Senior Theses
Mathematics majors at Pomona College all have to work on a year-long
project which culminates in the writing of a senior thesis. This is what
Pomona's mathematics department has decided will be the capstone senior
experience for our major. There is a wide range of the kinds of projects
that are acceptable: eg. original research, historical exposition,
expository writing on an advanced topic, etc. You can find more
information about the mathematics senior theses here.
Below is a list of students who worked with me on their theses with brief synopses of what their work was about:
- Benjamin Greenberg (Pomona College Class of 2009)
Thesis title: Godel's Incompleteness Theorems: Proofs and
Implications
Synopsis:
(from the abstract:)
In this paper, I explore Godel's Incompleteness Theorems with a
particular focus on how they apply to the formal theory Peano Arithmetic
with Exponentiation (PE). I prove that the First Incompleteness Theorem
holds for PE (PE is incomplete) using the notion of truth in PE and the
fact that PE is sound, and I discuss how this proof relates to the liar
paradox as well as Cantor's diagonal proof that the reals are
uncountable. I then give an outline of a proof that the Second
Incompleteness Theorem holds for PE (PE cannot prove its own
consistency).
- Ian Cunningham (Pomona College Class of 2008)
Thesis title: Anti-Groups
Synopsis:
(from the abstract:)
The numerous results contained in any group theory textbook are
ultimately obtained from three simple properties applied to a set and a
binary operation. The purpose of this paper is to explore the results of
changing these initial properties and comparing the results to those of
group theory. These new algebraic objects, called "anti-groups", are
less innately structured than groups, but their resulting properties
frequently complement the well-known properties of group theory in
interesting ways.
- Andreea Nicolae (Pomona College Class of 2008)
Thesis title: The Algebra of Tropical Matrices, and their
Application in Tropical Geometry
Synopsis:
(from the abstract:)
The purpose of this thesis is to offer an introductory exposition of
the applications of tropical algebra. I focus mainly [on] its
applications to linear algebra and the study of matrices, paying
particular attention to different methods of approaching matrix
ranks.
- Kimberly Walters (Pomona College Class of 2008)
Thesis title: Groebner Bases and their Applications to Integer Programming
Synopsis:
(from the abstract:) Linear programming problems involve optimizing a
linear objective function subject to a ļ¬inite set of linear constraints.
If the variables must take integer values, it is called an integer
program. These types of problems have many applications for practical
planning and decision making in mathematics, economics, business, and
engineering. This paper presents an approach to integer
programming problems that utilizes the properties of Groebner bases.
To go back to Gizem Karaali's main webpage, click here.