2009 Talks

Here is the list of courses that were offered at cSplash 2009.

Icons




indicate difficulty rating, in ascending order.

Description of the icon system.

Colloquium Speaker

To Flip or Not to Flip

Teacher info: Sinan Gunturk (math prof)

Notes: cSplash2009.pdf

Period 1


Finding Meaningful Science Projects

Finding a meaningful science project is difficult. In this session, we will discuss using accessible scientific literature to identify and develop novel lines of impactful research. First, we will review a
recent scientific article regarding the effects on memory of odor during sleep. Next, I will explain what I was looking for when I identified this particular article as a promising base for a high
school research project. Finally, we will develop follow-up studies for use in publications and research competitions, such as the Intel Science Talent Search. After class, students should have one concrete project design and the ability to find their own meaningful science project.

Length: 1 Hour 15 minutes

Prerequisites: No specific mathematical knowledge necessary

Teacher info: Will Findley (findley [at] gmail.com)



How do you measure reliability?

Variability is everywhere. It can cause your friend's iPod to have a longer battery life than yours. It can also cause a baseball pitcher to have an off day. In order to study variability and reliability, we must have a way to measure it. In the first part of this class, you will invent several measures of variability in order to solve a practical problem. Then, we will present and analyze the most common measure of variability, the standard deviation.

Length: 1 Hour 15 minutes

Prerequisites: None.

Teacher info: Paul Hand (hand [at] cims.nyu.edu)



The Ancient History of Odd Numbers, Even Numbers and Primes

We will talk about odd and even numbers, prime numbers, and a little bit about modulus. Each concept will come with some history, a theorem, and a fun problem for each!

Length: 1 Hour 15 minutes

Prerequisites: Addition and multiplication using fractions and variables

Teacher info: Robert Simione



Special Relativity

What does the world look like when traveling close to the speed of light? Very different, it turns out. Our intuition about space and time from everyday life breaks down. In this class we will see how simple logical reasoning forces us to accept strange phenomena such as slowdown of time and contraction of lengths.

Length: 1 Hour 15 minutes

Prerequisites: Familiarity with the formula speed=distance/time together with some trigonometry.

Teacher info: Jens Christian Jorgensen



The Epidemics of Contagious Diseases

Understanding the spread of diseases such as Flu, AIDS, Plague and smallpox is of paramount importance to humanity. Will the epidemics spread or die out? Do vaccinations help? We will study these using a simple mathematical model that can explain many different epidemics that have occurred in history, like the influenza epidemic of 1918-1919 or the Great Plague. How can one model represent so many different epidemics?
Most of the time in this class will be spent on developing the model and looking at different solutions of the model. Most importantly we will study how do we know when the epidemic spreads and when does it die out. Depending on the allotted time, we will learn how to make the model more specific, possibly for a specific disease, how to include vaccinations and so on.

Length: 1 Hour 15 minutes

Prerequisites: None.

Teacher info: Enkeleida Lushi (lushi [at] cims [dot] nyu [dot] edu)



Cigar Smoke, Coffee with Milk, and the Global Circulation

While reading this blurb, you are immersed in a fluid, but because it is transparent, you don't see the extraordinarily complicated and beautiful motions you generate with every move. How do you see the way a fluid moves? If the light is just right, sometimes you can tell by looking at its surface, but sometimes there is no surface! (think of scuba diving.) A better way to see a fluid is with a 'tracer' - something that follows the flow - like milk in your coffee, smoke from your uncle's cigar, clouds on Jupiter or even plankton in the ocean. There is a near infinite variety of possible motions, but there are some big categories. A really important category separates how fluids behave when they are rotating (such as the circulation of our atmosphere!) compared to when they are not rotating (such as the cigar smoke, assuming your uncle doesn't smoke while spinning on a sit-n-spin). In this mini-course I'll introduce you to how fluids move, especially with and without rotation, demonstrate examples in a rotating and non-rotating tanks, and even show you how we model such flows on a computer.

Length: 1 Hour 15 minutes

Prerequisites: Massive amounts of curiosity, plus a little patience. Also, please know that you live on a rotating planet, that mass is conserved, and, if you know what a derivative is, that would be super (but not necessary!).

Teacher info: Shafer Smith (shafer [at] cims.nyu.edu)



Combinational logic circuits

Every second, you computer processor performs an incredible
number of computations involving basic arithmetic. So how exactly does
it know how to perform these calculations? The answer lies in
combinational circuits - circuits which are built entirely of logic gates.

In this class, we'll cover the basics of boolean algebra, logic, and
circuit design as we design rudimentary combinational logic circuits.

Length: 1 hour and 15 minutes

Prerequisites: basic familiarity with circuits (ie, know what
approximately voltage and current are)

Teacher info: Tony Valderrama ( tvald [at] mit.edu)



The CPU, Revealed

This course reveals the inner workings of the CPU, the heart of every computer. We will design a simple CPU, down to the (zero or one-carrying) wires. Then we'll trace the execution of a computer program, leaving no mysteries unrevealed. Time permitting, we'll also talk about high-level programming languages. This course is for everyone unsatisfied by explanations that don't reach to the very bottom.

Length: 1 Hour 15 minutes

Prerequisites: Please know what binary (base 2) numbers are.
Please know basic boolean logic (and, or, not).

Teacher info: Yotam Gingold (gingold [at] cs.nyu.edu)



Quantum Mechanics and You

How do you detect something without looking at it? How do you do thousands of calculations simultaneously? How does teleportation work? Where are the other universes? This is a class in pragmatic quantum mechanics, with some real examples, and applications to quantum computation.

Length: 1 Hour 15 minutes

Prerequisites: Basic (really basic) probability. Waves and super-position principle a plus.

Teacher info: Seth Cottrell (cottrell [at] cims.nyu.edu)



An Introduction to Differential Equations

Differential equations are everywhere in the mathematical sciences. Theories in quantum mechanics, heat flow in your car's engine, the curvature of space-time, water currents in the ocean, flow around airplane wings, and the motion of the galaxies are all described by differential equations. In fact, the last three are described by essentially the same differential equation. They are the primary framework in which we state fundamental laws and formalize our descriptions of complex behavior. Newton's Second Law is one differential equation that you may be familiar with, although it might not have been stated as such. In this talk, we will answer a few introductory questions like "What is a differential equation?" and "What is a solution to a differential equation?". We will also look at some qualitative aspects of differential equations and many applications, which will be drawn primarily from biology and physics.

Length: 1 Hour and 15 minutes

Prerequisites: Some knowledge of calculus would be useful. If you know that the derivative of the exponential is itself then you will be more than fine.

Teacher info: Adam Stinchcombe (ars522 [at] cims.nyu.edu)



Let's Go For a Walk on a Donut

Imagine you land on a strange alien planet. You start walking in a straight line, and eventually you get back to where you started. Can you figure out what the planet's shape is? Is it a sphere? Is it a donut? Is it some weird-looking surface without an inside or an outside? What if you walk forever and never get back to the same place? In this class, we'll answer these questions, and other questions about different surfaces using a branch of mathematics called algebraic topology. We'll be able to find out what the shape of a surface is, just by figuring out simple invariants, such as the fundamental group. Donuts will be provided.

Length: 1 Hour 15 minutes

Prerequisites: Knowledge of complex numbers would be good, but not essential. Also, being able to imagine surfaces bending and twisting and stretching in space (think of Play-doh).

Teacher info: Karol Koziol (karolkoziol [at] nyu.edu)



Optimization in Infinite Dimensional Spaces (part 1)

Optimization arises in many application areas. A simple example is that of finding the radius and height of a cylindrical container such that the cost of manufacturing it are minimized subject to the constraint of a given volume. This is an example of low-dimensional optimization, because the number of design parameters is small (i.e. the height of the cylinder and its radius). In many industrial level optimization problems, the number of design parameters can be large, even infinite. Examples are finding the shape of a wing such that the drag is minimized; specifying the shape requires an`infinite number' of design parameters, hence the term infinite-dimensional optimization. Furthermore, the constraints can depend on complicated models, such as for fluid flow past the wing. These models are also infinite dimensional and come from Partial Differential Equations (PDEs).

Part I of this lecture will discuss concepts of optimization according to the following outline:

  1. Motivate optimization with examples.
  2. State what a function is and what local maxima and minima of a function look like.
  3. Review the derivative and the gradient from calculus.
  4. Show how to optimize a function of many variables with constraints; review Lagrange multipliers.
  5. Introduce ideas from the calculus of variations (for part II); relate to directional derivative.

Part II of this lecture will go over concepts of infinite-dimensional optimization according to the following outline:

  1. State an example infinite-dimensional optimization problem.
  2. Discuss Partial Differential Equation (PDE) constraints; show example.
  3. State a general way (in low dimension) to solve an optimization problem.
  4. Restate the method in infinite dimensions. Will show some simulations/movies illustrating the optimization process.
  5. Conclude with some discussion of other optimization methods.

Length: 2 Periods, 1 hour 15 minutes each

Prerequisites: Calculus

Teacher info: Shawn Walker


Period 2


LaTeX, a introduction to Mathematical Typesetting

LaTeX is the most popular document preparation tool in the mathematical and scientific world. Knowledge and execution of LaTeX as an alternative to Microsoft Word or like programs become a powerful and useful tool to mathematicians at any level. This class will illustrate the basics of creating a PDF file using LaTeX. Also, learn some helpful tricks and useful codes to produce sharp and clear articles and homeworks. LaTeX and accessory packages make it incredibly easy to typeset any mathematical equation from matrices to analytical proofs to more complicated figures and diagrams. LaTeX has environments quickly organizing sections, chapter, table of contents with way less work than WYSIWIG programs can provided.

Length: 1 hour 15 minutes

Prerequisites: Some programming experience would be helpful, but is not necessary.

Teacher info: Alec Jacobson (alecjacobson [at] nyu.edu)



The Physics of Superheroes

Most of us have at least heard of the exploits of Superman and Iron Man either on the pages of a comic book or in the movies. And while they managed to capture our imaginations, how plausible are they in the real world? In his book, The Physics of Superheroes, James Kakalios explores this exact question. Could Superman really leap tall buildings in a single bound? Is it even possible that the X-men's Kitty Pryde could walk through a solid wall? What about the Fantastic Four's Invisible Woman? You've probably never realized that comic books sometimes actually get their physics right. In this class, we will explore some of these claims and along the way learn some of the basics of topics in physics ranging from electromagnetism to quantum mechanics.

Length: 1 Hour 15 minutes

Prerequisites: High School Algebra

High School Physics is encouraged but not required

Teacher info: John Thompson (jdt257 [at] nyu.edu )



Subdivision Algorithms in Computer Graphics

In computer graphics, the Catmull Clark algorithm is used in subdivision surface modeling to create smooth surfaces. It is used in Pixar movies. We will understand this algorithm and related ones and how it is used in computer graphics.

We will start by understanding the simple concepts in 1 dimension where this is approximation theory. We will then understand the different choices we can make by generalization to two dimensions. Finally we will describe one or two special subdivision algorithms in a little more detail and why they are so important.

Length: 1 Hour and 15 minutes

Prerequisites: None.

Teacher info: Sara Grundel (sara.grundel [at] gmail.com)



Upstart Puzzles

The writer of puzzles often invents puzzles to illustrate a principle.
The puzzles, however, sometimes have other ideas.
They speak up and say that they would be so
much prettier as slight variants of their original selves.

The dilemma is that the puzzle inventor sometimes can't solve those variants.
Sometimes he finds out that his colleagues can't solve them either,
because there is no existing theory for solving them.
At that point, these sassy variants deserve to be called upstarts.

We discuss a few upstarts inspired originally
from the Falklands/Malvinas Wars, zero-knowledge proofs,
and hikers in Colorado, and city planning.
They have given a good deal of trouble to a certain
mathematical detective whom I know well.

Length: 1 hour 15 minutes

Prerequisites:

Teacher info: Dennis Shasha (shasha [at] cims.nyu.edu)



Keeping it (Approximately) Real

When you compute with a calculator or a computer, or even when you
keep only a few digits in a pen-and-paper calculation, you are doing an
approximate calculation. In this class, we discuss some of the
surprising differences between the behavior of mathematically equivalent
formulas when they are used for computation. Come learn how to trick
your computer or calculator into saying 1 = 2 -- and how to avoid being
tricked yourself!

Length: 1 hour 15 minutes

Prerequisites: Good algebra skills. A little calculus will be helpful,
but not necessary.

Teacher info: David Bindel



Knot Theory

Take a piece of string, tangle it up any way you like, and then glue the ends together, what you have is a knot! If your friend tangles up another string and glues its ends together, how can you tell if you've created the same knot (a.k.a. can you tangle and untangle your knot so it looks like your friend's knot)? It turns out that there are lots of ways to associate numbers, polynomials, and other mathematical objects to knots to be able to tell them apart. These mathematical objects are called invariants of the knot. We will learn about some of these invariants, learn to compute some by just looking at pictures of knots, and learn about some of the new developments and cool results in the field. Beginner's Knot theory is remarkably accessible, since it often relies only on your intuition about what a knot looks like. However, if you really like the topic, there is a lot more to Knot theory once you learn college math.

Length: 1 hour 15 minutes

Prerequisites: Basic algebra (polynomials)

Teacher info: Laura Starkston (lstarkst [at] gmail.com)



To Infinity, and Beyond!

The concept of infinity has always fascinated and sometimes troubled mankind. However, thanks to Cantor and others, we now know how to work with different kinds of infinities, and have access to even deeper mysteries! If you are not afraid of paradoxes, come and discover infinity's true nature!!!

Length: 1 Hour 15 minutes

Prerequisites: An inquisitive mind is needed! Also, some knowledge on functions and calculus might help, but it is not strictly necessary.

Teacher info: Eduardo Corona



Welcome to the Symmetric Group

Is it possible to alphabetize a shelf of randomly-arranged books if you are only allowed to switch them a pair at a time? How about if you have to switch two pairs of books at a time? How about if your only legal moves are switching the first two books, or putting the first book all the way at the end of the shelf, but you can do either of these as many times as you want?

In general, what is the nature of rearrangement?

These questions are at the heart of group theory, one of the deepest unifying themes of modern mathematics. In this class, we will explore some of the building blocks of the theory via questions like these.

The class will put you in the position of a research mathematician - you will be spending your time actually attacking problems, discussing them with your peers, and possibly posing and attacking new ones as your work reveals new avenues to explore. There will be no lecturing; the teacher will provide interesting questions to start with, and just enough information for these questions to make sense. The primary goal is the excitement of exploring the questions. The specific content you learn will be based on the direction your research takes you when you get to work - but it is sure to be something you didn't expect.

Length: 1 Hour 15 minutes

Prerequisites: No prerequisite knowledge is required. However, come prepared to think very hard!

Teacher info: Ben Blum-Smith ( bbs247 [at] nyu.edu)



Taylor Series

Ever wonder how four of the most important numbers in mathematics relate to each other? How about if you can write a function as infinitely many polynomials? In this class, you will learn the basics of Maclaurin and Taylor series. These series are extraordinarily useful when approximating functions in computer science as well as in physics and in engineering. At first, we will review power series. Then we will cover all of the fundamentals of Taylor series. At the end of the class, we will go over how to derive the relationship between e, i, pi, and -1. The series are an integral part of the derivation. You must know basic differentiation rules for this class.

Length: 1 Hour 15 minutes

Prerequisites: Basic calculus and a love of math!

Teacher info: Svetlana Shneyderman (ss4546 [at] nyu.edu)



The Mathematics of Dating

If you embark on a quest to find your true love, what would be your odds of finding him or her? Can numbers predict and improve your odds of finding a good date? As Einstein once asked: "How on earth are you ever going to explain in terms of chemistry and physics so important a biological phenomenon as first love?" With math, of course! We shall cover strategies that dramatically raise the odds of finding the best people 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!

Length: 1 hour 15 minutes

Prerequisites: Basic calculus and some basic probability.

Teacher info: Alex Rozinov



Simulating the Jet Stream

Ten kilometers above our heads is the Jet Stream, a fast-flowing river of air. What causes this mysteriously strong and permanent air current? In this class you will see a physical demonstration (using water instead of air) of the principles behind the Jet Stream. You will also learn some of the mathematics used by researchers here at Courant to understand the physics of the atmosphere.

Length: 1 Hour 15 minutes

Prerequisites: Some knowledge of physical concepts (density, pressure, temperature, velocity) is necessary. Some basic calculus (derivatives) would be helpful.

Teacher info: Carl Gladish (gladish [at] cims.nyu.edu)



Smashing the Stack for Fun and Knowledge

We hear about computers being hacked all the time, but have you ever wondered what "being hacked" really amounts to? In this class, we'll go over some basics of how the C programming language and the operating system organize your programs and attempt to provide a secure system. We'll then walk through the details of an example "hack" called stack smashing which is possible when a C programmer writes buggy code. This will end with a real live demo of the hack. The point of this exercise is to illustrate one of the many ways that computers can be made insecure through human error, and thus to motivate the need for programmers to be security-conscious.

Length: 1 hour 15 minutes

Prerequisites: Background in programming, especially in C, will be useful for understanding the material.

Teacher info: Eric Hielscher (hielscher [at] gmail.com)



Optimization in Infinite Dimensional Spaces (part 2)

See part 1 above for details.



Mix it Up: An Introduction to Ergodic Theory

If I'm a baker kneading dough, how do I know if it's been kneaded enough? More generally, how can I measure the amount of mixing done by an arbitrary transformation on an arbitrary space? These are the typical questions that are asked in the field of ergodic theory. Ergodic theory is the study of dynamical (time-dependent) systems from a measure-theoretic perspective. Although the field of ergodic theory originated from statistical mechanics and physics, results in ergodic theory have proven to show up everywhere, from number theory, to serious analysis, to information theory. In this class, we will start with a brief introduction to measure theory, and then move on to study some classical topics in ergodic theory, such as the Baker's transformation, Weyl's equidistribution criterion, and the Birkhoff ergodic theorem.

Length: 1 Hour 15 minutes

Prerequisites: You need to know about limits and integrals, so you probably need to have some background in calculus. Also, be ready to think abstractly!

Teacher info: Jessica Lin ( jessicalin [at] nyu.edu), Mathematics, Undergraduate Senior


Period 3


Statistics and Smoking

You've heard for years that smoking is bad for your health because it causes lung cancer and other health problems. By what logic can we make such a claim? In studies on smoking, how do we know it isn't a random fluke that smokers develop lung cancer more often than non-smokers? In this class, we will discuss basic statistical reasoning as it applies to a
real study on smoking done in Finland. You will then get additional practice in thinking like a statistician.

Length: 1 Hour 15 minutes

Prerequisites: None.

Teacher info: Paul Hand (hand [at] cims.nyu.edu)



Mathematics of Consonance and Dissonance

Math is fun and beautiful. Music is fun and beautiful. Let's do both! There are a number of aspects of music theory and perception that can be understood using a little bit of mathematics. We will talk about things things like:

  • Why does a clarinet's A sound different from an oboe's A?
  • Why do some pairs of notes sound great together, and others sound
    terrible?
  • Why and when do we hear notes that don't really exist?
  • Musical paradoxes - how we can construct a scale that always goes up, yet repeats itself infinitely many times (like Escher's staircase).

And we won't just talk - we'll listen to these things too!

Length: 1 hour 15 minutes

Prerequisites: Familiarity with basic trigonometry, such as understanding sine and cosine. It will help to know a little bit about music theory, such as what is a perfect 5th, but the only real 'musical' prerequisite is an interest in listening to strange musical experiments!

Teacher info: Miranda Holmes-Cerfon



Induction On Everything

Induction is a basic mathematical technique that everyone should know about! If you don't know it yet, then I'm here to tell you about it.

We'll talk about induction on numbers and structures like squares. A false proof involving horses. Stamps. The Towers of Hanoi. An assortment of interesting problems solved with induction.

Length: 1 Hour 15 minutes

Prerequisites: None.

Teacher info: Evan Chou (chou [at] cims.nyu.edu), NYU Math Department, 1st Year PhD Student



Introduction to Number Theory and Cryptography

Have you ever wondered how websites keep your credit card number secure when you buy something online, or how you can send a secret message over the internet? It's simple, public key cryptography, or RSA. In this class we will learn the math behind keeping your identity safe. We will start by learning some basic properties of the integers and prime numbers and apply them to learn how the RSA algorithm works. We will then explore why RSA is so safe and possibly talk about some of the weaknesses of public key cryptography. This will be
different than the course taught last year.

Length: 1 hour 15 minutes

Prerequisites: Basic algebra, and you should also know what a prime number is.

Teacher info: Dan Mitchell (dam444 [at] cims.nyu.edu)



Steiner Triple Systems

Take a few numbers (1 through n) and start making sets of 3 numbers out of them. Like (1, 2, 4), (2, 3, 5), and so on. But you can't make two triples using the same pair, for example: (1, 2, 3) and (1, 3, 4) can't exist at the same time. Also, every pair has to be in some triple. (Try it for some small n. See if you can come up with a set of triples that follows the rules.) These things get incredibly complicated when n is big... and there's a lot that we don't know about them. But we can still play some fun games with it. I'll also tell you how this relates to Sudoku.

Length: 1 Hour 15 minutes

Prerequisites: Nothing really. If you've seen graphs or combinatorics before you will see a connection, but you shouldn't need it to follow this class.

Teacher info: Juliana Freire



Introduction to Neuronal Modeling

Neurons, the excitable cells of the nervous system, are often called the brain's basic computational units. Each neuron communicates with many others via electrical and chemical signals, allowing animals to create an internal representation of their surroundings. To understand how brains work, we often build simplified mathematical models and implement them on a computer, stringing together individual neurons into networks that perform rudimentary brain-like functions. As a first step, it makes sense to design an individual neuron. This already raises a whole series of questions. What does it mean to mathematically "model" a neuron, anyway? What assumptions can we make? How many annoying biological details can we throw away before we lose touch with reality?

We'll start with a discussion of how real neurons communicate and what features we might want to build into a model neuron. Then we'll implement those features, using physical analogy, simple mathematical expressions, and computer code. If time permits, we will compare our model to others that are more sophisticated and biologically realistic, and discuss the advantages and disadvantages of each.

Length: 1 hour 15 minutes

Prerequisites: The concepts will be quite basic and we will aim for an intuitive approach. However, we will touch on a rather diverse range of subjects: animal biology, simple electrical circuits, differential equations, numerical approximation, and computer programming. Familiarity with any of the foregoing, then, would be helpful but is not assumed.

Teacher info: Robert Levy (rbl2 [at] nyu.edu)



Non-Euclidean Geometry

In (Euclidean) geometry we learn that for every line and point not on that line there is exactly one line passing through the point that is parallel to the given line. This is sometimes called Euclid's fifth postulate or the "parallel postulate." When we don't assume this postulate, all sorts of crazy things become possible. For example, triangles no longer need to have 180 degree angles and the Pythagorean theorem can fail! In this class, we'll look at a few examples of non-Euclidean geometries to see what falls apart without this important postulate.

Length: 1 Hour 15 minutes

Prerequisites: No specific knowledge is required, but you will get more out of the class if you've seen triangle geometry and trigonometry.

Teacher info: Michael Burr



Uncertainty, Entropy, and Information!

What do communications channels, compression algorithms, betting on horse races, statistical mechanics, and the bell curve have in common? They are all linked to the simple formula -x log(x), which allows mathematicians to quantitatively measure information. We'll explore where this idea comes from and how it is applied to various fields.

Length: 1 Hour 15 minutes

Prerequisites: An understanding of logarithms, some probability, and some familiarity with integrals.

Teacher info: Spencer Greenberg



Mathematics in a Flash - View from In and Out

This is an overview class that is giving an impressionistic portrait of mathematics. Its aim is to help students orient in a neighborhood of
mathematics, answer many of the questions often asked while studying mathematics and create additional motivation for their study. Each section of this talk will be sparkled with some interesting problems and counterexamples.

Length: 1 Hour 15 minutes

Prerequisites: An open mind - nothing else; of course, it helps to know how to add.

Teacher info: Dmytro Karabash



The Riemann Hypothesis

In his 1859 paper investigating the distribution of the prime numbers, Bernhard Riemann made a casual guess about the function he was investigating. Today, proving that Riemann's guess was right has become one of the greatest - if not THE greatest - unsolved problems in mathematics. In 2000, the Clay Institute listed the Riemann Hypothesis as one of the seven Millennium Prize Problems, and is offering a prize of one million dollars to whoever can prove (or disprove) it.

This course will give you a basic understanding of the Riemann Hypothesis - that all non-trivial zeros of the Riemann zeta function have real part 1/2. Don't worry if that sounds like gibberish right now! You will learn what the Hypothesis is, why it's so important, and why it has baffled the greatest mathematicians of out time. This could be your first step towards claiming the million-dollar prize!

Length: 1 Hour 15 minutes

Prerequisites: If you're familiar with the complex numbers and logarithms you should be fine. Trigonometry and calculus will help, but are not necessary.

Teacher info: Corey Everlove (corey.everlove [at] nyu.edu)


Period 4


Writing Math in Ancient Languages

Have you ever wondered how to do math in Egyptian hieroglyphs or Babylonian cuneiform? Or tried to multiply using Roman numerals? In this course we will learn how to write numbers in several ancient number systems. We will discuss the context in which each society used numbers and
the innovations of each number system. We will also discuss the origins of our modern system and how aspects of some of the ancient number systems have survived to the present day.

Length: 1 hour

Prerequisites:

Teacher info: Meredith Burr (meredith.burr [at] tufts.edu)



A neuroscience sampler

Studying the brain seems complicated. How do you understand the outside world? How does your brain process and distill information from your environment? What does it even mean to "perceive" something? This course will investigate some interesting phenomena (observed in humans and other animals) which relate to sensation, perception and understanding.

Length: 1 hour

Prerequisites:

Teacher info: Adi Rangan



Logic in Computer Science

People want to express and convey the results of their reasoning, and prove if the results are logically acceptable as knowledge. Logic supports their desire to achieve. Furthermore, we have adopted Logic into computers to verify information to be used in various industries and researches. We will have a discussion about related issues.

Length: 1 hour

Prerequisites:

Teacher info: Mina Jeong (mjeong [at] cs.nyu.edu )



Why Wood Burns

The ability to control fire was one of the most important discoveries of prehistoric man. This class will go into the science behind fire - thermodynamics, reaction kinetics and conservation of energy. Why do only some things burn? What is fire anyway? The class will include discussion of different forms of fuel, and why some work better than others. Why can't cars run by turning water into hydrogen and burning the hydrogen? Why are fossil fuels so hard to replace? Why does water put out fire? These questions will be answered by looking at the chemical nature of fire, and why molecules release heat when they burn.

Length: 1 Hour

Prerequisites: A basic understanding of what molecules and chemical bonds are.

Teacher info: Abraham Katz ( ank236 [at] nyu.edu)



Skeptical Empiricism

Here there be monsters. What do aliens, ghosts, yetis, psychic readings, financial models and climate simulations have in common? We will use mathematics only to the extent of better meditating on the known and the unknown.

Length: 1 Hour

Prerequisites: High school mathematics.

Teacher info: Trishank Karthik



Modeling Biological Systems

This class will be a broad overview of all the different ways you can model biological systems. We will cover the basics of predicting protein folding and ligand/receptor dynamics.

Length: 1 Hour

Prerequisites: Differential equations, high school biology and chemistry.

Teacher info: Roshini Zachariah ( roshini [at] mit.edu)



Geometry as a Clue to Computation

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. What can be computed? How can computers prove theorems about Geometry? What ruler-and-compass constructions tell us about geometry.

Length: 1 Hour

Prerequisites: High School Geometry

Teacher info: Chee Yap (yap [at] cs.nyu.edu)



Does God Play Dice?

One of the principles of quantum mechanics states that we cannot precisely know the position of a particle - only the probability that it will be somewhere - until we measure it. So, does that just mean that our knowledge about this particle is limited? Or does that particle actually not have a specific location? Does it even matter? In this class, we will talk about arguably the biggest physics debate of the 20th century that pegged Einstein against Pauli and Bohr, and basically split the scientific community into three camps. We will find out who turned out to be right, and whether our world is in fact uncertain on a fundamental level.

Length: 1 Hour

Prerequisites: none

Teacher info: Dina Genkina (dinkazavr [at] gmail.com)



Biomechanics: simple consequences of scaling

How high can an animal jump? Why can a small animal survive a high drop? How come an ant can lift up to twenty times its own body weight, but we struggle with half of ours? These are the kinds of questions that we will try to answer in this class, using simple mechanical arguments to explain (and explore!) the consequences of size in nature. If you have an interest in applying maths and physics to biology, then this class is for you (just one installment in a million-part series in using mathematics to understand the natural world)!

Length: 1 hour

Prerequisites: Basic understanding of the concepts of force, energy, and power.

Teacher info: Ken Ho (ho [at] cims.nyu.edu)



Particle Physics and the Large Hadron Collider

The Large Hadron Collider, or LHC, is part of the biggest, most complex experiment ever attempted by mankind. It is located at CERN, an international physics laboratory in Geneva, Switzerland, and will begin running any day. Physicists at the LHC will study the fundamental components of matter by colliding particles at very high energy, and making observations from the resulting "debris." The aim of this course will be to discuss the goals of the LHC experiment, some of the challenges involved in building it, and to give an introduction to the fascinating subject of particle physics.

Length: 1 Hour

Prerequisites: none

Teacher info: Thomas Fai



Fibonacci Numbers Everywhere

Ever wonder why some numbers seem to pop up everywhere? The Fibonacci numbers are a sequence of numbers named after Leonardo of Pisa, known as Fibonacci, though they were first studied in ancient India thousands of years ago. The first number of the sequence is 0, the second number is 1, and each subsequent number is equal to the sum of the previous two numbers, yielding the sequence 0, 1, 1, 2, 3, 5, 8, etc.

Amazingly, this sequence occurs in many places in nature, from the leaves on ferns to the spirals on the heads of sunflowers, and we'll find out why. We'll also explore some of the unique properties of the Fibonacci numbers which make them so interesting to mathematicians. Plus we'll see some cool applications of the sequence, such as a tiling with squares whose sides are successive Fibonacci numbers in length, the Fibonacci spiral, and Fibonacci numbers in the stock market.

Length: 1 hour

Prerequisites: basic algebra

Teacher info: Fernando Shao and Ezra Winston



New York Math Circle - Problem Solving

This is a sample of New York Math Circle classes -- which
incidentally take place at the same building on other Saturdays! The class
will focus on problem solving, selecting intriguing and beautiful math
problems from contests such as AMC 10, AMC 12, and AIME. Come sharpen your
reasoning and problem solving skills, and enjoy fun problems.

Length: 1 hour

Prerequisites: The class assumes familiarity with some material outside
standard curriculum, e.g. you should be able to do the first 10 questions of
the AMC 12 (available at http://tinyurl.com/daevac).

Teacher info: David Hankin, Distinguished teacher and Chair of AIME



General Relativity and Black Holes

General Relativity is considered the most elegant physical theory to date. Together, we will explore some of the fundamental concepts of GR, such as spacetime, curvature, metric and causality. If time permits, we will explore the nature of singularities and maybe even fall into a Schwarzschild black hole!

Length: 1 Hour

Prerequisites: Basic trigonometry, knowledge of vectors and matrices would be helpful.

Teacher info: Abhishek Kumar


Period 5


So Much More Than Just Dots and Lines

If there are five people in a room, and every person has to shake every other person's hand exactly once, how many hand shakes occur? Sure, we could count. But we could make a pretty picture to go along with it too... namely, a graph! (Note: "graph" here means a "bunch of dots" all connected to each other by a "bunch of lines" -- not the kind you plot in algebra class.) And once we find said graph, we could do all sorts of things to it. For instance, how many colors would we need in order to color all the edges of the graph and have no same-colored edges touching? What about if we colored all the dots? Could we find a path going through the graph so that by the time we're done, we've traced out the entire thing? There are lots of questions like these whose answers can be used to characterize any graph you want. In this class, we'll look into these questions for some very special graphs. If you are a fan of the "less numbers, more pictures" side of math, you'll enjoy this. There will be lots of pictures. Pictures and colors and graphs, OH MY!

Length: 1 Hour

Prerequisites: None.
*Note: I do realize that most, if not all, high school students have not seen any graph theory. That being said, elementary topics in graph theory are not hard to grasp, and I think most students will be fine with just the introduction I will provide.

Teacher info: Kristin McNamara (kristin.m.mcnamara [at] gmail.com), James Madison University, Department of Mathematics & Statistics



Prions

When first discovered, prions were so novel and controversial that those who stood by the prion theory were shunned by much of the science community.

Why? Because prions, the infectious proteins that cause Mad Cow Disease, break the fundamental genetic dogma of biology.

How do prions work? Where are prions found? Why do prions exist? What do prions have to do with cannibalism? We'll attempt to answer all these questions and more during this class.

Length: 1 Hour

Prerequisites: High school biology or an equivalent understanding of genetics, proteins and cell structure.

Teacher info: Stephanie Bachar ( sbachar [at] mit.edu)



Fashionably Tech. Technically Fashionable.

This class will discuss data mining techniques and recommendation systems (in general) and how they 'fail' in fashion (and anywhere where we quantify 'tastes'.

Length: 1 hour

Prerequisites:

Teacher info: Froilan Mendoza (froilan [at] nyu.edu )



New York Math Circle - Metrocard Origami Polyhedra and Euler's Formula

Reusable metrocards offer a resilient material for unit origami.
In this hands-on workshop, we will fold metrocards into a dimpled
cuboctahedron, mathematically known as a cubohemioctahedron, or even as a
triacontahedron, and verify that Euler's formula holds for this solid.
The instructor will provide used metrocards (cleaned in a dish soap
solution).

Length: 1 hour

Prerequisites: Participants should have basic origami experience,
eagerness to assemble a three-dimensional model, and strong fingers.

Teacher info: Japheth Wood, Bard College and New York Math Circle



Computer Programming is for Lazy People

Do you hate doing boring, repetitive work? Would you like to harness the power of a computer to do work for you? This course will show you how with a gentle introduction to computer programming geared towards problem-solving. It will cover why programming is useful, a brief history, and the basic language constructs that so many languages use. We will use these constructs to implement solutions to common problems and mathematical functions, culminating in the solution to a classic programming problem.

Length: 1 Hour

Prerequisites: This course assumes very little or no prior programming experience. There are no other prerequisites except very basic math.

Teacher info: Paul Gazzillo (pcg234 [at] nyu.edu)



Stochastic (Random!) Models of Climate Variability

What does a random walk have to do
with the wintertime weather in Europe, Greenland, and eastern North America? I will describe how climate scientists characterize atmospheric variability, and introduce three patterns in particular: the North Atlantic Oscillation and the Northern and Southern Annular Modes. I'll then construct a simple model based on random walks to demonstrate that basic geometric and physical constraints lead to the dominance of these patterns in the winter atmosphere.

Length: 1 Hour

Prerequisites: basic probability theory

Teacher info: Edwin Gerber



Geometry of the Quantum World

Was Euclid wrong?! The two major physics discoveries of the 20th century rely on deep questionings of his geometry: general relativity shows that gravity makes our world obey not Euclidean, but Riemannian geometry, and quantum physicists proved that at the small scales the geometry of our world in non-commutative! What are these strange new geometrical beasts? Come to the class, and we'll find out!

Length: 1 hour

Prerequisites: Basic knowledge of Euclidean geometry

Teacher info: Antoine Cerfon (antoine.cerfon [at] gmail.com )



Physics of Rock Climbing

Modern rock climbers climb thousands of feet up steep cliffs. How do they stay on the rocks when the cliffs are vertical and one slip could lead to a massive fall? Through physics and mathematics climbers have developed climbing techniques and equipment that allow them to reach fantastic heights safely. We'll look at videos of climbers making their way up cliffs and taking some large falls. We'll take a close look at the gear they use and understand the physics that make it all work. The forces applied on the gear, the rock, the rope, and the climber's body can be calculated using basic physical concepts. The discussion will consist mainly of understanding how Newton's Second Law can be applied in these situations.

Length: 1 Hour

Prerequisites: Some basic trigonometry.

Teacher info: Shilpa Khatri



Modeling with Regression Analysis

Regression analysis is a statistical method that allows researchers to explore relationships between dependent and independent variables. These could range from the relationship between SAT scores and college admissions, or between natural resource abundance and likelihood of civil war. We will touch on research methods, basic statistics, conceptual mathematics, econometrics, and the study of social issues.

Length: 1 Hour

Prerequisites: Some basic intuition about data, models, correlations, and best fit lines.

Teacher info: Brenda Jimenez (bjg294 [at] nyu.edu)



Seeing the end or the bend of the line: reading brainwaves in the visual cortex

Continuity is a basic concept in mathematics but it is also important in visual perception. Discontinuities help define the shapes of objects. Recently we found that continuity and discontinuity are represented in local circuit activity in the visual cortex, the part of the cerebral cortex that analyzes visual images and leads to perception. We will look at visual phenomena called Illusory Contours that are affected by image discontinuities, and then how we can analyze brain activity to find the brain areas that sense discontinuity. We'll also discuss mathematical tools that are very helpful in finding out how the mind's eye sees things.

Length: 1 Hour

Prerequisites: None.

Teacher info: Robert Shapley, Neural Science, Professor



Automata and Language Models

It's often desirable to craft algorithms that can label documents or emails written by people. Other times we might actually want a computer to generate text in a style which looks believably human in origin.

In this class we'll explore basic formalisms which are useful for dealing with text.

We will start by looking at formal languages and how some of them can be described using mathematical machines called "finite automata".

We will then enhance these automata to make them more suitable for recognizing and generating natural language.

Length: 1 Hour

Prerequisites: Basic understanding of probability

Teacher info: Alex Rubinsteyn ( ar1738 [at] nyu.edu)



Love dynamics

Would your life be easier if you could predict the future? You will
learn how to do just that in some simple cases -- when only two types
of objects interact. For example, Romeo and Juliet are in love. Will
they live happily ever after? If sheep and rabbits compete for food
during a 10 month drought, which species will survive? You will learn
how to introduce two variables, write the dynamics in terms of
equations, and determine the final state of a system through math
analysis.

Notice: no rabbits will die during this course.

Length: 1 hour

Prerequisites: Be able to solve a quadratic equation in 30
seconds, know what a 2D vector is (we will deal with vector-fields). Knowing about matrices would help, but is not necessary.

Teacher info: Lyuba Chumakova



Patterns in nature

They say in Africa that if you have been bitten by a leopard, you see spots everywhere. Ever wondered where those spots come from? Mathematics is the study of patterns and biology is the study of the most complicated patterns we know. Come and hear about what they can do for each other.

Length: 1 Hour

Prerequisites: Calculus

Teacher info: Jan Morup ( jorgenj [at] cims.nyu.edu)


About the difficulty icons:

We have delopved a color grading system in an attempt to indicate the overall difficulty of each talk. A green icon indicates that
anyone with a standard high-school mathematics background should be able to follow. A black icon indicates that the talk will be fast-paced, and that students without extra-curriculuar exposure to more advanced mathematics---through math camps, college courses, competition preparations, and so on---are likely to find the talk challenging. These are the two extremes, and blue and purple icons indicate the midpoints of the difficulty spectrum. It is, of course, impossible to determine the objective difficulty of a talk, and the icons should only be taken as a crude approximation. The best way to figure out whether the talk is at the right level for you is to talk to the lecturer. Instructors' emails are listed on this page, so ask away!