Each senior major gives a 35-40 minute colloquium on new material, after a month’s preparation with a faculty advisor

**Patrick Anderson ‘17
**“Game, Set, Match – On the Mathematics of the Card Game Set”

**Yoonsang Bae ‘17
**“Voting in Agreeable Societies”

**Kathryn Barnitt ‘17
**“An Exploration of K-Nearest Neighbors”

**Bridget Bousa ‘17
**“Julia Sets and the Mandelbrot Set”

**Osama Brosh ‘17
**“Dimension of Crowns”

**Melissa Caplan ‘17
**“Three Proofs of the Ballot Theorem”

**Richard D. Chen ‘17
**“Wave Breaking in the Hunter-Saxton Equation”

**Jaeho Choi ‘17
**“The Wave Equation”

**Dana Cohen ‘17
**“The Dynamics of a Multi-Strain Disease with Cross Immunity”

**Marcus Colella ‘17
**“The Chinese Remainder Theorem”

**Alyssa Crain ‘17
**“Synchronizing Biological Oscillators”

Duncan** Cummings ‘17
**“Optimal Strategy for he Best Choice Problem”

**Catherine Dickinson ‘17
**“The Algebra of Wallpaper”

**Tyler Duff ‘17
**“A Bayesian Approach to the Multinomial”

**Jack Ferguson ‘17
**“Life in the Permuting Lane”

**Ross Flieger-Allison ‘17
**“Pseudorandom Number Generation”

**Max Friend ‘17
**“Squares, Damned Squares, and The Gaussian Integers”

**Stetson Futterman ‘17
**“Rejection Sampling and Adaptive Rejection Sampling”

**Patrick Gainey ‘17
**“Geodesics on the Tetrahedron”

**Joseph Glass-Katz ‘17
**“A Comparison of Bayesian and Frequentist Statistics”

**Alexander Gonye ‘17
**“The German Tank Problem”

**Hans Halvorson ‘17
**“A Winning Strategy in Dots-and-Boxes”

**Matthew Hayes ‘17
**“Global Solutions to the Camassa-Holm Equation”

**Matthew Hennessy ‘17
**“The Banach-Tarski Paradox”

**Stephanie Horan ‘17
**“Cayley Digraphs and Hamiltonian Paths”

**Intekhab Hossein ‘17
**“Discrete Bidding Games”

**Christian Hoyos ‘17
**“The ABCs of Approximate Bayesian Computation”

**George Hunkele ‘17
**“Conway’s Napkin Problem”

**Grant Johnson ‘17
**“Exploring GARCH Models”

**Patrick Kane ‘17
**“Gambler’s Ruin”

**D.H. Lee ‘17
**“Simultaneous Confidence Intervals for Benford’s Law”

**Janice Lee ‘17
**“Expanding e into a Continued Fraction”

**Lia Lee ‘17
**“Using Conditional MLE for Logistic Regression Models”

**Kathryn Leinbach ‘1**7

“The Stable Marriage Problem”

**Jamie Lesser ‘17
**“An Afternoon with Buffon”

**Jilly Lim ‘17
**“What Do Fluids and Fractional Linear Transformations Have in Common?”

**Olivia Lima ‘17
**“Clustering Methods”

**Benjamin Lin ‘17
**“Why Count Von Count Should Study Abstract Algebra”

**Paul Lindseth ‘17
**“The Stable Marriage Problem and Generalizations”

**Jieming Liu ‘17
**“Closed Testing Procedures for Multiple Comparisons”

**Steven Louis-Dreyfus ‘17
**“A Function Without an Anti-Derivative”

**Si Young Mah ‘17
**“Map Coloring for Mathematicians”

**Chinmayi Manjunath ‘17
**“Seven Bridges of Königsberg”

**Samuel Manzi ‘17
**“The Basketball Problem”

**Gabrielle Markel ‘17
**“A Finite Talk on Infinite Sets”

**William McGuire ‘17
**“The Geometry of Musical Harmony and Counterpoint”

**Jonathan McLean ‘17
**“Valid Configurations of the Rubik’s Cube”

**Schuyler Melore ‘17
**“Using PDE to Model Tumor Growth”

**Frank Mork ‘17
**“Multiple Comparison Testing Analysis in ANOVA”

**Connor Mulhall ‘17
**“The NTRU Public Key Cipher”

“Controlling the False Discovery Rate”

**Johnson Nei ‘17
**“Negative Binomial Regression Modeling”

**KimThanh Nguyen ‘17
**“VWEIrgvctxmsr”

**Gabriel Ngwe ‘17
**“Continuous but Nowhere-Differentiable Functions”

**Sein Oh ‘17
**“Handling Imbalanced Datasets Using SMOTE”

**Philip Oung ‘17
**“Lagrange Four Square and Waring’s Problem”

**James Pappas ‘17
**“Game Theory and the Brouwer Fixed-Point Theorem”

**Rohan Paranjpe ‘17
**“How Abelian Are Non-Abelian Groups: What is the Probability that Two Elements in a Group Commute?”

**Robin Park ‘17
**“Bézout in the Tropics”

**Dylanger Pittman ‘17
**“Double Bubbles in Borell Space”

**Brooks Rao ‘17
**“Exploring Double Lasso Variable Selection”

**Reidar Riveland ‘17
**“Polynomial Interpolation”

**Ariana Ross ‘17
**“The Fold-and-Cut Problem”

**Nicole Salani ‘17
**“Longitudinal Data Analysis Using GLMs”

**Michael Samayoa ‘17
**“The Power of the Exact Test”

**Aaditya Sharma ‘17
**“Wigner’s Semicircle Law and Random Matrix Ensembles with Split Limiting Spectral Distributions”

**Anne Sher ‘17
**“Cauchy’s Corollary to Sylow’s First Theorem”

**Molly Siebecker ‘17
**“Straightedge and Compass Constructions”

**Troy Sipprelle ‘17
**“The Art (and Math) of Illumination”

**Benjamin Solis-Cohen ‘17
**“Ramsey Theory and the Probabilistic Method”

**Sean Spees ‘17
**“Discrete Fourier Transform Encoding”

**Stephanie Stacy ‘17
**“Random Graphs: The Erdos-Renyi Model”

**Madeline Swarr ‘17
**“A Bayesian Approach to A/B Tests”

**Elena Teaford ‘17
**“Blue-Red Hackenbush and Surreal Numbers”

**Matthew Thomas ‘17
**“The Isoperimetric Inequality”

**Dvivid Trivedi ‘17
**“Regression Transformations”

**Anthoney Tsou ‘17
**“Compression Algorithms”

**Stephen Tyson ‘17
**“Continued Fractions and the Twelve-Tone Musical Scale”

**Vidya Ventkatesh ‘17
**“Prime Decomposition in Number Rings”

**Eleanor Wachtell ‘17
**“A Global Problem: Projecting Spheres onto Flat Surfaces”

**Wendy Wiberg ‘17
**“The Mathematics of Tetris”

**Fan Zhang ‘17
**“Nonparametric Regression”

**Jenny Zheng ‘17
**“Modeling Car Traffic Flow: Burgers Equation”

**Miller Zhu ‘17
**“HMM Inference with Baum-Welch Algorithm”

**Prof. Shawn Rafalski** returned to Williams to see former colleagues and students and spoke on “If You Are Wandering Through a Hall of Hyperbolic Mirrors, How Small Can You Be?” on March 6, 2009:

**Natee Pitiwan ’09** talked about Sperner’s Lemma in colloquium March 4, 2009:

**David Aitoro** talked about the Monster Group, which has about 8×10^53 elements, in the math colloquium today (February 23). He constructed it as a Coxeter group based on the 13-point projective plane.

**Maggie Tucker ’09** gave several interesting proofs of the irrationality of √2 in the math colloquium today (February 9). The comments below by Professor Miller conclude with a challenge to all students.

*Comments on Margaret Tucker’s colloquium by Steven Miller:*

On 2/9/09, Margaret Tucker gave a nice colloquium talk about proofs of the irrationality of √2. Among the various proofs is an ingenuous one due to Conway. Assume √2 is rational. Then there are integers m and n such that 2m^2 = n^2. We quickly sketch the proof. Let m and n be the smallest such integers where this holds (i.e., we have removed all common factors of m and n). Then two squares of side m have the same area as a square of side n. This leads to the following picture:

We have placed the two squares of side length m inside the big square of side length n; they overlap in the red region and miss the two blue regions. Thus, as the red region is double counted and the area of the two squares of side m equals that of side n, we have the area of the red region equals that of the two blue regions. This leads to 2x^2 = y^2 for integers x and y, with x < m and y < n, contradicting the minimality of m and n. (One could easily convert this to an infinite descent argument, generating an infinite sequence of rationals….).

Professor Morgan commented on the beauty of the proof, but remarked that it is special to proving the irrationality of √2. The method can be generalized to handle at least one other number: √3. To see this, note that any equilateral triangle has area proportional to its side length s (and of course this constant is independent of s). Assume √3 is rational, and thus we may write 3x^2 = y^2. Geometrically we may interpret this as the sum of three equilateral triangles of integral side length x equals an equilateral triangle of integral side length y. Clearly x < y, and this leads to the following picture:

Above we have placed the three equilateral triangles of side length x in the three corners of the equilateral triangle of side length y. Clearly x > y/2 so there are intersections of these three triangles (if x <= y/2 then 3x^2 ≤ 3y^2/4 < y^2). Let us color the three equilateral triangles formed where exactly two triangles intersect by blue and the equilateral triangle missed by all by red. (There must be some region missed by all, or the resulting area of the three triangles of side length x would exceed that of side length y.) Thus (picture not to scale!) the sum of the three blue triangles equals that of the red triangle. The side length of each blue triangle is 2x-y and that of the red triangle x – 2(2x-y) = y-3x, both integers. Thus we have found a smaller pair of integers (say a and b) satisfying 3a^2 = b^2, contradiction.

This leads to the following question: for what other integers k can we find some geometric construction along these lines proving √k is irrational?

**Deividas Seferis ’09** spoke in the math colloquium today (January 27) on “Isoperimetric Sequences.” Photo and video clip feature Seferis and audience, including members of his track team and coach Fletcher Brooks.

**Hristo Milev ’09** spoke in colloquium January 13 on “Confidence Intervals for Binomial Proportions.” Check out this short video take on Hristo and the audience.