# Madison Math Circle Abstracts 2015-2016

## September 14 2015

 David Sondak Title: How to SEE Sound The idea is to give a simple overview of sound waves by introducing sines and cosines and some of their basic anatomy (amplitude and frequency). We will then have a computational component where the students create their own sound waves by fiddling with parameters in the sines and cosines (again, amplitude, frequency and different superpositions of the sines and cosines). They will actually be able to see plots of their waves AND listen to their waves. Finally, if time permits, the students will use their own sound waves to make Oobleck dance. This will bring the exercise full circle in that they will be able to see their very own sound waves in action.

## September 21 2015

 Prof. Uri Andrews Title: Guarding Mona Lias You have gotten a tip that a famous art thief is going to steal something from the Louvre. It is your task to organize a security team that can watch all the works of art. The problem is that the Louvre is really big and has a strange layout. Where do you put your guards? And how many do you need?

## September 28 2015

 Eva Elduque Title: Pick's Theorem In this talk, we will a very easy formula that allows us to quickly compute the areas of polygons whose vertices are points of a grid, and we will prove that this formula works. (Solutions to the worksheet distributed during the circle can be found File:Pick.pdf.)

## October 5 2015

 Jessica Lin Title: The Math of Sudoku Have you ever sat next to someone in the airport or airplane who plays sudoku? Have you ever tried to play yourself? When you play, do you have some strategies that help you to complete the puzzle? It turns out that there is some deep mathematics behind this simple game. Come to math circle this week to learn about it, and maybe you can help the person next to you solve his/her sudoku!

## October 12 2015

 Ryan Julian Title: The Geometry of Hockeysticks and Eight Dimensional Oranges Like most people, I've often considered opening an eight dimensional grocery store. Of course, the main difficulty with this plan is that I'd need some way of neatly stacking all of the eight dimensional fruit that I'd be selling. In this talk, we'll explore a variety of elementary counting problems, discover that nearly all elementary counting problems are really the same problem, and we'll apply these new insights to determine how to stack 8 dimensional fruits into neat 8 dimensional pyramids.

## October 26 2015

 Megan Maguire Title: Aperiodic tilings: Beyond your parents' bathroom floor A tiling is a way of covering the plane with geometric shapes such that there are no overlaps or gaps. If you have any tile in your home (maybe in your kitchen or bathroom) that is most likely an example of a tiling. Come learn about the cool math behind tilings and about the coolest tiling of all, the Penrose tiling.

## November 2 2015

 Marko Budisic Title: Mathematics of GPS Satellites GPS is a system of satellites circling the Earth at a height 12,500 miles. That means you could easily fit both Mars and Venus in the distance between your phone and each car-sized satellite hovering in space. Once considered science fiction, GPS is now a part of our everyday life: we can use it through our phones, through our car navigation, and even some watches. Simple math equations lie at the heart of this system, and we will write them down, understand what they mean, and figure out how to solve them.

## November 9 2015

 Tess Anderson Title: Gold Coins and Goast What do pulling gold coins out of a a hat have to do with the famous Monty Hall "Goat Problem" in which you are a game show contestant trying to pick out the one prize hidden behind one of three doors? Come and find out while savoring some chocolate gold coins. We will also discuss a jailer problem in which an infinite number of jailers try to free an infinite number of prisoners. If time permits, other fun problems will be discussed.

## November 16 2015

 DJ Bruce Title: To Infinity and Beyond? 1, 2, 3,..., infinity? What is infinity? Is infinity plus one bigger than infinity? Beginning by figuring out what we mean when we say to collections of objects have the same number of things we will slowly work our way deep into the world of infinity. This world is often weird and counterintuitive, and we shall explore it!

## November 23 2015

 Prof. Tullia Dymarz Title: To Infinity and Beyond? TBA

## February 1 2016

 Will Mitchell Title: Are these networks the same? The question of deciding whether two things are the same comes up in many different places in math. In this session we'll consider the problem of deciding if two networks or "graphs" are the same. This leads to some entertaining and challenging puzzles. We will also learn a bit about how people try to solve similar problems using computers. This problem has applications in the design of electronic circuits and in searching for organic chemical compounds within large databases. The handout, presentation, and solutions, can be found here, here, and here respectively.

## February 8 2016

 Lalit Jain Title: Large amounts of small money This week we will talk about coins, change and some of the surprising mathematics and computer science behind a very simple problem: How many ways are there to make change for a dollar?

## February 15 2016

 Daniel Erman What is the biggest number? The title is something of a joke, though a more serious question would be: what are the biggest numbers that have actually been used in mathematics or science? We will use this question as a launching off point for exploring the use of exceptionally large numbers in math and science. Along the way, we will touch on ideas related to the prime numbers, chemistry, astrophysics, and graph theory.

## February 22 2016

 Soumya Sankar What would you do if you had only nickels and dimes? This is an instance of the classical Frobenius Coin problem: If you are given two denominations of coins and asked to combine them to get different values, after what point, if at all, can you get all subsequent values of money? Starting from this simple problem, one can ask a variety of questions. I will talk about how one can play around with some of these.

## February 29 2016

 Alexandra (Sashka) Kjuchukova How big is your infinity? If a "box" A fits inside a "box" C, with room to spare, is A "smaller" than C? Oddly, if A and C are infinite, the answer may be negative. In this talk, we generalize the concept of "counting" to sets with infinitely many elements. We then find examples of infinite sets which are "equal in size" even though one contains "several copies" of the other. We also show that not all infinite sets have the same size.

## March 7 2016

 Solly Parenti 101-meter Dash This talk will address the question of the fastest prime number. If you don't know too much about prime numbers, that's okay, we'll learn about prime numbers. We'll see why the prime numbers are important, how many primes there are, and we'll even have a race between the primes. Come find out who wins!

## March 14 2016

 Chiara Franceschini PRIME NUMBERS: Why (not only) mathematician care about them? Mathematicians have studied primes since before antiquity, and the desire of finding the biggest prime can be compared to the desire of winning an Oscar for an actor. However, if in the past they did it ‘for the glory’, now primes are everywhere in our everyday life. For example every time we use our credit card. In this talk I’ll give some historical references, some curiosity about primes and I’ll try to explain their modern use in cryptography.

## April 4 2016

 Becky Eastham Become a Mastermind at Codebreaking and Logic Puzzles! Mastermind is a two-player codebreaking game in which Player One creates a secret code (a sequence of four pegs, each of which is one of six different colors) and Player Two must guess the correct code within a certain number of guesses. For each sequence of four colored pegs guessed by Player Two, Player One indicates both how many of the pegs are both the right color and in the right spot, and how many of the pegs are the right color but in the wrong spot. We will play through some examples of this game, solve some logic puzzles related to the game, and learn algorithms to guess codes efficiently.

## April 11 2016

 Micky Soule Steinberg Groups: An Introduction We discuss the definition of groups along with some familiar examples. We also present the group of symmetries of an equilateral triangle.

## April 18 2016

 Alisha Zacharia The Power Tower If we allow ourselves to define what it means to perform an arithmetic operation infinitely many times, we open the doors to answering a slew of cool questions. We will discuss how we can rigorously make sense of this, and use it to answer a question about "power towers" that we might have otherwise dismissed as not meaningful.

## April 25 2016

 Betsy Stovall Title: N to N+1. The art of induction. Mathematical induction is a tool that lets us prove statements about the integers from surprisingly little information. In this talk, we will see a brief explanation of the principle, and then do lots of examples. We will also "prove" that every horse is the same color.

## May 2 2016

 Jen Beichman Title: Using Bubbles to Solve Math Problems There are several questions in mathematics about finding the shortest path between points and on surfaces. We'll discuss two of these problems and find solutions using bubbles! Come prepared to work with your hands and possibly get a little damp. There will also be pen and paper activities if you don't want to play with bubbles. Special location: 9th Floor of Van Vleck Hall.

# High School Meetings

## September 28 2015

 Prof. Daniel Erman Title: How to Catch a (Data) Thief I will discuss some surprising statistical facts that have been used to catch companies that lie about data.

## October 19 2015

 Carolyn Abbott Title: Donuts and coffee cups: the topology of surfaces A classic problem in topology is to decide whether one surfaces can be deformed into another, without creating any holes or connecting any new points (stretching and bending is allowed!). If you can do so, such surfaces are considered 'the same.' We will formalize this notion and classify all closed surfaces, along the way answering such questions as whether a coffee cup is the same as a donut.

## February 22 2016

 Jordan Ellenberg Title: The Game of Set TBD

## March 31 2016

 Daniel Erman Title: How to catch a (data) thief I will discuss some surprising statistical facts that have been used to catch companies that lie about data.

## April 18 2016

 DJ Bruce Title: To Infinity and Beyond 1, 2, 3,..., infinity? What is infinity? Is infinity plus one bigger than infinity? Beginning by figuring out what we mean when we say to collections of objects have the same number of things we will slowly work our way deep into the garden of infinity. A garden that is often profoundly strange and filled with quite a few surprising snakes.

## April 21 2016

 DJ Bruce Title: Can you untie a know with a knot Is it possible to tie two knots on a rope such that when you slide them together they unknot themselves? The answer turns out to be interesting, and related to the sum 1-1+1-1+1-1+...

## April 21 2016

 DJ Bruce Title: Can you untie a know with a knot Is it possible to tie two knots on a rope such that when you slide them together they unknot themselves? The answer turns out to be interesting, and related to the sum 1-1+1-1+1-1+...

## May 2 2016

 DJ Bruce Title: Is any knot not the unknot? You're walking home from school, and you pull out your head phones to listen to some tunes. However, inevitably they are a horribly tangled mess, but are they really a knot? We'll talk about what exactly is a knot, and how we can tell when something is not the unknot.