Learn a new theorem related to the course material and communicate it
as a written document and via a video presentation.
Minimum Requirements
Each student will focus on a different result.
The written document will introduce/motivate, correctly state, and
prove a theorem. It will also include at least one interesting
example, construction, or special case illustrating the theorem. The
article will be as self-contained as possible. The new document must
be typed, be at most eight pages in length (with one inch margins and
a 12pt font), and be available in the PDF format.
The video presentation must introduce and state the theorem. It
should also include at least one example, construction, or special
case illustrating the theorem. This new video must be at most
20 minutes in length and available in a common format such as the MP4
file type.
By design, this project is very open-ended. Students are strongly
encouraged to create their own examples. Consider what was the original
motivation or historical context for your theorem. Does your theorem
have any interesting specializations or important applications?
Potential Topics
The following are natural candidates:
Alexander duality
— SANTIAGO GUTIERREZ
Automatic theorem proving
— JOHN ALAJAJI
Bernstein theorem
— JINGJING MAO
Budan-Fourier theorem
— COLE GIGLIOTTI
Computations in local rings
— DOMINIC AUSTRIA
Conditional independence models
Descartes rule of signs
— SUNNIE ZHANG
Elliptic curves
— GRAYSON PLUMPTON
Fröberg theorem
Grassmannians
Generic initial ideals
Going-up theorem
— SOPHIA SHEN
Gröbner fan
Hilbert syzygy theorem
— JULIA MCCLELLAN
Integer programming
— TREVOR SHILLINGTON
Invariant theory
— ANGUS MCGREGOR
Kushnirenko theorem
Lexsegment ideals
Linear partial differential equations
— HOWIE HONG
Local-global principle
— YUXIAN AN
Multivariate polynomial splines
— BORIS ZUPANCIC
Multivariate resultants
— JAMES CONNELLY
Newton polytopes
— ALEXANDER KAPTY
Noether normalization
Puiseux series
Quillen–Suslin theorem
Real nullstellenstaz
— XINYI LIANG
Sagbi basis
Secant varieties
Semidefinite programming
— ALICE PETROV
Solving equations via eigenvectors
— CLAIRE VANDESANDE