2006 Talks

Here is the list of courses that were offered at cSplash 2006.


indicate difficulty rating, in ascending order.

Description of the icon system.

Period 1

The Reeb Foliation of the 3-Sphere

First, you get a circle. Go up a dimension and you get a "normal" sphere. Go up another dimension and you get the 3-sphere. This is a really interesting ! object: it has to sit inside four dimensions, because it doesn't fit in three-dimensional space, and it has a number of really interesting properties. We're going to study those properties, first by figuring out exactly what this 3-sphere thing is, and then by analyzing it by taking a "foliation." If that doesn't make sense, don't worry about it --- we'll go over it in class. But if you want to start to visualize things in four dimensions, this is a great class to do so.

Length: 1 hour

Prerequisites: None

Teacher info: Daniel Zaharopol, danz at alum.mit.edu, 2nd year PhD student at University of Illinois-UC (former director of MIT Splash)

Proof without Words

Did you know many of the mathematical results can be proved without even writing a word? This class will be an introductory session to a different method of proving & approaching mathematical results; visually, those that do not involve any word comments in the proof. We will study common Mathematical results along with their visual proofs and study how various results could be represented visually. We will also attempt to use this technique to solve mathematical problems. The topics covered will include high school algebra, calculus, geometry & combinatorics.

Length: 1 hour

Prerequisites: (none)

Teacher info: Rajesh Vijayaraghavan, rv448 at nyu.edu, grad student

An Introduction to financial Mathematics (part 1)

Financial Products, so-called derivatives, have been created decades ago; however, it has only been a few years that they have efficiently entered the market. They are now a fast developing breed, also helped by the exponential growth of powerful computers. In this class, we will see different sorts of financial products, starting with an introduction to the market, to the need of derivatives, and then some more complex products and ways to give them a fair price.

Length: 2 classes, 1 hour each (2nd is optional, but depends on the first)

Prerequisites: basic undergrad math

Teacher info: Simon Leger, Yann Renoux, simon.leger@nyu.edu,yr366@nyu.edu, first year math finance master's students

The math behind Google

Ever wonder how Google knows which web sites are better than others? Well, part of the recipe is a company secret, but a very important aspect of Google search
is the idea of PageRank, named after Larry Page (co-founder of the company).

We start by introducing elementary graph theory, which is one very useful way to model the internet.
One way to think about PageRank is as a random walk on a graph.
Next we will see how to combine this idea with standard matrix multiplication.
The result is a huge system of simultaneous equations, but we can't solve it in the usual way!
This is because the variables to solve for are on both sides of the equation --
so how can we solve it?
We'll see a very clever algorithm which might surprise you with its simplicity
and effectiveness.

And in the end, the solution to this crazy equation gives us our ranking
of all the net's web pages!

Length: 1 hour

Prerequisites: what a matrix is (even better if you know how to multiply them)

Teacher info: Tyler Neylon, neylon at cims.nyu.edu, 5th year math phD student

Graph Theory and the Donut

Draw five dots on a page. Can you connect them without crossing any lines? We'll (rigorously) solve this puzzle --and answer lots of other questions about dots and lines-- in this 1-hour investigation into Topological Graph Theory. And the answers will not be limited to the plane: can you connect the same five dots if they're sprinkled on a donut instead of drawn on a piece of paper? You'll come away with a new understanding of the old joke about a mathematician being "someone who can't tell the difference between a donut and a coffee cup."

Length: 1 hour

Prerequisites: No specific content, but a reasonable level of mathematical maturity. Should be comfortable with deductive and inductive proofs.

Teacher info: Marisa Debowsky, mad464 at cims.nyu.edu, math graduate student

Ordinary Differential Equations

Calculus is the foundation needed for solving all kinds of problems in science. This is because many problems in science can be stated as differential equations. In this class, we'll solve a few examples of differential equations. The examples might be population growth, electric circuits, mixing chemicals, or the pendulum. (With ordinary differential equations, we deal with functions of 1 variable, such as f(x). With partial differential equations, the function depends on more than 1 variable, such as f(x,t). We'll see why they're called partial differential equations in another class called "Partial Differential Equations".)

Length: 1 hour

Prerequisites: Students should know the basics of calculus (how to do some derivatives and integrals).

Teacher info: Sam Stechmann, stechman at cims.nyu.edu, math graduate student

Period 2

Computer Modeling of Climate

Predicting the earth's future climate represents an enormous mathematical and scientific challenge, and one with profound societal ramifications. In this presentation we focus on one particular aspect of climate change, namely that of global sea level rise. Computer models that constitute the basic tool for future prediction of sea level are presented and discussed. The key areas of ongoing research are highlighted.

Length: 40 min

Prerequisites: (none)

Teacher info: David Holland, holland at cims.nyu.edu, Professor of Mathematics

Traffic flow, rivers and oceans

The same mathematical framework: systems of conservation laws, describes a number of physical scenarios, including the flow of cars through roads, and of water through rivers and oceans. Through this unifying mathematical theme, we'll find analogies between traffic jams, river bores and tsunamis.

Length: 1 hour

Prerequisites: Some calculus.

Teacher info: Esteban G. Tabak, tabak at cims.nyu.edu, Professor of mathematics.

Finding Your Match: Exact and Inexact String Matching Techniques

The ability to find a target word in a large pool of text can be very difficult, but it is also a key issue in Computer Science and Mathematics. It's often compared to finding a needle in a haystack, or finding your one true love. String matching applications range from finding a word in a newspaper article, to finding the correct website for a topic, to identifying submatrices of a matrix used to render objects in a video game, to finding a critical gene in your DNA, to identifying the enemy on a military radar grid, to catching key sound effects in a sound-byte of a criminal police report.

Length: 1 hour

Prerequisites: Basic math knowledge (high-school algebra), Basic knowledge of Computer Algorithms and Data Structures

Teacher info: Ofer Gill, gill at cs.nyu.edu, 5th year Computer Science Ph.D. student

An Introduction to financial Mathematics (part 2)

See part 1 above for details.

From secured emails to elliptic curves in an hour

The number (129 digits): 114 381 625 757 888 867 669 235 779 976 146 612 010 218 296 721 242 362 562 561 842 935 706 935 245 733 897 830 597 132 563 958 705 058 989 075 147 599 290 026 879 535 541 is the product of two prime numbers, which are they? The robustness of the RSA cryptographic protocol, which is used in particular to secure our emails, is based on this type of questions. Let's see if we can find a clever way to solve this math problem, and thus break this famous and widely used protocol!
To do so, we will have a fast and beautiful journey through many very important fields of mathematics: from a quick introduction to the principles of public key cryptography, we'll jump to the basics of number theory on which it is mainly based, and come across some fundamentals of group theory on the way. We'll then use what we just learnt about groups to launch our assault on the final topic, elliptic curves (famous in particular thanks to Andrew Wiles), with which we'll hopefully be able to solve our question!

Length: 1 hour

Prerequisites: No special knowledge of the topics we'll go through is required, but please, come with tons and tons of mathematical open-mindedness!
Students having already heard about algebra and binary relations on different fields than simply Q, R or C, will probably feel a bit more comfortable in this class.

Teacher info: Antoine Cerfon, antoine.cerfon at gmail.com, first year physics PhD candidate, MIT

Period 3

What does dimension 1.5849625 look like? An Introduction to Fractals

Lines are 1-dimensional and surfaces are 2-dimensional and volumes are 3-dimensional, but is there anything in between? In this class we will learn how to make spaces that have dimensions that are not integers: fractals! Fractals are beautiful, self-symmetric objects that appear frequently in nature. We will look at many examples of fractals, learn what the concept of dimension means, and see how we can use this to calculate the fractal dimension of patterns found in nature, such as clouds and coastlines.

Length: 1 hour

Prerequisites: (none)

Teacher info: Miranda Holmes, mirandaholmes at yahoo.ca, first year math graduate student

So, why is Einstein so cool?

You always heard that Einstein is, almost undoubtedly, the most influential physicist of the 20th century. But why is that so? What did he do that was so novel?
In this talk I will try to enlighten the processes that led him to propose ultimately his Theory of General Relativity. Starting with why he postulated that the light has a fixed speed, with immediate result that our world is 4 dimensional, I will then attempt to explain the philosophical ideas that led him to think our world was in fact curved. I will give different examples (black holes, worm holes, twins paradox, etc.) of the extraordinary consequences of his theory.

Length: 1 hour

Prerequisites: Having seen the movie Contact might help.

Teacher info: Frederic Laliberte, grad student

Information Theory (part 1)

Seven horses are participating in a horse race, but they have different chances to win. After the race is over, the result (the winner) shall be transmitted using a digital channel, so just using 0s and 1s. We are trying to design a procedure that, so that, on average, one needs to transmit the smallest possible number of these symbols. In the first part of the class, we will try to optimize our encryption hands on using this and other examples. In the second half, we will discuss the mathematical theory behind this type of problems, information theory, that will allow us to solve this problem in great generality.

Length: 2 hours

Prerequisites: logarithms and good equation manipulation skills, some intuitive probability is helpful

Teacher info: Felix Krahmer, krahmer at cims.nyu.edu, 2nd year Ph.D. student in Mathematics

Finding order in chaos -- an introduction to Ramsey theory

Take a grid -- a very very big grid. Now color all the points red, green, blue or purple. Then I can always find (no matter how you color it) a square in your grid, where all the corners have the same color. In fact, not just a square, but any reasonable shape you want...
This type of result -- no matter how you do the coloring, I can find something very ordered inside it (a square, say) is an example of Ramsey theory. We will discuss this and other examples, including, maybe, tic-tac-toe in high dimensions, or the game of Set.

Length: 1 hour (probably)

Prerequisites: None

Teacher info: Akshay Venkatesh, Professor

Measure Theory & Lebesgue integration

How could we integrate a fuction (x in (0,1)) which is 1 when x is a rational number, and is 0 when x is an irrational number or a function which is 0 when x is a irratinoal number, and is 1 when x is an irrational number. As you know from Riemman integration (usual integration), the Riemann integrals of these functions are not defined well. You will find out the answer to this question. We will disucuss simple functions, measurable functions,and Lebesgue integration. Actually, Lebesgue integration is just an extension of Riemann integration. It plays an important role in the theory of Real analysis.

Length: 1 hour

Prerequisites: Calculus(Especially, Riemann integration)

Teacher info: Youngseok Lee, ysl238 at cims.nyu.edu, 1st year math phD student

Introduction to Abstract Algebra (part 1)

"Algebra," you say. "That thing I learned in 8th grade?"
When a mathematician! talks about algebra, they don't mean "solve for x." For us, it's a way to generalize the notions of addition and multiplication to things *other* than numbers. What we'll get are operations that look like addition, or look like multiplication, but aren't limited to sets like the real or complex numbers. It's a whole new way of looking at things, and one of the fundamental things everyone who studies math has to know.

Length: 3 hours

Prerequisites: None

Teacher info: Daniel Zaharopol, danz at alum.mit.edu, 2nd year PhD student at University of Illinois-UC (former director of MIT Splash)

Period 4

Ask the mathematician!

Do you have a question about math, but never knew who to ask? Here is your chance! A real, live, mathematician will be here to answer all your questions. Ask him anything you want: about math, what it's like to be a mathematician, the purpose of math, and he even knows things about astrophysics, earth science, chemistry, and molecular biology. There is no such thing as a silly question! If nobody has a question, then he will ask YOU!

Length: 1 hour (most likely)

Prerequisites: None

Teacher info: Fred Greenleaf, Professor of Mathematics

Probability Theory

What are the chances of winning in a Casino Roulette? Have you heard about the Martingale Method for beating the Casino? (There is such a way) What do Casinos do against this method (Trust them - they have a counter strategy)? What is the chance that two students in a class of 23 share the same birthday? (more than 50%). How does McDonald's predict how many Quarter Pounders to prepare so that people will get their order fast? Take a sponge-like stone and put it in water. What is the probability that its center gets wet? Come tame "chance" and learn all the basics of the "Theory of Probability".

Length: 1 hour

Prerequisites: (none)

Teacher info: Oren Louidor, louidor at cims.nyu.edu, 1st year PhD Math

Mathematics and Origami (part 1)

Take a piece of paper and fold it flat using origami (mountain and valley) folds. Then cut across the folded paper. What shapes can be cut out through this process? This problem is called the fold and cut problem with the amazing result that any polygonal object can be cut out! We will explore, using a hands on approach, topics in rigidity, origami, and unfolding with applications in robotics, proteins, satellites, sheet metal manufacturing, and many more. For more information on the fold and cut problem see: http://theory.csail.mit.edu/~edemaine/foldcut/examples/

Length: 2 hours

Prerequisites: Good knowledge of basic geometry (e.g. polygons, angle bisectors, perpendicular bisectors) + a good spacial orientation skills.

Teacher info: Michael Burr, burr at cims.nyu.edu, grad student

Partial Differential Equations

Ordinary differential equations (see the cSplash class called 'Ordinary Differential Equations') are fine if the function you care about only depends on one variable x, such as f(x). But what if we had f(x,t) -- a function that depends on the 2 variables x and t? Then, instead of ordinary derivatives and ordinary differential equations, we have to deal with partial derivatives and partial differential equations. In this class, you'll learn how to find partial derivatives, and we'll solve a few examples of partialdifferential equations. The examples might be the wave equation, the heat equation, Laplace's equation, or others. Partial differential equations (PDEs) are important because problems in science often depend on more than 1 variable. For instance, lots of problems in science depend on the variables space and time. When this happens, scientists need to use PDEs instead of ordinary differential equations (ODEs). For example, we could use a PDE to understand how cocoa mixes in milk to give you hot chocolate or to understand how the weather changes.

Length: 1 hour

Prerequisites: Students should know the basics of calculus (how to do some derivatives and integrals) and the basics of ordinary differential equations. The cSplash class 'Ordinary Differential Equations' will provide enough background on ordinary differential equations.

Teacher info: Sam Stechmann, stechman at cims.nyu.edu, math graduate student

Information Theory (part 2)

See part 1 above for details.

Introduction to Abstract Algebra (part 2)

See part 1 above for details.

Period 5

Hands-on Sea Ice Modeling

We will have a look at how sea ice is currently being represented in global climate models.
After going quickly through the fundamentals of sea ice physics ("from Newton to Rubber bands"), we play around with an implementation of the equations in Matlab. You will change winds and ocean currents, spin the Earth or stop it, build islands and walls, change the ice strength etc. and will see how the modeled ice will react to our maltreatment.
On the side I will also point out some of the 'cheats' that modelers (have to) make these days and also problems associated with trying to represent reality in a computer.

Length: 1 hour

Prerequisites: Math knowledge: none needed really. Precalc?
Physics: You should have heard of Newton's law: F = ma
Computers: You should be able to use a simple text editor like 'notepad'.

Teacher info: Christof Konig, konigc at cims.nyu.edu, 5th year PhD student in "Atmosphere and Ocean Sciences and Mathematics"

Markov Chains with (interesting) applications

This course will present an introduction to the theory of Markov Chains, which are random processes whose evolution depends only on the current state of the process and not where the process was before. One way to think about it is a mouse that runs around the house, having forgotten where it has been and randomly choosing doors to go through. Students will first learn the basics of Markov Chains with finitely many possible states and then more advanced topics such as the long-term behavior. Students will then see interesting applications of Markov Chains such as why a mouse in a house with finitely many rooms will always find the cheese, why Google works so well, and how a monkey can recreate Shakespeare on a typewriter.

Length: 1 hour (probably)

Prerequisites: A basic knowledge of matrix multiplication will be helpful.

Teacher info: Yevgeny Vilensky, vilensky at cims.nyu.edu

Introduction to Number Theory

This course will explore unobvious properties and consequences of odd numbers, even numbers, and prime numbers. Applications to the solutions of problems, including but not limited to connections with irrational numbers and Pythagorean triples.

Length: 1 hour

Prerequisites: (teacher has not specified)

Teacher info: Robert Simione, (smart) undergraduate math major

Mathematics and Origami (part 2)

See part 1 above for details.

Plato, Fourier, and i

Learn some crazy stuff about complex numbers, Taylor series, and Fourier series.Complex numbers let you solve any polynomial, even ones that force you to take square roots of negative numbers. Taylor series give a nice perspective to functions, which lets us see some surprising connections. Finally, Fourier series let us analyze periodic functions in terms of frequencies, such as turning an airwave into a set of musical tones. If time permits, we'll also touch on how the Cauchy integral theorem reveals a deep connection between Taylor and Fourier series. A slinky will be used at some point in the talk.

Length: 1 hour

Prerequisites: basics of calculus, a little bit about sine and cosine

Teacher info: Tyler Neylon, neylon at cims.nyu.edu, 5th year math phD student

Introduction to Abstract Algebra (part 3)

See part 1 above for details.

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!