Archive
2017-18 Academic Year Activities
September 16, 2017
Beginner Group – Amudhan Krishnaswamy-Usha on A Few Easy Tests for Divisibility
Presenter: Amudhan Krishnaswamy-Usha
Department of Mathematics, Texas A&M University
Abstract: I will present some tests for divisibility by small numbers, and introduce modular arithmetic and congruence to explain why they work.
Intermediate Group – Frank Sottile on The Five Color Theorem
Presenter: Frank Sottile
Department of Mathematics, Texas A&M University
Abstract: Doodling on a map of England in 1852, Francis Guthrie noticed that only four colors were needed to color the counties. He conjectured that any map could be colored with only four colors. Several mathematicians tried and failed to prove this; notably in 1879 Kempe published a proof and only in 1890 was the flaw found by Heawood. This four color conjecture evaded a proof until 1972, when Appel and Haken gave a proof that required a computer. While there is as yet no Human readable proof, Kempe’s argument suffice to prove that five color suffice, and this gives a flavor of known proofs of the four color theorem. I will sketch this history and prove the five color theorem.
Advanced Group – Alex Sprinston on Cracking the Code
Presenter: Alex Sprintson
Department of Engineering, Texas A&M University
Abstract: We will provide a brief overview of the fundamentals and applications of the coding theory. First, we will focus on efficient error and erasure correcting codes. Then, we will discuss network codes and codes for distributed storage.
September 23, 2017
Beginner Group – David Sykes on Euclid’s Algorithm
Presenter: David Sykes
Department of Mathematics, Texas A&M University
Abstract: We will explore properties of common divisors. In particular we will discuss how to find greatest common divisors using the Euclidean algorithm, and we will investigate why the Euclidean algorithm works.
Intermediate & Advanced Groups – Dr. Zuming Feng on An Example on Math Learning via Classroom, Extra Extracurricular, and Contest Activities
Presenter: Dr. Zuming Feng
Phillips Exeter Academy
Proof School Board member
Cogito, part of Johns Hopkins University’s CTY SET program Board member
Former coach of the USA International Mathematics Olympiad (IMO) team
September 30, 2017
Beginner Group – Kagan Samurkas on Mathematical Games of Strategy
Presenter: Kagan Samurkas
Department of Mathematics, Texas A&M University
Abstract: Some mathematical games that one part has a winning strategy.
Intermediate & Advanced Groups – Sherry Gong on Algebra Tricks for Math Contests
Presenter: Sherry Gong
Massachusetts Institute of Technology
October 14, 2017
Beginner Group – Amudhan Krishnaswamy-Usha on Primes
Presenter: Amudhan Krishnaswamy-Usha
Department of Mathematics, Texas A&M University
Intermediate Group- Philip Yasskin on Axiomatic Finite Geometries
Presenter: Philip Yasskin
Department of Mathematics, Texas A&M University
Abstract: We will study geometries with a finite number of points and lines satisfying a set of axioms.
Advanced Group – Igor Zelenko on Sums of kth Powers
Presenter: Igor Zelenko
Department of Mathematics, Texas A&M University
Title: Sums of kth powers: from telescopic sums and Lagrange interpolations to Bernoulli numbers and Euler-Maclaurin formula
Abstract: The formula for the sum of first n positive integers is taught in school. What is the sum of their squares, cubes etc? During this class, we will learn various methods to derive the formulas for these sums from more elementary to more advance.
October 28, 2017
Beginner Group – Philip Yasskin on Playing with Toilet Paper
Presenter: Philip Yasskin
Department of Mathematics, Texas A&M University
Abstract: We will solve a series of problems associated with folding toilet paper.
Intermediate Group- Maurice Rojas on Guessing, Sorting, and Optimizing
Presenter: Maurice Rojas
Department of Mathematics, Texas A&M University
Abstract: The log function is something we should all know. In this activity, we’ll see how log pops up in the game of “high-low”, and in algorithms for sorting. We’ll then see log again appears in an interesting geometric problem: How do you find the rectangle with axis-parallel sides of largest area inside a polygon? We’ll see how this geometric problem is practically important in architectural design.
Advanced Group – Edriss S. Titi on What is mathematics? A journey through examples.
Presenter: Edriss S. Titi
Department of Mathematics, Texas A&M University
Abstract: Why the honeycomb has hexagonal cell shapes? Is it because bees are lazy, unlike what is commonly believed!! Remarkably, a new mathematical framework has to be invented, every now and then, in order to answer intriguing, yet simple, questions of the kind mentioned above. In this lecture, I will provide few other simple examples, that have played fundamental role in advancing mathematics, as an additional support of this observation.
November 11, 2017
Beginner Group – Amudhan Krishnaswamy-Usha on The Euclidean Algorithm
Presenter: Amudhan Krishnaswamy-Usha
Department of Mathematics, Texas A&M University
Abstract: We will explore the GCD, LCM, and the Euclidean algorithm.
Intermediate Group- Abraham Martin del Campo on Probability and Algebra
Presenter: Abraham Martin del Campo
Department of Mathematics, CIMAT
Abstract: We will explore some basic probability concepts through a coin tossing game and use a little bit of algebra to find if we can play a fair game.
Advanced Group – Nathan Green on Polya Counting
Presenter: Nathan Green
Department of Mathematics, Texas A&M University
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.
November 18, 2017
Beginner Group – David Sykes on The Fibonacci Sequence
Presenter: David Sykes
Department of Mathematics, Texas A&M University
Abstract: We will learn about the golden ratio while solving problems associated with Fibonacci Numbers.
Intermediate Group- Jennifer Whitfield on Using Euler and Hamiltonian Paths to Get Around
Presenter: Jennifer Whitfield
Department of Mathematics, Texas A&M University
Abstract: In this session, we will investigate the different paths that exist on a given graph. We will also discover some properties of Euler and Hamiltonian paths and then apply the properties to solve problems.
Advanced Group – Doug Hensley on Putnam Problems
Presenter: Doug Hensley
Department of Mathematics, Texas A&M University
Abstract: The “Putnam” is the William Lowell Putnam mathematical competition. It’s famous for being both challenging and fun. The hard part is that the problems always have a twist to where you never “know how to work that type”going in. The fun part is that there is always a sweet solution.
November 25, 2017
Beginner Group – Amudhan Krishnaswamy-Usha on Fibonacci and Other Numbers
Presenter: Amudhan Krishnaswamy-Usha
Department of Mathematics, Texas A&M University
Abstract: Continuing the discussion led by David Sykes’ last math circle, we will look at Fibonacci numbers and a few of their properties. If time permits, we will look at a few other sequences coming from ‘recurrence relations’.
Intermediate Group- Phil Yasskin on When do 4 or more Points Lie on a Circle?
Presenter: Phil Yasskin
Department of Mathematics, Texas A&M University
Abstract: We will first answer the question: “When do 4 points lie on a circle?”. Then we will prove the Nine Point Theorem and maybe some others.
Advanced Group – Frank Sottile on The Five Color Theorem
Presenter: Frank Sottile
Department of Mathematics, Texas A&M University
Abstract: Doodling on a map of England in 1852, Francis Guthrie noticed that only four colors were needed to color the counties. He conjectured that any map could be colored with only four colors. Several mathematicians tried and failed to prove this; notably in 1879 Kempe published a proof and only in 1890 was the flaw found by Heawood. This four color conjecture evaded a proof until 1972, when Appel and Haken gave a proof that required a computer. While there is as yet no Human readable proof, Kempe’s argument suffices to prove that five color suffice, and this gives a flavor of known proofs of the four color theorem. I will sketch this history and prove the five color theorem, and then discuss the coloring theorem for other surfaces (torus, projective plane, Klein bottle…).
December 9, 2017
Elementary Group – Dean Baskin on Towers of Hanoi
Presenter: Dean Baskin
Department of Mathematics, Texas A&M University
Abstract: This activity is aimed at students in second grade. We will introduce the Towers of Hanoi game, work out how to solve it in small cases and try to find a pattern in the number of moves required. If there is time left at the end, we will play some mathematically oriented games.
Beginner/Intermediate Group- Jane Long on Factors and Primes
Presenter: Jane Long
Department of Mathematics, Stephen F. Austin University
Abstract: Prime numbers, those counting numbers with exactly two distinct factors (themselves and one), are very special in mathematics. We’ll discuss ways to find prime numbers and other factors of counting numbers, and investigate perfect numbers, amicable numbers, and some really big numbers.
Intermediate/Advanced Group – Nicholas Long on Counting Rectangles with Integer Sequences
Presenter: Nicholas Long
Department of Mathematics, Stephen F. Austin University
Abstract: When you look at a chess board, you can see lots of squares. How many do you see? How many rectangles can you make with the blocks on a chess board? What if the chess board was a different size? Slightly different figures can lead to wildly different patterns when you think flexibly about where squares and rectangles come from. We will also look at how the sequence of integers generated by these other questions can lead to a lot of related problems on the OEIS (Online Encyclopedia of Integer Sequences).
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.
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.
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.
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 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.
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.
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.
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.
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 28, 2018
Speaker: David W. Gent, P.E. (Founder and Chairman of SofTest 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.
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.
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.