### March 3, 2018

**Beginner Group:**

**Speaker: **David Sykes , Texas A&M University

**Title: Planar Graphs**

**Abstract:** Given three cottages and three wells, can we find non-intersecting paths so that every cottage is connect to each well by a different path? We will consider this problem along with others that introduce the topic of planar graphs.

**Intermediate Group:**

**Speaker: Nathan Green**, Texas A&M University

**Title: Polya Counting**

**Abstract: Polya counting theory allows us to count how many ways there are to arrange objects taking symmetry into account. For example, how many different bracelets can we make using only 3 colors of beads? How many ways can we color a cube using n colors? This counting technique has been used to count the number of different molecules which can be formed from certain sets of atoms and many other important applications.**

**Advanced Group:**

**Speaker: John Weeks**, Texas A&M University

**Title: Relations, Equivalence Classes, and Langrange’s Theorem**

**Abstract:** We will give a few definitions related to the study of relations and introductory group theory, inquire into some examples of equivalence classes, and utilize this information to analyze the nature of subgroups in finite groups.

### February 17, 2018

**Beginner Group:**

**Speaker: **Amudhan Krishnaswamy-Usha , Texas A&M University

**Title: **Fractions and bases

**Abstract:** We will look at decimal expansions of fractions and try to determine when they terminate. We will then try to expand fractions in other bases (such as binary).

**Intermediate Group:**

**Speaker: **Frank Sottile, Texas A&M University

**Title: **Shape of Space

**Abstract: **In mathematics and science, we often need to think about high (3 or more) dimensional objects, called spaces, which are hard or impossible to visualize. Besides the question of what such objects are or could be, is the problem of how we can make sense of such spaces.

The goal of this discussion is to give you an idea of how mathematicians manage to make sense of higher-dimensional spaces. We will do this by exploring the simplest spaces, and through our explorations, we will begin to see how we may tell different spaces apart. Along the way, we will dissect donuts, and I ask that at least half of the participants bring a belt.

**Advanced Group:**

**Speaker: **Parth Sarin, Texas A&M University

**Title: **Pancakes, Ham Sandwiches, and Topology

**Abstract:** Topology is a field of mathematics that tries to understand the shape of things without regards to distance or angles. In this Circle, we’ll explore some famous and surprising concepts from Topology. For example, we’ll consider whether there are two opposite points of the Earth that have the exact same temperature and pressure. And, we’ll explore how these questions are related to things you wouldn’t expect, like whether or not you can cut two pancakes in half with a very large knife using a single cut.

### February 10, 2018

**Beginner Group:**

**Speaker: **David Sykes , Texas A&M University

**Title: **Patio Planning Problems

**Abstract:** How many ways can we configure non-overlapping square tiles to build a patio with a given shape and a given perimeter? We will explore variations of this problem by drawing patio designs with small perimeters and using what we will find to make informed guesses about the answer for larger perimeters and that we then prove or disprove.

**Intermediate Group:**

**Speaker: **Valentin Zakharevich, The University of Texas at Austin

**Title: **Symmetry and Affine Transformations

**Abstract:** One of the most important ideas in geometry is that of symmetry. Understanding the symmetry of a problem can often significantly simplify finding a solution. In this presentation, we will be considering the affine symmetries of the plane, i.e. the symmetries which preserve straight lines. We will apply these ideas to understand theorems of Ceva and Menelaus.

**Advanced Group:**

**Speaker: **Tom Gannon, The University of Texas at Austin

**Title: **How to Make Friends with Graph Theory

**Abstract:** We’re going to learn about a subject called graph theory, which will be sure to impress all your friends. Graph theory is a subject about dots and lines and the various ways you can draw them. We’ll talk about complete graphs and about how friendship can be modeled by graph theory. We’ll also discuss a problem that no one on earth knows the answer to!

### February 3, 2018

**Beginner Group:**

**Speaker: **Philip Yasskin , Texas A&M University

**Title: **Cell Phone Dropping

**Abstract:** You work for a cell phone company. For advertising purposes, you are assigned the task of testing a new model of phone protector by dropping a phone from various floors of a 100 story building to determine the highest floor from which it can be dropped and not break. What is the most efficient way to perform this task if you are given 1, 2, or 3 phones?

**Intermediate Group:**

**Speaker: **Frank Sottile, Texas A&M University

**Title: **Balls and Boxes: Common shapes in uncommon dimensions

**Abstract:** Today, we will explore how common shapes- balls, cubes, triangles, and others –behave strangely in high-dimensional space. This is also an explorations of regular solids in dimensions greater than four. This is independent of last week’s circle activity.

**Advanced Group:**

**Speaker: **Alex Sprintson, Texas A&M University

**Title: **Fun with Finite State Machines

**Abstract:** We will continue our discussion about design and analysis of Finite State Machines (FSM). We will talk about minimization and equivalence problems. Towards the end, we will attend to write a program that plays a short five note song. *Please bring your computer if at all possible.*

### January 27, 2018

**Beginner Group:**

**Speaker: **Alex Sprintson, Texas A&M University

**Title: **Pi Math Contest (PiMC)

**Abstract:** We will work on the Pi Math Contest (PiMC) written by an expert committee, many of whose members are from MIT/Harvard/Stanford (see pimathcontest.com). All students in 4th and 5th grade students will officially participate in the contest. However, all students in the beginner group will work on the problems. We will discuss the problem and their solutions later in the circle. Top scoring students in this round will be invited to a Final Round in Bay Area, California on April 28th 2018.

Rulers and compasses are allowed. Calculators are not allowed (no problem on the test will require the use of a calculator). The students are strongly encouraged to visit https://alphastar.academy/event/pimc/#PiMC_2017 to see prior year tests and solutions. Top students in the first round are invited to the final round.

**Intermediate Group:**

**Speaker: **Frank Sottile, Texas A&M University

**Title: **Regular solids in all dimensions

**Abstract:** While we are all familiar with regular polygons (equilateral triangles, squares, …), and many of us know about the Platonic, or regular solids (tetrahedron, cube, octahedron,….), few are familiar with their analogs in dimensions four and higher. Of course, this is because we are not equipped to perceive four-dimensional space directly.

Nevertheless, the description of the regular solids in all dimensions has been known for a long time. The purpose of my talk will be to introduce you to these objects, with an emphasis on how to think about them. This presentation will be spiced up with some models of four-dimensional regular solids, some of which you can build yourself. This is independent of last week’s circle activity.

There is a link to an animation:

http://www.math.tamu.edu/~sottile/talks/17/4D/index.html

**Advanced Group:**

**Speaker:** Philip Yasskin, Texas A&M University

**Title: **Point Set Topology

**Abstract:** We will discuss the basic definitions of point set topology.

### January 20, 2018

**Beginner Group:**

**Speaker: **Amudhan Krishnaswamy-Usha, Texas A&M University

**Title: **Counting with Aliens

**Abstract:** We will discuss different number system and discover how to convert numbers from one base to another.

**Intermediate Group:**

**Speaker: **Frank Sottile, Texas A&M University

**Title: **Archimedean Solids

**Abstract:** Most of us know the five Platonic (or regular) solids; next to the sphere, they are the most regular and beautiful objects in our three-dimensional world. Less well-known are the Archimedean or semi-regular solids. In this math circle activity, we will recall the Platonic solids, and then explore the Archimedean solids and some relations between them. We will be building them and then studying our constructions. If time, I will explain their relation to fair dice.

**Advanced Group:**

**Speaker:** Alex Sprintson, Texas A&M University

**Title:** Introduction to Finite State Machine Design

**Abstract:** We will discuss mathematical aspects of logic design. We will start with a review the fundamentals of boolean algebra and design of K-maps. Next, we will discuss the fundamentals of design and implementation of Finite State Machines (FSM) that solve engineering problems. If time permits, we will discuss the capabilities and limitations of FSM.