LC 207 Mathematical Experiments in Computer Science

Meetings:  Wednesday  2:30 - 5:00,            Room 308 Stanger.

Description: Computers provide us with tools to explore mathematics in deeper ways than ever before.  They allow empirical testing of mathematical conjectures with elusive proofs.  Computers enable us to experimentally analyze algorithms whose performance defies theoretical analysis.  Mathematics provides a structure and formalism that turns the study of computers and their applications into a science and not just an engineering discipline.  This learning community focuses on the delicate balance between theory and practice in computer science, revealing the dual and sometimes contradictory nature of computer science as both an engineering and a mathematical discipline.

Goals:  To appreciate the symbiotic relationship between mathematics and computer science.  To understand how mathematics is used to formalize computer science, and how computer science can be used as a tool to explore mathematics.

Labs and Expectations:  The semester is divided up into approximately five different self-contained labs as time allows.  Each lab is divided up into three parts:

• Preparation and Programming
• Experimentation and Exploration
• Enrichment.
  

Everyone in class will work in groups of two or three.  Before each lab, each group will prepare by doing the appropriate readings and meeting in advance to discuss them.  In the first meeting of each lab, the topic of the lab will be introduced and reviewed by Professors Bravaco and Simonson, after which the groups will break out to work on their programming and problem solving.   This time will be intense but interactive.  It will be a collaboration as the groups use the problem sets to focus on learning the topic explored in that lab.  The programs are used to explore open questions about the lab and related topics.  The style will be fun and flexible.  The idea is to reap the rewards of all the programming work.  The explorations will be guided by Professors Bravaco and Simonson.

Finally, each lab has an  enrichment component that examines an aspect of the lab's topic relating to matters outside of computer science and mathematics.  It may involve videos, group games, history, theatre, field trips, or simulations.

Here is an example of one of our labs and its three parts.

This lab studies the abstract mathematics of Greece and the Renaissance and how it relates to modern cryptography.  Before the first meeting the students read and study a brief history of cryptography from ancient times to the present.

During the first two meetings, the students write programs to cement their understanding and implement various cryptographic methods.  They simultaneously write solutions to problem sets on number theory, to make sure their understanding of the foundations is clear.

During the third meeting, the programs are used to encrypt and decrypt various information.  A contest to see who can crack whose codes will commence.  Experiments as to the practicality of breaking codes and the safety of e-commerce will be performed.  Discussions of the future of the internet and its dependence on results in this area will complete the day.

Enrichment for this lab might discuss the life and times of Alan Turing, a famous pioneer computer scientist and mathematician who worked in England during World War II helping to crack German codes.  Turing is also famous for being an outspoken homosexual, who was tormented by his own country (for his sexual preferences) despite his heroic contributions to science and the war.  He eventually committed suicide at young age.  There is an excellent PBS video on Alan Turing and his life, as well as an acclaimed biography, and a Broadway play based on the biography.  Another possible area of discussion is the issue of government control over encrypted information, regulation versus freedom.

• A title page and introduction to the lab including how the math and computer science interact and assist each other.
  
• Summarize the purpose and results of each program, and display and analyze the results and output. Respond to any explicit questions, problems and/or essays.
  
• Impressions of the lab and/or enrichment: Discuss what you liked best, what you liked least, and what you learned.
  

Write with enough detail so that an educated reader can reconstruct the details of the lab, without having read through the web site's description.

• Lab 1   Fractals, Chaos, and Theatre
  

An exploration into complex numbers, the scientific theory of chaos, and computer graphics, through the award winning Tom Stoppard play Arcadia.

(If you want to do this lab using C++, you should  also download directions for using the CMU graphics package.  If you are using Java, you do not need any special graphics package).

• Lab 2  Cryptography
  

 An exploration into the world of abstract number theory and its application to the modern world of e-commerce.

• Lab 3  Compression
  

An exploration of the techniques used to send information in compressed formats.  Applications to the online music sharing craze.  (Here is the lab as a Word document with better layout -- Lab 3 ).

• Lab 4  The Four-Color Theorem 

• Interlude  Man Versus Machine

Comparing an algorithm to human performance in SameGame.

• Lab 5  A Mathematical Card Trick
  

A computational exploration of a trick where mathematical analysis falls short of a complete solution.