Discovering Mathematics Through Computer Experiments
The course goal is to introduce students to the use of computing in mathematics to explore mathematical problems containing rich behaviors and including unsolved problems at the edge of research. The topics covered often have deep mathematical theory and structure, well beyond what can be covered in the summer course, but have interesting behavior that can be discovered through simulation.
We will start by learning Python skills, which will be immediately applied to exploring problems. We will explore new mathematical concepts working together in small groups, spending one or two days per topic, covering the background but quickly diving into simulations. Students will observe the behavior, formulate their own questions and computational experiments, and show off their ideas to the rest of the group.
Problems explored include the study of dynamical systems, probability, mathematical physics, and math biology, seeing examples of simple systems that exhibit chaotic behavior as well as extremely complex systems that exhibit formation of coherent patterns.
This course is offered at the UMass Amherst campus as a residential program. Local students may apply to attend as a commuter.
- Lectures and demonstrations of mathematical concepts
- Problem-solving exercises
- Small group work
- Python coding practice
Pre-Requisites
Students should have strong algebra skills.
Materials
Students should bring:
- a laptop
- notebook and pens/pencils
By the end of the course, students will be able to:
- Turn a mathematical algorithm into a Python program
- Carry out simulations to explore a math problem and formulate hypotheses
- Use plots and graphics to help present their findings
Class time is Monday-Friday from 9 am - 4 pm.
|
Time |
Activity |
|
9:00–9:40 AM |
Card Trick 1, students learn trick, discover how it works, and generalize |
|
9:40–10:10 AM |
Card Trick 2, explain the trick and talk about randomness |
|
10:10–10:40 AM |
Random number generation in Python notebook |
|
10:40–10:50 AM |
Break |
| 10:50–11:50 AM |
Combinatorics and counting poker hands activity |
| 11:50AM–1:00 PM |
Lunch |
| 1:00–1:40 PM |
Shuffle techniques and poker hands activity with real decks |
| 1:40–2:30 PM |
Plotting empirical data and deck shuffling in Python |
| 2:30–2:40 PM |
Break |
|
2:40–4:00 PM |
Experimentation, presenting data and presentations |
In the evenings and on weekends resident counselors will run a series of social activities. Students are encouraged to join in, relax and have fun with new friends! With social events on campus and in the surrounding Amherst area, and access to the UMass Recreation & Wellness Center, there's something for everyone to enjoy.
Learn more about student life at UMass Amherst Summer Pre-College
Meet the Faculty
Matthew Dobson, Associate Professor, Department of Mathematics & Statistics
Matthew Dobson researches scientific computing of micro-scale physical systems where the interaction of randomness and deterministic laws govern the behavior. He enjoys making visualizations that are easy to share with a wide audience. His hobbies include strategy card games and hiking on the nearby trails in the Pioneer Valley.
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