Topology
- Grade 9+
- CTY-Level
-
Residential
- Mathematics
Topology is the mathematical study of shapes and space that considers questions such as, “What objects that visually seem quite different share the same properties?” One of the major fields of mathematics, topology possesses wide-ranging applications and beautiful theorems with far-reaching consequences. This course will introduce you and your classmates to point-set topology as you delve into bizarre notions of "space" and develop skills with rigorous, proof-based mathematics. You’ll begin by tackling the core concepts of sets, topologies, and continuous mappings before moving on to topological invariants such as compactness, connectedness, and the separation axioms. With these tools in hand, you will explore how to deform shapes and spaces without altering their fundamental properties. This knowledge allows you to see why it took 100 years for mathematicians to prove Poincaré’s 1904 conjecture about the nature of a sphere. Finally, you’ll survey different applications of topology, such as how the study of knots influenced our understanding of proteins, or how the study of manifolds led scientists to a deeper understanding of the topological shape of the universe.
Typical Class Size: 16-18
Summer Dates & Locations
Session One
Session Two
Testing and Prerequisites
| Math | Verbal | |
|---|---|---|
| Required Level | CTY-Level | Not required |
Students must achieve qualifying scores on an advanced assessment to be eligible for CTY programs. If you don’t have qualifying scores, you have several different testing options. We’ll help you find the right option for your situation.
Sign up for Testing Learn MoreCourse Prerequisites
Topology requires:1 prerequisite
Geometry and Algebra II
Cost and Financial Aid
Tuition
- Varies
Application fee
- Nonrefundable Application Fee - $50 (Waived for financial aid applicants)
- Nonrefundable International Fee - $250 (outside US only)
Financial Aid is available
We are committed to serving all talented youth regardless of financial circumstances. Financial assistance is available based on need.
Course Materials
Please acquire all course materials by the course start date, unless noted as perishable. Items marked as “perishable” should not be acquired until the student needs them in the course. If you have questions about these materials or difficulty locating them, please contact [email protected].
Sample Reading
These titles have been featured in past sessions of the course, and may be included this summer. CTY provides students with all texts; no purchase is required.
- Introduction to Topology, by Bert Mendelson
Technical Requirements
Students must bring a tablet, laptop computer, or Chromebook for use during the session. A smartphone will not be sufficient.
About Mathematics at CTY
Explore the study of shapes
Many of our courses allow students to describe the world around them in basic and profound ways. Our younger students learn about shape, scale, and proportion in Geometry and Spatial Sense. Middle School students explore beautiful real-world applications of lines; analyze data based on curves that fit a uniform, symmetric and bell-shaped, or skewed pattern in Data and Chance. And advanced students explore the underlying mathematics and fundamental characteristics of shapes, distance, and continuous deformations in our proof-based Topology course.
Dive deep into logic and reasoning
Our courses in formal logic give you the tools to question the world around you. Inductive and Deductive Reasoning introduces younger students to different types of reasoning, as well as the strengths and weaknesses inherent in various forms of critical analysis. Older students explore how logical reasoning can explain (or fail to explain) counter-intuitive results in Paradoxes and Infinities, or take a more rigorous approach to formal logic in Mathematical Logic.
Meet our instructors and staff
When I was a student at CTY, I was excited to be in a place where all of the instructors and program assistants loved learning as much as I did. It wasn't until I came back as a PA that I realized CTY instructors feel just as excited to work with such dedicated and curious students.
Nick Smith
Program Assistant and CTY Alum
There is nothing better than seeing that 'eureka' moment and the extension of that moment as students exuberantly share their thoughts.
Dan Sievers
Math Instructor
The students are so bright, interesting, and often willing to try some of the most silly and unique activities that the RAs create.
Rebecca Somer
Resident Assistant