Part of the final exam requirement for each course in the MASS program is the presentation of a topic chosen by the student, researched by him or her in advance. Here are some of the topics students have presented in their final exams.

MASS 2014

Finite Fields and Applications

An Introduction to Dynamical Systems

Affine and Projective Geometries

Semester-long Research Projects

MASS 2013

Number Theory in the Spirit of Ramanujan

Elements of Functional Analysis

Winding Number in Topology and Geometry

MASS 2012


Random Walk and Brownian Motion

An introduction to geometric topology in dynamics

MASS 2011

Introduction to Ramsey Theory

Spaces: from geometry to analysis and back

From Euclid to Alexandrov: a guided tour

MASS 2010

Function field arithmetic

Differential equations from an algebraic perspective

Dynamics, mechanics and geometry

MASS 2009

Groups and their connections to geometry

Complex analysis from a fluid dynamics perspective

Explorations in convexity

MASS 2008

Elliptic Curves and Applications to Cryptography

Elements of Fractal Geometry and Dynamics

Introduction to Symplectic Geometry

MASS 2007

Computability, Unsolvability, and Randomness

Topics in Probability Theory

Surfaces: Everything You Wanted to Know about Them

MASS 2006

Finite Fields and Their Applications

Mathematical Theory of Waves

Aspects of Symmetry: From Representations of Quantum Field Theory

MASS 2005

Integer Partitions

P-adic Analysis in Comparison with Real

Geometry and Billiards

MASS 2004

Finite Fields and Applications

Introduction to Dynamical Systems

Differential Geometry and Topology of Curves and Surfaces

MASS 2003

Number Theory with Applications to Communication Networks

Topological Dynamics

Geometry and Relativity: An Introduction

MASS 2002

Number Theory with a Tilt at Elliptic Curves

Intuitive Topology

MASS 2001


Geometry and Relativity

Fluid Dynamics

MASS 2000

Finite Groups, Symmetry, and Elements of Group Representations

Projective and Non-Euclidean Geometries

Real and P-adic Analysis

MASS 1999

Topics in Number Theory

Geometric Structures, Symmetry, and Elements of Lie Groups

Mathematical Methods in Mechanics

MASS 1998

Number Theory

The Exponential Universe

Functions and Dynamics in One Complex Variable

MASS 1997

Arithmetic and Geometry of the Unimodular Group

Real Analysis

Explorations in Geometry

MASS 1996

Explorations in Number Theory

Introduction to Dynamical Systems

Linear Algebra in Geometry

Stories about Research in MASS

[ Photograph of Dumas and Kouchnirenko]
Director of MASS Program, A. Kouchnirenko, presents David Dumas an award for his research project, MASS-96

As a student in the MASS-96 program at Penn State, I was presented with a number of challenging problems in the three principal areas of coursework that were intended to spawn in-depth research into these fields on the part of MASS students. One such problem that I studied over the course of the semester involved the trinomial analogue of Pascal's triangle. It was then presented as a problem to show that the center column of the triangle had a particularly simple rational generating function. After some research into combinatorics and generating-functionology, augmented by discussion and consultation with the MASS teaching assistants, I was able to prove the identity in a rather unusual way.

David Dumas, MASS-96
currently a graduate student at Harvard University

For my final exam presentation in Geometry, I started to investigate the Hausdorff metric. With the help of Dr. Burago and Eric Johnson, I delved into proving Blaschke's Selection Theorem, which states that given a bounded sequence of convex bodies, there exists a subsequence which converges to a convex body. I presented that topic for my final exam. When I returned to Juniata College the following semester, I was asked to present this topic as part of the series of mathematics seminars given by invited professors. I was honored, and I'm thankful to both Juniata and to the MASS program for the experience and confidence I received.

David Shoenthal, MASS-97
currently a graduate student at Penn State

The quadratic (logistic) family ƒ(x, λ) = λx(1 − x) of the maps of the unit interval is probably the most extensively studied model in the whole area of dynamical systems, both rigorously and numerically. Existence of a period 3 orbit for an interval map implies existence of orbits of all other periods. This is a particulat case of the celebrated theorem by A. Sharkovsky which was re-discovered by Li and Yorke who called their paper Period three implies chaos, and thus coined the famous term chaos. In the quadratic family a periodic orbit of period 3 appears at the value of λ = 1 + √8. This was known since 1950's but the original proof by P.J. Marburg uses sophisticated methods of the theory of analytic functions. In his popular textbook Nonlinear dynamics and chaos (p. 363) Steve Strogatz tells the story how he suggested to find this value in his advanced class at MIT and how several (rigorous) solutions, some of them using computer algebra, were produced. I repeated Strogatz's experiment in my MASS dynamical systems class. An Nguyen, a participant in MASS Program from University of Texas, Austin not only found the sacramental value, but he did something apparently not known before: he found (also rigorously) the value at which the second period four orbit appears (the first one appears at λ = 1 + √6). The Nguyen's value of λ is 1 + √(4 + ∛108) ≈ 3.9601… and this is quite possibly the last major bifurcation value which can be expressed in radicals.

Anatole Katok, MASS-96 instructor
An Nguyen is presently a second year graduate student in computer science at Stanford.

The learning experience that I gained by working with Dr. Kouchnirenko showed me how to apply abstract ideas in mathematics to very specific examples, especially in a computer programming environment. I was a bit disappointed that I didn't have time to finish the program entirely, but I was able to hack the code enough to work a few nontrivial problems. The program attempts to find roots to polynomials with two variables using a homotopy proceedure (creating a family of polynomials from the given polynomial and a polynomial with known roots), Newton polyhedrons, convexity, and Newton approximation (using only the linear term / first derivative of the given polynomial). While working with Dr. Kouchnirenko I also learned his very unique style of solving problems, something of which I hope to develop in my own style some day.

Chris Staskewicz, MASS-97
currently a graduate student at the University of Utah

Homotopy Program
Homotopy Program Description