# Fall 2019

In Fall 2019 we had four student-mentor pairs working on four different topics! The undergraduate participants gave talks on their projects in a department colloquium in December.

- Category TheoryStudent: David Tu

Mentor: Rylee Lyman

Resources: Category Theory in Context by Riehl

- CryptographyStudent: Katherine Harkness

Mentor: Matt Friedrichsen

Resources: Introduction to Mathematical Cryptography by Hoffstein, Pipher, and Silverman

- Elliptic CurvesStudent: Alex Wei

Mentor: Daniel Keliher

Resources: Rational Points on Elliptic Curves by Silverman

- p-adic NumbersStudent: Natalie Bohm

Mentor: Curtis Heberle

Resources:p-Adic Numbers by Gouvea

# Summer 2018

The students for Summer 2018 gave presentations in the department colloquium on October 12th, 2018.

- An Application of Clustering to Socioeconomic DataStudent: Eva Sachar

Mentor: Casey Cavanaugh

Resources: Murphy, Machine Learning: A Probabilistic Perspective

Abstract: When analyzing socioeconomic data we wish to uncover its inherent and underlying structure. We will be presenting a few approaches to clustering and discussing their advantages and disadvantages when applied to a housing dataset, and see if the results of clustering on property characteristics and census block demographics accurately reflect tiering in housing prices. - Fractal GeometryStudent: Carter Silvey

Mentor: Matthew Friedrichsen

Resources: Falconer, Fractal Geometry: Mathematical Foundations and Applications

Abstract: Fractals are some of the most beautiful and mysterious things to come out of mathematics. I’m going to discuss the geometry behind these fractals, such as how they are created and their dimensions. Specifically, I will talk about the Middle Third Cantor Set, Julia Sets, and the Mandelbrot Set as well as some applications that fractal geometry has in both the realm of mathematics and the real world.

# Summer 2017

The students for Summer 2017 gave presentations in the department colloquium on October 6th, 2017.

- Cantor Sets and the Cantor Surjection TheoremStudent: Glenn VanWinkle

Mentor: Curtis Heberle

Resources: Pugh, Real Mathematical Analysis

Abstract: An introduction to the standard Cantor set and its properties, followed by a demonstration of the proof of the Cantor Surjection Theorem, which states that every compact metric space is the continuous image of a Cantor set. - Spectral Graph TheoryStudent: Eduardo Barrera

Mentor: Joanne Lin

Resources: Chung, Spectral graph theory; Babai, Linear algebra notes

Abstract: Starting with one of the earliest problems in graph theory, the bridges of Königsberg, we will introduce the history and basics about Graph Theory. With an understanding of graph Laplacian, we will discuss some popular applications that derive from solving graph Laplacian problems. Finally, we focus on one particular area, Matchings in Bipartite Graphs, and walk through the theories, algorithms and applications that apply to this subject. - Linear ProgrammingStudent: Wenbin (Astrid) Weng

Mentor: Nate Fisher

Resources: Chvatal, Linear Programming

Abstract: Were you to cut different sizes of papers out from large pare rolls for your clients, how would you cut to minimize the waste? If you play the game of Morra with your friend, is there a way to secure your gain or protect you from losing? This talk is to demonstrate the practical implications of linear programming on real-world problems, including cutting stock problem and matrix games, and modification required to solve practical challenges efficiently.

# Summer 2016

The students for Summer 2016 gave presentations in the department colloquium on October 7th, 2016.

- The Banach-Tarski ParadoxStudent: Ruth Meadow-McLeod

Mentor: Rylee Lyman

Resources: S. Wagon, The Banach-Tarski Paradox

Abstract: The Banach-Tarski paradox asserts the striking fact that one can chop up the unit ball in R^3 into a few pieces and, just by rotating the pieces, create two new balls with the same volume. We will walk through the construction and touch on the connection to actions of free groups. - Ehrhart TheoryStudent: Zach Munro

Mentor: Murphy Fields

Resources: M. Beck and S. Robins, Computing the Continuous Discretely

Abstract: Given a polygon, we can compute its “discrete area” by counting the number of integer lattice points that it contains. In this talk, we look at Pick’s theorem, which gives a relation between this notion and our standard, continuous definition of area. We’ll also discuss some problems for which finding discrete areas is a useful method, and finally, touch on how this can be generalized to higher dimensions. - p-adicsStudent: Ben Hansel

Mentor: Jue Wang

Resources: F. Gouvea, p-adic Numbers

Abstract: It is well known that the real numbers are constructed via “filling in the gaps” between rational numbers based on the absolute value of their difference. If we instead consider a new kind of distance with respect to a prime p called “p-adic absolute value” then repeat such completion process, we will enter the realm of “p-adic numbers”, where the information about divisibility is encoded in the distance. They are wildly different from the real numbers yet analogous to them in many ways. In modern number theory, p-adic numbers play a central role by allowing analytic methods to be used in algebra and number theory. In this talk, I will explain the construction of the p-adic numbers in an elementary and natural way, and introduce some interesting basic properties and applications. - The Riemann HypothesisStudent: Ryan Kohl

Mentor: Nariel Monteiro

Resources: B. Mazur and W. Stein, Prime Numbers and the Riemann Hypothesis

Abstract:The Prime numbers have fascinated mathematicians for millennia, eluding any attempt to describe them in simple terms. In the 19th century, a powerful new technique to analyze the behavior of the prime numbers was introduced: the complex valued Zeta function. The Zeta function has been used to prove important results about the prime numbers and is key to the Riemann Hypothesis, perhaps the most famous unsolved problem in mathematics. This talk will explore how developments in number theory motivated the investigation of this peculiar function, and what its behavior implies about the distribution of the primes.

# Summer 2015

The students for Summer 2015 gave presentations in the department colloquium on September 18, 2015.

- Continued Fractions and Transcendental NumbersStudent: Freddy Saia

Mentor: Michael Ben-Zvi

Resources: A. Ya. Khinchin, Continued Fractions and Allen Hatcher’s Topology of Numbers

Abstract:We will begin by talking about continued fractions — what they are, some of their properties, and the notation we will use to discuss them. We will then venture into seemingly unrelated territory, presenting and proving Joseph Liouville’s theorem on diophantine approximation. This theorem was used by Liouville in the 1840s both to prove the existence of transcendental numbers (real numbers which are not roots of any non-zero polynomials with rational coefficients) as well as to give specific examples of such numbers, dubbed Liouville numbers (only a proper subset of the transcendentals, though still uncountable). In this vein, we will use the properties of continued fractions we had previously discussed to display a quick, neat construction of Liouville numbers. - Gödel's ProofStudents: Matt DiRe and Ryan Hastings-Echo

Mentor: George Domat

Resources: E. Nagel, Gödel’s Proof

Abstract:At the turn of the last century, mathematicians like Hilbert and Russell attempted to unify all branches of mathematics within a single logical calculus. But, at the time, there were no tools available to settle questions of inconsistency (contradictions derived within a system) or incompleteness (existence of true, expressible statements with no formal proof) within a formal system. In 1931, Kurt Gödel introduced a method, now know as Gödel numbering, which allowed him to translate meta-mathematical statements into well-defined arithmetic relations expressible within the calculus. Using Gödel numbering, he was able to explicitly express the statement “this statement is not expressible” within the calculus, thus deriving a critical contradiction. The nuanced logic of this proof was complex yet irrefutable, and it provided a definitive answer to the problem of inconsistency and incompleteness.