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 [5, Theorem 5.24] or
[4, Subsection 1.5.3]
Automatic theorem proving; see [1, Proposition 6.4.8]
Computations in local rings; see [2, Proposition 2.11]
Conditional independence models; see [8, Proposition 8.1] —
Harley Easton
Descartes rule of signs; see [6, Theorem 2.1] or
[8, Theorem 1.5] — Emily Cullingworth
Fröberg theorem; see [4, Theorem 9.3.3]
Generic initial ideals; see [4, Theorem 4.1.2] or
[3, Theorem 15.18]
Going-up theorem; see [3, Proposition 4.15] — Ben Syms-Wilson
Gröbner fan, see [2, Theorem 4.1]
Hilbert syzygy theorem; see [2, Theorem 6.2.1]
or [3, Theorem 15.10] — Gabby Wolfe
Integer programming; see [2, Theorem 8.1.11] or [7, Theorem 5.5]
— Junhee Park
Invariant theory; see [1, Theorem 7.3.5] — Curtis Wilson
Kushnirenko theorem; see [6, Theorem 3.2] — Konstantin Uvarov
Lexsegment ideals; see [4, Theorem 6.3.1] or [5, Theorem 2.22]
Linear partial differential equations; see [8, Theorem 10.3]
Multivariate polynomial splines; see [2, Proposition 3.7]
— Hayden Pfeiffer
Newton polytopes; see [7, Lemma 2.2]
Noether normalization; see [3, Theorem 13.3] — Neil Utom
Puiseux series; see [3, Corollary 13.15] or [8, Theorem 1.7]
Quillen–Suslin theorem; see [2, Theorem 5.1.8]
Real subvarieties; see [8, Theorem 7.2]
Sagbi basis; see [7, Theorem 11.4]
Strickelberger theorem; see [2, Theorem 2.4.5] or
[8, Theorem 4.6]
Symmetric polynomials; see [1, Theorem 7.1.3] — Sara Stephens
Triangulations and toric ideals; see [7, Theorem 8.3]
Universal Gröbner bases; see [7, Theorem 7.1] — Victor Bubis