### May 14, 2016

##### Beginner and Intermediate Groups: Kaitlyn Phillipson on Math Games

Presenter: Kaitlyn Phillipson

Department of Mathematics, Texas A&M University

Abstract: We’ll discuss some games and try to come up with winning strategies.

##### Advanced Group: Riad Masri on the Arithmetic of Integer Partitions

Presenter: Riad Masri

Department of Mathematics, Texas A&M University

Abstract: The goal of this project is to explore some arithmetic aspects of integer partitions. In particular, we will focus on Ramanujan’s famous congruences for the partition function, and study how the Dyson rank and the Andrews/Garvan crank can be used to give a combinatorial explanation for these congruences.

### May 7, 2016

##### Beginner Group: Philip Yasskin on Balance Beams

Presenter: Philip Yasskin

Department of Mathematics, Texas A&M University

Abstract: We use a meter stick as a balance beam with a pencil at the 50 cm mark. In each problem, we put weights at the locations indicated and experiment to figure out where to put the extra weights. We will progress to using equations to figure out where to put the weights.

##### Intermediate Group: Gregory Berkolaiko on Icosahedron Made from Scratch

Presenter: Gregory Berkolaiko

Department of Mathematics, Texas A&M University

Abstract: The task is to make an icosahedron from scratch using only paper and glue (plus compass, ruler, scissors and pencil). Along the way we will need to solve the problem of dividing the circle into the equal parts lengthwise.

##### Advanced Group: Timo de Wolff on Public Key Cryptography

Presenter: Timo de Wolff

Department of Mathematics, Texas A&M University

Abstract: Cryptography handles with the secure transmission of secret

messages. More precisely, a third party is supposed to be unable to

understand the content of an intercepted message if it is encrypted.

Classically, both parties exchange secret keys for a secure en- and

decryption. Nowadays, however, a lot of communication happens via

insecure channels like the internet. Thus, secret keys often cannot be

exchanged securely. Thus, one needs a new type of crypto system, in

which parts of the keys do not need to be hidden anymore. This is called

public key cryptography.

In this talk we will first review a couple of classical symmetric crypto

systems like the Ceasar cipher. In the second part I will explain and

show the RSA crypto system, which is the current industry standard for

public key cryptography.

### April 30, 2016

##### Beginner Group: Frank Sottile on Meet the Cube

Presenter: Frank Sottile

Department of Mathematics, Texas A&M University

Abstract: We will investigate the familiar cube, using it

to study three-dimensional geometry.

##### Intermediate Group: Christopher O’Neill on When Can You Draw a Picture Without Picking Up Your Pencil?

Presenter: Christopher O’Neill

Department of Mathematics, Texas A&M University

Abstract: Suppose someone hands you a picture and asks you to trace it in one continuous motion, that is, without picking up your pencil or backtracking. When is it possible to succeed? How should you decide where to start tracing?

##### Advanced Group: Eric Rowell on Mathematical Knots and Links

Presenter: Eric Rowell

Department of Mathematics, Texas A&M University

Abstract: Knots and links have been used as decorations for centuries, but their mathematical study only began in the 19th century. For a brief period it was believed that atoms were just knotted bits of swirling ether, and physicists set to work to tabulate them. It turned out they were completely wrong, but this led to the development of topology. More than 100 years later, knots may again be useful in physics though Topological Quantum Computation. We will explore important questions surrounding knots and links, such as: how do we know when two knots are actually the same? How can we tell that they are genuinely different?

### April 23, 2016

##### Beginner Group: Roger Howe on Around the Pythagorean Theorem

Presenter: Roger Howe

Department of Mathematics, Yale University

Abstract: This session will discuss a few of the many applications of the Pythagorean Theorem in the real world. Among the questions to be considered will be, why do ladders work, taking shortcuts, and how far can we see?

##### Advanced Group: David Dynerman on Normal Mapping and Video Games

Presenter: David Dynerman

Department of Mathematics, University of California, Berkley

Abstract: Normal mapping is a way of increasing surface detail when rendering 3D graphics and has become a standard technique in the video game industry. Normal mapping sneaks in higher quality lighting detail over a lower-quality polygonal model. This talk will give an overview on how this interesting application of math, computer science and physics creates better looking video games.

### March 26, 2016

##### Beginner Group: Ola Sobieska on Weighings and Counterfeit Coins

Presenter: Ola Sobieska

Department of Mathematics, Texas A&M University

Abstract: This session will focus on problems about balance scales and weights. The students will learn to identify counterfeit coins, discover tricky ways to weigh objects, and solve other puzzles.

##### Intermediate Group: David Sykes on Dinner Party Problems and Graph Coloring

Presenter: David Sykes

Department of Mathematics, Texas A&M University

Abstract: We will be discussing the Dinner Party Problem along with some of its generalizations while exploring the concept of graph coloring. The problems are special cases of a theorem established by Frank Ramsey around 1930. The discussion will build towards the solution to a challenging Ramsey Theory problem along with the statement of problems that remain unsolved today.

##### Advanced Group: Matt Young on Which Numbers are Sums of Two Squares?

Presenter: Matt Young

Department of Mathematics, Texas A&M University

Abstract: Some numbers are the sum of two squares, and some numbers aren’t. For example, 5 is (since 5 = 1 +4) but 7 isn’t. Numbers that can be expressed as the sum of two squares have many amazing properties, and we will discover many of these patterns in this math circle.

### March 5, 2016

##### Beginner Group: Phil Yasskin on Splitting Piles and Handshakes

Presenter: Phil Yasskin

Department of Mathematics, Texas A&M University

Abstract: We will consider 2 problems and ultimately see how they are related.

1) Take a pile of coins, say 10 coins. Split it into two piles, with say 4 and 6 coins. Write down the product 4*6=24. Split each of those piles into two piles, with say 1 and 3, and say 2 and 4. Write down those products 1*3=3 and 2*4=8. Continue in this way until you have ten plies each with 1 coin. Then add all the products, say 24+3+8+… What are all possible sums?

2) If 10 people are in a room, how many ways can they shake hands?

##### Intermediate Group: Eviatar Procaccia on The Gambler’s Ruin and a Disoriented Bird

Presenter: Eviatar Procaccia

Department of Mathematics, Texas A&M University

Abstract: Probability theory is the mathematical framework to study

randomness in the universe. We will learn how to use one source of

randomness to create another and why a disoriented bird will never find its

nest.

##### Advanced Group: Volodymyr Nekrashevych on Binomial Coefficients and Their Properties

Presenter: Volodymyr Nekrashevych

Department of Mathematics, Texas A&M University

Abstract: We will discuss Pascal’s triangle, binomial coefficients,

combinations, triangular numbers, and different interesting facts

about them.

### February 27, 2016

##### Beginner and Intermediate Group: Michelle Pruett on Code-Breaking Through the Years

Presenter: Michelle Pruett

Texas State University at San Marcos

Abstract: A variety of codes have been used throughout history. We will discover how

to code and decode messages using several techniques.

##### Advanced Group: Konrad Wrobel on Dealing with Infinite Sets

Presenter: Konrad Wrobel

Department of Mathematics, Texas A&M University

Abstract: The roots of modern day set theory stem from Georg Cantor’s work in 1874, when he introduced several concepts that many mathematicians of the time found disconcerting. We’ll delve into his notion of size, or cardinality, and what it means when applied to infinite sets.

### February 20, 2016

##### Beginner Group: Anneliese Slaton on The Bridge Problem: A Puzzle That Changed the Mathematical World

Presenter: Anneliese Slaton

Undergraduate Student at George Mason University

Abstract: We will be discussing The Koningsberg Bridge Problem, a seemingly simple problem that was solved by Euler and opened the door to the development of graph theory as we know it. We will not only look at Euler’s original proof, but will explore variations of the problems in physically get up and try to walk to the path!

##### Intermediate Group: Igor Zelenko on Domino Puzzles, Invariants, and Walking Along the Grids and Bridges

Presenter: Igor Zelenko

Department of Mathematics, Texas A&M University

Abstract: During the activity we will try to solve various problems regarding covering grids by dominos, trominos, transforming the tables of numbers according to certain rules, moving along grids without raising a pencil, or walking along the bridges of cities with many bridges. In all these problems we will ask whether we can complete certain tasks, the answer will often follow from certain nontrivial observations or properties of certain quantities, called invariants, that are preserved by natural transformations allowed in the problem.

##### Advanced Group: Ramalingam Saravanan on Predictability of Weather and Climate

Presenter: Ramalingam Saravanan

Department of Mathematics, Texas A&M University

Abstract: The discovery of the limits to weather predictability by Edward Lorenz was a seminal event both in theoretical meteorology and in nonlinear dynamics. The mathematical and physical basis for the predictability of weather and climate will be discussed in the context of this discovery. Topics to be covered will include trigonometric functions, limit cycles, and chaotic attractors.

### February 13, 2016

##### Beginner Group: Frank Sottile on Word Problems and Common Sense

Presenter: Frank Sottile

Department of Mathematics, Texas A&M University

Abstract: While we are taught to use algebra to solve word problems, many can be solved just using common sense. In this circle, we will use our common sense to solve word problems.

##### Intermediate Group: Maurice Rojas on Hats, Codes, and Lattice Points

Presenter: Maurice Rojas

Department of Mathematics, Texas A&M University

Abstract: We will see how a puzzle involving hats relates to codes that help protect data from noise. We’ll then see how lattice points come up in many different mathematical puzzles, as well as the modern study of secret codes.

##### Advanced Group: William Rundell on A 5,000 Year History into Mathematical Innovation

Presenter: William Rundell

Department of Mathematics, Texas A&M University

Abstract: Here are a series of questions. How does your calculator come up with it’s approximation to the square root of, say, 2? How were the square roots calculated in antiquity? Is there anything new to say about the problem? This talk will explore some of the answers.

### February 6, 2016

##### Beginner Group: Frank Sottile on Mathematical Auction

Presenter: Professor Frank Sottile

Department of Mathematics, Texas A&M University

Abstract: We will be doing a team contest called ‘mathematical auction.’

##### Intermediate Group: Phil Yasskin on The a+b+ab Problem

Presenter: Professor Phil Yasskin

Department of Mathematics, Texas A&M University

Abstract: Write down the numbers from 1 to 100. Randomly select 2 numbers from the list, say a and b, and cross them off, but add to the list the number a+b+ab. You now have 99 numbers. Repeat this process until you have only 1 number left. What are all possible final numbers?

##### Advanced Group: Maurice Rojas on Polygons, Lattice Points, and Equations

Presenter: Professor Maurice Rojas

Department of Mathematics, Texas A&M University

Abstract: We’ll see how geometric series and the Triangle Inequality allow us to understand hard equations with simple pictures. We’ll then see how counting lattice points in polygons leads us to some beautiful and unexpected applications of mathematics.