Past Projects

Summer 2018

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

  • An Application of Clustering to Socioeconomic Data
    Student: 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 Geometry
    Student: 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 2017 Tufts Directed Reading Project students and mentors with one of the organizer and one of the faculty sponsors

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

  • Cantor Sets and the Cantor Surjection Theorem
    Student: 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 Theory
    Student: 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 Programming
    Student: 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 Paradox
    Student: 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 Theory
    Student: 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-adics
    Student: 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 Hypothesis
    Student: 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 Numbers
    Student: 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 Proof
    Students: 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.