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

**Stephen Ai ‘18
**“Cardinality and Variety in Music: A Mathematical Investigation”

**Spencer Alpaugh ‘18
**“Intervention Analysis”

**Nathan Andersen ‘18
**“What Does it Really Mean to be Random?”

**Prosper Atukwatse ‘18
**“An Exposition of Fejer’s Theorem”

**Jackson Barber ‘18
**“The Spatial Dynamics of WNS”

**Andrew Barry ‘18
**“Violating the Parallel Postulate: A Crash Course in Hyperbolic Geometry”

**John Beirne ‘18
**“Properties of Multivariable Complex Functions”

**Isaac Benioff ‘18
**“Euclid’s A Liar! A Brief Introduction to Spherical Triangles”

**Marit Bjornlund ‘18
**“How Do We Know What’s Unfair?: Using Markov Chains to Evaluate Gerrymandered Political Districts”

**Cody Cao ‘18
**“A Numerical Investigation of the Traveling Wave in EGT”

**Granger Carty ‘18
**“Can a Monohedral Tiling be Aperiodic?”

**Jacques Chaumont ‘18
**“Exploring Random Draws”

**Frankie Chen ‘18
**“Developing an Essential Tool for Dealing With Polynomials and Their Roots: The Resultant”

**Eugene Choe ‘18
**“Intro to Regression Trees”

**Jack Cloud ‘18
**“Marketing Mix Models to Optimize Ad Spending”

**Trevin Corsiglia ‘18
**“A Proof for the Completeness of Propositional Logic”

**Christopher D’Silva ‘18
**“Quantum Computers, Mixed Drinks, and Friendship Bracelets: Searching for the Braid Index”

**Kevin Deptula ‘18
**“Kenneth Arrow’s Impossibility Theorem”

**Benjamin Drews ‘18
**“The ABC Conjecture and Fermat’s Last Theorem”

**Agastya Easley ‘18
**“Disease in Metapopulation Models”

**Emily Eide ‘18
**“Analysis of Spatial Data Using Kriging”

**Mason Elizondo ‘18
**“Evolutionary Dynamics and Rocks Paper Scissors Lizards”

**Madeleine Elyze ‘18
**“On Higher Distance Commuting Matrices”

**Alyssa Epstein ’18
**“The Fibonacci Game”

**Daniel Fisher ‘18
**“Hyperreals and the Transfer Principle”

**Ioannis Florokapis ‘18
**“Local Network Effects in Stochastic Graph Structures”

**Jahangir Habib ‘1**8

“Digit Bias in Data Sets”

**Beatrix Haddock ‘18
**“Weak Mixing in Infinite Measure”

**Helene Hall ‘18
**“Buffon’s Needle: How to Use a Random Experiment to Approximate Pi”

** Caroline Hogan ‘18
**“Architecture Frieze Patterns: The Seven One Directional Symmetry Groups”

**Isabella Huang ‘18
**“You’ve Peaked!: Combinatorial Problems on Peaks, Pinnacles, Descents, and Derangements”

**Sumun Iyer ‘18
**“Nonsingular Rank One Transformations”

**Richard Jin ‘18
**“Latent Semantic Analysis”

**Arjun Kakkar ‘18
**“Modeling and Analysis of Vegetation Patterns in Semi-Arid Regions”

**Molly Knoedler ‘18
**“Topology and Agent-based Modeling of Pollination Networks”

**Julianna Kostas ‘18
**“The Seasonal Influenza A Epidemic: SIRC Model and Bifurcation Analysis”

**Timothy Kostolansky ‘18
**“Hilbert’s Basis Theorem”

**Kiran Kumar ‘18**

“What is the Shape of Brexit?”

**Ryan Kwon ‘18
**“Axiom of Choice and Zorn’s Lemma”

**Henry Lane ‘18
**“Manipulation and Preferential Interaction in the Envelope Game, or How to Politely Dodge Favors”

**Edward Lauber ‘18
**“Pedagogical Approaches to Mathematics Instruction: Finding a Balance Between Lectures and Discovery”

**Haley Lescinsky ‘18
**“Population Dynamics of a Host Parasitoid System”

**Stephanie Li ‘18
**“Nim Games and the Sprague-Grundy Theorem”

**Andrew Litvin ‘18
**“Real-Time Win Probability in Texas Hold ‘em Poker”

**Tanner Love ‘18**

“Throwing Darts at the Real Line: A Probabilistic Intuition Against the Continuum Hypothesis”

**Ziqi Lu ‘18**

“The Game Theory of Social Norms”

**Calvin Ludwig ‘18
**“Mathematical Modeling of HIV Pathogenesis and Treatment”

**Wei Luo ‘18
**“Ramsey Numbers and Computational Methods”

**Dalia Luque ‘18**

“Juggling, Combinatorics, and Worpitzky’s Identity”

**Eleanor Lustig ‘18
**“Modeling the Impacts of Climate Change on Malaria Transmission”

**Daniel Maes ‘18
**“A Financial Statistical Approach for Leveraged Exchange-Traded Funds (LETFs)”

“Assessing Critical Mass at UC-Berkeley: Creating Predictive Models for Affirmative Action Policies in Undergraduate Admissions in the United States”

**Jacob Marrus ‘18
**“Can Pac-Man Ever Escape?”

**Eliza Matt ‘18
**“The Density of States of RNA”

**Jonathan Meng ‘18
**“The Mathematics Behind Heisenberg’s Uncertainty Principle”

**James Millstone ’18
**“The Catalan Numbers and the Wigner Semicircle Distribution”

**Ian Mook ‘18
**“An Exploration of Turing Patterns and Their Presence in Nature”

**Matthew Morris ‘18
**“Modeling the Entire Distribution: An Introduction to Quantile Regression”

**Anna Neufeld ‘18
**“Longitudinal Regression Trees”

**Daishiro Nishida ‘18
**“Multi-Crossing Braids”

**Francesca Paris ‘18
**“Missing Data in Public Health and the EM Algorithm”

**Ashay Patel ‘18
**“Fundamental Solutions for PDEs and a Generalized Calculus”

**Chetan Patel ‘18
**“The P vs. NP Problem: A Short Exploration”

**Sohum Patnaik ‘18
**“Statistical Approaches for Automatic Text Summarization”

**Seth Perlman ‘18
**“Fun With Partition Numbers”

**Timothy Randolph ‘18
**“Proving the Existence of Monocolor Arithmetic Progressions (Van Der Waerden’s Theorem)”

**Claudia Reyes ‘18
**“Points of Visibility”

**Andrew Robertson ‘18
**“Boosting: Improving Learning Algorithm Performance”

**Thomas Rosal ‘18
**“The Gaussian Copula in Practice: Modeling Dependence and Where Assumptions Fail”

**Jake Savoca ‘18
**“Continued Fractions and Quadratic Irrationals”

**Andrew Scharf ‘18
**“Representing Tropical Intersection Curves”

**Lev Schechter ‘18
**“Optimal Strategies in a Basketball Shooting Contest”

**Robert Schneiderman ‘18
**“NFL Survival Analysis”

**Alex Semendinger ‘18
**“Noetherian UFDs with Weird Prime Spectra”

**Nohemi Sepulveda ‘18
**“Happy Numbers on a Happy Day”

**Emily Sundquist ‘18
**“Mapping Measles: Temporal and Spatial Epidemic Patterns”

**Greg Szumel ‘18
**“Gale and Stewart’s Theorem on Games”

**Spencer Thomas ‘18
**“Is This the Real Deal?: Using Discrete Wavelet Transforms to Detect Forgeries in the Art World”

**Karan Tibrewal ‘18
**“The Twin Prime Conjecture and Brun’s Sieve”

**Darla Torres ‘18
**“Quadratic Reciprocity”

**Minh Tuan Tran ‘18
**“Fast Fourier Transform and Applications”

**Austin Vo ‘18
**“Spatial Point Pattern Analysis”

**Sean Wang ‘18
**“The Ideal Number of Shuffles for a Deck of Cards”

**Emilia Welch ‘18
**“Modeling and Predicting Ocean Tides With Fourier Analysis”

**Colin Williams ‘18
**“Proving the Existence of a Nash Equilibrium”

**Hallee Wong ‘18
**“Using Data to Predict Hospital Readmissions at Berkshire Medical Center”

**Zihan Ye ‘18
**“Generalized Additive Models”

**Benjamin Young ‘18
**“Generating Functions and the Coin Problem of Frobenius”

**Yiheng Zhang ‘18
**“Liouville Numbers: Category vs. Measure”

**Fangyuan Zhao ‘18
**“A (Gentle) Introduction to Causal Inference – With a Focus on Propensity Score Methods”

**Weitao Zhu ‘18
**“A Local Ring With an Unusual Ideal Structure”

**Michael Zuo ‘18
**“Secure Secret Sharing for Strangers”

**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.

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