Madison Math Circle Abstracts 2021-2022

Meetings for Fall 2021

Fall 2021
Date	Speaker	Topic
September 20th	Daniel Erman	Number Games We’ll play some math-based games and then try to understand some of the patterns we observe.
September 27th	Evan Sorensen	The fastest way to travel between two points Given two points, we know the shortest distance between the points is a straight line. But is that always true? We will talk about how to build the best track for a toy car to travel between two points. We’ll start by trying a few different options together and having a race. We’ll then talk about how two brothers thought about how to solve this problem using interesting examples from physics.
October 4th	Yandi Wu	Do you wanna build a donut? Topology is a field of math that deals with studying spaces. This math circle talk is an introduction to a concept in topology called “cut-and-paste” topology, which is named that way because we will build spaces out of cutting and gluing pieces of paper.
October 11th	Ivan Aidun	Words, Words, Words We'll play a game where you have to guess a secret word that I choose. We'll figure out how to use logic to improve our guesses. Then, we'll explore some questions like: is there a best way to guess? or, what happens when I change the rules slightly?
October 18th	Allison Byars	Sheep and Wolves In this math circle talk, we'll look at placing sheep and wolves on a grid so that none of the sheep get eaten. We'll find different arrangements and try to figure out the maximum number which can be placed on a board of given size and generalize it for an arbitrary board. We will also discuss how this relates to a field of mathematics called combinatorics.
October 25th	Jacob C Denson	Proofs in Three Bits or Less How many questions does it take to beat someone at Guess Who? How long should it take for you to figure out how to get to this math talk from your house? How many questions do you have to ask your classmate before you know they're telling the truth to you? Let's eat some pizza, and talk about how mathematicians might reason about these problems.
November 1st	Qin Li	How do we describe the world? The physical world consists of everything from small systems of a few atoms to large systems of billions of billions of molecules. Mathematicians use different languages and equations to describe large and small systems. Question is: How does mother nature use different languages for different systems and scales? Let us see what these languages look like, talk about their connections and differences, and see how they are reflected in our day-to-day life.
November 8th	John Yin	River Crossings Here's a classic puzzle: A farmer needs to move a wolf, a sheep, and a box of cabbages across a river. He has a boat that can fit only one object other than himself. However, when left alone, the wolf will eat the sheep, and the sheep will eat the cabbages. How can the farmer move the wolf, the sheep, and the box of cabbages across the river without anything being eaten? I will discuss this problem by connecting it to graph theory, then give a generalization.
November 15th	Erik Bates	How big is a cartographer’s crayon box? Have a look at a world map. If you are looking at one with borders and colors, notice that no border has the same color on both sides. That is, no neighboring countries are colored the same. So how many different colors are needed to make this possible? Does the answer change for a map of the U.S., when we try to color its fifty states? What about a map of Wisconsin with its 72 counties? We will explore these questions---and uncover some very deep mathematics---by doing the simplest and most soothing activity: coloring.
November 22nd	Robert Walker	Lagrange's Four Square Sum Theorem How many perfect squares are needed to represent each nonnegative integer n as a sum of perfect squares? This talk will answer that precise question -- students will get to the bottom of this.

Meetings for Spring 2022

Spring 2022
Date	Speaker	Topic
February 7th	Aleksandra Cecylia Sobieska	Mathematical Auction We will play a game called “Mathematical Auction,” where teams have the opportunity to solve and steal problems for points by presenting solutions that build on one another.
February 14th	Jake Fiedler	Fractals in Math and Nature If you've ever had to clean up branches after a storm, you may notice that the branches look surprisingly like the whole tree they fell from, just at a smaller scale. Similarly, lightning bolts during that storm probably had numerous "arms", each appearing similar to the entire bolt. In this talk, we'll investigate this behavior more closely through objects called fractals. We'll see how fractals are made, where they appear in the real world, and then you'll get a chance to build your own.
February 21st	Mikhail Ivanov	Elevator with just 2 buttons. There are two buttons inside an elevator in a building with twenty floors. The elevator goes 7 floors up when the first button is pressed, and 9 floors down when the second one is pressed (a button will not function if there are not enough floors to go up or down). Can we use such elevator? We'll play with this elevator found math behind it.
February 28th	Michael Jesurum	Bubbling Cauldrons Place our numbers into the cauldrons in ascending order – you can choose which cauldron each one goes in. However, if two numbers in one cauldron add up to a third number in that same cauldron, they bubble up and cause an explosion! This means that all the numbers leave the cauldrons, and you must start all over again. Our goal is to find the largest number we can place in our cauldrons without them exploding… do you think you’re up for this daunting task?
March 7th	Erika Pirnes	Reconstructing Graphs A graph is a "picture" with dots (called vertices) and lines (called edges). From a graph, we can extract information called the deck. In this talk, we will explore the connection between a graph and its deck, and how we can move from one to the other. We will do a lot of examples! There is a famous conjecture (unproven result) that stays that a graph can always be reconstructed (recovered) from its deck. This is called the reconstruction conjecture. (There are some small restrictions on what the graph can be)
March 14th	SPRING BREAK	NA NA
March 21st	Ian Seong	Center of a triangle? But which center? It is easy to locate the center of a circle, or regular polygons. How do we define the center for an arbitrary triangle? In fact, for each triangle, there are many points that can be entitled the "center". We will investigate a few of them (classic examples are circumcenter and incenter) and learn how they are constructed.
March 28th	Caitlin Davis	Math and voting: Can math help us make decisions more fairly? We are often faced with decisions we must make as a group. For example, a city might need to decide on a new mayor, or you and your friends might need to decide on a movie to watch or a type of pizza to share. We often use voting to try to make a fair choice. The voting method which you’re probably used to is called “plurality,” but it turns out there are many other possible voting methods. Could one of them be more fair than plurality? We’ll talk about how math can be used to study questions like this.
April 4th	BREAK	NA
April 11th	Aleksander Skenderi	Happy Numbers In many areas of mathematics, we look for patterns to describe or model a particular problem. However, sometimes these patterns occur in some late stage of some process, and sometimes not at all! For instance, if a particle of gas is moving around in a container, it may be that, after some time, the gas particle follows an easily described trajectory. It may also be, depending on the initial trajectory, that the gas particle moves totally randomly. In this talk, we'll describe a class of numbers called "happy numbers," and explore some of their properties and patterns.
April 18th	John Cobb	Chip-Firing Games on Graphs We will play a game called the dollar game, where we will try to clear out debt among a group of people in a funny way. Then, we’ll investigate ways to see when this is possible and how to do it, leading to some unexpected conclusions and a look into a very active area of math called tropical geometry.
April 25th	Chengxi Wu	Non integer bases and Paths on colored graphs We can count in base 10 (decimals) or base 2 (binary), but how about counting in base 5/3, or the golden ratio? We will investigate that via the question of finding possible paths on a graph with colored vertices, and also look at some of the interesting self similar patterns we can get from it!