2010 Talks

Here is the list of courses that were offered at cSplash 2010.

Link to Lecture Notes


indicate difficulty rating, in ascending order.

Description of the icon system.

Colloquium Speaker

Computer Arithmetic

Teacher info: Michael Overton (math prof)

Abstract: Whenever you use C++, Java, Python, Matlab, Excel or any other computer system to do arithmetic operations you are almost certainly using floating point arithmetic. I will explain this concept, which has been used since the earliest days of computers and is the basic workhorse for scientific computing, computer graphics and much more, giving an idea of its power and its limitations.

Period 1

Some Proofs in Geometry (part 1)

What is the philosophy of mathematics? How does one actually DO math, i.e. what are the general methods? In other words, how does a mathematician think?

This class will try to illustrate some of these aspects by proving some facts in high school geometry. The main purpose here is to show the method of proving something, and the thinking process behind it, by motivating examples from geometry.

Part I of this lecture will show some elementary concepts:

  1. State some basic axioms for doing geometry.
  2. Prove some basic facts for areas of simple geometric figures, i.e. triangle, parallelogram, trapezoid.
  3. Prove the Pythagorean theorem.
  4. Hint at how to compute areas for more general geometric shapes.

Part II of this lecture will show some slightly more advanced concepts:

  1. Prove the formula for the area of a circle
  2. Prove the formula for the circumference of a circle
  3. Prove the formulas for the volumes of some simple shapes, such as a pyramid, a parallelepiped, etc.
  4. Prove the formula for the volume of a sphere

Length: 1 Hour 15 minutes

Prerequisites: Part I: A 1-year course in high school algebra is enough. Some intuitive notions of geometry are a plus. If you have already taken a full course in high school geometry, then you may be bored with this class (see part II).

Teacher info: Shawn Walker (walker [--a-t--] cims.nyu.edu), Courant Math, Postdoc

Bertrand's Paradox

If you throw a stick at a circle, it intersects the circle at two points and makes a chord. What is the probability that the length of this chord is greater than the side of an inscribed equilateral triangle?

I will try to convince you that the answer is 1/2, 1/3, AND 1/4. How can these all be right? This question has puzzled many famous mathematicians and philosophers, and has to do with how we define the notion of "randomness". In this class we will answer the question by experiment (yes, we will throw sticks at circles!), and then we will use our results to try to figure out why one answer is actually better than the others. You'll learn why randomness is so hard to define, and how sometimes simple ideas from physics can help mathematicians solve difficult problems.

Length: 1 Hour to 1 Hour 15 minutes

Prerequisites: A little bit of geometry, such as how to find the distance from the midpoint of a chord on a circle to the center of the circle. You should also be comfortable with basic notions of probability, although we won't use anything formal. Most importantly, you should come prepared to participate!

Teacher info: Miranda Holmes-Cerfon (holmes [--a-t--] cims.nyu.edu), Courant, Mathematics

Course notes available here

Puzzles for Science and Profit

I solve puzzles for a living. Over the last few years,
I've tried to make this activity useful to biologists, finance,
and third world countries.
This talk will give an overview of some of those attempts, involving the use of
combinatorial design to reduce the size of experimental search spaces,
visualization of
experimental data,
probabilistic puzzles for finance and gambling, and an algorithm for banking between mutually "suspicious agents".

Length: 1 Hour 15 minutes

Prerequisites: None

Teacher info: Dennis Shasha (shasha [--a-t--] cims.nyu.edu), Courant/Computer Science, Professor

Course notes available here

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 (jcbjorgensen [--a-t--] gmail.com), Courant/Math, 2nd Year/PhD

Course notes available here

Physics of the Big and Small

Physics today is all about extremes. We're at a point where we're trying to understand the smallest, biggest, oldest and most energetic things in the universe. We often have to use the biggest things (the universe itself) to tell us about the smallest things (particles) and vice versa.

I will start by discussing how the biggest and smallest scales in the universe are related and give some examples on how one extreme can shed light on the other. I will then discuss in depth how physicists have used information about tiny particles and atoms to figure out that the entire universe is expanding. This discussion requires no prerequisite knowledge--I will explain all the things you need to know.

Length: 1 Hour 15 minutes

Prerequisites: None

Teacher info: Rob Morris (rem368 [--a-t--] nyu.edu), Physics, 3rd Year PhD

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? Some 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

Length: 1 Hour 15 minutes


Teacher info: href="http://www.math.nyu.edu/~lushi/">Enkeleida
Lushi (lushi [--a-t--] cims.nyu.edu), Courant, 4th Year/PhD

Mathematics in Poker

Explore the mathematics behind optimal strategy in poker. Answer such questions as how often should I bluff? Should I check or bet? Am I playing too passively?

Length: 1 Hour 15 minutes

Prerequisites: Basic Algebra, basic familiarity with poker, and experience in some two player game (i.e. poker, chess...)

Teacher info: Maciej Tomczyk (mpt253 [--a-t--] nyu.edu), Courant, M.S. Mathematics in Finance

Calculus and Probability

Basic calculus. Differentiation and Integration. Develop problem solving skills.
Some Probability Theory.

Length: 1 Hour 15 minutes

Prerequisites: Trigonometry.

Teacher info: Unnikrishna Pillai (pillai [--a-t--] hora.poly.edu), NYU Poly/Electrical Engineering

Course notes available here

Two 50 Dollar Problems of Combinatorial Geometry

Can you cover a given regular hexagon of side 1+Ɛ by 7 equilateral triangles of side 1? Given a convex figure of area 1, how many points can you place so that the area of any triangle formed by these points is greater than 0.25?...and a few other interesting problems and their history.

Length: 1 Hour 15 minutes

Prerequisites: Knowing a triangle...personally.

Teacher info: Dmytro Karabash (karabash [--a-t--] cims.nyu.edu), Courant/Math, 2nd Year/PhD

Elliptic Curves and You!

Everyone remembers the Pythagorean theorem: a^2 + b^2 = c^2. What if I asked you to find 3 numbers (a,b,c) such that a^n + b^n = c^n for some integer n greater than 2? The punchline is, there are no solutions (other than the easy ones)! In this talk we'll discuss elliptic curves, an essential tool in solving this problem. For example, y^2 = x^3 + x + 1 is an elliptic curve. We'll show how you can "add points" on them, how to find ALL points with integer coordinates, and how they are related to the problem of finding solutions to a^n + b^n = c^n.

Length: 1 Hour 15 minutes

Prerequisites: You should know what complex numbers are (like a + bi), as well as prime numbers (you know, like 2, 3, 5, 7, 11, etc).

Teacher info: Karol Koziol (karol [--a-t--] math.columbia.edu), NYU Alum/Columbia Grad Student/Math, 1st Year/PhD

Course notes available here

The Art and Science of Digital Audio

This lecture will give you a chance to discover the fascinating world of science that stands behind the art of digital sound. This class will cover the basics of sound effects and sound synthesis with live demonstrations.
During this class you will learn how sound works and how engineers can use simple but clever mathematics to manipulate it, not only for practical purposes, but also for aesthetic ones. The following topics will be covered:
- Basic physics of sound
- What does it mean for sound to be digital?
- Sound effects
- Sound synthesis
- Current research in digital audio processing and what to expect down the road

Length: 1 Hour 15 minutes

Prerequisites: Algebra, trigonometry; basic concepts of calculus recommended but not necessary.

Teacher info: Andrei Krishkevich (andrey_krishkevich [--a-t--] ieee.org), NYU Poly/Electrical Engineering

A Taste of Computational Geometry

In this hands-on class, you'll have a chance to learn about a subject that mixes mathematics and computer science. Together, we'll look at a few problems dealing with points, triangles, and polygons and try to come up with algorithms to solve them on our own.

Computational geometry is a subject which is studied with both mathematics and computer science. It can be thought of as teaching a computer how to work with simple geometric objects. One place where computational geometry shows up is in computer graphics, because graphics require keeping track of many different polygons so that objects look right (we won't talk about these applications in class though).

Length: 1 Hour 15 minutes

Prerequisites: High school geometry including triangle and circle theorems.

Teacher info: Michael Burr (burr [--a-t--] cims.nyu.edu), Courant/Mathematics, 5th Year/PhD

Period 2

Unix 101

Linux and Mac OS X bring the Unix world to personal computers. Unix-like operating systems are widespread in colleges and universities, industry, and government.

We will learn the basic commands to navigate these powerful systems through the simple command-line interface available on all Unix-like operating systems, even Mac OS X.

We will briefly go over processes, memory management, the file system, and input/output management, all features of the most common OSes, including Windows, Mac OS X, and Linux.

Length: 1 Hour 15 minutes

Prerequisites: This class is for those with no or very little experience using the unix command-line.

Optionally bring a laptop to follow along with the tutorial. A Unix emulator, Cygwin, is available for Windows. Mac OS X and Linux already have all the tools installed. For directions setting up your machine, go to http://www.cs.nyu.edu/~pcg234/csplash/ (available in April a couple weeks before cSplash).

Teacher info: Paul Gazzillo (pcg234 [--a-t--] nyu.edu), Courant/Computer Science, Master's

Course notes available here

Some Proofs in Geometry (part 2)

See part 1 above for details.

Interesting Problems in Probability

This is not your ordinary probability class! Perhaps you've heard of the Monty Hall problem. This problem is popular because it has publicly fooled many people from gamblers to professors in mathematics [1]. In this class we will explore other exciting problems in probability and uncover intuitive and analytical approaches to solving these problems.

1: http://www.marilynvossavant.com/articles/gameshow.html

Length: 1 Hour 15 minutes

Prerequisites: Algebra II

Teacher info: Yalu Wu and Michael Axiak (yalu [--a-t--] alum.mit.edu), MIT Affiliated, Alum/Junior

How to Solve The Cubic

Ever wanted to go beyond the quadratic? Let's tackle the cubic! We'll wrestle with some stubborn equations, but by the end of class, they'll be putty in our hands. Special bonus: the bizarre history and intrigue behind the cubic formula.

Length: 1 Hour 15 minutes

Prerequisites: Algebra, the quadratic formula, completing the square, solving simultaneous equations, substitution, graphing parabolas, graph transformations.

Teacher info: Japheth Wood (japheth [--a-t--] nymathcircle.org), Bard MAT Program, Mathematics Faculty

Course notes available here

Machine Learning / Machines That Can Learn

Fifteen years ago, the content of this class would have been considered science fiction. Cameras that detect and recognize human faces. Self-driving robots finding their way in a forest. Telephones that seem to understand spoken language. How was that made possible? Well, by using "intelligent" algorithms that can learn from examples, and of course, lots of data. From Google search to washing machines, machine learning is now all the rage. By the way, did you know that information about who you are "friends" with on a social network could be used for advertising?

We will learn about "features", "linear classifiers", "neural networks", and applications of the Bayes' Theorem. We will see how it all works on two relatively simple problems: spam filtering and learning to play tic-tac-toe. Then, we will take a peek at more difficult problems like computer vision.

Length: 1 Hour 15 minutes

Prerequisites: Vectors, equations and basic probability

Teacher info: Piotr Mirowski (mirowski [--a-t--] cs.nyu.edu), Courant/Computer Science, 5th Year PhD

Course notes available here

Introduction to Quantum Electronics

The past two decades have seen a revolution in the field of electronics. The size of electronic components is quickly shrinking and approaching molecular and even atomic dimensions. The laws of classical physics, that govern the current generation of electronic devices, are becoming obsolete. The new generation of ultra-small electronic components is governed by a fundamentally different set of physical laws--the laws of quantum physics. This class is a brief introduction to the application of quantum physics to the field of electronics. We will begin with a brief overview of the physics of contemporary electronics. We will then turn to a discussion of quantum physics and how it is being utilized in the design of future electronic circuits and devices. Some examples of novel electronic devices that we will discuss include quantum wells, quantum wires, and quantum dots, as well as single molecule and single atom transistors. A brief synopsis of modern fabrication techniques which are being utilized to make these novel electronic components will be presented at the end.

Length: 1 Hour 15 minutes

Prerequisites: This is a comprehensive course, as such there are no prerequisites. However, a basic understanding of introductory physics and chemistry will be helpful.

Teacher info: Milan Begliarbekov (mbegliar [--a-t--] stevens.edu), Stevens Institute of Technology/Department of Physics & Engineering Physics

How to Calculate Sunset, Sunrise and the Local Time

Do you want to know how people calculate the sunset/sunrise times by pure thought?
How Calenders evolved?
What is Time?
Actually, you can do it yourself, using basic trigonometry knowledge.

We will learn basic spherical trigonometry and its use for the celestial sphere.

Length: 1 Hour 15 minutes

Prerequisites: Trigonometry.

Teacher info: Naftali Cohen (naftalic [--a-t--] gmail.com), Courant/Math, 2nd Year/PhD

Course notes available here

Geometric Invariants: How We Knew the Earth was Round Before Magellan

Contrary to popular belief, civilizations as old as the Greeks and Persians were aware that the Earth was spherical and not flat. But how did they discover this before circumnavigating the earth and using telescopes, let alone flying in planes and space shuttles? Even more surprising, if the world was flat or even donut-shaped, they still would have been able to figure out the shape of the Earth. We will talk about "geometric invariants": the techniques used to calculate the shape and size of objects as large as the Earth or as small as an atom.

Length: 1 Hour 15 minutes

Prerequisites: The only math required is a good understanding of high-school geometry. However, it should be noted that the material will likely be new to students with all levels of knowledge, as the class runs parallel to classical high-school math.

Teacher info: Russell Posner (posner [--a-t--] cims.nyu.edu), Courant Math, 2nd Year PhD

Paradoxes Abound!

The concept of infinity has baffled mathematicians for centuries. How can a point traverse a line, when it has to go through an infinite number of midpoints? How can you add up an infinite number of terms? Was Hilbert really a hotel manager? Through an historic analysis of infinities, infinitesimals, and paradoxes we'll answer these questions and pose more.

But warning! We will count to infinity in a finite amount of time!

Length: 1 Hour 15 minutes

Prerequisites: No mathematics is assumed except high school algebra. Calculus and work with infinite series is a plus, but not necessary in the slightest!

Teacher info: Josh Fry (jpfry [--a-t--] nyu.edu), Courant

Strategies and Decision Making

In real life, problems sometimes just emerge out of nowhere. It requires a lot of creativity and experience to come up with possible solutions. However, there are often many ways to solve a problem and the issue is what is the best strategy according to the situation. This course is designed to help you solve these kinds of problems through the corporation of teamwork in a fun and slightly competitive atmosphere. It also requires a lot of thinking and solving the problems on your own rather than me teaching. So be prepared to be active, have a lot of fun, and possibly sweat a little bit.

Length: 1 Hour 15 minutes

Prerequisites: Very, very basic knowledge of game theory
(Not a big deal at all)

Teacher info: Shawn Ning (yn415 [--a-t--] nyu.edu), CAS Math, Junior Undergraduate

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 General Relativity, 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 15 minutes

Prerequisites: Knowledge of vectors and matrices. Basic trigonometry and geometry.

Teacher info: Abhishek Kumar (abhishek [--a-t--] nyu.edu), GSAS/Physics

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: Familiarity with any of the following would be helpful but is not assumed: simple electrical circuits, differential equations, numerical approximation, animal biology, and computer programming.

Teacher info: Robert Levy (rbl2 [--a-t--] nyu.edu), CNS, Postdoctoral fellow

Course notes available here

p-adic Numbers

All of the math you learn in high school - algebra, geometry, trigonometry, calculus - is done using real numbers. The real numbers are obtained by taking the rational numbers (the fractions) and "filling in" the holes between them to give you numbers like pi and the square root of 2 that can't be written as fractions. It turns out that the real numbers are only one of infinitely many possible sets of numbers you can get by "completing" the set of fractions! While the real numbers seem to best describe our physical universe, these other numbers - called "p-adic numbers" - have found many uses in areas like number theory and quantum mechanics. In this class, we'll construct the p-adic numbers and explore some of their properties, and we'll compare and contrast them to the real numbers we are used to using.

Length: 1 Hour 15 minutes

Prerequisites: Nothing more than high school algebra will be assumed. Previous exposure to some formal (proof-based) mathematics will be helpful. This class will move quickly and will present many new concepts in a short period of time.

Teacher info: Corey Everlove (corey.everlove [--a-t--] nyu.edu), Courant/Math

Period 3

The Nature of Infinity

This talk will look at the nature of infinity; what it means to be infinite, how to deal with infinity and all the strange consequences of dealing with infinite objects.

Length: 1 Hour 15 minutes

Prerequisites: No math experience necessary

Teacher info: Alon Stern (sternalon [--a-t--] gmail.com), Courant

Physics in Medicine

What use is a particle accelerator to a doctor? Why is cholesterol so bad for your heart? What really happens inside an MRI machine?

This class will explore medical techniques and ideas that come from physics. We will do mostly imaging techniques, such as MRI and ultrasonography, but also think about the heart and lungs and talk about some neat interventions.

Length: 1 Hour 15 minutes

Prerequisites: Basic physics and biology (anatomy).

Teacher info: Abraham Katz (ank236 [--a-t--] nyu.edu), CAS/Physics, Senior/Undergraduate

The Logistic Map: the Simplest Route to Chaos

The world is often nonlinear and chaotic, but we can understand part of it using the theory of dynamical systems. In this course we will talk about one-dimensional maps, which is a particular example of dynamical systems, and use the logistic map to illustrate how a very simple nonlinear map can produce variety of complex effects and, ultimately, exhibit chaotic behavior. Also we will talk about more general concepts such as stability, periodic orbits, and how they relate to real-life problems.

Length: 1 Hour 15 minutes

Prerequisites: Knowledge of functions, derivatives, graphs of functions, and graphs of derivatives is necessary. Knowledge of Matlab would be useful, but is not required.

Teacher info: Lyuba Chumakova (lyuba [--a-t--] math.mit.edu), MIT, PostDoc

Information Technologies in the Real World

The world economy uses both BIG MACHINES and little machines to process all our data. These machines must play nice together (integration). We will explore and discuss these various technologies and their impact on tomorrow's data processing.

Length: 1 Hour 15 minutes

Prerequisites: Bring an open mind!

Teacher info: Vincent Careri (vcareri [--a-t--] gmail.com), Stevens Institute of Technology, MSIS; Adj. Prof Stevens Tech

Primes: The Periodic Table of the Integers

Number Theory is one of the oldest areas of mathematics and yet there is still a lot of research to be done. In my opinion, the results of number theory are some of the most surprising and interesting results in all of mathematics. Most of the results involve the use of prime numbers. In fact, they can be considered a sort of "periodic table" for the integers. This will be a short introduction to elementary number theory and its many elegant and simple yet quite powerful results. These will include the proof that there an infinite number of primes, that every integer is the product of primes, and that the square root of 2 is irrational. We will also try to discuss the greatest common divisor and some of the results from it. If time permits, we will prove that every non-square integer has an irrational square root and talk about a few other interesting results. This course will not only give you a good idea of what number theory is about, but pure mathematics in general!

Length: 1 Hour 15 minutes

Prerequisites: Basic arithmetic involving fractions and variables.

Teacher info: Zachary DeStefano (zrd202 [--a-t--] nyu.edu), Courant/Math, 3rd Year/Undergraduate

Website Development 101

In this talk, we will learn how to create a website using HTML and CSS. You will come out with the knowledge needed to build your own personal website.

Length: 1 Hour 15 minutes

Prerequisites: No math knowledge is required.

Teacher info: Max Stoller (mbs422 [--a-t--] nyu.edu), Computer Science, Undergraduate Sophomore

Fermi Problems: How to Solve Anything by Educated Guessing

One of the most useful skills is the ability to make quick, but thoughtful estimates of just about anything. What is the energy of a quantum oscillator? How much energy is released by nuclear weapons? How many serial killers live in your town?

All of these questions can be answered just by thinking about what could be relevant to the problem.

Length: 1 Hour 15 minutes

Prerequisites: Be comfortable with scientific notation, logarithms and exponents.

Teacher info: Abraham Katz (ank236 [--a-t--] nyu.edu), CAS/Physics, Senior

Deeper Exploration of Contest Problems!

This talk is a sample lecture given every Saturday at Courant by the New York Math Circle (nymathcircle.org). We will be discussing interesting topics in number theory, algebra, and geometry, and address subtle mathematical ideas, including the nature and construction of proofs. Some material will come in the form of intriguing problems from contests such as AMC 10, AMC 12 and AIME. New York Math Circle is a non-profit organization formed by several ex-math team teachers and coaches offering year round math enrichment programs for both middle school and high school students. Check out our website for more information about many course offerings for spring and the summer!

Length: 1 Hour 15 minutes

Prerequisites: You should be enthusiastic about the study of mathematics, and comfortable with regular high school material, including at least basic algebra.

Teacher info: David Hankin (oana [--a-t--] nymathcircle.org), New York Math Circle, Professor

Amazing Text with The Gimp

Many students in high school are required to present using PowerPoint, multiple times a year. Proper use of graphics is required to make an effective presentation. In this class, you will learn the basics of The Gimp and how to make some cool, icy and fire text!

Length: 1 Hour 15 minutes

Prerequisites: - A working knowledge of computers and Open Office.
- Basic math skills.
- Basic art knowledge (primary colors, etc)

Teacher info: Pavel Tamarin (ptamarin [--a-t--] metlife.com), DeVry BioMed Engineering Tech, Junior

Course notes available here

Vedic Maths: Magic of Mental Math

Are you stuck in class doing calculations, and algebraic problems slowly?
Or do you want to sharpen your algebraic skills to the point that multiplication and division of any two numbers is done mentally?
Welcome to the wonderful world of Vedic Mathematics, a system far simpler and enjoyable than modern math.

Vedic Mathematics is the ancient system of mathematics developed in India.
According to the vedas, all of mathematics is based on sixteen sutras or formulae.
These formulae describe the way the mind naturally works and are therefore a great help in directing the student to the appropriate method of solution.
The simplicity of Vedic Mathematics means that calculations can be carried out mentally (though the methods can also be written down).
This leads to more creative, interested and intelligent pupils.
Even complex problems involving a good number of mathematical operations, the time taken by the Vedic method will be a third, a fourth,
a tenth or even a much smaller fraction of the time required using modern methods.

Come join us and get acquainted with the wonderful technique of doing complex math in your mind!

Length: 1 Hour 15 minutes

Prerequisites: Students should be acquainted with current methods of factorization, multiplication, division, solving equations simultaneously.

Teacher info: Ankit Ashok Parekh (aparek01 [--a-t--] students.poly.edu), NYU-Poly Math

How Does a Search Engine Work?

Find out how a search engine goes through all the billions of pages on the web to give an answer to your query in a second or two.

Length: 1 Hour 15 minutes

Prerequisites: None.

Teacher info: Ernest Davis (davise [--a-t--] cs.nyu.edu), Courant/Computer Science

Course notes available here

Modeling with Ordinary Differential Equations

Sometimes, we know how a function changes in time, but we don't actually know what the function is! These types of problems are called differential equations, and they arise in all sorts of disciplines from physics to biology to psychology. We'll look at a geometric description of ordinary differential equations, and learn a few techniques for solving them. Then, we'll apply these skills to some interesting problems, such as dating with radioactive carbon isotopes, the way pollution spreads out among the Great Lakes, and how to find the curve along which a bead under the influence of gravity travels in the shortest time from point A to point B.

Length: 1 Hour 15 minutes

Prerequisites: Basic Calculus (knowing the derivatives and integrals of elementary functions, like polynomials, sin, cos, e^x, is enough)

Teacher info: Thomas Fai (tfai [--a-t--] cims.nyu.edu), Courant/Math, 2nd Year/PhD

Differential Calculus - The Heat Equation and Vibrating String

This course will be an in depth look at the treatment of solving the classical Heat Equation and Vibrating String Equation. Most students will not learn this in a differential equations class or are simply given formulas to remember, but not in my course. Come to this course and get an in depth review of two of the most fundamental equations of partial differential equations.

Length: 1 Hour 15 minutes

Prerequisites: Trigonometry, Calculus I and II, Algebra, Mechanics - Basic Physics

Teacher info: Ronen Peled (rpeled [--a-t--] citco.com), Courant Math, Graduate - Masters Degree

Information, Data Compression and Transmission

Do you know what the two fundamental questions in communication theory are? First, what is the ultimate data compression limit? Second, what is the ultimate transmission rate of communications? In this talk, we will try to find the answers to these questions by introducing the concept of entropy and mutual information. Examples in data compression (e.g., source coding) and transmission are provided for better understanding.

Length: 1 Hour 15 minutes

Prerequisites: Logarithms and probability (e.g., discrete random variables, probability mass function, expectation). I will also give a brief intro to the knowledge of probability.

Teacher info: Jialing Li (jialing.li.phd2 [--a-t--] gmail.com), NYU-Poly/Department of Electrical and Computer Engineering, 5th Year PhD

Course notes available here

Calculus 101

We will discuss derivatives and antiderivatives (integrals) of calculus.

We will work with simple equations such as x^2, x^4, ln x, e^x, x, and 1.

Length: 1 Hour 15 minutes

Prerequisites: Algebra and Pre-Calculus.

Teacher info: Joel Guardia (jguardia87 [--a-t--] gmail.com), Math/Physics/Electrical Engineering, Undergraduate

Markov Chains: Theory and Examples

Markov chains are a very important subject and useful tool in probability theory. We will discuss the theory of discrete-time Markov chains, including definitions, class structures, hitting times, absorption probabilities, strong Markov property, recurrence and transience as well as examples including virus mutation, gamblers' ruin and random walks.

Length: 1 Hour 15 minutes

Prerequisites: Basic probability

Teacher info: Ling Jiong Zhu (ling [--a-t--] cims.nyu.edu), Courant, NYU

Period 4

The Quantum Physics of Chemistry

Have you taken chemistry and wondered what those letters s, p, d and f mean? Those letters have physical meaning and come from Schrodinger's Equation (the equivalent of Newton's laws to non-relativistic Quantum mechanics). This crash course will describe the interesting phenomena that occur in quantum systems as a result of Pauli's exclusion principle, Heisenberg's uncertainty principle, energy and momenta quantization. For the curious minded who seek knowledge, come have a glance at the fundamentals governing chemistry.

Length: 1 Hour

Prerequisites: Addition, subtraction, multiplication and division.

Teacher info: Michael Hazoglou (mhazog01 [--a-t--] students.poly.edu), Polytechnic Institute of NYU/Chemical Engineering, 3rd Year/Undergraduate

The Black Magic of Computer Architecture

When you turn on your computer and start using it, did you ever wonder what is going on inside? How does the software interact with the hardware? What is a multicore? What is the difference between Intel, Microsoft, and Dell? We will discuss these questions and many more ... This is the black magic: Computer Architecture!

Length: 1 Hour

Prerequisites: Just need to be a computer user :)

Teacher info: Mohamed Zahran (mzahran [--a-t--] acm.org), Computer Science Dept, NYU

Course notes available here

Cultural History in Understanding Mathematics

Learning mathematics and then looking at cultural history puts a lot of the knowledge we acquire into context! Did you know that Cambridge University in the long nineteenth century had the same format of textbooks we had today?

Length: 1 Hour

Prerequisites: None

Teacher info: Sirazum Islam (sublimeglow [--a-t--] gmail.com), NYU Poly, Humanities

The Basic Theorem of Calculus

The class is mainly focused on the basic theorem of Calculus. The subject of mathematics is quite difficult for many students in high school or college. This class is designed to help high school students prepare for college-level math. In addition, this class will show the students how to use the graphing calculator and all of the useful functions that can help them solve difficult problems.

Length: 1 Hour

Prerequisites: Algebra, Trigonometry, and some Pre-Calculus knowledge.

Teacher info: Mr. Pak (pho03 [--a-t--] students.poly.edu), NYU Poly Math, 1st Year Undergraduate

The Game of Life and other cellular automata

Cellular automata are very simple programs which produce very complex results, and can also be used to make pretty images and animations. They are somewhat related to fractals and illustrate how unimaginably complex systems can arise from as little as three simple rules that a ten year old could understand and carry out.

Length: 1 Hour

Prerequisites: Knowledge of logic gates an how computers work is suggested but not required.

Teacher info: Jacob Hickey (jdh380 [--a-t--] nyu.edu)

Boolean Logic on Functions

"I finished my homework AND I ate lunch." Do you know whether that statement is True? If it is, you're probably feeling well fed. But you're also familiar with Boolean logic!

Boolean logic is the study of all the mathematical operators that you can use to compare things that could be true and could be false. For the first part of this class, we'll take a quick look at Boolean algebra, and at what kinds of problems you can solve with it.

But after that, things will start getting weird: Boolean operators, like "AND" and "OR", typically take statements or variables as arguments. "True OR False", "a AND b". But there's a branch of Boolean algebra where they instead take other functions as arguments. "And AND Or" is a perfectly valid expression. What does it mean? We'll be spending the second half of class figuring that out!

Since we'll be messing around with functions in all kinds of strange ways, we need an environment that makes this easy. So, we'll be showing examples of this logic using the Scheme programming language.

Length: 1 Hour

Prerequisites: A basic grasp of high-school Algebra; having heard of Boolean Algebra will be useful but isn't required.

Teacher info: Adam Seering and Yuri Lin (aseering [--a-t--] mit.edu and rye [--a-t--] mit.edu), MIT/Computer Science

Superconductivity and Superfluidity

Sure most of us have heard of superconductivity and superfluidity, but what exactly are they? How does something become a superconductor or superfluid? How do they behave? What are some real world applications of these materials? In our journey to learn about these materials, we?ll touch on the DeBroglie wavelength, fermions and bosons, the Pauli-Exclusion Principle, Schrodinger?s Equation, and more. Also, look forward to a special guest appearance on the related subject of Bose Einstein Condensation! Take this course and dive into the quantum mechanics and mathematics necessary to understand one of the coolest (literally) phenomena in physics.

Length: 1 Hour

Prerequisites: Algebra, Pre-calculus, Calculus (helpful but not required)

Teacher info: Dorothy Lee (dl1291 [--a-t--] nyu.edu), CAS - Physics, 3rd Year Undergrad

Iterative Maps Iterative Maps Iterative Maps ...

Put your calculator in radian mode. Take the cosine of any number and then repeatedly take the cosine of the result. The results always approach 0.7390851. Where does this number come from? In our study of iterative maps, we will shed light on this phenomenon and discuss several other topics, including stability of fixed points, cobweb diagrams, period-doubling, chaos, and fractals.

Length: 1 Hour

Prerequisites: Some exposure to complex numbers or calculus would be helpful, but is not necessary.

Teacher info: Adam Stinchcombe (stinch [--a-t--] cims.nyu.edu), Courant Mathematics, 2nd Year PhD

The Radon and Fourier Transforms: the Mathematics of X-Rays and CT-Scans

How exactly do these medical imaging devices work? We will introduce some surprisingly sophisticated and beautiful mathematics to answer this question. Along the way, we will survey related problems from calculus, signal processing, geometry, etc.

Length: 1 Hour

Prerequisites: Calculus is desirable, multivariable calculus would be good but is not necessary. Both are not really needed, and I will quickly cover the basics of calculus at the beginning upon request.

Teacher info: Steven Heilman (heilman [--a-t--] cims.nyu.edu), Courant/Math, 1st Year/PhD

Course notes available here

Period 5

Prove it!

This session will serve as an introduction to proofs beyond the high school geometry level. Does 1+1=2? Can you prove it? We will first look at proofs-gone-wrong that attempt to disprove this statement. Then we will explore how to communicate intuition with mathematics, and learn how to work through some of the sophisticated proofs behind everyday mathematics.

Length: 1 Hour

Prerequisites: Some background knowledge of series is helpful, but not necessary.

Teacher info: Tatyana Shestopalova (tas362 [--a-t--] nyu.edu), CAS, Economic Theory and Mathematics

Biomechanics: Consequences of Size in Nature

"A grasshopper can jump up to ten times its size, so a human-sized 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 human-powered flight is so difficult; and why deep-diving mammals are so large.

Note: this class covers essentially the same material as a class of similar name given last year (so you should probably only take it if you didn't take it then).

Length: 1 Hour

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

Teacher info: Ken Ho (ho [--a-t--] courant.nyu.edu), Courant/Computational Biology

Course notes available here

Exploring Fractals with Maple

You may have seen fractals appear on t-shirts, desktop graphics or calendars, but those chaotic patterns appear in many complex branches of science that are important to us, like weather prediction, fluid dynamics, celestial mechanics, biological systems, financial markets and many more. In this course, we will not do anything that complicated, but we will learn what fractals are, how they arise and how to calculate their dimensions. We will then use the Maple computer software to generate a few nice fractal graphics. No prior knowledge of Maple or programming is needed since we will work through all the steps together.

Length: 1 Hour

Prerequisites: Not much. Just the will to learn new (and cool) things!

Teacher info: Enkeleida Lushi (lushi [--a-t--] cims.nyu.edu), Courant Math, 4th year PhD student

Fun with Computer Security

Computers and networks are becoming more complex every day. It's the job of a computer security expert to understand and find flaws whenever possible. By thinking outside the box and asking ?what if?, cyber security experts are trained to locate and fix possible exploits. This course attempts to show students why it's important to play the role of the bad guy to find and fix vulnerabilities before they can exploit them. This crash course will demonstrate vulnerabilities and exploits, and then walk students through the process of how they were discovered. I'll conclude with some information about professional and competitive opportunities in the world of cyber security. Students should leave with a heightened interest in computer security and a set of ethics to guide them through their research.

Length: 1 Hour

Prerequisites: General interest in puzzle solving, computers and technology, and security.

Teacher info: Theodore Reed (treed [--a-t--] stevens.edu), Stevens Institute of Technology - Computer Science, 4/5 Undergrad

Programming Robots

In this class, we will be demoing two robots. One is an iRobot Roomba vacuum cleaner that was modified. The other robot, we will be able to program and observe interactively as a class.

You will learn some programming in this class.

Length: 1 Hour

Prerequisites: None.

Teacher info: Rebecca Davidson (becca.davidson [--a-t--] gmail.com), Courant

Nutrition and the Functions of the Brain

This topic revolves around the importance of understanding your body and most importantly understanding nutrition and the functions of the brain.

Length: 1 Hour

Prerequisites: No prerequisites necessary.

Teacher info: Ximena Aristizabal and Jason Ko (xarist01 [--a-t--] students.poly.edu), NYU Poly/Chemical and Biomolecular Engineering, Freshman/Undergraduate

How do little creatures swim?

Have you ever wondered why crabs seem so awkward in water or why bacteria have such a long and flexible flagella ? Drag is the resistance to movement of a body through a fluid and we will learn how nature figured out a way to overcome the drag due to the water, when the creature's inertia is small. The equations we will use are the Stokes equations. The answer lies in the way these little creatures use their shape to effectively swim in water or break the symmetry. Movies of real life creatures or simulated swimmers will be used to illustrate the intuition and a glimpse into the mathematical argument will also be given. We will conclude by some examples of mechanical ratchets (robots) built using the same principles.

Length: 1 Hour


Teacher info: Christel Hohenegger (choheneg [--a-t--] cims.nyu.edu), NYU/CIMS, Post-doc

Overcoming Your Fear of Computer Science

This class will throw light on basic concepts of Computer Science. It is to encourage students to be not scared of math and computer subjects. Encouraging students to take up Computer Science in their Undergraduate studies.

Specifically, my talk will throw light on one of the core courses in Computer science - Data Structures and show how easily you can understand such concepts without extensive math knowledge. You don't need to have extensive math background to become a computer engineer!

Length: 1 Hour

Prerequisites: Basic math.

Teacher info: Apoorva Vadi (vadiapoorva [--a-t--] nyu.edu), Courant Computer Science Department, 1st Year Graduate Student

Course notes available here

Algorithmic Game Theory

We will start by describing simple games that they can relate to rock, paper, scissors (and others) and gradually explain the abstraction of formally defining 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 games in class that will help students understand how players think and which strategies are good

Length: 1 Hour

Prerequisites: Capability of grasping abstract notions and some basic discrete math (sets and functions).

Teacher info: Vasilis Gkatzelis (gkatz [--a-t--] cims.nyu.edu), Courant/Computer Science, 2nd Year/PhD

Video Game Programming: Collision Detection with Separating Axis Theorem

If you're making a video game, then it's usually pretty important to know when stuff hits other stuff. How can we tell if two shapes are overlapping based only on their position and a mathematical description of their shape, and how can we tell where they overlap? More importantly, how can we make a computer do it FAST? We'll cover collisions between rectangles, circles, and triangles, and talk a bit about how to apply this technique to a physics engine in a computer or video game.

Length: 1 Hour

Prerequisites: Knowledge of vectors and geometry.

Teacher info: Jacob Hickey (jdh380 [--a-t--] 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!

Lecture Notes:

Talk Name Teacher Link to Notes
The Radon and Fourier Transforms: the Mathematics of X-Rays and CT-Scans Steven Heilman 2010-49.pdf
Unix 101 Paul Gazzillo 2010-27.pdf
Elliptic Curves and You! Karol Koziol 2010-50.pdf
Calculus and Probability Unnikrishna Pillai 2010-57.pdf
Amazing Text with The Gimp Pavel Tamarin 2010-09.pdf
How Does a Search Engine Work? Ernest Davis 2010-39.pdf
Introduction to Neuronal Modeling Robert Levy 2010-23.pdf
How to Calculate Sunset, Sunrise and the Local Time Naftali Cohen 2010-45.pdf
The Black Magic of Computer Architecture Mohamed Zahran 2010-02.pdf
Bertrand's Paradox Miranda Holmes-Cerfon 2010-14.pdf
Puzzles for Science and Profit Dennis Shasha 2010-05.ppt
Special Relativity Jens Christian Jorgensen 2010-42.pdf
Overcoming Your Fear of Computer Science Apoorva Vadi 2010-48.ppt
Biomechanics: Consequences of Size in Nature Ken Ho 2010-31.pdf
How to Solve The Cubic Japheth Wood 2010-01.pdf
Machine Learning / Machines That Can Learn Piotr Mirowski 2010-22.pdf
Information, Data Compression and Transmission Jialing Li 2010-25.pdf