II - July 21–August 7, 2020
One year of algebra as well as a familiarity with how to perform mathematical proofs. Participants should be comfortable with math and logical reasoning.
“I have gained a greater understanding of mathematics and was exposed to challenging and interesting problems.” — From a program course evaluation
In this curricular option, which is intended for students who enjoy math and logical reasoning, participants develop a deeper appreciation for the richness of mathematics while further developing their thinking and problem-solving skills. The combination of subjects and methods enables participants to experience math in a way high schools are often unable to present it, approaching problems as open-ended opportunities for creativity, independent thinking, and intellectual excitement.
The course is divided into three units:
- Logic: An exploration of logic builds the groundwork for further mathematical reasoning. What is mathematical logic? How does one structure a proof? How can this framework be applied to philosophical thought? One topic covered is game theory (how mathematicians study decision-making), which has applications in fields such as economics, biology, and psychology.
- Counting and Probability: Abstract counting, in the form of combinatorics, serves as an introduction to set theory, which in turn sets the stage for probability theory. As an application, the class delves into cryptography, looking at how math can be used to create and read secret messages.
- Advanced Topics: In the final week, building on what we have done already and depending one students’ interests, we look at a few advanced topics. Potential topics include planetary orbits, symmetry, topology, common statistical fallacies, and Markov chains.
After exploring these various applications of mathematics, participants select and complete a group project based upon their own interests within the field. The course concludes with group presentations on those projects. As collaboration and communication are essential in modern science, the presentations offer a valuable opportunity to practice and receive feedback.
Andrew Lipnick is currently working towards a Ph.D in mathematics at the Courant Institute of Mathematical Sciences at New York University. He earned a B.S. degree summa cum laude with highest honors in mathematics and neuroscience from Brandeis University. Andrew enjoys applying mathematics to real-world problems and helping others do the same.
Joe Quinn holds a PhD in mathematics from The Graduate Center, CUNY, and publishes in the field of geometric topology. He has taught and designed math curricula for Hunter College, Johns Hopkins Center for Talented Youth, Bronx Community College, Matemorfosis CIMAT (Guanajuato), MoMath, and Bridge to Enter Advanced Mathematics. He is passionate about transmitting knowledge between the math research community and the rest of the world, for the benefit of both parties.