August 18, 2018

# Sample Projects

Based on ideas from graduate students at UW, here are some sample projects you could be working on through the WDRP. The official project proposals for Fall 2018 will be visible in the application, which will be open to to undergraduate students from 9/18 to 9/24. Note that you do not need to have prior experience with a topic or even fully understand the project description in order to work on it!

**Convex Neural Codes**

Motivated by patterns of neural activities, use some geometry and combinatorics to explore neural codes from a mathematical perspective.

*Background needed: Any level!*

**Representation Theory**

Learn about understanding abstract algebraic structures using linear algebra techniques. For ideas on potential topics, take a look at Bruce Sagan’s book.

*Background needed: Math 308 (Or equivalent linear algebra experience)*

**Ideals, Varieties and Algorithms**

Learn about computational algebraic geometry by understanding systems of polynomial equations (ideals), their solutions (varieties), and how to manipulate these objects (algorithms), with material coming from Cox, Little and O’Shea’s book.

*Background needed: Calculus sequence and Math 308*

**Chaos, Fractals, and Dynamics**

Depending on background and time commitment, the goals for this project could range anywhere from notably interesting and elementary ideas like understanding the difference between countable and uncountable infinities and how to apply a Cantor diagonal argument to prove Cantor’s set is uncountable (with lots of pictures along the way), to much more involved ideas like beginning to understand the meaning behind the Mandelbrot set (google it, pretty pictures!) and the behavior of cycles of the doubling map in the complex plane.

*Background Needed: Minimum of Math 125. Your background will help you determine your project of choice.*

**Hyperbolic Geometry**

Learn the difference between Euclidean and Hyperbolic geometry, which is the geometry behind surfaces of constant negative Gaussian curvature. We would explore this topic using the book by James Anderson.

*Background Needed: None specified*

**Number Theory**

Topics explored could include Fermat’s sum of squares and solving Diophantine Equations, construction of the p-adic Rationals and examples of the Hasse Principle, the Futurama Theorem (yes proved for the show Futurama!), counting the orbits of a Rubik’s Cube, and many more exciting ideas.

*Background Needed: Math 125.*

**Computability Theory**

Follow along in this book by Rebecca Weber. The intro of the book states: “What can we compute—even with unlimited resources? Is everything within reach? Or are computations necessarily drastically limited, not just in practice, but theoretically? These questions are at the heart of computability theory.”

*Background Needed: An interest in the topic!*

**Convergence Spaces**

These are a slight generalization of topological spaces, where you can work on understanding similar tools.

*Background Needed: Math 441*

**Linear Algebra (Done Right!)**

Learn about Linear Algebra from the view of linear transformations, using the book of Sheldon Axler. This provides a great way to solidify Linear Algebra knowledge and a perfect way to start working with proof-based mathematics.

*Background Needed: Math 308, Math 300 preferred.*

**Geometry and Geometric Flows**

Follow a book by Sibley called “Thinking Geometrically: A survey of geometries” or take some time learning about geometric flows.

*Background Needed: Math 124-5-6 and 324.*

**Have a different topic you want to explore? **

You can always find a graduate student or contact us to make suggestions and add preferences and we can match you with a graduate student of similar interests.