Schedule 2013-2014

September 14, 2013 – Adithya Ganesh – “Pólya-Burnside Enumeration in Combinatorics”

Burnside’s lemma from group theory has a broad scope of application in combinatorial enumeration problems.  Pólya’s enumeration theorem, which generalizes Burnside’s lemma using generating functions, provides a remarkable framework to easily solve counting problems in which we want to regard two entities as equivalent under some symmetry.

September 21, 2013 – Dr. Titu Andreescu – “Why Math on a Saturday Afternoon?”

September 28, 2013 – Dr. Branislav Kisačanin – “A Tour of Mathematical Functions”

In this talk we will learn about the art of protecting information, some incredible (but true) stories from the history of cryptography, and how the all-important RSA code ( http://en.wikipedia.org/wiki/RSA_(algorithm) ) works. To get there, we will need a bit of Number Theory, in particular the Fermat’s Little Theorem ( http://en.wikipedia.org/wiki/Fermat%27s_little_theorem ) and the Euler’s Theorem ( http://en.wikipedia.org/wiki/Euler%27s_theorem ). In the process we will also learn how to solve a class of problems that might be seen at AMC competitions, such as:

• What is the remainder when 2^1000 is divided by 997?
• Determine the last digit of 1^1 + 2^2 + 3^3 + … + 2009^2009.

October 05, 2013 – Dr. Jonathan Kane – “Rose of Roses”

Let’s talk about the sine and cosine functions. One does not need to use very much information about these commonly seen functions in order to understand a large number of curves which can be drawn by graphing sine and cosine in Cartesian and polar coordinates. We will see sine curves, sums of sine curves, Lissajous figures, cycloids, hypocycloids, epicyclodes, and, of course, many rows of roses.

October 12, 2013 – Cosmin Pohoata – “Non-standard Induction”

We prove a few combinatorial gems by using induction on unexpected quantities.

October 19, 2013 – No Math Circle

October 26, 2013 – Austin Marstaller – “Cantor Sets”

This week’s math circle presenter will be MMC participant and UTD student, Austin Marstaller.  He’ll be discussing Cantor Sets, a set of points lying on a single line segment that have a number of interesting properties.  Learning about Cantor sets is a great way to introduce general topology.  Announcements will also be made about how to register for AMC 8 if your school does not offer the test.  We look forward to seeing you!

November 2, 2013 – Adrian Andreescu – “AMC 8 Problems”

Please join us this Saturday as Adrian Andreescu leads the group through a selection of some of the most interesting AMC 8 problems.

November 9, 2013 – Titu Andreescu – “More AMC 8 Problems”

Please join us this Saturday as Titu Andreescu leads the group through a selection of some of the most interesting AMC 8 problems.

November 16, 2013 – Ivan Borsenco – “The Study of Prime Numbers: from Euclid till present days”

Why do prime numbers fascinate us? Why prime numbers is still an active area of research?What advances have mathematicians made in this field this year?  You will find answers to these questions and many more if you come to the lecture.

November 23, 2013 – No Math Circle – Thanksgiving

November 30, 2013 – Dr. Ivor Page – “How fast can we do arithmetic?”

Addition is the most elementary arithmetic operation required of computers. Their invention and development has almost entirely been to do arithmetic extremely fast. I will present the simplest binary adder circuits (ripple carry adder), show some surprising results about carry propagation, develop carry look-ahead circuits for log-time addition, and then talk about redundant number systems and constant time addition and, consequently, log-time multiplication.

I will also present residue number systems that enable constant time addition and multiplication.

It’s all integer math, modulo arithmetic, and some simple logic circuits.

December 14, 2013 – Dr. Vladimir Dragovic – “Unusual Billiards”

We will present an attractive crossroad of mechanics and mathematics. We will talk about interesting problems of billiards on triangular and elliptical tables. Video clips illustrating the subject will be demonstrated. We will provide basic definitions and some historical background.

About the lecturer: Vladimir Dragovic is a professor of Mathematics at UT Dallas. He has been a full research professor and the Head of the Department of Mechanics of the Mathematical Institute of the Serbian Academy of Sciences and Arts. He served as the Director of the Mathematical High School in Belgrade (2004-2008), known as one of the world’s most successful schools in the IMO competitions ( http://en.wikipedia.org/wiki/Matemati%C4%8Dka_gimnazija).

December 21, 2013 – No Math Circle

December 28, 2013 – No Math Circle

January 4, 2014 – No Math Circle

January 11, 2014 –Mathew Crawford – “Mathematical Research as Problem Writing”

Research is one of the more difficult aspects of education to provide for students. New research, by its nature, often involves large time commitments, and there are additional skills involved in communicating research that take time to develop. As a result, many students find the nature of research to be mysterious and simply don’t know how to get started. That’s a shame, because mathematical exploration rewards us on many levels.

One solution is to create problems (pun intended). In this talk, we will take a look at a few cool problems (some of the really cool ones, in fact) and discuss how they were or might have been the result of “miniature research projects”. We will then ourselves participate in the process. We will start with some brainstorming, playing, and doodling, just to see where it leads. We will synthesize some of the ideas into original problems and challenges of our own. It is even possible that some of the ideas lead to more interesting projects.

This talk is friendly for a very wide audience. If you know some math, you can play, and playing is part of the research process!

January 18, 2014 – Dr. Titu Andreescu and Mathew Crawford – “First Metroplex ARML Team Practice”

Please join Dr. Titu Andreescu and Mathew Crawford as they offer an American Regions Mathematics League (ARML) session for students interested in learning more about the contest and wishing to be exposed to the types of problems they would encounter.  This will be a very interactive math circle where students will have the opportunity to work together to solve interesting problems. Here is the plan for this first practice as detailed by Mathew Crawford:

• We plan to talk about our goals for the team.
• Discuss the rules and format for ARML, the primary event for the Metroplex Math Team
• Practice three of the four rounds from a past ARML contest (Individual, Team, and Relay)

We will practice the fourth ARML round (the Power Round) as the Metroplex Math Team participates in the ARML Power Contest the following weekend.

For this first contest, we’re picking an ARML contest that isn’t from the past few years both in hopes that fewer students have seen the problems and also because the problems from older contests are a little more accessible to beginners. We plan to use more recent practice problems as the team progresses.

Veterans of ARML are encouraged to show up to this first practice to help make new students comfortable with the process and also because leadership matters—and we will practice like we play, as a team.

If you know of talented high school students or extremely motivated younger students in the Dallas Metroplex area who may want to attend this event, please pass this link along.

January 25, 2014 –Emily Herzig – “Matrix Operations and Applications”

Matrices are an extremely useful concept – they are widely used in many areas of mathematics, science, engineering, and data analysis.  This talk will provide a peek into what can be done with matrices in a few of these contexts.  We will start by discussing what a matrix is, how the basic arithmetic operations are defined, and some useful properties, and then we’ll explore some simple applications in linear algebra, geometry, and data analysis.

February 1, 2014 – AMC 10/12 Preparation

The AMC 10A/12A is on February 4th and the AMC 10B/12B is on February 19th.  If you’d like a chance to learn some interesting tips and techniques to help with these competitions, then please join us for this enlightening Math Circle.  Adrian Andreescu, Vinjai Vale, and Dr. Titu Andreescu will be presenting problems to challenge and delight.

If you have yet to sign up for these tests and your school does not offer them, there is still room for both dates (these testing sites are not sponsored by Metroplex Math Circle).  Registration will be held outside the math circle room, ECSS 2.201.

February 8, 2014 –Dr. Andreescu and Adithya Ganesh – ARML Training

February 15, 2014 – No Math Circle (UTD Programming Contest)

February 22, 2014 – No Math Circle (MATHCOUNTS Chapter Competition at UTD)

March 1, 2014 – Ivan Borsenco – “Introducing Polynomials”

There are many times when we come across polynomial equations. In school, we certainly learn how to solve linear equations, then we learn how to solve quadratic and cubic equations, and finally we find out that we can view them as one mathematical object! That object is called polynomial, it is simple and has very nice properties.

In our lecture we present a unified view of polynomials. This will help you understand concepts covered in school much better and much faster. There will be plenty of tricky Olympiad problems related to them and it will be fun!!

March 22, 2014 – Austin Marstaller – “Abstract Algebra”

Abstract Algebra is the set of advanced topics of algebra that deal with abstract algebraic structures rather than the usual number systems. The most important of these structures are groups, rings, and fields. In the following introduction to this topic, we will discuss Binary Operations, Groups, Subgroups, Cyclic Groups, Cayley Digraphs, and how they relate to each other. With a thorough understanding of these topics, students will have the basis to further examine the subject that is Abstract Algebra.

March 29, 2014 – Mathew Crawford – “ARML Training”

Metroplex Math Team students practice for ARML. We plan to practice Team, Individual, and Relay rounds and discuss solutions to some of these challenging problems. Join us!

April 5, 2014 – Branislav Kisačanin and Milena Djordjević–Kisačanin – “Overview of Combinatorics”

The first hour of this talk will be targeted primarily at the younger part of our audience (roughly in grades 4-7) – we will explain the fundamental rules of counting that can be used to solve even very hard problems: rule of sum (addition principle), rule of product (multiplication principle), pigeonhole principle, and the inclusion-exclusion principle. We will use them to solve a number of interesting problems from various competitions. For those students already familiar with these concepts, we will have a set of problems to keep them busy during the first hour.The second hour will be targeted at the more experienced members of the Math Circle community (roughly grades 8-12). First, we will discuss several techniques used to prove combinatorial identities (combinatorial arguments, algebraic manipulations, and the method of generating functions). We will prove several important combinatorial identities using all of these techniques, illustrating the diversity of approaches found in combinatorics. After that we will look at applications of combinatorics in number theory, geometry, and graph theory, illustrating them with more interesting and challenging problems. While the younger part of our audience may not be able to follow everything during the second hour, it will be a great exposure to advanced mathematical topics, to show them they have a lot more to learn.Throughout the talk we will highlight the mathematicians who developed this beautiful mathematical field, from its beginnings in gambling to modern applications in medicine, science, and technology.

April 19, 2014 – Jacob Cordeiro– “Eigenvalues and Dynamical Systems”

A dynamical system is a set of functions that depend on each other. For example, dynamical systems are often present in ecosystems: more plants mean more plant-eaters, and more plant-eaters mean fewer plants. These systems can be difficult to predict, as one thing affects another, which affects the first, which affects the other, and so on.Fortunately, dynamical systems can be simplified. This lecture focuses on an important type of dynamical system that can be solved with some creative usage of matrices. We’ll also discuss the related concept of “eigenvalues,” a simple concept with some very interesting implications.

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