Survey of Modern Mathematics

Welcome!

Syllabus: Click here for a PDF

Course Calendar

Day 1: Monday, June 25

Introduction to the Course

Homework: Finsih the in-class worksheet from the afternoon, find somewhere on campus to access Mathematica, and come to class tomorrow with something cool that Mathematica can do (ie Google is your friend!)

Day 2: Tuesday, June 26

Differential Geometry

Homework: For the spiral curve in Example 5 from the lecture slides, do all of the following things:

  1. Draw the curve on the x-y plane.
  2. Compute the velocity vector and speed. Draw a few velocity vectors on the graph you drew in number 1.
  3. Compute the curvature.

Day 3: Wednesday, June 27

Differential Geometry

Homework: Complete this worksheet. You do not have to print the worksheet, and you will not have to hand in your work. However, we will be using the answers to these problems in class tomorrow, so it is imperative that you complete the worksheet before the beginning of the morning session. You are particularly encouraged to compare your answers with your classmates to check your work.
You should also choose a Mathematician for your presentation and let me know who you've chosen as soon as possible. No two people can present on the same person.

Day 4: Thursday, June 28

Differential Equations
  • Morning: Introduction to Differential Equations
  • Afternoon: Solving ODEs and Modeling Soutions using Mathematica (Click Here for the Worksheet)

Homework: Complete this worksheet. You should hand in your solutions at the start of the morning's session tomorrow.

Day 5: Friday, June 29

Differential Equations

No Homework! Enjoy the weekend!

Day 6: Monday, July 2

Algorithms
  • Morning: Introduction to Algorithms
  • Afternoon: Programming Algorithms in Mathematica
    • Implement the algorithms you designed in the morning. Don't forget to test that your algorithms work! Mathematica has a random number generator that you can use to create inputs to test your algorithm.
      1. Input an array of numbers from the user. Output the smallest number in the list.
      2. Input a number from the user. If the number is negative or 0, ask the user to input a positive number. If the number is positive, output the number of times the input can be divided by 2 before it is less than 1.
      3. Insertion sort: Input an array from the user. Put each element of that array into a new array in ascending order. Output the new sorted array. (Challenge: can you do this without creating a new array?)
      4. Selection sort: Input an array from the user. Take out the smallest number and insert into a new array. Then repeat with the next smallest number, and the next, and repeat until the old array is empty. Output the new sorted array. (Challenge: can you do this without creating a new array?)
      5. Merge sort: Input an array from the user. Split the array into two halves. Sort each half. Merge the two sorted halves so that the final array is sorted. Output the sorted array
    • Here are some useful links for implementing algorithms into Mathematica:
    • Here are some examples of other algorithms you can try to implement and test:

Homework:

  1. Watch this YouTube video about the fibonacci numbers.
  2. Compute the 3rd, 7th, and 12th Fibonacci numbers.
  3. Write an algorithm in pseudocode to input a positive integer n and then compute the nth Fibonacci number.
We will be using the algorithms you write for homework in an activity at the beginning of class tomorrow. Your algorithms don't need to be perfect, as we'll be editing them during the activity.

Day 7: Tuesday, July 3

Algorithms Continued
  • Morning: Working on Presentations
  • Afternoon: Sorting Algorithms and Mathematica
    • Implement the following algorithms into Mathematica:
      1. Insertion Sort (if you didn't already complete this yesterday)
      2. Selection Sort (if you didn't already complete this yesterday)
      3. Merge Sort (if you didn't already complete this yesterday)
      4. Quick Sort
      5. Bubble Sort
      6. Bucket Sort

No Homework! Enjoy the holiday!

Wednesday, July 4: No classes, enjoy the holiday!

Day 8: Thursday, July 5

Computational Geometry and Computational Complexity

Homework: Complete this worksheet. You'll be handing it in at the start of the morning class. You can either print the worksheet or write the answers on a separate piece of paper. We will be discussing the solutions at the start of tomorrow's morning session, so it is very important that you've completed it before then.

Day 9: Friday, July 6

Number Theory
  • Morning: Introduction to Number Theory (You'll want a calculator for this session. It doesn't need to be fancy - just something than can multiply, divide, add, and subtract)
  • Afternoon: Fast Modular Exponentiation in Mathematica
    • Implement the Fast Modular Exponentiation algorithm into Mathematica. You may find the following Mathematica functions useful:
    • If you finish early, work on your presentation

Homework: Finish the Fast Modular Exponentiation algorithm in Mathematica. You'll need this algorithm to be working correctly for the activity we're doing during Monday's class.

Day 10: Monday, July 9

Number Theory and Cryptography

Day 11: Tuesday, July 10

Fractals and MoMath

Day 12: Wednesday, July 11

Complex Numbers, Iteration, and Fractals

Day 13: Thursday, July 12

Fractals and Random Walks

Day 14: Friday, July 13

Wrap Up
  • Morning: Presentations
  • Afternoon: Creating Websites in Mathematica and Fun with Photos

Student Web Pages

At the end of the course, students compiled some of their favorite parts of the course into one Mathematica document which we exported as html files. Click on each student's name below to see their page.

List of Mathematicians

At the end of the course, each student will give a 3-5 minute presentation on a famous mathematician. The following is a list of potential subjects. Students should explore the list below and choose a person whom the student finds interesting. Students may choose to present about a mathematician not on this list, with approval from the instructor.

* indicates a female Mathematician.

In your presentation, you should tell us where and when your chosen Mathematician lived. Tell us a little bit about their life (many of these people had very interesting lives). Finally, tell us about some of the mathematics that they are known for - what made them famous? Why is their name still known today? I can help you with that last part, as some of the math might require further background than you currently have.

Other Resources

  • Codingame
  • Books
    • Letters to a Young Mathematician by Ian Stewart
    • Flatland by Edwin A. Abbott
    • Flatterland by Ian Stewart
    • Love and Math by Edward Frenkel
    • A Mathematician's Appology by G. H. Hardy
    • Great Feuds in Mathematics by Hal Hellman
    • How Round is Your Circle? by John Bryant and Chris Sangwin
    • The Man Who Knew Infinity by Robert Kanigel
    • The Man Who Loved Only Numbers by Paul Hoffman
  • Videos