Sonia Kovalevsky Day at North Dakota State University

Past Workshops and Keynote Addresses

The Workshops and Keynote Addresses present mathematical concepts and applications of mathematics to real life problems.
Here are some of the topics we have covered in previous years: 

Cracking the Enigma Cipher: How Mathematics played a role in History

The Enigma was a machine used by the German military during World War II to encode and send ciphered messages. The encrypted messages were broken by the Allies using mathematical techniques. In this workshop, each student was given a paper Enigma simulator. After the explanation of how the encryption works, there was a discussion on the flaws of the system, the mistakes of the officers setting up the daily code, and on ways to decipher a secret message whose beginning we know.  The students were able to decipher the message, and at the end of the day they could take the Enigma simulator home to continue encrypting messages. The photo on the right shows the paper version of the plugboard and three rotors of an Enigma machine.

Our 2012 Keynote Speaker was Catherine Micek, Senior Consultant in Analytics and Research at Travelers (an insurance company) in St. Paul, Minnesota. She holds a Ph.D. in Applied Mathematics from the University of Minnesota, and in her Ph.D. thesis, Katy developed mathematical models for polymer gels used in artificial bone implants and drug-delivery devices. How does an insurance company determine what price they should charge in order to make a profit on the policies they sell? To help answer this question, students used cards, dice, and probability to conduct a hands-on simulation of pricing a car insurance policy. The students formed groups, each group representing a car insurance company, and were given a deck of cards, dice, and a calculator, which they used to calculate the probabilities of car accidents and claims submitted to their insurance company. Based on the results, each group came up with a price for car insurance. However, upon comparison of the results, some groups had priced their insurance higher than others, and they had to decide whether to adjust their quotes in order to offer a more competitive price.

On an election year, we presented a workshop about the mathematics of voting. We wish to have a fair procedure to elect one winner from several (more than 2) candidates. A number of different voting schemes were considered (namely, plurality voting, absolute majority voting, Borda count, Condorcet's method, and finally the dictatorship). The main conclusion of the workshop was that there exists no election procedure, different from dictatorship, that satisfies several very natural and "fair" properties (this statement is called Arrow's theorem).

Students explored Kepler's laws by constructing their own ellipses using the gardener's method and using them to approximately compute the swept areas. A computer simulation was also presented to illustrate the Three Body Problem.

Our 2013 Keynote Speaker was Deepa Mahajan, a Mathematics Ph.D from the University of Minnesota, who works in for Boston Scientific, Inc, a company that develops and manufactures medical devices. Deepa is a Principal Scientist in Cardiac Rhythm Management Division. She gave a presentation on the basics of interpreting Electrocardiography (ECG) rhythm using a graph paper, in order to detect possible irregularities in the heart rhythm.

Drawing an n-dimensional cube

How can we visualize objects in dimension higher than 3? This workshop presented a way to generalize what we know about squares on the plane and cubes in 3 dimensional space into higher dimensions. Students were able to draw representations of 4 and 5 dimensional cubes.

Game Workshops

Many familiar games have features or winning strategies that can be used to illustrate mathematical concepts. Here are some of our past Game Workshops and the math topics they covered: Rubik's Cube: Combinatorics, Group Theory. The Geometry of Tetris: Symmetries, Rotations, Combinatorics. Rock-Paper-Scissors and the Monty Hall game: Transitive relations, Graph Theory, Probability.
The Game of Nim: Graph Theory, Base two numbers and calculations. The Math of Mastermind: Strategy, Combinatorics. Non-transitive Dice The game of SET

Topology

Topology is an area of Mathematics that studies the properties of objects that are preserved under deformations such as stretching and bending (but cutting, tearing, or gluing are not allowed). Our past workshops on topology have explored the following topics: Cylinders and Moebius bands. Both can be made by appropriately gluing of strips of paper, but while the cylinder has an inside and an outside, the Moebius band has only one side. Funny things happen when we cut a Moebius band at different distances from its edge. Playing Tic-Tac-Toe on a Torus and a Klein Bottle. The relation between the torus and the Klein bottle is similar to the relation between the cylinder and the Moebius strip: Each can be formed from a square whose edges are glued to each other in a particular way. Rather than constructing them, we represent them as flat squares, with arrows on their sides to indicate how the gluing is done. By drawing a 3x3 grid on the square, it becomes a Tic-Tac-Toe board where additional winning combinations are allowed.

Knots and Braids. First, students were shown examples of knots appearing in Chemistry (knotted molecules), Biology (DNA) and Art (Celtic knots, Islamic patterns, knitting). Then they studied when two seemingly different knots are the same, meaning that one can be transformed into the other by a series of Reidemeister moves. Since the Reidemeister moves preserve the tricolorability of the knot, we can distinguish two different knots if one can be tricolored and the second one cannot.  Another workshop on knots used Tangle toys (see the photo on the right) to study planarity properties of knots.  Hexaflexagons. Hexaflexagons are constructed by folding strips of paper, that can be flexed or folded in certain ways to reveal faces besides the two that were originally on the back and front. There is a topological method that can reveal all the hidden faces. Students were taught how to construct hexaflexagons with 3 and 6 faces.

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