Each senior major gives a 35-40 minute colloquium on new material, after a month’s preparation with a faculty advisor.
Liam Abbott ’13
“The EM Algorithm: Using Hidden Variables to Find MLEs”
Elise Baker ’13
“Calculating Power in Multicameral Voting Systems”
Dhyan Adler-Belendez ’13
“The Infinitude of Primes and Bertrand’s Postulate”
Andrew Bishop ’13
“College Football Rankings: Using Directed Networks and Random Walks to Solve the Dilemma of a Multi-Billion Dollar Industry”
Kyle Bolo ’13
“Picking up Where the Greeks Left Off: Confronting the Unsolved Geometrical Questions of Their Time”
Ryan Brand ‘13
“Evolutionary Dynamics and Universal Grammar”
Lucas Casso ‘13
“Convex Polytopes in Higher Dimensions”
Christopher Corbett ’13
Swimming Upstream: The Benefits of Fish Schooling”
Evan DeDominicis ’13
“Game Theory, Fixed Points, and Football?”
Tara Deonauth ’13
“How to Guard an Art Museum”
Carlos Dominguez ’13
“Quaternions and Lagrange’s Four-Square Theorem”
Jack Ervasti ’13
“Using Convexity to Approximate Bond Price Change”
Gregory Eusden ‘13
“History and Concepts Behind the Max-Flow, Min-Cut Theorem in Graph Theory”
Kushatha Fanikiso ‘13
“Mathemagic: What Happens at the Intersection Between Magic and Math?”
Jalynne Figueroa ’13
“Authentication in the Digital Age”
Christopher Fogler ’13
“Curve Reconstruction: From Triangles to Pixar”
Jeffrey Fossett ‘13
“Introduction to Latent Semantic Analysis”
Kevin Garcia ‘13
“God’s Number: Fewest Moves to Solve the Rubik’s Cube”
Katy Golvala ’13
“Intervention Analysis and Its Applications”
Jennifer Gossels ‘13
“Linear Programming and Baseball Elimination Numbers”
Alexander Greaves-Tunnell ’13
“Elliptic Curve Cryptography”
Yiming Guo ‘13
“Optimizing Blackjack Playing Strategy With ‘Lucky Bucks’”
Charles Hammond ’13
“An Introduction to Survival Analysis”
Wen Han ‘13
“Buffon’s Needle”
Julian Hess ‘13
“The Exterior Algebraic Formulation of Maxwell’s Equations“
Kam Shan Ho ‘13
“Hall’s Marriage Theorem and Matchings in Graphs”
Joy Jing ‘13
“Magic and Math: Seeing Through Lies“
Casey Jones ‘13
“Lottery Tickets, Green Cards, and Random Generators“
Daeus Jorento ’13
“The U.S. Treasury, Ebay and Craigslist: Auctions and the Revenue Equivalence”
Andrew Kelly ‘13
“Kronecker’s Theorem and the Question of the Reflected Ray”
Christina Knapp ‘13
“Curvature”
Meghan Landers ‘13
“The Infinitude of Primes”
Shirley Li ’13
“Extensions to the Carpenter’s Rule Conjecture”
Joe Long ‘13
“From Counting Eggs to Keeping Secrets: How the Chinese Remainder Theorem Can Help You”
Guannan Lu ‘13
“The Friendship Paradox”
Julio Luquin ‘13
“Land Ho! Choosing the Perfect Map”
Zane Martin ‘13
“Elementary, My Dear Bertrand”
Madeleine Mitchell ‘13
“Cooperation Within Competition: Match-Rigging in Sumo Wrestling”
Becky Miller ‘13
“Congruent Numbers: From Right Triangles to Elliptic Curves”
Eugene Murphy ‘13
“What Are the Chances? Using Bayesian Statistics to Estimate a Proportion”
Kristine Nakada ‘13
“Topology and Combinatorics of the Soccer Ball”
Nicholas Neumann-Chun ’13
“Life: Discrete or Continuous”
Mai Okimoto ‘13
“Tetrahedra in Polyhedra”
Michael Ormsbee ‘13
“A Brief Introduction to Elliptic Curves”
Christopher Picardo ‘13
“Order Statistics and the German Tank Problem”
Tejesh Pradhan ‘13
“The Wallet Paradox”
Alexander Rich ‘13
“Four Degrees of Separation: Small World Networks and Why They Matter”
Eric Robinson ‘13
“The Fitch and Sankoff Algorithms for Plant Phylogenies”
Scott Rodilitz ’13
“A Two Card Cover-Up Game”
Chance Rueger ‘13
“Pythagoras at the Bat”
Scott Sanderson ’13
“Hilbert’s Nullstellensatz: An Introduction to Algebraic Geometry”
Roshan Sharma ‘13
“The Weierstrass Representation Always Gives a Minimal Surface”
Benjamin Seiler ‘13
“Fractals in Finance: Price Jumps and a Peculiar Smile”
Sandra Shedd ‘13
“Lost in the Woods: Phylogenetic Trees, Splits, and Applications”
April Shen ’13
“Of Groups and Graphs”
William Speer ’13
“Musical Actions of Dihedral Groups”
Wei Sun ’13
“Twenty Questions, Huffman Code and Youtube”
Kaison Tanabe ‘13
“Sponges (Coxeter-Petrie)”
David Taylor ’13
“Machine Learning and Bootstrap Random Forests”
Philip Tosteson ‘13
“Counting Primes (L-Functions)”
Philip Treesh ‘13
“Chaos, Waterwheels, and the Lorenz Equations”
Erich Trieschman ’13
“Lost in a Forest”
Thomas Vieth ’13
“Traffic Flow”
Rhys Watkins ‘13
“A Proof of Sphere Packing in 2D”
Peter Watson ‘13
“The Two Envelopes Paradox: Is There a Correct Solution”
Alexander Wheelock ‘13
“A Lattice-Based Cryptosystem for a Quantum Force”
James Wilcox ‘13
“Sequences of Convergents of Continued Fractions”
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.