineering, etc.);
ii. MATH 266 Linear Algebra I (proof-based, for pure math);
iii. MATH 366 Linear Algebra II (based on Axler's Linear Algebra Done Right; assisted with this course twice);
iv. MATH 362 Rings and Fields (Dummit & Foote);
Students and professors' feedbacks were exclusively highly positive. I had frequently offered lessons to friends for free, all of which were highly successful. I am exceptionally good at giving lectures and explaining things. In addition, I have also worked as TA for a number of first-year courses in probability and statistics, some of which were done physically at the University of Saskatchewan, while others done remotely for the University of Regina.
2019- DPhil in Physical and Theoretical Chemistry, University of Oxford
2015-2019 BSc Triple Honours in Mathematics (with Concentration in Pure Mathematics), Physics (with Specialization in Theoretical Physics) and Biochemistry, with High Honours, University of Saskatchewan, SK, Canada
2014-2018 BSc Honours in Biological Science, Honours College of Capital Normal University, Beijing, China
2008-2014 Beijing National Day School, Beijing, China
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