Winter 2021
This Winter 2021, the important dates and times (tentative) for WDRP are as follows:
Wednesday, January 6th, 5:00 pm: Start-of-quarter kick-off event via Zoom.
Wednesday, February 10th, 5:00 pm: Mid-quarter event (for undergraduate students only) via Zoom.
Wednesday, March 10th, 5:00 pm: End-of-quarter presentations via Zoom.
Outside of these times, undergraduate students and graduate students are expected to meet for 1 hour per week at a time of their choosing. In between meetings, undergraduate students are also expected to spend 4 hours per week reading and working on their own. Graduate students are expected to take the requisite time prepare for meetings on their own time as well. All meetings will be virtual.
Projects and Participants
Mathematics for Sustainability
Participants: Rachel Ren
Mentor: Sam Fairchild
Description: We will follow the book “Mathematics for Sustainability” which is written for a broad audience. Regardless of your background this book is designed to build mathematical knowledge and reasoning to help us better engage in studying, advocating, and embracing sustainability.
Let’s Talk About Grad School
Participants: Titus Kariuki, Kheng Rowen Lapastora, Carey Zhou, Kylee Hoffman
Mentor: Paige Helms
Description: Applying to graduate programs in math and math-adjacent areas can be a scary and sometimes mystifying process. In this project, we will have weekly meetings to discuss what grad school is like, flesh out the details of the application process, and workshop materials. The culminating presentation should synthesize the most important things you learn about the process, and will help communicate what you learned to all of our other WDRP friends!
Vector Calculus and Geometry
Participants: Fran Herr, Chengyuan Liu
Mentor: Kuan-Ting Yeh
Description: Vector calculus is extremely useful in many fields of science. We will introduce the notion of differential forms to study the basic theorems in vector calculus like Green’s theorem, Stokes theorem, and Gauss theorem. Our goal is to apply those theorems to some surfaces and understand their geometric behavior in a higher dimension!
Error Correcting Codes
Participants: Alex Reiswig
Mentor: Catherine Babecki
Description: Information all around us is sent as strings of 0’s and 1’s, but sometimes the process of sending it corrupts the information. How can we detect or even correct errors in a message sent through a noisy channel? We will start with Chapter 1 of “Strange Curves, Counting Rabbits, and Other Mathematical Explorations” by Keith Ball, and see where that takes us.
M.C. Escher and Hyperbolic Tessellations
Participants: Haley Riggs
Mentor: Josh Southerland
Description: In the middle of the twentieth century, M.C. Escher produced a series of prints entitled “Circle Limit”. These fascinating prints consist of rather bizarre looking tessellations: the tessellations shrink and seemingly vanish along the perimeter of a circle. In this reading project, we will study the mathematical underpinnings of these tessellations. Surprisingly, the ideas behind these tessellations are connected to many topics in modern mathematics.
Structural Balance in Signed Social Networks
Participants: Seoyoung Cho
Mentor: Megan Morrison
Description: Structural balance theory posits that nodes in certain types of social networks are pressured to have balanced relationships with other nodes — “the enemy of my enemy is my friend”. Balanced node triads are stable and consist of either one or three positive edges while unbalanced triads are not stable and consist of either one or three negative edges. We will explore how to use tools from graph theory to find balanced versus imbalanced triads in a network, measure balance levels, and create a differential equation that describes how edge weights change in response to balancing forces. As an application, we will analyze balance levels in the European international system during the 19th and 20th centuries. This analysis may help us understand the contribution balancing forces made to the formation of alliances and rivalries between European countries during certain time periods.
Algorithms
Participants: Kasper Lindberg, Kevin Kim
Mentor: Sami Davies
Description: We’ll study algorithms from the perspective of theoretical computer science. The specific types of algorithms that we will explore are completely up to the student’s interest though; some topics that might be a good place to start include graph theory, resource allocation, and learning theory.
Ill- posed problems, Inverse Problems, and numerical application
Participants: Jason Miller, Zhiyang Li
Mentor: Kirill Golubnichiy
Description: We are doing computational ill-posed problems using minimization of Tikhonov functional and machine learning afterwards. Supervised Machine learning has been applied to the binary classification neural network for the logistic probability loss function with regularization.
Category Theory
Participants: Keyan Ding
Mentor: Eric Zhang
Description: Category theory is about structures and structure-preserving functions. For instance, group is an algebraic structure and group homomorphism is the corresponding structure-preserving function. In another context, continuous maps are structure-preserving functions for topological spaces.
Don’t be intimidated by the word “structure.” Here is an intuitive example: in a plane, a line is mapped to another line by a continuous function. It can’t be transformed to two disconnected pieces of line segments. The abstruse wording “structure” is captured by the visually intuitive concept of connectedness.
We will read together on Paolo Aluffi’s Algebra: Chapter 0 and teach each other. The focus will be on group theory.