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—Derek Youngman
Automatic theorem proving; see [2, Proposition 6.4.8]
Bernstein theorem; see [3, Theorem 5.4] or [10, Theorem 3.2]
Computations in local rings; see [3, Proposition 2.11]
Conditional independence models—Brandon Johnson
Descartes rule of signs—Taylor Crandles
Fröberg theorem—Cyril Ji
Grassmannians—Dom Kim
Generic initial ideals—Timofey Knyazhevskiy
Going-up theorem—Sylvie Skelton
Gröbner fan; see [3, Theorem 4.1]
Hilbert syzygy theorem—Antonio Nigro
Integer programming; see [3, Theorem 8.1.11] or [9, Theorem 5.5]
Invariant theory—Jeremy Hare-Chang
Lexsegment ideals; see [6, Theorem 6.3.1] or [7, Theorem 2.22]
Linear partial differential equations— Josef Naus
Multivariate polynomial splines; see [3, Proposition 3.7]
Multivariate resultants; see [3, Theorem 2.3] or [10, Theorem 4.4]
Newton polytopes—Bernd Zwanziger
Noether normalization; see [1, Theorem 10.2.1] or [4, Theorem 13.3]
Puiseux series; see [4, Corollary 13.15] or [10, Theorem 1.7]
Quillen–Suslin theorem; see [3, Theorem 5.1.8]
Real nullstellenstaz; see [10, Theorem 7.2]
Sagbi basis; see [9, Theorem 11.4]
Secant varieties—Hussam Hayek
Solving equations via eigenvectors; see [3, Theorem 2.4.5] or [10, Theorem 4.6]
Sums of squares—Yanbin Xu
Symmetric polynomials—Jingzhuo Zhang
Triangulations and toric ideals; see [9, Theorem 8.3]