Math 398: Research Experience in Mathematics
Summer 2026 Theme: Numerical invariants in prime characteristic

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Short Description

**satisfies Experiential Learning (EL) requirement **minimal enrollment 8

GOAL: To learn how to do research on a novel mathematical question; to develop an understanding of how algebra, number theory, and analysis combine to create the sub-field of “numerical invariants in prime characteristic”.

  1. Students will attempt to make progress on unanswered questions about the F-signature and the limit F-signature, which are numerical invariants that capture the geometry and number theory behind the graph of a polynomial f(x) whose coefficients are all integers.

  2. Students will also gain understanding of the logistics of mathematical research:

    1. finding and reading mathematical research papers,

    2. using computers to guide your intuition by generating examples and counterexamples,

    3. what goes into a good research presentation, and

    4. writing professionally and mathematically using LaTeX.

  3. A possible (but not guaranteed!) outcome is that students may coauthor an original research paper to be submitted for peer review and publication at the end of this course.

  4. Finally, this course will prepare students to do further research in mathematics, with the goal of being able to funnel their experiences into successful UCARE or external REU applications.

Prerequisites: knowledge on what a proof is (MATH 309, MATH 310, or MATH 325)

References:

  1. Measuring singularities with Frobenius: the basics by Benito, Faber, Smith
    This doesn’t actually discuss the F-signature or limit F-signature specifically, but does give a flavor of how analysis, algebra, and number theory are all related for a similar numerical invariant called the F-threshold!

  2. Course Webpage: https://anna-brosowsky.github.io/unl/math-398.html
    We will also use Canvas, but this page will contain some public info about the course that you don’t have to be enrolled in to read.

Long Description

“Hasn’t all of math already been figured out?”  No!  In this experiential learning class, you will work on an open question at the intersection of algebra, analysis, and number theory.  Your work this summer might lead to coauthoring original research papers!

In this course, students will undertake an original, collaborative research project. The main objective is for students to gain an in-depth understanding of and experience doing research in pure mathematics. In a typical mathematics class, exercises are pointed and concise, the instructor already knows and understands the solutions, and they expect students to solve the problems in a straightforward way. In contrast, in conducting mathematical research, the problems can be vague and open-ended, and no one has ever found a solution to these problems. Doing research can be intensely frustrating, but it can also be incredibly rewarding.

Students will also gain understanding of the logistics of mathematical research, and at the end of the course, they will know how to find and read mathematical research papers, they will understand what goes into a good research presentation, and they will write professionally and mathematically using the LaTeX package. They will also learn how to use computer tools (such as the computer algebra system Macaulay2 and our local high performance computing cluster, the HCC) to help generate examples.

A possible (but not guaranteed!) outcome is that students may coauthor an original research paper to be submitted for peer review and publication at the end of this course. Finally, this course will prepare students to do further research in mathematics, with a goal of being able to funnel their experiences into successful UCARE or external REU applications.

This summer, students will research the limit F-signature, a numerical invariant for singularities in prime characteristic. So what does that mouthful of words really mean? An F-signature takes in a polynomial and a prime number, and spits out a number between zero and one that measures something about the geometry of the graph of that polynomial. A limit F-signature turns a polynomial into a number by taking a limit of the possible F-signatures as the prime numbers used go to infinity. Mathematicians are hoping that this number will again measure something about the geometry of the polynomial... but the problem is that we know almost nothing about it! Does this limit even exist? When it does exist, can we give a formula for it? Our job this summer will be to see what we can discover about it.

This question can be tackled from many angles: analysts might want to study the limit, number theorists might focus on understanding how the F-signature itself is affected by the choice of prime number, and algebraists might dig into how polynomials are used to actually compute the value.