### MATH 401 – Fall 2022

This website is a simplified version of the Sakai site for this course.

Here is the **Syllabus**!

Lecture 1

Lecture 2

Lecture 3

Lecture 4

Lecture 5

Lecture 6

Lecture 7

Lecture 8

Lecture 9

Lecture 10

Lecture 11

Lecture 12

Cycles and Transpositions

Lecture 13

Lecture 14

Lecture 15

Lecture 16

Cosets and Lagrange’s Thm

Lecture 17

Lecture 18

Lecture 19

Lecture 20

Lecture 21

Lecture 22

Ideals and Quotient Rings

Lecture 23

Lecture 24

Lecture 25

#### Lecture 9

Group Theory

Cayley Tables

Group or Not Group?

#### Lecture 10

#### Lecture 11

#### Lecture 12

Array and Cycle Notations + Reading: pg 105 – 111 Gallian’s book

#### Lecture 13

#### Lecture 16

#### Lecture 21

#### Topics

- Symmetries of the platonic solids.
- Cayley digraphs, Hamiltonian circuits/paths, and applications.
- The group structure of the Rubik’s cube.
- The Chinese remainder theorem for rings, and applications of the version in the integers.
- Constructible numbers, doubling the cube, squaring the circle, and trisecting an angle.
- Plane isometries and wallpaper groups.
- Abstract algebra behind the bitcoin protocol

#### Audience

Just Professor Arcila-Maya!

#### Time

The presentation will be 30 minutes long. Each student in the group must speak for 10 minutes on the topic.

**Presentation Rubrics**

Your presentation must include the following:

- Introduction: objectives and motivation.
- Addressing of the questions:
*what is being done*,*how it’s being done*, and*why it’s being done*. - Examples: trivial and nontrivial.
- Conclusions.

**Grading Criteria**

Below is how group presentations will be graded; grading will be based on both technical and content issues.

Technical issues, in short, are how well the audience (Prof. AM) understand the words and/or slides that your group is using (are you making sense, and speaking clearly, and are you acting as a group (good), or 4-5 individuals (not so good)?).

Content issues, in short, are based on the quality of content, and if the presentation showed a clear structure (introduction of topic, understanding of subtopics, conclusion/wrap-up).

| Grading Criteria | Total Points | Score |
---|---|---|---|

| Information is presented in a logical sequence (i.e. Introduction – Body – Conclusions). | 5 | |

| 5 | ||

Graphs/figures are clear and understandable. | 5 | ||

| Introduction is attention-getting, lays out the problem well, and establishes a framework for the rest of presentation. | 5 | |

Technical terms are well-defined. Examples are clear and relevant. | 5 | ||

Presentation contains accurate information. | 10 | ||

Material included is relevant to the overall message/purpose. | 5 | ||

Appropriate amount of material is prepared, and points made reflect well their relative importance. | 10 | ||

Propositions, proofs, examples, etc were obtained from reliable sources. Sources are cited. | 5 | ||

Conclusion “sums up” the goal that the presentation was focusing on. | 5 | ||

The presentation shows an understanding of the issues addressed. | 5 | ||

Mastery of the Subject(12 points) | Each speaker is able to answer questions professionally, rigorously and understandably. | 12 | |

| Each speaker maintains good eye contact with the audience and is appropriately animated (e.g. gestures, moving around, etc). | 6 | |

Each speaker uses a clear, audible voice. | 6 | ||

Transition between speakers is ordered. | 3 | ||

There is team collaboration and coherence throughout the presentation. | 3 | ||

Length of presentation is within the assigned time limits. | 5 | ||

Score | Total Points | 100 |

### Asking questions in class meetings by Professor Clark Bray

I encourage students to ask questions in class. This can be hard to do for some students, due perhaps to a combination of shyness and/or self-consciousness. Certainly all students should try to overcome these impediments, as these should never be the reason that you don’t ask a question in class. But there are also some related difficult questions — about what sorts of questions are the best to ask, and what the best venue is for which kinds of questions. __Here are some bits of advice that might help you when you are thinking about asking a question.__

- If your question is about a big idea that we have just introduced, or a subtle or complex step that your instructor has just done in an argument, keep in mind that probably other students are wondering about it too! If everyone waits for someone else to ask the question, then it never gets asked…

- In addition to the students that are wondering about the same question you are, note also that there are probably other students who may also have not understood, but who might not even have realized that they missed something!

- If you have a question about how an idea we are discussing in the lecture may relate to something from an earlier lecture, you might be on to a great observation. Very often these are wonderful questions! Depending on the situation your instructor might not be able to elaborate fully on such a possible connection, but they might either note that the full explanation is coming later in the course, or suggest that they could give you a full discussion in office hours.

- If it has been a while since anyone has asked a question, note that asking a question can be an appreciated change of pace for the lecture. So not only might it not be a distraction from the presentation, it might even help the presentation!

- Of course there are limitations on how many questions can realistically be asked in a class meeting; and time spent on questions is time that then cannot be spent on other things. A lecture is unfortunately different from private tutoring, in that all students have this shared interest. So some judiciousness with questions is appropriate.

The instructor will try to manage this somewhat; but your judgment in this is still important — if everyone filters themselves and asks only their best questions, then that leaves time for other students’ best questions, and the entire class benefits accordingly!

- If your question is about something relatively low-level (perhaps a point of algebra, in the computation of a triple integral), then you might consider not asking in class. These sorts of things can be clarified after class or in office hours, and missing a low-level point should not impede you from following the rest of the lecture.

Also, waiting for office hours will give you a chance to think about it on your own first, and possibly even figure it out for yourself. If you can do this, that’s the ideal — there is great benefit in working through something on your own. If you can do this, that is ultimately better for your learning.

On the other hand, if you think the instructor might have made an oops, and are pretty confident about it, it is entirely possible that you might be right! Something like that can be a great question, as correcting the record will help the rest of the class avoid future confusion when going over their notes.

- If you didn’t hear something that the instructor said, for whatever reason, ask the instructor to repeat what they said.

- Similar to the above, if you find yourself with only a partial understanding of something that your instructor said, and if you think that hearing it again would be all that you need to understand the rest of it, again ask the instructor to repeat what they said.

Keep in mind that it is not expected that students will fully understand absolutely everything immediately in a lecture. It is a healthy and important part of the learning process to grapple with confusion and overcome it; this sort of experience is essential to becoming better at learning math!

- If your question is a digression (for example, in a class on multivariable calculus, asking how the tools we study in that course are used to understand the heat equation in physics), then even though it might be a great question it is probably something that the rest of the class does not need to hear about. Such a question then might be a great choice to ask in office hours.

Of course when you are thinking about asking, you might not be entirely sure which of the above might apply to your question… No problem! If you ask, and if the instructor thinks there is a better venue for that question, they will just say so — no harm done!