### May 12, 2018

**Beginner Group:**

**Speaker: **Max Warshauer, Texas State University

**Title: **Modular arithmetic and divisibility

**Abstract:** In this Math Circle, participants will learn about modular arithmetic and use this to discover divisibility properties. This will be a hands-on exploration where participants investigate problems, make conjectures, and then give careful arguments to prove or disprove their conjectures.

**Intermediate 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 of the fundamentals of boolean algebra and design of K-maps. Next, we will discuss the fundamentals of design and implementation of the Finite State Machines (FSM) that solve engineering problems.

**Advanced Group:**

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

**Title: **Rigid motions and symmetries in 2 and 3 dimensions

**Abstract:** Rigid motions are transformations which preserve angles and distances (for example, rotations). We will explore what these look like in two and three dimensions and attempt to classify them. If time permits, we will also look at rigid motions which preserve a given figure (the symmetries of the figure), and see what happens in the case of regular polygons.

### May 5, 2018

**Beginner Group:**

**Speaker: **Erica Metheney, Texas A&M University

**Title: **Intro to Data: What’s the Story?

**Abstract:** We will begin with discussing what statistics is and why it is useful. During the session, students will learn about different data types, summary statistics, and data visualization techniques then apply their new skills on data they will collect themselves. After the session, students should be able to identify data types, summarize different types of data, and create appropriate data visualization graphics.

**Intermediate Group:**

**Speaker: **Theodora Chaspari, Texas A&M University

**Title: **How is mathematics used in artificial intelligence and machine learning applications?

**Abstract: **Artificial intelligence (AI) is the field of study that focuses on machines or computer programs that are capable of thinking, acting, and learning like humans. Machine learning is the sub-field of AI that develops computer programs able to access and automatically learn from data without human assistance or intervention. AI and machine learning applications are found in every aspect of everyday life from health and well-being to entertainment and military applications. Examples of these include online games, self-driving cars, chatbots, fitness monitors, and others. Mathematics is one of the most important tools for developing AI and machine learning applications since they allow representation of real-world objects into numbers and derive meaningful interpretations from these numbers. For example, an image taken by our camera can be represented as a matrix of integers, whose structure can convey meaningful information regarding the content of the image (e.g., whether an image contains a cat or a dog). This lecture will cover basic AI and machine learning concepts and will discuss how mathematics is tightly connected to these, inspired by two applications from the computer vision and biomedical health informatics domains.

**Advanced Group:**

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

**Title: **Groups, Cosets, and Lagrange’s Theorem

**Abstract:** We will explore the concepts of an algebraic group and subgroup and discuss some natural partitions of a group called cosets. If time allows, we will use modulo arithmetic to investigate how the number of elements in these groups compares with that of any subgroup contained within them. This will be an elementary talk with a goal of introducing students to the nature of upper-level mathematics.

### April 28, 2018

**Speaker: **David W. Gent, P.E. (Founder and Chairman of **S**of** T**est Designs)

**Title:**Amateur Radio Satellite Orbital Mechanics

**Abstract:**There are countless objects currently orbiting Earth. Some of these are Communication Satellites built by Amateur Radio operators like me.

To talk through these Satellites, we must chart their current locations in space and predict their future passes above our horizon. This tracking depends upon various concepts developed by several brilliant mathematicians over the years. This is the story of that development. Example calculations using the International Space Station (ISS) are given.

### April 21, 2018

**Beginner Group:**

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

**Title: The Icosian Game**

**Abstract:** We will play variants of William Rowan Hamilton’s icosian game and explore its relation to other mathematical puzzles.

**Intermediate Group:**

**Speaker: Pedro Morales**, The University of Texas at Austin

**Title: Internet privacy: How secure communication workss**

**Abstract: We will explore one of the most used encryption methods, known as the RSA. This is one of the most important applications of number theory and prime numbers. By applying basic concepts of modular arithmetic, we can develop a public key system which enables the secure exchange of information between two strangers.**

**Advanced Group:**

**Speaker: Eviatar Procaccia**, Texas A&M University

**Title: Simulations of random walks**

**Abstract:** We will learn basic MATLAB programming and then simulate a process called “simple random walk”, which models fundamental natural phenomena. We will use these simulations to compute some probabilities and verify the numerical results with theoretical calculations.

### April 14, 2018

**Beginner/Intermediate Group:**

**Speaker: **Josiah Coad, Texas A&M University

**Title: Everyone Can Code!**

**Abstract:** We are excited to announce a new opportunity for the students in TAMU Math Circle to learn how to code. We will be using a website code.org, which has developed many fun and educational programs for kids of all ages and skill levels to learn how to code, no previous experience necessary. In an hour, the students will learn through a drag/drop interface the basics of coding logic, all while programming their own game! With many opportunities to rank up fast, we hope to inspire students to a self-discovery of this exciting field.

**Advanced Group:**

**Speaker: Roger Howe**, Texas A&M University

**Title: The medial triangle, the Euler line, and the nine-point circle.
**

**Abstract:**Studying the relationship between a triangle and the triangle formed by the midpoints of its sides (known as the medial triangle) gives deeper insights into the triangle geometry than were found by the Greeks. A key part of this development is the Euler line, which exhibits a beautiful relationship between circumcenters, orthocenters, and centroids. This in turn sheds light on the nine-point circle and beautiful related configurations.

### March 24, 2018

**Beginner Group:**

**Speaker: **Josiah Coad , Texas A&M University

**Title: Everyone Can Code!**

**Abstract:** We are excited to announce a new opportunity for the students in TAMU Math Circle to learn how to code. We will be using a website code.org, which has developed many fun and educational programs for kids of all ages and skill levels to learn how to code, no previous experience necessary. In an hour, the students will learn through a drag/drop interface the basics of coding logic, all while programming their own game! With many opportunities to rank up fast, we hope to inspire students to a self-discovery of this exciting field.

**Intermediate Group:**

**Speaker: Dimitar Grantcharov**, The University of Texas at Arlington

**Title: Invariants**

**Abstract: Invariants are special “mysterious” tools that play an important role in various mathematical fields. We will solve several problems from number theory, game theory, combinatorics, and geometry that use invariants.**

**Advanced Group:**

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

**Title: Counting Polyominoes**

**Abstract:** We will be counting the number of polyominoes sharing certain properties. The activity will present several fun combinatorial problems, some of which remain unsolved; for example, to date, there is no explicit formula for the number of polyominoes having a given size.

### 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.