If you are an undergraduate, I am happy to mentor your senior project whether you are doing an expository paper or a research paper. I am also happy to work with you on a separate Research Experience for Undergraduates -- this is also true even if you are not an NDSU student.
In the vaguest terms, I am especially interested in any topic involving polynomials, or more generally algebra or geometry. These topics can range from very theoretical mathematics to very applied topics related to computer science or economics. Below is a (very non-exhaustive) short list of possible projects; if you are interested in something not on this list, I am probably still happy to mentor you -- just reach out, and let's talk about it!
Potential Expository Projects
Bézout's theorem or The Multi-homogeneous Bézout theorem
These theorems formalize and generalize the idea that two plane curves of degree d and e should intersect in de points, e.g., a line should intersect a parabola in two points.
This theorem states that although you can generalize the quadratic formula to degree 3 or 4 polynomials, it is impossible to generalize to degrees 5 or higher, e.g., there is no general quintic formula.
This slightly more abstract theorem proves the polynomial version of the famous ABC conjecture.
Potential Research Projects
The structure of lines on surfaces with a lot of lines
There are certain complex surfaces which are known to have "a lot" of lines, sometimes even hundreds or thousands, but there are many open questions about the structure of the lines on these surfaces. In this project, you would work out this structure by working out patterns in how the lines intersect each other.
Studying lines on surfaces over finite fields
Most surfaces of degree at least four do not contain any lines. However, special surfaces do have (lots of) line(s).
In this project, you could classify how many lines lie on each surface of a fixed degree over a fixed finite field. After working out examples, you could make and potentially prove conjectures about what happens in general. This project would be especially amenable to a student with a bit of programming experience.
Alternatively, you could work out fewer examples but study them in more depth by understanding the structure of the lines one each surface, e.g., which intersect which, how big is the largest set of lines which don't intersect, etc.
Automorphism groups of surfaces over finite fields
Similar to the previous project, you would consider all the surfaces of a fixed degree over a fixed finite field. Instead of considering lines, you would consider the matrices, i.e., linear transformations of three space, which fix the surface -- these are called automorphisms of the surface, and the group they form is called the automorphism group of the surface. Similar to the first version of the previous project, some coding experience would help.
The effective cones of nested Hilbert schemes
This slightly more abstract problem would require a bit more time and background. In the project, you would compute a geometric invariant called "the effective cone" of certain spaces which each parameterize collections of points on a surface.