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 2012

### Polynomials

- Cyclotomic Polynomials - Barger, W
- Straight-Line Linkages - Chao, M
- Euler Pentagonal Number Theorem - Cheung, W
- Nonnegative Polynomials and Sums of Squares - Copenhaver, M
- Location of Zeros of the Derivatives of a Hyperbolic Polynomial - Dai, Y, Erhmann, D
- The Fundamental Group of the Complement of the Discriminant is the Braid Group - Harris, R
- Kontsevich-Ghys Theorem - Jeong, I
- Muirhead’s Inequality - Kraisler, J
- Irreducible Polynomials in Integral Domains - Lai, T
- The converse to the 4-vertex theorem - Paulson, E
- Spherical Harmonic Polynomials - Rustad, J
- Hilbert’s Basis Theorem and Nullstellensatz - Vuchkov, R

### Random Walk and Brownian Motion

- The Hastings Algorithms - Barger, W
- Martingale Proof of the Law of the Iterated Logarithm - Chao, M
- Erdos-Kac Theorem - Cheung, W
- Bernstein’s Inequality and Compressed Sensing - Copenhaver, M
- Page rank - Dai, Y
- Harmonic Functions and the Random Walk - Erhmann, D
- Young Diagrams and their Convergence - Harris, R
- The Aztec Diamond Theorem - Jeong, I
- Brownian Motion and the Laplacian - Kraisler, J
- Martingales and the Strong Law of Large Numbers - Lai, T
- Wigner’s Semicircle Law of Symmetry for Random Matrices - Paulson, E
- Random Linear Expanding Maps and Random Walk in Random Environment - Rustad, J
- Doob’s Optional Stopping Time - Vuchkov, R

### An introduction to geometric topology in dynamics

- Topological Entropy - Barger, W, Chao, M
- Translations on p-adic Numbers - Cheung, W
- Pontryagin Duality for Locally Compact Abelian Hausdorff Groups - Copenhaver, M
- Markov Partitions for Toral Automorphisms - Dai, Y
- The Billiard Problem - Erhmann, D
- Hamiltonian Dynamics and the Liouville-Arnold Theorem - Harris, R
- The Rotation Set - Jeong, I
- The Seifert-Van Kampen Theorem - Kraisler, J
- p-Adic Analysis vs Real Analysis - Lai, T
- Furstenburg’s Theorem for Commuting Expanding Maps of the Circle - Paulson, E
- Translations on p-adic Numbers - Rustad, J
- The Seifert-Van Kampen Theorem - Vuchkov, R
- Covering Maps of Graphs - Zhang, X

## MASS 2011

### Introduction to Ramsey Theory

- Expander Graphs, Ramanujan Graphs, and the Cheeger Inequality - Al-Gattas, S
- The Banach-Tarski Paradox - Caine, A
- Ramsey-Type Properties and Categories - Canton, E
- Ultrafilters and Van der Warden's Theorem - Jeong, I
- Euclidean Ramsey Theory: Coloring of triangles in the plane - Lishke, A
- Graph Ramsey Theory: Calculating Ramsey Numbers - Peoples, C
- The Green-Tao Theorem - Russell, M
- Dual Ramsey Theorem - Smith, S
- The Paris-Harrington Theorem - Taylor, B
- Roth's Theorem - Zanazzi, J

### Spaces: from geometry to analysis and back

- The Fourier Transform, Related Theorems, and Applications - Al-Gattas, S
- The Carleson-Hunt Theorem - Caine, A
- Compact Groups, Dual Groups, Haar Measure and Pontryagin-van Kamper Duality - Canton, E
- E_m games - Jeong, I
- Fourier Analysis: The Heisenberg Uncertainty Principle - Lishke, A
- Spectral Theory in Banach Algebras - Peoples, C
- Weierstrass Approximation Theory - Russell, M
- Banach-Mazur theorem - Smith, S
- Open mapping theorem - Taylor, B
- Entropy - Zanazzi, J

### From Euclid to Alexandrov: a guided tour

- A Special Subset of the Plane; Coverings and Non-coverings of Euclidean space - Al-Gattas, S
- Showing bounded orbits in a complete, proper, metric space - Caine, A
- The F-signature Map of Homogeneous Polynomials in Dimension Two - Canton, E
- Straight Geodesics - Jeong, I
- Convex Polyhedra - Lishke, A
- Triangulation of a Torus - Peoples, C
- Maerodimension - Russell, M
- Polyhedral Surfaces - Smith, S
- The Existence of 2-point sets - Taylor, B
- Projections of polyhedra - Zanazzi, J

## MASS 2010

### Function field arithmetic

- Mertens’ Theorem for Finite Fields - Berkowitz
- Laplacians on Graphs over Finite Fields - Bydlon
- Newton Polyons - Chou
- Continued fraction representation of Carlitz’s exponential - Hamel
- Quantum Calculus and Finite Fields with Combinatorics - Mitchell
- Cyclotomic Theory and Weddeburn’s Theorem - Ramirez
- Comparing the Quadratic Reciprocity Laws over Q and F_ q [T] - Reiner-Roth
- The Frobenius Map as an Analog of Differentiation - Warner
- Classical Modular forms and Modular Forms in Characteristic p - Wright

### Differential equations from an algebraic perspective

- Toeplitz Operator Algebras - Berkowitz
- Fourier Transform - Bonet
- Primer on Clifford Algebras - Bydlon
- Clifford Algebras - Chou
- Classification of real finite-dimensional Clifford algebras - Hamel
- The Algebra of Formal Pseudodifferential Operators - Hom
- Quaternions and Octonions: Background, History, and Applications - Immendorf
- Canonical Anti-commutation Relation - Kc
- Clifford Algebras - Mitchell
- The Riemann sphere and Differential Operators - Ramirez
- The Completion and Localization of C [x] - Reiner-Roth
- Weyl Algebra of a Symplectic Form - Warner
- Kashiwara’s Theorem - Wright

### Dynamics, mechanics and geometry

- Birkhoff’s Proof of Poincare’s Geometric Theorem - Berkowitz
- Birkhoff-Lewis fixed point theorem - Bydlon
- Lyapunov Functions - Chou
- Optimal Control through Physical Examples - Fuller
- Stability of the inverted pendulum and the topology of SL (2,R) - Hamel
- A Tale of two Cones… and a Bicycle Wheel - Hom
- A Review of the Lagrangian Top - Immendorf
- Bohlin’s Treatment of Kepler’s Problem - Kc
- Kepler’s laws of Motion from a Newtonian perspective - Ramirez
- Hamiltonian Systems and Symplectic Topology - Warner
- Optimal Control through Physical Examples - Wright

## MASS 2009

### Groups and their connections to geometry

- The First Homology Group and Fundamental Group - Aria Anvia
- Hyperbolic Tessellations - Patrick Brandt
- SL(k,Z) acts on torus T^k - Chu Yue Dong
- A construction of the Convex Regular Polytopes in all Dimensions - Thomas Eliot
- Inequivalence of the unit ball and polydisc - Marriott Grimes
- Meschke’s Theorem - Brandon Hepola
- Veech Surfaces - My Huynh
- Topics in SL (2,Z) - Alexa Kottmeyer
- Projective Geometry over Finite Fields - Michael Lengel
- The Maurer-Cartan Equations for the Euclidean group - Kaloyan Marinov
- Deducing the Hyperbolic Law of Sines and Cosines - Alexander Montoye
- Cosmic topology space time Manifold and Fundamental polyhedra - Jun Yong Park
- SO (3) and the Tits alternative - David Prigge
- Braids, Braid Groups and their Quotient Groups - Keith Shusterman
- Tiling the Hyperbolic Plane - Jacob Turner
- Higher Homotopy groups of Spheres - Kurt Vinhage
- Introduction to Ideal class groups - Haining Wang
- Regular Tessellations on Surfaces - Zipeng Wang
- Galois Theory of Linear Differential Equations - Ray Zaiter

### Complex analysis from a fluid dynamics perspective

- A Proof of the Jordan Curve Theorem - Aria Anvia
- Complex Analysis on the Riemann Sphere - Patrick Brandt
- Duality of two forces laws: Newton law of Gravitational and Hooke law in complex plane - Chu Yue Dong
- The Karman Vortex Street in the Complex Plane - Thomas Eliot
- Topics in ℂ^n - Marriott Grimes
- Differential Forms and Complex Analysis - Brandon Hepola
- Microscopic Swimmers and a Viscous Equivalent of the Blasius Theorem - My Huynh
- Lift on Two Wings: A Two-Circle Theorem - Alexa Kottmeyer
- Semi-holomorphic Functions and Compressible Flows - Michael Lengel
- The Generalized Stokes Theorem - Kaloyan Marinov
- Blood Flow through a Tube - Alexander Montoye
- Frisbee Flight - Jun Yong Park
- Properties of the Villat function - David Prigge
- Non-irrotational Flows and the Complex Plane - Todd Regh
- Non-Holomorphic Fractals - Matthew Roberts
- The Existence of the Complex Plane - William Thomas
- The Fractional Calculus - Jacob Turner
- Extending the Cauchy Integral Theorem to Fractal Curves - Kurt Vinhage
- Burger Equation, Complex-Holomorphic-Functions and strange solutions to some PDE-non-linear System - Ray Zaiter

### Explorations in convexity

- Segments of constant area - Aria Anvia
- Recent 4-vetrex theorems - Patrick Brandt
- Origami hyperbolic paraboloid - Chu Yue Dong
- Regular polyheda in dimension 4 - Thomas Eliot
- Noninscribable polyhedra - Marriott Grimes
- Differential forms in differential geometry - Brandon Hepola
- Spherical geometry - My Huynh
- Totally skew discs - Roman Kogan
- Quasi-crystals and Penrose tilings - Alexa Kottmeyer
- Lorentz geodesics and Cayley-type theorem - Michael Lengel
- Cartan’s method in differential geometry - Kaloyan Marinov
- Models of hyperbolic geometry - Alexander Montoye
- Steiner symmetrization - Jun Yong Park
- Schwarzian derivative and Ghys theorem - David Prigge
- Classical inequalities - Todd Regh
- Alexander’s horned sphere - Matthew Roberts
- Classical and hyperbolic geometries: a proof-based comparison - William Thomas
- Hilbert’s geometry - Jacob Turner
- Crofton formula and its ramifications - Crofton formula and its ramifications
- Tropical geometry - Haining Wang
- Classical inequalities - Ray Zaiter
- Curves similar to their evolutes - Adam Zydney

## MASS 2008

### Elliptic Curves and Applications to Cryptography

- Endomorphism rings and supersingular elliptic curves - Charles Baker
- Supersingular elliptic curves - Chu Yue Dong (Stella)
- Factoring integers with elliptic curves using Lenstra’s algorithm - Christopher Grieves
- Computing in the Jacobian of a hyperelliptic curve - Michael Harrison
- The Tate-Shafarevich group of an elliptic curve - Sean Howe
- The Tate-Shafarevich group of an elliptic curve - Stephen Kleinberg
- Point counting on elliptic curves over finite fields - Nicole Kroeger
- Counting the number of points on an elliptic curve defined over a finite field - Kaloyan Marinov
- Complex multiplication on elliptic curves - Matthew Pancia
- Zeta Functions of elliptic curves - Jonathan Root
- Elliptic curves in Poonen’s undecidability proof of Hilbert’s Tenth Problem for large subrings of Q - Shane Sicienski
- Elliptic curves and Hilbert’s Tenth Problem - Jonathan Taylor
- The Tate pairing and efficient cryptosystems - Timothy Tusing
- The Weil pairing and its application to cryptography - Lee Walmach
- Complex Multiplication - Valentin Zakharevich

### Elements of Fractal Geometry and Dynamics

- Diophantine Approximation: Well-And Badly-approximated Numbers - Charles Baker
- The Lotka-Voltera Model and Its Dynamics - Chu Yue Dong (Stella)
- The Correlation Dimension and Its Applications - Christopher Grieves
- A Dynamical System of a Square - Michael Harrison
- The Dynamics of the Gauss Map - Sean Howe
- The Dynamics of the Doubling Map - Stephen Kleinberg
- The Logistic Map - Nicole Kroeger
- The Dynamics of the Vegetation Patterns Model - Kaloyan Marinov
- The Dynamics of The Julia Set - Matthew Pancia
- S-sets In Fractal Geometry - Jonathan Root
- The Weierstrass Function - Shane Sicienski
- The Baker Transformation - Jonathan Taylor
- The Collatz Conjecture - Timothy Tusing
- Newton’s Method for Solving Polynomial Equations - Lee Walmach
- Oscillators and Forced Oscillators - Valentin Zakharevich

### Introduction to Symplectic Geometry

- Hofer’s metric on the group of Hamiltonian diffeomorphisms - Charles Baker
- Hyperbolic paraboloids andorigami - Chu Yue Dong (Stella)
- Lagrangian and Hamiltonian Mechanics - Christopher Grieves
- Poincare-Birkhoff fixed point theorem and its applications to billiards - Michael Harrison
- Hofer’s metric on the group of Hamiltonian diffeomorphisms - Sean Howe
- Kepler’s n-body prroblem - Stephen Kleinberg
- Topology of paths Lagrangian subspaces - Nicole Kroeger
- Symplectic Manifolds and Darboux Theorem - Kaloyan Marinov
- Fixed points of Hamiltonian symplectomorphism of the standard torus T2n - Matthew Pancia
- Poincare-Birkhoff fixed point theorem and its applications to billiards - Lior Rennert
- Lagrangian and Hamiltonian Mechanics - Jonathan Root
- Contact Manifolds - Shane Sicienski
- Fixed points of Hamiltonian syplectomorphism of the standard torus T2n - Jonathan Taylor
- Poincare-Brikhoff fixed point theorem and its applications to billards - Timothy Tusing
- Contact Manifolds - Valentin Zakharevich

## MASS 2007

### Computability, Unsolvability, and Randomness

- Integration in Finite - Dory Deines
- An Overview of Algorithmic Randomness Weaker than Martin-Lof - Bryan Gillespie
- Hilbert’s 10th Problem. Study of Exponential Diophantine Sets. - Jorge Gonzalez
- Hoeffding’s Inequality and Strong Law of Large Numbers - Alison Green
- A practical application of the Enumeration and Parameterization Theorems - Sara Jensen
- Effective Hausdorff Dimension - Chor Hang Lam
- On Modular Machines and the Unsolvability of the Word Problem for Groups - Vincent Martinez
- The Domino Problem - Daniel McDonald
- Embedding lattices in Turing degrees - Timur Nezhmetdinov
- On the Statistical Properties of Martin Lof Random Sequences - Matthew Pancia
- The Lebesque Differentiation Theorem - Noopur Pathak
- A Comparison of Information Theory and Computability Theory with an Emphasis on Conditional Kolmogorov Complexity - Stephanie Tougas
- Axiomatic Theories of Arithmetic: Two Proofs of Essential Undecidability - Robin Tucker-Drob
- Turing Relations Under the Jump Operator - Ashley Wheeler

### Topics in Probability Theory

- The Quasispecies Model of Simple Evolution - Ryan Bradley
- Proofs of Polya’s Theorem with Electric Networks - Dory Deines
- The Arcsine Law - Bryan Gillespie
- Study of Regular Variation - Jorge Gonzalez
- Conditional Expectation and the Martingale Convergence Theorem - Alison Green
- Applications of Murkov Chains to Mathematical Biology - Sara Jensen
- Law of the Iterated Logarithm - Chor Hang Lam
- The Existence and Continuity of Brownian Motion - Vincent Martinez
- The Traveling Salesman Problem - Daniel McDonald
- Valuations on parallelotopes and polyconvex sets - Timur Nezhmetdinov
- Statistical Mechanics and the Fsing Model - Matthew Pancia
- Markov Chains and the Page Rank Agorithm - Noopur Pathak
- Proofs of Cartheodory Extension Theorem, Fubini-Tonelli Theorem, and Radon-Nikodyn Theorem - Shehryar Sikander
- A Probability Model of Olber’s Paradox - Stephanie Tougas
- Proof’s of the Caratheodory Extension Theorem and Fubini’s Theorem - Robin Tucker-Drob
- Ehrenfest Urn Model - Ashley Wheeler

### Surfaces: Everything You Wanted to Know about Them

- The Pseudosphere - Ryan Bradley
- Morse Functions and Morse Inequalities - Dory Deines
- The Classification of the Plane Crystallographic Groups - Bryan Gillespie
- Study on Modular Curves - Jorge Gonzalez
- The Four Color Problem and Coloring Maps on Surfaces - Alison Green
- An Alternate Proof of the Anti-Pasch Conjecture - Sara Jensen
- Geodesic Flow on the Upper Half Plane - Chor Hang Lam
- Fractals and Dimensions: A Brief Overview - Vincent Martinez
- Counting Polyominoes - Daniel McDonald
- Elliptic curves over C - Timur Nezhmetdinov
- An Elementary Survey of Morse Theory - Matthew Pancia
- Axioms of Degree Theory - Noopur Pathak
- The principle of Contractions and eventually Contracting Maps - Shehryar Sikander
- Tessellations of the Hyperbolic Plane - Stephanie Tougas
- Finite Projective Geometries - Robin Tucker-Drob
- Hamiltonian Systems with One Degree of Freedom - Ashley Wheeler

## MASS 2006

### Finite Fields and Their Applications

- Permutations of Finite Fields and Dickson Polynomials of the Second Kind - Jesse Barbour
- A New Conjecture for Completing Lat Hyper Cubes - Jonathan Beagley
- Studying Multiplicative Subgroups of Finite Fields Which Are Closed under Additive Inverses - Maria Burago
- Optimizing Sets of Partially Orthogonal Sets of Latin Squares of Order 6 - Michael Caputo
- Permutation Polynomials of a Given Degree - Rachel Davis
- Polynomial Wight and Irreducibility - Arthur Friend
- Composition of Polynomials over Fq with a Value Set of Cardinality K<q - Douglas Hogue
- Studying Squares Produced by the Polynomials ax
^{2}+y^{2}over Finite Fields - Jessica Jou - Discrete Logarithms - Erica Miller
- From Neofields to Latin Squares - John Mangual
- Lipschitz Automorphisms of Cantor Space - Nathan Ordansky
- Latin Squares and Polynomial Constructions - Benjamin Petersen
- The Number of Primitive Normal Bases over Finite Fields - Carl Sutherland
- Maximal Sets of Mutually Unbiased Bases and Complete Sets of Mutually Orthogonal Latin Squares - Danial Volmar
- A New Lagrange Interpolation Formula for Permutation Polynomials over Finite Fields - Bogdan Vioreanu
- Polynomial Analogs of the 3N+1 Problem - Tad Whitenight

### Mathematical Theory of Waves

- From Representation Theory to the Feynman Diagrams - Jesse Barbour
- Characterization of Graphs with Norm Less than 2 - Jonathan Beagley
- Confinement of Propagating Fronts in Forest Fires by Construction of 1-Dimensional Barriers - Maria Burago
- Elementary Symmetric Functions and the Galois Group - Michael Caputo
- The Braid Group - Rachel Davis
- A Proof Concerning Verticies of Alternating Knots - Arthur Friend
- Understanding Why Guintic Equations Are Not, in General, Solvable with Radicals - Douglas Hogue
- Optimal Confinement of Fire Spreading - Jessica Jou
- Knot Invariance - Erica Miller
- Plotting and Proving: On the Complex Roots of the Bernoulli Polynomials - John Mangual
- The DNA Group - Nathan Ordansky
- Young Orthogonal Representation of the Symmetric Group on 4 Elements - Benjamin Petersen
- Distribution of Energy Among Overtones of a Vibrating String, Depending on the Point Where the String Is Initially Plucked - Carl Sutherland
- The Introduction of Group Theory into Quantum Mechanics - Daniel Volmar
- Standard Bases for the Ring of Symmetric Functions in Infinitely Many Variables - Bogdan Vioreanu
- Galois Theory and Roots of Polynomials of Degree at Most 4 Using the Tetrahedron - Tad Whitenight

### Aspects of Symmetry: From Representations of Quantum Field Theory

- Several Rarefaction Wave Solutions to the Riemann Problem for Isentropic Gas Dynamics - Jesse Barbour
- Traffic Flow Models Using Conservation Laws - Jonathan Beagley
- Confinement of Propagating Fronts in Forest Fires by Construction of 1-Dimensional Barriers - Michael Caputo
- Modeling the Spread of Disease - Rachel Davis
- Strategies for Containing Forest Fires - Arthur Friend
- Shock Wave Solutions to the Riemann Problem for Systems of Conservation Laws - Douglas Hogue
- Knots and the Jones Polynomial - Jessica Jou
- On the Shape of Avalanches - Erica Miller
- Harmonic Functions on the Sierpinski Gasket - John Mangual
- Uniform Convergence of Fourier Series - Nathan Ordansky
- Equal Temperament VS Just Scale - Benjamin Petersen
- Computational Complexity of Knot Problems - Carl Sutherland
- A Numerical Experiment on the Granular Materials Equation - Danial Volmar
- The Front Tracking Method for Systems of Conservation Laws - Bogdan Vioreanu
- Analysis of Gibbs Phenomenon - Tad Whitenight

## MASS 2005

### Integer Partitions

- A Composition Problem Related to Group Theory - Eugene Eyeson
- Modular functions - Ki Chul Kim
- Variants of Ferrer's Diagram - David Koslicki
- Ferrer's graphs and
Durfee Cubes

- Sarah Matz - Variations of Ferrer's Graph - Karen McCready
- Alternative Proof for Generating Functions on Some New Types of Partitions Associated with Ferrer's Graph - Aya Mitani
- Infinite Partitions - Toan Phan
- a-Gaussian Polynomials - Frances Worek

### P-adic Analysis in Comparison with Real

- P-adic Measures and Integration - Eugene Eyeson
- Algebraic Extensions of P-adic Numbers - Ki Chul Kim
- P-adic Analysis and Bernoulli Numbers - David Koslicki
- Modified Harmonic Series in R and Qp - Sarah Matz
- Euclidean Models of the P-adic Integers - Karen McCready
- G-adic Integers: When P Is Not a Prime - Aya Mitani
- P-adic Circle - Toan Phan
- P-adic Interpolation - Joe Roberts
- P-adic Solenoid - Leslie Ross
- Newton Polygons - Frances Worek

### Geometry and Billiards

- Cavity Rippling - Eugene Eyeson
- Hilbert's Fourth Problem and Crofton's Formula - Ki Chul Kim
- Substitutions and Fractals - David Kolsicki
- Periodic Billiard Orbits in Shapes that Tile the Plane - Sarah Matz
- Morse Theory - Karen McCready
- Four Vertex Theorem - Aya Mitani
- Minutes with Games - Toan Phan
- Derivation of Cylindrical Borr-Sommerfeld Equation - Joe Roberts
- Using Morse Theory to Determine the Configuration Spaces of Planar Linkages - Frances Worek

## MASS 2004

### Finite Fields and Applications

- kq-Orthogonal Latin Squares - Jenna Hammang
- Elliptic Curve Cryptography - Robert King
- Generating orthogonal frequency - type squares - Sarah Mall
- Permutation Polynomials and Latin squares - Zachary McCoy
- Positively Expansive Invertible Maps on Compact Manifolds - James Krysiak

### Introduction to Dynamical Systems

- Markov Partitions - Jenna Hammang
- Nielsen Fixed Point Theory - Robert King
- Coding Toral Automorphisms - Sarah Mall
- The Gauss Map and the Modular surface with Applications to Quadratic forms - Zack McCoy
- Partially Orthogonal Sets of Frequency Squares - James Krysiak

### Differential Geometry and Topology of Curves and Surfaces

- The Distortion of Curves - Jenna Hammang
- The Geometry of Minimal Surfaces bounded by convex planar curves - Rob King
- The DNA Inequality for a non-convex cell - Sarah Mall
- Alexandrov's Conjecture - Zack McCoy
- Alexandrov's Conjecture on Degenerate Surfaces and Surfaces of Revolution - James Krysiak

## MASS 2003

### Number Theory with Applications to Communication Networks

- Quadratic reciprocity law: different proofs and applications - Mark Horn
- Comparison between graphs and manifolds, Cheeger's constant and spectral gap - Russel Halper
- Abelian Ramanujan graphs and character sum estimates - Mike Willis
- Terras' construction of nonabelian Ramanujan graphs - Rebecca Swanson and Jeff Ginn
- Zeta functions of graphs - Vivek Srikrishnan and David Jordan
- Zeta function of curves defined over finite fields and application to character sum estimates - John Skukalek
- Quadratic reciprocity law: different proofs and applications - Jeff Paulsen
- Waring's problem over Z - Chris King and Tim Trudell
- Dirichlet's theorem on uniform distribution of primes in arithmetic progressions - Nathan Collins
- Low density parity check codes - Walter Chen

### Topological Dynamics

- Rates of convergence in 2
^{m}3^{n}- Marc Horn and Jeff Paulsen - Restricting iterates in van der Waerden's Theorem - Nathan Collins and Russel Halper
- The Ellis-Auslander Theorem - Chris King
- The topology of orbits under a real flow - Vivek Srikrishnan
- Dynamics of commuting transformations with fixed points - Mike Willis and Becky Swanson
- The proximal relation and regionally proximal relation - John Skukalek
- Sets of recurrence - Walter Chen
- Dynamics of the Cantor middle thirds set - David Jordan
- The entropy of DNA sequences - Jeff Ginn and Tim Trudell

### Geometry and Relativity: An Introduction

- Minimal surfaces and differential geometry - Jonathan Barry
- General relativity and GPS - Walter Chen
- Nonimmersion of the hyperbolic plane in 3-dimensional Euclidean space (Hilbert's theorem)
- Rotating black holes - Tom Essinger-Hileman, Matthew Pelc, and Matthew Yanoff
- Rotating black holes - Vivek Srikrishnan
- The twin paradox of relativity - Marc Horn
- Geometry of black holes - David Jordan
- Hilbert-Einstein action principle - Christopher King and Tim Trudell
- The Einstein-Podolsky-Rosen paradox and Bell's theorem - Nathan Kurz
- Einstein's three methods for determining Avagadro's number - Jeffrey Paulsen
- Gauss-Bonnet theorem and generalizations - John Skukalek
- Gromov's notion of hyprebolicity - Rebecca Swanson
- Gravitational lensing and dark matter - Michael Willis

## MASS 2002

### Number Theory with a Tilt at Elliptic Curves

- Rings and things; commutative algebra through Hilbert's Basis Theorem and the Nullstellensatz
- The solutions of X
^{3}+Y^{3}+Z^{3}and Krummer's Conjecture - The resultant and its applications
- The mathematics of the Rijndael Encryption Scheme
- An introduction to P-adic numbers and Hensel's Lemma
- Diophantine approximation theorem: Thue's approach to approximations of cube roots
- Rational conics: finding a rational solution
- Virtually everything you wanted to know but were afraid to ask about cubic and quartic reciprocity
- Partition analysis: generating functions of three-dimensional pyramids
- Minimal discriminants of elliptic curves
- Uniform distribution modulo 1 using Weyl's method
- Geometry of numbers
- Nagell-Lutz vs. the computer

### Intuitive Topology

- Topology of linkages
- Legendrian knots
- Contact vector fields
- Unknoting number and prime knots
- Jordan Curve Theorem
- Discrete Morse theory
- DNA and knots
- Path planning problem

Brady McCary, MASS 2002 participant, studied motion-planning algorithms in an environment with obstacles. See implementations here and here. - Topological problems of wave propagation
- Hyperbolic geometry models for classical surfaces
- Symbolic coding of curves
- Poincare sphere

## MASS 2001

### Combinatorics

- Partitions of
`n`and connected triangles `n`-Centripetal di-graphs- Dyson's adjoint of a partition
- The Ramanujan congruences for
`p`(`n`) - Polynomials of Knots

### Geometry and Relativity

- The Schwarzschild soluton to Einstein's equations
- Riemannian curvature of Lie groups
- The Jordan curve theorem
- Prescribing curvature on a torus
- The total curvature of knots
- Three proofs of Morley's theorem
- Zero Riemannian curvature
- Foundations of projective geometry, finite projective planes, and automated theorem proving
- Combinatorial Morse theory
- Closed geodesics in the fundamental region of
*Gamma (2)*

### Fluid Dynamics

- Poincare-Hopf Theorem
- Continuity of the flow map and free boundaries for objects entering the fluid domain
- Relativistic fluid dynamics
- Differential equations, strings, and knots
- Kidney flow
- Vortex shedding in the complex plane: time-dependent Milne-Thompson Theorem
- Pressureless Euler flows
- Non-radial flow in spherical coordinates: mappings to the complex plane
- Steady irrotational flow in particular bounded domains with N-point vortices
- Curl and potential flows in
*n*-dimensional space - Poisson bracket formulation of Euler's equations in the plane: the Lax pair approach

## MASS 2000

### Finite Groups, Symmetry, and Elements of Group Representations

- Knots and the Bracket Polynomial
- The Frobenius-Schur Theorem
- Visualizing the Binary Icosahedron Group
- Homfly and Jones Invariants

### Projective and Non-Euclidean Geometries

- Barycentric Coordinates in the Affine Plane
- Two-Dimensional Crystallography
- Proving the Cross Ratio in Mobius Geometry is Invariant
- Minkowski Geometry and Special Relativity

### Real and P-adic Analysis

- On Diagonal Cubic Equations
- The X-adic Norm of a Power Series
- Exploration of Isometries in the p-adic Field
- Estimations on the 2-adic Logarithm of Negative One

## MASS 1999

### Topics in Number Theory

- Recursive Relations and Lucas Primality Testing
- The Omega Operator and Directed Graphs and Partitions
- On Selberg's Asymptotic Formula
- Pseudo-primes, Carmichael, and Sigma-Phi Numbers

### Geometric Structures, Symmetry, and Elements of Lie Groups

- Crystallographic Restriction and Wallpaper Groups
- Random Walks and Plane Arrangements
- The Growth of Generalized Diagonals for Polygonal Billiards
- Geodesic Flow on Compact Factors of the Hyperbolic Plane

### Mathematical Methods in Mechanics

- The Antikythera Mechanism and Continued Fractions
- Geometric Consideration of Hill's Equations on the Solid Open Torus
- Fourier Series and Isomorphisms between Canonical Hilbert Spaces
- Exploration of the Cycloid

## MASS 1998

### Number Theory

- Congruences for Gaussian Hypergeometric Series
- Consecutive Primes in Arithmetic Progressions
- Kaplansky's Ternary Quadratic Form
- Multiplicative Eta-functions

### The Exponential Universe

- Gromov's Theory of Hyperbolic Groups
- The Isoperimetric Problem in the Hyperbolic Plane
- Hilbert's Third Problem
- The Proof of Poncelet's Porism

### Functions and Dynamics in One Complex Variable

- Attracting and Super-attracting Fixed Points
- Univalent Functions and Koebe's Distortion Theorem
- Lorentz Transformations on the Sphere at Infinity
- Picard's Theorem

## MASS 1997

### Arithmetic and Geometry of the Unimodular Group

- Coding of the Closed Geodesics
- Geodesics on the Cube
- Triangle Groups in Hyperbolic Geometry
- Proving 2
^{n}can Start with an Arbitrary String of Digits

### Real Analysis

- Baire Functions
- Transcendence of ⅇ
- Fubini's Theorem for Sums
- The Gamma Function and Stirling Formula

### Explorations in Geometry

- Proof of the Jordan Curve Theorem
- The Gauss-Bonet Formula for Curves in the Hyperbolic Upper Half Plane
- Mixed Volumes
- Curves and Surfaces in 3-Dimensional Real Space

## MASS 1996

### Explorations in Number Theory

- Super Exponential Sequence
- On Quadratic Irrationality and Continued Fractions
- Perfect Partitions
- Solovay-Strassen Method for Determining Primality

### Introduction to Dynamical Systems

- Sharkovsky Theorem
- Stability of continuous-time dynamical systems in 2 dimensions
- Topological Entropy and Topological Markov Chains
- Stable Manifolds

### Linear Algebra in Geometry

- Tensor Products and Matrix Equations
- Iteration Method
- Affine Transformations
- Engel's Theorem in Lie Algebra

## Stories about Research in MASS

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 termchaos. 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 textbookNonlinear 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

hackthe 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 veryuniquestyle 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