Fall 2020 : Math 100 Introduction to the Profession

Math 100 Introduction to the Profession

Instructor: Hemanshu Kaul
E-mail: kaul [at] iit.edu

Class Time: 9:40-10:55am, Monday and Wednesday
Place: Blackboard Live Classroom

Office Hours: Monday at 3:30-4:30pm and Tuesday at 4:30-5:30pm on Google Meet (link will be shared through IIT Email and Calendar).

Discussion Forums: Math 100 Discussion Forums at Blackboard.

TA Office Hours: Gunjan Sharma. Monday and Wednesday at 4:30-6pm. Accessible through Virtual Math Learning Center.

Course Information:

This course is an introduction to the nitty-gritty of work and study in the mathematical sciences. Fresh applied math majors are introduced to undergraduate studies and careers in applied math; to the effect of ethics, diversity, and related social issues to work and study in mathematics; to the study of mathematical statements, logic, and proofs; to experimentation and computation in mathematics; and they work on a semester-long group project that investigates a mathematical topic.

The Course Information Handout has extensive description of the course - topics, textbook, student evaluation policy, as well as other relevant information. Read it carefully!

The end-of-semester letter for students: What Next?

Advice for students:

Here is my Welcome Letter to students like you.

Excellent advice by Francis Su on good mathematical writing.

Why do we have to learn proofs?
Understanding Mathematics - a study guide
On a more abstract note, here is a discussion of Language and Grammar of Mathematics - which is what you are learning in a course like this.

Excellent advice for math majors, especially those planning to go on to graduate school, by Terry Tao, 2006 Fields medallist. Required reading.

Read this book on a variety of experiences in the journey to learn mathematics: Living Proof

Some of the primary sources of information/discussion for careers in Mathematical Sciences:
MAA - Careers
SIAM - Careers
INFORMS - Careers
AMS - Careers

Class Announcements:

Monday, 11/16 : The university has finally announced the final exam schedule. See below for updated deadlines for Project Submissions and Presentation.

Monday, 9/28 : Note the change in Office hour on Tuesday to 4:30-5:30pm.

Thursday, 9/24 : See below for Project topics and updated deadlines.

Wednesday, 9/23 : See below for Special Lecture dates and Speakers.

Monday, 9/7 : See above and in the Course Information Handout below for TA office hours information.

Sunday, 8/23 : Check this webpage regularly for weekly lecture topics, videos, and HW.

Course Topics:

Undergraduate studies in Applied Math - During and Afterwards : Careers - Documentary, discussions, presentations by instructor and by Career Management Center. Managing Courses, how to study, and making the jump from High School Math to Undergrad Math.
Ethics in Mathematical Sciences : How ethics, diversity, and related social issues affect work and study in Mathematics and its applications.
Mathematical Statements and Logic : Second order logic and simple mathematical statements, Comparison of Existential and Universal statements, the conjunction, disjunction, and their negations. Comparing Mathematical statements to real-life statements. Understanding Mathematical definitions and Statements of Theorems.
Mathematical Proofs : Classical examples of proofs (and non-proofs) from Geometry, Number Theory, Set Theory. Different types of proofs and how to find them and how to write them - Direct implications, Contradiction, Contrapositive, Mathematical Induction.
MATLAB : How to use MATLAB to learn experimentation and computation in Mathematics - Fibonacci Sequence, Matrices, etc.
Special Lectures : Lectures by faculty in Applied Math introducing different fields/topics within Applied Math and their connection to real-world, further courses in the curriculum.
Project : A semester-long group project on investigation of a mathematical topic.
Scavenger Hunt : (Virtually) Meeting faculty and learning through discussion.

Special Lectures:

Wed, 10/14: Professor Fred Hickernell (Computational Math)
Wed, 10/21: Professor Michael Pelsmajer (Discrete Math)
Wed, 10/28: Professor Lulu Kang (Statistics)
Wed, 11/4: Professor Igor Cialenco (Randomness and Probability)
Wed, 11/11: Professor Tom Bielecki (Mathematical Finance)

Projects:

The project requires you to work in teams of 2-3 students with the Professor offering the project. The project adviser will specify the details of what his/her project entails, the outlines are given in the descriptions below.
The successful completion of the project entails:

Before the project Starts -

1) Send me an email with your choice of project(s) on Monday, 9/28 after class. Projects will be assigned in first-come first-serve order.
2) After I confirm your project and team members by Thursday, 10/1, write an email to your chosen project adviser by Friday, 10/2 to set up an appointment with him/her.

After the project starts -

3) Regular meetings with the project adviser and following his/her instructions.
4) At the end of each week before Friday 11:59pm, each student will also write a short 2-3 line summary of what she/he did related to the project over the past 7 days. This will be directly written in the Google doc file for each project I will share with you.
5) A midterm report on the status of the project before Wednesday, 10/28 .
6) A final meeting with project adviser before Wednesday, 12/2 to confirm the completion of the project, and discuss the outcome of the project.
7) Submission of project report by email to me before the start of the Final exam week on Sunday, 12/6, 11:59pm.
8) Submission of project presentation file (as a PDF file) by email to me before the start of all presentations on Thursday, 12/10, 7:59am.
9) Final project presentation in front of your classmates during the Final exam week: Thursday, 12/10, 8-10am, on Blackboard.

Project Topics:

Project #1: Advisor: Prof. Shuwang Li.
Project #2: Advisor: Prof. Shuwang Li.
Project #3: Advisor: Prof. Hemanshu Kaul.
Project #4: Advisor: Prof. Hemanshu Kaul.
Project #5: Advisor: Prof. Hemanshu Kaul.
Project #6: Advisor: Prof. Hemanshu Kaul.

Weekly Class Log with Videos, Discussion Questions, and HW:

Week #1 :
- Topics and Readings:
  Introduction to the course and each other; President Garfield's proof of Pythagoras's Theorem; a foundational discussion of President Garfield's proof, and the geometric lemmas and axiom(s) underlying it.
  Lecture Recordings available at Blackboard Collaborate Ultra Recordings
- Homework:
  Reading HW (Be ready to discuss in class next week): Read Sections 3.1 to 3.7 of Alcock, How to Study as Math Major; Read Sections 2.1 and 2.2 of Devlin, Intro to Mathematical Thinking. Follow the HW instructions and suggestions in the Course Information Handout. When reading a mathematics textbook, you need to have a pencil and paper next to you so you can make note of any thoughts of confusion (or of clarity!) that strike you as you read, and you can scribble the details of examples and non-examples of the concepts you are reading.
  HW#1 for Submission. Due Wednesday, 9/2, by 11:59pm. Submit a PDF file through Blackboard Assignment.
  Questions? Ask on the Blackboard Discussion Forum. Or, send email.
Week #2 :
- Topics and Readings:
  Dissection of a square into smaller n squares; Axioms, Definitions, and related concepts; Truth table and comparing logical statements; Proof by contradiction - There are infinitely many primes; Universal vs Existential statements - how to prove and disprove such statements; All real numbers are irrational vs. There is a real number that is irrational; Proof of sqrt(2) is irrational.
  Lecture Recordings available at Blackboard Collaborate Ultra Recordings
- Homework:
  Reading HW (Be ready to discuss in class next week): Read Sections 4.1, 4.2, 4.5, 4.6, 4.7, 4.8 of Alcock, How to Study as Math Major; Read Sections 2.3 and 2.4 of Devlin, Intro to Mathematical Thinking. Follow the HW instructions and suggestions in the Course Information Handout. When reading a mathematics textbook, you need to have a pencil and paper next to you so you can make note of any thoughts of confusion (or of clarity!) that strike you as you read, and you can scribble the details of examples and non-examples of the concepts you are reading.
  HW#2 for Submission. Due Wednesday, 9/9, by 11:59pm. Submit a PDF file through Blackboard Assignment.
  Questions? Ask on the Blackboard Discussion Forum. Or, send email. Or, attend TA or Instructor's Office Hours.
Week #3 : Labor Day holiday.
- Topics and Readings:
  Discussion of proof and disproof of two statements with multiple quantifiers and interchanged "for all" and "there exists" quantifiers from Alcock. Discussion of HW problems from Devlin.
  Lecture Recordings available at Blackboard Collaborate Ultra Recordings
- Homework:
  Reading HW (Be ready to discuss in class next week): Read Sections 4.9-4.10, 5.1-5.6 of Alcock, How to Study as Math Major; Read Sections 3.1-3.3 of Devlin, Intro to Mathematical Thinking. Follow the HW instructions and suggestions in the Course Information Handout. When reading a mathematics textbook, you need to have a pencil and paper next to you so you can make note of any thoughts of confusion (or of clarity!) that strike you as you read, and you can scribble the details of examples and non-examples of the concepts you are reading.
  HW#3 for Submission. Due Wednesday, 9/16, by 11:59pm. Submit a PDF file through Blackboard Assignment.
  Questions? Ask on the Blackboard Discussion Forum. Or, send email. Or, attend TA or Instructor's Office Hours.
Week #4 :
- Topics and Readings:
  Discussion of Mathematical Statements, implications, truth tables, quantifiers, and HW problems from Devlin. Discussion of how to understand "product of three consecutive integers is divisible by 6" from Alcock through examples and non-examples, a non-proof, a case-by-case proof, and finally a short proof that captures the mathematical truth underlying the case-by-case analysis.
  Lecture Recordings available at Blackboard Collaborate Ultra Recordings
- Homework:
  Reading HW (Be ready to discuss in class next week): Read Sections 6.1-6.4 of Alcock, How to Study as Math Major; Read Sections 3.4-3.5 of Devlin, Intro to Mathematical Thinking. Follow the HW instructions and suggestions in the Course Information Handout. When reading a mathematics textbook, you need to have a pencil and paper next to you so you can make note of any thoughts of confusion (or of clarity!) that strike you as you read, and you can scribble the details of examples and non-examples of the concepts you are reading.
  HW#4 for Submission. Due Wednesday, 9/23, by 11:59pm. Submit a PDF file through Blackboard Assignment.
  Questions? Ask on the Blackboard Discussion Forum. Or, send email. Or, attend TA or Instructor's Office Hours.
Week #5 :
- Topics and Readings:
  Discussion of 5 types of proofs with examples: Direct argument, Case-by case analysis, Proof by Contrapositive, Proof by Contradiction, Principle of Mathematical Induction. Discussion of some old and new HW problems.
  Lecture Recordings available at Blackboard Collaborate Ultra Recordings
- Homework:
  Reading/Preparatory HW (for next week): We will begin our study of MATLAB as a tool for mathematical experimentation from next week. We will access MATLAB through the IIT's Virtual Computing Lab (VCL) which is accessible through your my.iit.edu account. Try using the VCL on your own before classes next week and email me/OTS if you face any issues. See FAQs regarding VCL. Refer to VCL Instructions here.
  Watch/ Read the following and be ready to try examples in class next week:
  - MATLAB Overview
  - Getting Started with MATLAB
  - MATLAB Development Environment
  - Preface of Moler's Textbook
  - Chapter 1 of Moler's Textbook (We will start with these examples next week.)
  HW#5 for Submission. Due Wednesday, 9/30, by 11:59pm. Submit a PDF file through Blackboard Assignment.
  Questions? Ask on the Blackboard Discussion Forum. Or, send email. Or, attend TA or Instructor's Office Hours.
Week #6 :
- Topics and Readings:
  Discussion of projects and their structure and expectations. Discussion of how to study an infinite tower of sqrt(2)^sqrt(2)^ ....: mathematical expression, proof of its properties using mathematical induction, and value of the expression. Discussion of HW problems and use of properties of integers, proof by contrapositive, and modeling a pigeonhole principle problem with a proof by contradiction.
  Lecture Recordings available at Blackboard Collaborate Ultra Recordings
- Homework:
  Reading/Preparatory HW (for next week): Review the HW problems all the previous HWs and be ready to discuss solutions in class for ones you find difficult/challenging with my help.
  Review the MATLAB reading HW from last week.
  Read the following and be ready to try examples in class next week:
  - Chapter 1 of Moler's Textbook (We will start with these examples and problems next week in class. See below for the problems you will have to submit by the end of the week.)
  HW#6 for Submission: Solve and submit the following problems from the Chapter 1 of Moler's Experiments with MATLAB 1.2, 1.3, 1.4, 1.6, 1.7, 1.8, 1.11.
  Due Monday, 10/12, by 11:59pm. Submit a PDF file or .m file through Blackboard Assignment.
  Questions? Ask on the Blackboard Discussion Forum. Or, send email. Or, attend TA or Instructor's Office Hours.
Weeks #7 & #8 : Fall Holiday on 10/12.
- Topics and Readings:
  - Statement of Pigeonhole principle and how it can be proved using contradiction, More examples of modeling a problem with pigeonhole principle - handshake problem with or without requiring at least one handshake, pigeonhole principle within case-by-case analysis; Discussion of recursive definitions in math (using our old example of tower of sqrt(2), and in MATLAB, Discussion of MATLAB chapter 1 and the related HW problems.
  - Special lecture on 10/14 by Professor Hickernell (Computational Math).
- Homework:
  Reading/Preparatory HW (for next two weeks; due to holiday on 10/12 and special lecture on 10/14): Work on your projects and send me weekly updates.
  Read the following and be ready to try examples:
  - Chapter 2 of Moler's Textbook (We will start with these examples and problems in the week after next in class. See below for the corresponding problems you will have to submit.)
  HW#7 for Submission: Solve and submit the following problems from the Chapter 2 of Moler's Experiments with MATLAB 2.3, 2.4, 2.6, 2.9.
  Due Monday, 10/26, by 11:59pm. Submit a PDF file or .m file through Blackboard Assignment.
  Questions? Ask on the Blackboard Discussion Forum. Or, send email. Or, attend TA or Instructor's Office Hours.
Weeks #9 & #10 :
- Topics and Readings:
  Discussion of
  - how to overcome roadblocks and difficulties faced as a student - asking for help from professors and colleagues, building a support network around you through your family, friends outside IIT, friends and acquaintances in IIT, mentors like Professors or senior students, attending academic and social events (like Math tea, SIAM chapter, AWM, ML, etc.).
  - Discussion of Research opportunities with professors and through REUs.
  - Discussion of MATLAB chapter 2 and the related HW problems.
  - Special lecture on 10/21 by Professor Pelsmajer (Discrete Math).
    Special lecture on 10/28 by Professor Kang (Statistics).
- Homework:
  Reading/Preparatory HW : Work on your projects and send me the required weekly updates.
  Read the following and be ready to try examples:
  - Read Chapter 3 and Chapter 4 of Moler's Experiments with MATLAB. And be ready to work on problems listed below in HW#8.
  HW#8 for Submission: Solve and submit any 5 out of the following 6 problems from the Chapter 4 of Moler's Experiments with MATLAB 4.3, 4.4, 4.5, 4.6, 4.7, 4.10ab.
  Due Monday, 11/16, by 11:59pm. Submit a PDF file or .m file through Blackboard Assignment.
  Questions? Ask on the Blackboard Discussion Forum. Or, send email. Or, attend TA or Instructor's Office Hours.
Weeks #11 & #12 :
- Topics and Readings:
  Discussion of
  - the speech that the outgoing President of MAA (Mathematical Association of America), Francis Su, gave at the Joint Mathematical meetings in January 2017, available at Math for Human Flourishing. His theme is the ideals of doing mathematics as a part of a complete and fulfilling life. And mathematics can not live up to those ideals unless it is inclusive to all regardless of their gender or cultural background. He discusses these issues further in an interview
  - Further topics of inclusiveness in Mathematics at this blog hosted by AMS (American Mathematical Society) that has frequent discussions on these issues: Inclusion-Exclusion Principle.
  - Role models:
    If you want to learn more about "ordinary" women who were part of the sciences many decades ago despite the tremendous social obstacles they faced, take a look at entries in the blog "Grandma got STEM". The math entries are listed here.
    February is the black History month and on that occasion four mathematicians created a list of inspirational black mathematicians.
    This website was inspired by an earlier one featuring inspirational Hispanic mathematicians from the Hispanic Heritage Month in September-October.
  - Discussion of MATLAB chapter 4 and the related HW problems (see above).
  - Special lecture on 11/4 by Professor Cialenco (Randomness and Probability).
    Special lecture on 10/28 by Professor Bielecki (Mathematical Finance).
    Lecture Recordings available at Blackboard Collaborate Ultra Recordings
  - Discussion of midterm Project reports with each team, and feedback for their current and future work on the project.
- Homework:
  Reading/Preparatory HW : Work on your projects and send me the required weekly updates.
  Read the articles linked above related to Francis Su's address and to Inspirational Mathematicians.
  Read and attempt the topics and HW problems from HW#8 above on Matlab.
  
  HW#9 for Submission: One 400-500 word essay on a topic related to Mathematical culture, inclusiveness, or ethics, inspired by one or more of the following articles/topics:
  - Math for Human Flourishing
  - Mathematicians from: "Grandma got STEM"; or inspirational black mathematicians; or inspirational Hispanic mathematicians; or another such inspirational mathematician you would like to suggest.
  - Seven Ethics Questions.
  - Ethics of Automated Prison Sentencing.
  - Any other issues related to ethics of mathematical/technological applications in real life. You can find useful resources at Markkula Center for Applied Ethics.
  Due Wednesday, 11/25, by 11:59pm. Submit a PDF file through Blackboard Assignment.
  Questions? Ask on the Blackboard Discussion Forum. Or, send email. Or, attend TA or Instructor's Office Hours.
Weeks #13 & #14 : Thanksgiving Holiday.
- Topics and Readings:
  Discussion of
  - (continuation) topics related to Mathematical culture, inclusiveness, and ethics. How Mathematical models and techniques underly modern advances in Technology and our responsibility for ethical applications of these techniques to real world problems. What is Ethical Decision making? Five perspectives on Ethics and Ethical Reasoning. Further resources related to Ethics applied to real world problems can be found at Markkula Center for Applied Ethics.
  - Useful Resources related to course planning, and mathematical internships available at IIT Applied Math Student Resources.
  - Visits by 3 current junior and senior students to share their social and academic experiences and advice with new students.
  - A Final Review of older topics on Logic and Different Proof techniques.
- Homework:
  Reading/Preparatory HW : Work on your projects and send me the required weekly updates.
  Be ready to discuss your draft essay for HW#9 above.
  Submission: Note the deadlines for submission of Project Report and Project Presentation, as listed in the Project section above.
  Questions? Ask on the Blackboard Discussion Forum. Or, send email. Or, attend TA or Instructor's Office Hours.
Weeks #15
- Topics and Readings:
  - Discussion of "Impossible Questions" in mathematics - How do we show something is impossible in mathematics? Question 1: Construction of an 8-gon with 4 interior angles of 90 degrees? Non-convex? Convex? Sum of interior angles of an n-gon? What about non-Euclidean Geometry? Question 2: Construction of an (Eulerian) walk through bridges of Konigsberg? Translation to Graph Theory and a necessary condition.
  - Visits by five more faculty members to introduce themselves and their work.
- Homework:
  Submission: Note the deadlines for submission of Project Report and Project Presentation, as listed in the Project section above. And the requirements for the two submissions as described in email to al students.

MATLAB Information:

Our textbook:
Moler's Experiments with MATLAB

MATLAB Access:
We will access MATLAB through the IIT's Virtual Computing Lab (VCL) which is accessible through your my.iit.edu account.
See FAQs regarding VCL.
Refer to VCL Instructions here.

Introduction to MATLAB:

Matlab Links for More Information

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