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Number of students Yi has taught since their arrival at Superprof

5 (2 reviews)

$118/h

1^st lesson free

Mathematics
Physics
Natural Sciences
Biochemistry

Oxford DPhil student offering maths and physics lessons up to university/MSc level

Name: Yi
Address: Oxford, US
Telephone: undefined
Price range: $118

Mathematics
Physics
Natural Sciences
Biochemistry

Lesson location

- online

About Yi

Listed below are subjects that I have taught before.

1. First course on 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 those ways, 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.

Highlights include:
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.

I have taught this subject twice in full and a few more times as part of other teaching plans. The main textbook that I follow is Dummit & Foote's Abstract Algebra, 3rd edition.

2. Linear Algebra

Rather than messing with solving systems of linear equations or manipulating matrices, I shall always put an emphasis on finite-dimensional vector spaces (as an abelian group with a field action) and linear maps between them (as vector space homomorphisms), accompanied with matrix representations of linear maps when a basis is chosen. This is possible and is in fact beneficial even for a first encounter with linear algebra. Motivations and intuitions become much clearer in this approach. Topics covered will include but not restrict to Sheldon Axler's Linear Algebra Done Right (I would add, for example, an introduction to tensor product spaces). I have taught this course once in full, and have worked as the grader for this course twice.

3. Group Representations and Applied Group Theory

This includes a full explanation of linear representations of finite groups. The basic belief is that knowing all the eigenvalues of all the group elements is sufficient to determine all possible irreducible representations of the group, leading to the use of characters. I shall treat a representation as a CG-module, and treat the so-called intertwining maps as CG-module homomorphisms. It is also possible to discuss representations of Lie groups and Lie algebras, as well as applications of group theory to physics and chemistry.

I taught these materials in the problem classes associated with the Quantum Chemistry Supplementary course given to 2nd year students at the University of Oxford, and the tutorials for the tutors of the above course, and also as part of a private pure maths tutoring at the MSc level.

For representation of finite groups I partly follow Fulton & Harris's Representation Theory, and partly follow Dummit & Foote Chapter 18; for Lie groups and Lie algebras I mainly follow Hall's Quantum Theory for Mathematicians, and Lie Groups, Lie Algebras, and Representations.

4. Ring Theory, Commutative Algebra and Classical 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 lead to an investigation of the Algebra-Geometry Dictionary. Topics include Hilbert basis theorem, Hilbert Nullstellensatz, Hilbert syzygy theorem, and the use of Grobner bases throughout; both affine spaces and projective spaces are covered, as well as the coordinate rings as finitely generated k-algebras with no nontrivial nilpotents. I have taught this subject once.

Textbooks that I follow are Dummit & Foote and Cox, Little & O'Shea.

5. 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.

I gave a seminar talk on this subject when I was an undergraduate student, and later on taught the materials in full twice.

6. Mathematical Analysis

This includes all standard materials that go into a first or a second course in analysis, e.g., ordered fields and construction of the real number system, sequences, series, differentiation and integration, etc.

Since mathematical analysis is often part of a students' first encounter with pure mathematics, it is often difficult for the student to get started/get used to the language/grasp the motivation and intuition behind the proofs at the very beginning. I therefore have designed a series of special lessons that gives a beautiful and highly intuitive introduction to ordered fields and the real number system, highlights in which include
i. the advertisement that a mathematical system is often a set equipped with some extra structure (e.g., binary operations, binary relations, outstanding collection of subsets), leading up to ordered fields;
ii. filling holes in the rational line, leading to real numbers (fill one point into each cut) or hyperreal numbers (fill infinitely many points into each cut);
iii. a full explanation of why the real line should be regarded as a truly continuously drawn straight line that has no holes in the middle, covering the least-upper-bound property, Archimedean property, connectedness, which eventually opens up a window to general topology.

I have taught this subject three times, each covering slightly different parts of the materials.

7. General Topology and Topological Manifolds

This include both a first and a second course in general topology, featuring a satisfacgtory motivation to the idea that the open subset structure of a set controls lots of behaviours (namely, topological properties), a patient explanation through all basic concepts (e.g., open/closed sets, interior/closure, limit points, bases for a topology, continuous maps and homeomorphisms, connectedness and compactness), a thorough investigation into the countability axioms and separation axioms, and how all these eventually help us shape the most useful definition of a topological manifold (in particular, emphasis is given to the equivalence between normality, Urysohn lemma, and Tietze extension theorem, together with associated metrization theorems, and is also given to the process of upgrading from normal spaces to paracompact Hausdorff spaces to make sure that every open cover admits a partition of unity subordinate to it, which is crucial for constructions on manifolds). Ideas of and structures on smooth manifolds may also be introduced.

I have taught this subject twice. I follow Munkres' Topology and John Lee's Topological Manifolds.

8. Classical Mechanics

I used to help another tutor with preparation of their tutorials on classical mechanics. I have also given a general introduction to classical mechanics to a mathematician. Emphasis was given to the equivalent ways of formulating the equations of motion, namely, through Newton's second law, Lagrangian, Hamiltonian, Poisson brackets, and more. I mainly follow Morin's Introduction to Classical Mechanics.

9. Quantum Mechanics/Quantum Chemistry

I've taught this as part of the problem classes associated with the Quantum Chemistry Supplementary course given to 2nd year students at the University of Oxford. The lecture course covers all fundamentals of quantum mechanics and various approximation techniques. I have also taught the mathematical background of quantum mechanics to a mathematician. Emphasis was given to the Hilbert space structure and an attempt of explaining why these mathematics are needed for quantum mechanics. I shall advertise the viewpoint that quantum mechanics is incomplete.

10. A General Introduction to Pure Mathematics and Theoretical Physics

I have been constantly giving introductory lessons on serious pure mathematics and theoretical physics, to both beginning undergraduate students and mature students. These lessons emphasise the thoughts behind the theory and the ideas that are intrinsic to the backbone of maths and science. Most applied aspects of the subjects are omitted or left as exercises. The goal is to give students a general taste of the true contents of modern mathematics and physics. Topics could range over everything mentioned on this page, depending on the needs and background of the student. And yes, it is possible to teach higher-level courses such as manifold theory and quantum field theory to beginners, although some proper preliminary discussions are required.

__________________

I have also taught A-Level Further Maths once in the past.

Previously when I was an undergraduate student, I worked as a 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.

About the lesson

Having trouble following your lectures in university/high 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 makes things extra intuitive and extra clear; for most of the time, you just won't 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.

Courses that I can teach include:
(1) Anything at the pre-university level, e.g., A-level further maths and A-level physics;
(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;
(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);
(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., Maxwell's equations and 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 see the "About Yi" section for courses that I have taught before. If you ask me to teach a university-level course that I have not taught before, I might offer you a discount.

Please note that technically I belong 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 the 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).

Review

All of our reviews are collected by us and are 100% reliable. They correspond to a real experience lived by students with Yi.

5 (2 reviews)

J

Joe
- 5
Perfect! Very clear explanations and offered good insight into problems.
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J

Joe

Perfect! Very clear explanations and offered good insight into problems.
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Recommendations

Recommendations come from relatives, friends and acquaintances of the teacher

M

Mia

Yi’s explanations of topology penetrate deep into the concepts. I’ve now acquired an extremely intuitive picture of everything, which I will never gain by attending university lectures or by reading textbooks. Yi’s lessons are highly insightful, and he’s able to connect everything that I’ve learned previously. Highly recommended.
Read more

M
Mia
Yi’s explanations of topology penetrate deep into the concepts. I’ve now acquired an extremely intuitive picture of everything, which I will never gain by attending university lectures or by reading textbooks. Yi’s lessons are highly insightful, and he’s able to connect everything that I’ve learned previously. Highly recommended.
View more recommendations

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Oxford DPhil student offering maths and physics lessons up to university/MSc level

Lesson location

Recommended

About Yi

About the lesson

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Recommendations

Rates

Rate

Pack rates

online

free lessons

Other tutors in Mathematics

Sofia

Jose

Josiah

Lyle

Joe

Fadil

Philip

Reza

Natalie

Matias Ignacio

Yannik

Jody

Seren

Sevan

Cynthia

Usman

Drew

Bryan

Laura

Steve

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