Colloquium Speaker

Some Intuition about Fermat's Last Theorem
Teacher info: Larry Guth (math prof)
Abstract: We give some intuition why Fermat's last theorem is plausible based on a probabilistic model. Then we compare the model to some computer experiments. 
Period 1 
Take a Tour with Euler
Tour the city as we attempt to take an Euler tour of the
bridges and tunnels of New York City. In the process, you
will learn the basics of graph theory and encounter the
Chinese Postman Problem. You will also learn about
Hamiltonian circuits and the the famous Traveling Salesman
Problem. Discover a vibrant and exciting area of discrete
mathematics  graph theory! Length: 1 Hour Prerequisites: none Teacher info: Amro Mosaad (amro_mosaad [at] yahoo [dot] com), Middlesex County Academy (HS)  Math, Teacher of Mathematics Course notes available here

Infinity and The Diagonal Argument
What do you mean by infinity? Suppose we had ''infinitely'' many people coming to a picnic. Suppose that every prime numbered visitor brought one slice of bread, every 2nth visitor who wasn't a prime numbered visitor brought a bottle of peanut butter, and every 2n+1st visitor who wasn't a prime numbered visitor brought a bottle of jelly, and every prime numbered visitor brought one slice of bread, then we'd have just the right amount of everything to feed everyone as many peanut butter jelly sandwiches as they'd like without wasting any food. We'll talk about why this is possible by clarifying how we decide how big any given ''infinity'' really is using a neat trick called the diagonal argument. Length: 1 Hour Prerequisites: None Teacher info: Aukosh Jagannath (aukosh [at] cims.nyu [dot] edu), Courant Math, 1st year PhD

How does a Web Search Engine Work?
In a fraction of a second, Google and Bing can find and rank the web pages that match your search query, from among the billions of pages on the Web. Find out how it all works!
Length: 1 Hour Prerequisites: None. Teacher info: Ernest Davis (davise [at] cs.nyu [dot] edu), Courant, Computer Science, Professor

Concepts of Calculus
Before taking a course in the subject, Calculus seems mysterious and intimidating. However, Calculus deals with some of the most intuitive ideas in mathematics: real systems and how they change. Through examples in physics, algebra, and everyday life, we will seek a conceptual grasp of the central problems in Calculus; limits, differentiation, and integration will each be treated. A special emphasis will be placed on applying Calculus to realworld problems, as well as grounding other fields (particularly physics) in mathematics. Length: 1 Hour Prerequisites: Comfortability with high school algebra. The concepts will be challenging, but they will be presented in an extremely intuitive way to maximize understanding of the processes at work. Teacher info: Marcus Levine (marcusianl [at] gmail [dot] com), Columbia University, Astrophysics, 1st Year Undergraduate

So You Think You Can Count? The Mathematics of Games
Counting things is not always so easy. How many spades can you get by withdrawing 10 from a deck of cards? What is the probability of getting the total sum of 12 by throwing 3 dice? All this can be intriguing and help you understand the world around you in a different way. Length: 1 Hour Prerequisites: None. Teacher info: Weilun Du (wd387 [at] nyu [dot] edu), Courant Math, Undergrad 2nd Year

Fundamental Identities in Trigonometry
I will try to prove three fundamental trigonometry identities. The first one that I will prove is sin^2(angle)+ cos^2 (angle) = 1, and from proving this first one, I will try to prove a second and third trigonometry identity. Length: 1 Hour Prerequisites: Trigonometry (unit circle, and trigonometry functions), pythagorean theorem Teacher info: Jong Woo (John) Yoon (jyoon0529 [at] gmail [dot] com), Courant Math, 3rd year undergraduate Course notes available here

Rigid Origami
When a satellite is launched into orbit, the solar panels must fold up to fit inside
a rocket, but when it gets into orbit, the panels are unfolded. These unfoldings
are done in a rigid way because solar panels can't bend! In this class, we will
discuss the mathematics behind how objects can fold rigidly and the models we use to
understand these foldings. Length: 1 Hour Prerequisites: Good spacial orientation skills are a must for this course! Teacher info: Michael Burr (burr [at] cims.nyu [dot] edu), Fordham Mathematics, Professor

Period 2 
Genetic Algorithms and Their Applications in Problem Solving
Genetic Algorithms are algorithms which imitate evolutionary processes allowing for the accurate approximation of solutions to a wide range of problems. These problems can be as simple as finding solutions to equations or as complex as solving the Traveling Salesman Problem, for which no "good" solutions exist. This class will discuss the theory of Genetic Algorithms and their applications in problem solving. Length: 1 Hour Prerequisites: No prerequisite knowledge is needed or assumed for this class. Teacher info: Nick Lesniewski (nick.lesniewski [at] gmail [dot] com), NYU CAS, Computer Science, Freshman

Three applications of Euler's formula
This is a chapter from "the proofs from the book" which gathers some very beautiful yet simple proofs from the mathematical world.
A graph is planar if it can be drawn in the plane R^2 without crossing edges. Euler's formula states that:
ne+f=2 where n,e and f are the number of vertices, edges and faces of the graph, respectively. You'll be surprised to know that this is even true in more generality, as in the torus or R^3. However, we only work in 2 dimensions, and we will prove three interesting application of this formula:
1 The SylvesterGallai theorem:
Given any set of n>2 points in the plane, not all on one line, there is always a line contains exactly two of the points.
2 Monochromatic lines:
Given any configuration of “black” and “white” points in the plane, not all on one line, there is always a “monochromatic” line: a line that contains at least two points of one color and none of the other.
3 Pick's theorem:
The area of any (not necessarily convex) polygon Q in R^2 with integral vertices is given by
A(Q)=n_int + 1/2(n_bd)1,
where n_int and n_bd are the number of integral points in the interior and on the boundary of Q, respectively. Length: 1 Hour Prerequisites: Induction Teacher info: Behzad mehrdad (mehrdad [at] cims.nyu [dot] edu), Courant math, 3rd year PhD student

Observing the Constellations
Come learn all you need to know about the night sky and how to get the most out of a stargazing experience!
I will present a quick and easy introduction to observational astronomy using technology or just a simple planisphere.
I will also try to answer any questions about stellar structure and evolution of the universe! Length: 1 Hour Prerequisites: None Teacher info: Gladys Velez Caicedo (gbv2105 [at] live [dot] com), Columbia University, Department of Astronomy and Astrophysics, FirstYear Undergraduate Student

Puzzle Magic
Some puzzles are so surprising that they seem magical. I will try a few on you and then teach you how to do them. They are inspired by mentalism, cryptography, and geographical politics. Length: 1 Hour Prerequisites: High school algebra and geometry at a 10th grade level.
You will have to think. Teacher info: Dennis Shasha (shasha [at] cs.nyu [dot] edu), Courant Computer Science, Prof

The Mathematics of Dating
Can numbers predict and improve your odds of finding a good date? Of course! We shall cover strategies that dramatically raise the odds of finding your sweetheart in such settings as dance clubs, bars, schools or any other place ripe for romantic encounters. In the end, we shall see that even though there isn't a clear cut formula for love, there are strikingly profound predictions we can make with a little math. You will be convinced that Cupid carries a calculator with him! If time permits we shall also look at ways of matching groups of people into happy couples. Length: 1 Hour Prerequisites: Some familiarity with basic probability is recommended. Teacher info: Alex Rozinov (alexrozinov [at] nyu [dot] edu), Courant Institute of Mathematical Sciences, 3rd Year PhD

Intro to Programming
A gentle, fun introduction to programming. We will be working in the computer lab to design a simple graphics application in the Python programming language. Length: 1 Hour Prerequisites: No programming experience necessary. Algebra, trigonometry, and analytic geometry (i.e., precalculus) will be useful. Teacher info: Paul Gazzillo (pcg234 [at] nyu [dot] edu), Courant Computer Science, 2nd Year PhD

What is Geometry? The View from Computing
Geometry has been a wellspring of profound ideas in every branch of mathematics, from number theory to algebra, to analysis, to combinatorics, and even to computation. I will talk about this last connection. Many books, articles and talks have been entitled “What is Geometry?” One is reminded of the parable of the Elephant and the Blind Men. In this lecture, the Computer Scientist joins the Blind Men to probe this Geometric Elephant. Length: 1 Hour Prerequisites: high school geometry Teacher info: Chee Yap (yap [at] cs.nyu [dot] edu), Courant CS, Professor

Period 3a 
Mathematics in Finance and Economics
The class will begin with simple ideas and concepts in Economics, and we will quickly go into the concept of indifference curves and budget constraints. Then we will teach the idea of optimization and the use of Lagrangian. After this, we try to drive towards optimizing across two time periods. This is the idea of the discount factor. After this, depending on class performance, we can derive the Sharpe Ratio and Beta, or we can explain the Time Value of Money. Length: 1 Hour Prerequisites: Calculus and basic differentials. The Lagrangian concept is supposed to be Calculus III but can be introduced during the class. Teacher info: Jeremiah Leong (ghl230 [at] stern.nyu [dot] edu), Stern and Courant Math, Junior Course notes available here

Mathematics: Its Past, Present, and Future
Have you ever been sitting in math class and wondered "When am I ever going to use this?" or "How did we ever come up with this stuff?" Well then, this class will at least try to answer those questions. From likes of Archimedes to recent mathematicians like Mandelbrot and Courant, this class gives a bird's eye view of the context within which modern day mathematics operates and influences the world. Philosophers often place ideas within the context of a period of time, so why shouldn’t we, as mathematicians, do the same? Just as human thought flowed from Platonic philosophy to existentialism to postmodernism, so too did mathematics from Euclid to Newton’s Calculus to modern day Chaos Theory. The placement of ideas within this historical context helps to illuminate why these mathematical methods were invented in the first place, and what they can help us with on a day to day basis, whether pondering the philosophical truths of this universe or building an econometric model. In essence, uncovering our mathematical past will frame our modern day condition and adjust our expectations for the future of mathematics. Length: 1 Hour Prerequisites: One should be familiar and comfortable with mathematical reason. High school algebra and precalculus would be a big plus, but we will be brushing through everything from basic trigonometry to more complex topics. Really all that is required is a logical mind and willingness to think. Teacher info: Ador Michael Cristofi (amc738 [at] nyu [dot] edu), Stern Finance and Courant Math, 2nd Year Undergrad, Double Major in Math and Finance

The Mathematics of Games
Combinatorial game theory studies games. Which games have winning strategies? Can we describe them? Do games like chess and checkers have optimal strategies?
In this class, we will see how to *prove* that games have optimal strategies, and we will see how to give explicit descriptions of optimal strategies for some very special games. Length: 1 Hour Prerequisites: none Teacher info: Wesley Pegden (pegden [at] math.nyu [dot] edu), Courant Math, NSF Postdoctoral Fellow Course notes available here

Intro to Programming (II)
A gentle, fun introduction to programming. We will be working in the computer lab to design a simple graphics application in the Python programming language.
Length: 1 Hour Prerequisites: No programming experience necessary. Algebra, trigonometry, and analytic geometry (i.e., precalculus) will be useful.
Teacher info: Paul Gazzillo (pcg234 [at] nyu [dot] edu), Courant Computer Science, 2nd Year PhD

Counting with Algebra
I will present a way that lets you use tricks from algebra to let you count things. This idea is known as "generating functions". I will also develop an explicit formula for the nth Fibonacci number. Length: 1 Hour Prerequisites: Familiarity manipulating algebraic equations; know how to sum geometric sequences Teacher info: MihaI Nica (nica [at] cims.nyu [dot] edu), Courant Math, 1st Year PhD Course notes available here

Period 3b 
Math made Difficult: Equivalence Relations
It's the next best thing to equality  it's equivalence! Join us for the formal mathematical definition of an equivalence relation, and then test out your new knowledge by discerning some motivating examples from a few nonmotivating nonexamples. Length: 1 Hour Prerequisites: The willingness to learn from examples, and the ability to make great mistakes. Material will be pulled from many levels of mathematics. Teacher info: Japheth Wood (japheth [at] nymathcircle [dot] org), New York Math Circle, President Course notes available here

From Leibniz to a Little Big Planet (TM): Using FirstPrinciples Physics to Simulate Complex Motion
Have you ever wondered how character movements in animated movies or video games look so realistic? What concepts in physics, mathematics and computer science are engineers using to create these graphics? It turns out that basic laws of physics, such as velocity and acceleration, can be integrated with computational methods to generate these results. We will begin with a discussion on what derivatives and differential equations are, and how they can be used in modeling the dynamics of movement. Commonly used computational algorithms for solving these equations will be introduced, and we will define areas where these approaches have proven useful, such as providing entertainment and even solving scientific problems. Length: 1 Hour Prerequisites: None. Teacher info: Loretta Au (lau [at] ams.sunysb [dot] edu), Stony Brook University, Dept. of Applied Mathematics, 4th year PhD student

A Few Great Problems
Classes at the New York Math Circle are problem solving sessions. During this talk I'll discuss a small number of my favorite problems, some from combinatorics, some from probability, and some from geometry. Hopefully we'll solve them all! Length: 1 Hour Prerequisites: Geometry, basic combinatorics and probability Teacher info: David Gomprecht (davidgomprecht [at] gmail [dot] com), New York Math Circle and the Dalton School, Math Teacher

Robotic Navigators and Cartographers
What do Phoenician merchants, Claudius Ptolemy, Christopher Columbus, certain brands of vacuum cleaners and the selfdriving Google car have in common? All need to navigate under uncertainty or to map the unknown. They guess their next location and correct their predictions thanks to (always noisy) observations of some kind, be it from the North Star, Jupiter’s satellites, a compass, a Kinect camera or a laser LIDAR system. They define coordinate systems and piece together small maps. We will see how robots can do this. A little mobile robot will make a guest appearance. Length: 1 Hour Prerequisites: Trigonometry, probabilities. Teacher info: Piotr Mirowski (mirowski [at] cs.nyu [dot] edu), Bell Labs, Statistics and Learning group, Research Scientist Course notes available here

Period 4 
Stars and their Physical Properties
We wish upon them, gaze upon them on a clear night, but what do we really know about stars? Together we will explore exactly how stars are formed, how they die, and everything in between (even their afterlife!). We will talk about the big bang initiating the contents of the universe today, but how the stars do everything else. We'll even discuss how stars can either burn out to be harmless and docile, or turn into super massive black holes. Don't just marvel at their pretty twinkling appreciate the delicate physics behind this incredible phenomenon! Length: 1 Hour Prerequisites: A high school level understanding of physics is helpful, but not mandatory. Most physics concepts will be quickly reviewed in class. The course will primarily center around abstract concepts and relationships rather than hard numbers. Teacher info: Isabel baransky (isabelbaransky [at] yahoo [dot] com), Columbia Engeering School Applied Phyics, 1st Year Undergraduate Course notes available here

Biomechanics: consequences of size in nature
"A grasshopper can jump up to ten times its size, so a humansized grasshopper can jump up to 60 feet in the air." Is this true? We'll show, in fact, that it is not, by demonstrating that there are fundamental consequences of size in nature that determine what is physically possible. By using simple mechanical arguments, we will explain, among others: why elephants have such thick legs; why ants can lift so much weight; why humanpowered flight is so difficult; and why deepdiving mammals are so large. Length: 1 Hour Prerequisites: Some physics (you need to be familiar with the concepts of force, energy, and power). Teacher info: Ken Ho (ho [at] courant.nyu [dot] edu), Courant, Computational Biology, 5th year PhD Course notes available here

Matrix Partitions and Extensions
We will extend matrix partitions to basic operations, including matrix multiplication, the vector dot product, and elementary row/column operations. Length: 1 Hour Prerequisites: Knowledge of vectors and basic matrix operations (Algebra II) Teacher info: Daniel Zhou (dfz211 [at] nyu [dot] edu), CAS, 1st year Undergraduate Course notes available here

Welcome to the Alternate Reality of Frequency Space! Fun with Fourier Series
We spend most of our lives in the physical world, thinking about how various quantities change in space. However, there is another, less intuitive way to think: in terms of frequency! That is, we can decompose a signal into waves of different frequencies and compare their relative strengths. In this course, we will introduce the theory of Fourier series and view some applications in climate science and music, using computers to do big calculations for us! Length: 1 Hour Prerequisites: You should know how to integrate basic trigonometric functions like sine and cosine Teacher info: Thomas Fai (tfai [at] cims.nyu [dot] edu), Courant Math, 4th Year PhD

Algorithmic Game Theory
We will start by describing simple games that students can relate to (e.g. rock, paper, scissors) and gradually explain the abstract formal definition of a game in strategic normal form. We will explore notions such as the Nash Equilibrium and the Price of Anarchy of a Game. We will also be playing some fun games in class that will help students understand how players think and which strategies are good. Putting yourself into other people's shoes is the first thing that you want to do here before you choose your course of action. If students bring laptops we can also play some games online. Length: 1 Hour Prerequisites: Capability of grasping abstract notions and some basic discrete math (sets and functions). Teacher info: Vasilis Gkatzelis (gkatz [at] cims.nyu [dot] edu), Courant Computer Science, 4th Year PhD

Analysis: Calculus done Carefully
Are some parts of calculus troubling you? Perhaps little details (and certain proofs) were glossed over that make you uneasy? Then analysis is the class for you!
We will learn whether or not the real numbers are really just made up, (finally!) see the proof for the Intermediate Value Theorem, and think carefully about integrals! Hopefully by the end of this talk you'll see that math isn't just about getting the exact numbers, but also about saying exactly what you feel. Length: 1 Hour Prerequisites: Familiarity with proofs and high proficiency in calculus is strongly suggested. Teacher info: Benjamin Xie (bjx201 [at] nyu [dot] edu), NYU College of Arts and Science, 2nd year undergraduate

Period 5 
Fibonacci Numbers in Nature
Most of you have probably heard about the Fibonacci numbers (1, 1, 2, 3, 5, 8, 13, ...) and how they are found practically everywhere in nature. But has anyone ever given you a convincing reason why? In this class, we will go through the history of the Fibonacci numbers and see for ourselves their allure through the ages, point out common misconceptions regarding their ubiquity, and finish up by giving a very "rational" reason for their emergence in the context of flower patterns and plant growth. No background necessary!if you can reason about numbers, then you, too, can learn how nature "knows" math. Length: 1 Hour Prerequisites: Numbers. Maybe fractions? Teacher info: Ken Ho (ho [at] courant.nyu [dot] edu), Courant, Computational Biology, 5th year PhD Course notes available here

Why the bell curve?
The bell curve occurs frequently in nature. Heights, weights, IQs and many other values have a bell shaped distribution. In this class, we will do a handson activity and discuss the mathematics behind this surprisingly common phenomenon. Length: 1 Hour Prerequisites: None Teacher info: Meredith Burr (mburr [at] ric [dot] edu), Rhode Island College Math Department, Professor

Stirling Numbers
A teacher wants to divide four students into groups. How many such divisions are possible? We will start with this simple counting question, and explore some of its ramifications. Along the way, we will solve more challenging problems and develop a few fantastic formulas. Length: 1 Hour Prerequisites: Preferably students should be familiar with combinations, the InclusionExclusion Principle and recursive equations. Teacher info: David Hankin (oana [at] nymathcircle [dot] org), New York Math Circle, Mathematics Teacher

An Introduction to Memory Corruption and Exploit Development
Computer security is always big news, but it's hard to get started without a proper introduction. This talk will be a brief immersion into one of the most technical areas in information security. We'll quickly cover the basics of C and x86 and talk in depth about memory corruption vulnerabilities on modern platforms. We will analyze several real vulnerabilities and write exploits for them. Many topics will be quickly covered as we discuss arbitrary code execution including: compilers, optimization, reverse engineering, smashing the stack, ASLR, NX, ROP, clowns and sorcery. Length: 1 Hour Prerequisites: Computer programming experience, lower level languages are better. Teacher info: Julian Cohen (hockeyinjune [at] isis.poly [dot] edu), NYU Poly, ISIS Lab, Junior Undergraduate Course notes available here

Exploring Infinity (and Beyond?)
I want to give a short talk about the cardinality of the natural numbers and the real numbers, and show the students the diagonal argument. When I was first shown this proof it "blew my mind" and really sparked a long interest in Math, and I hope if I showed it to others, it could inspire them as well. If there is time, I would also go into the density of the rationals and their cardinality as well. Length: 1 Hour Prerequisites: A suggested prerequisite of Precalculus Teacher info: Patrick Song (patsong [at] nyu [dot] edu), Courant Math, 4th year undergraduate

Fractals In the Limit! Exploring Fractals Using Iterated Function Systems.
Fractals are amazing forms that appear throughout the natural and mathematical worlds. How can we understand, define, and create them? In this class we’ll look at one way of understanding fractals, as the result of iterated function systems. This will allow us to explore some of the defining features of fractals in more depth. We’ll dive into the math behind why Iterated Function systems produce fractals. We’ll end up talking about things like the “distance between two pictures”, and what a “sequence of pictures” might be. Then, we’ll see how Iterated Function Systems produce sequences of pictures that converge to fractals! Awesome. Length: 1 Hour Prerequisites: We’ll be thinking about things fairly abstractly. Familiarity with sequences and limits is a plus. But we’ll also be seeing a lot of neat fractals that require no prerequisites to appreciate. Teacher info: Arjun Kataria and Ezra Winston (ezrawinston [at] gmail [dot] com)

