Students majoring in mathematics complete a capstone project during their senior year to fulfill the Senior Comprehensive Requirement. The capstone project is either a Senior Comprehensive Project or, for honors students, an Honors Project in mathematics. The capstone project is an opportunity for the student to deeply explore a mathematical topic under the guidance of a faculty member and share their insights with others.

Senior Comprehensive Projects

For a Senior Comprehensive Project, the student works with a professor to select a topic that relates to but goes beyond the usual scope of a 300- or 400-level mathematics course, writes a paper on the topic, and gives a talk based on the paper to students, faculty, and guests. 

Students often select topics that relate to their interests beyond mathematics. For example, a student minoring in finance may choose a mathematical topic that relates to finance.

The student takes the one-credit course MATH 492 during their graduation semester.

Honors Projects

The Honors Project is a year-long endeavor in which an honors student works intensively on the definition and solution of a scholarly problem, or on the development of a creative work. The honors student’s work is guided by an adviser and submitted to two readers for approval. Upon completion, it is publicly presented to interested faculty, students and guests.

An Honors Project in mathematics with a grade of C or better fulfills the Senior Comprehensive Requirement.

An honors student takes the three-credit course HON 498/499 during their senior year. Further details about a Degree with Honors are provided in the Honors Program website.

2025–2026

• Madigan Collver: “The Structure of Solutions: A Concise Take on the Unsolvability of the General Quintic Polynomial by Radicals.”
  Mentor: Dr. Christopher Hill.

  Using John Stillwell’s paper on Galois Theory as inspiration, Madigan provided a concise proof of one of the great “impossibility" results in mathematics: general polynomial equations of degree five or higher cannot be solved using the four basic arithmetic operations and radicals, despite the existence of such formulas for solving quadratic, cubic, and quartic equations. The proof provided a tour through many topics of abstract algebra, including groups, rings, and fields.

• Danielle Folland: “Using the Cauchy Residue Theorem to Sum the Basel Series.”
  Mentor: Dr. Christopher Hill.

  Danielle's project was a nice companion to Keon Cruz's project the year before. Danielle showed how complex analysis can be used to compute the sums of certain challenging real series. She used the Cauchy Residue Theorem to prove a powerful "master theorem," which provides the sums of many series of a certain form. Danielle applied the theorem to sum the famous Basel series: $$\sum_{n=1}^{\infty}\frac{1}{n^2} = \frac{\pi^2}{6}.$$ The Basel series was named after the hometown of Leonhard Euler, the first person to find its sum. Danielle included Euler's original argument, which, while making certain unjustified assumptions, yielded the right answer.

• Robert Hermansen: “Proving the Bertrand–Chebyshev Theorem.”
  Mentor: Dr. Christopher Hill.

  Robert proved the Bertrand–Chebyshev Theorem, also known as Bertrand’s Postulate. This elegant fact states that for any integer \(n>1\), there exists a prime number \(p\) such that \(n < p < 2n\). Initially conjectured in 1845 by Joseph Bertrand, Pafnuty Chebyshev provided the first full proof in 1852. Robert's proof follows a shorter argument published by Paul Erdős in 1932. Robert also demonstrated a logical connection between the Bertrand–Chebyshev Theorem and the powerful Prime Number Theorem.

  

• Faith Pucci: “Exploring the Cantor Set.”
  Mentor: Dr. Christopher Hill.

  Faith investigated the remarkable Cantor set, which is a certain subset of the interval [0,1]. She proved that the Cantor set \(C\) is both uncountable and has measure zero. To prove the uncountability of \(C\), Faith used base-3 and base-2 representations of real numbers. Faith's proof also used the uncountability of the interval [0,1]. For completeness, she included Georg Cantor's original proof of this fact, which is widely regarded as one of the most beautiful proofs in all of mathematics.

2024–2025

• Keon Cruz: “Using the Cauchy Residue Theorem to Solve Complicated Real Integrals.”
  Mentor: Dr. Christopher Hill.

  Keon's project was inspired by his interest in problem solving. He showed how the Cauchy Residue Theorem (CRT) can be used to evaluate certain real definite integrals that are challenging to compute using traditional methods. His project included a sketch of a proof of the CRT.

• Julia Marinaro: “Cruising Cubes: Minimizing Turns of Hamiltonian Cycles in 3D.”
  Mentor: Dr. Christine Uhl.

  Julia sought the minimum number of turns that can occur for a Hamiltonian cycle in an \(m \times n \times \ell\) grid. Julia's project builds on the work in a paper appearing on arXiv. Julia gave exact answers for special cases including \(2 \times 2 \times 2\) and \(3 \times 3 \times 3\) cubes. She then described her progress in generalizing her findings to any \(m \times n \times \ell\) grid.

2023–2024

• Walter Kinder: “On Lebesgue’s Theorem: A Mathematical and Historical Exploration.”
  Mentor: Dr. Christopher Hill.

  Walter proved part of Lebesgue's theorem: If a bounded function on a closed, bounded interval has a set of discontinuities of measure zero, then the function is Riemann integrable. He illustrated the result with an example of a function that is Lebesgue integrable but not Riemann integrable. Walter's project included the fascinating historical context for Lebesgue's work.

• Nathan King: “Redistricting from a Mathematical Standpoint.”
  Mentor: Dr. Christine Uhl.

  Nathan explored some of the promising mathematical approaches to detecting and reducing gerrymandering in setting up congressional districts. Nate first discussed partisan symmetry and the efficiency gap. He then described the Markov Chain Monte Carlo (MCMC) method, which takes advantage of the enormous computing power at our disposal.

• Kaylee Middaugh: “Mini-Sudokus and Equivalencies.”
  Mentor: Dr. Christine Uhl.

  Kaylee's project concerned Sudoku grids, which are solutions to Sudoku puzzles. The number of "essentially different" \(9 \times 9\) Sudoku grids has been determined — it's over five billion. Kaylee's project described the work in a paper that used group theory to count the essentially different mini-Sudokus, which are analogous \(4 \times 4\) grids.

• Theodore Murphey: “Bayesian Updates in Golf Matches.” (Honors Project)
  Mentor: Dr. Michael Klucznik.

  Teddy combined his love of mathematics with his love of golf by exploring how Bayesian methods could be used to address the problem of "sandbagging" in the use of golf handicaps. Sandbagging refers to a player artificially inflating his handicap index in order to improve his chances of winning matches. Teddy's work allows sandbagging to be addressed within a single match, thanks to a program he wrote in Python based on Bayesian techniques.

2022–2023

• Brett Chiodo: “Gauss’ Proof of Fermat’s Last Theorem for \(n = 3\).”
  Mentor: Dr. Christopher Hill.

  Brett's project related to one of the most famous problems in all of mathematics: Fermat's Last Theorem (FLT). Pierre de Fermat stated in 1637 that the equation \(x^n + y^n = z^n\) has no positive integer solutions when \(n > 2\), a claim that became known as his "last theorem." It would take over 350 years for a proof to be found for Fermat's claim. Along the way numerous special cases of FLT were proved, including the case \(n = 3\) by Euler. Brett followed the work of Carl F. Gauss by proving more generally that the equation \(x^3 + y^3 = z^3\) has no nontrivial solutions in the Eisenstein integers, which is a certain ring of complex numbers containing the ordinary integers.

• Vincent Pasquale: “Applications of Survival Analysis.”
  Mentor: Dr. Michael Klucznik.

  Vinnie's project focused on survival analysis, which is an area of statistics concerned with time-to-event data for the deaths of living creatures or the failures of machines. Vinnie provided a survival analysis for a COVID-19 study and a Primary Biliary Cholangitis (PBC) study. His analysis involved statistical methods such as Kaplan-Meier curves and Cox Proportional hazards and the programming tool R. Vinnie's discussion included insights into the risk factors associated with death for COVID-19 patients and PBC patients.