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; see [7, Theorem 5.24] or [6, Subsection 1.5.3]
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; see [10, Proposition 8.1]
Descartes rule of signs; see [8, Theorem 2.1] or [10, Theorem 1.5]
Fröberg theorem; see [6, Theorem 9.3.3]
Grassmannians; see [1, Theorem 13.2.2] or [5, Proposition 11.30]
Generic initial ideals; see [4, Theorem 15.18] or [6, Theorem 4.1.2]
Going-up theorem; see [4, Proposition 4.15]
Gröbner fan; see [3, Theorem 4.1]
Hilbert syzygy theorem; see [3, Theorem 6.2.1] or [4, Theorem 15.10]
Integer programming; see [3, Theorem 8.1.11] or [9, Theorem 5.5]
Invariant theory; see [2, Theorem 7.3.5]
Lexsegment ideals; see [6, Theorem 6.3.1] or [7, Theorem 2.22]
Linear partial differential equations; see [10, Theorem 10.3]
Multivariate polynomial splines; see [3, Proposition 3.7]
Multivariate resultants; see [3, Theorem 2.3] or [10, Theorem 4.4]
Newton polytopes; see [9, Lemma 2.2]
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; see [5, Proposition 4.18]
Solving equations via eigenvectors; see [3, Theorem 2.4.5] or [10, Theorem 4.6]
Sums of squares; see [10, Theorem 7.3]
Symmetric polynomials; see [2, Theorem 7.1.3]
Triangulations and toric ideals; see [9, Theorem 8.3]
