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Here is the list of courses to be offered at cSplash 2014, which will take place on April 26th.

2014 Schedule of Classes

Icons indicate difficulty rating, in ascending order. Description of the icon system.

       
Period 1

Mathematics of the Multiverse

The first part of the talk will be a general introduction to inflation, eternal inflation, and the multiverse. The second part will describe some of the mathematics behind the multiverse: exponential growth, the measure problem, binary trees, and conformally invariant stochastic models.

Prerequisites: I will structure the talk so that all but the last few minutes are accessible to high-school students. In the last few minutes I will connect what I have said to current research.

Teacher info: Matthew Kleban (kleban [at] nyu [dot] edu), Physics Department, Associate Professor

Class Size: 50 (enrolled: 27)


Introduction to Machine Learning

Machine learning is used all around us: in Microsoft's Kinect, Apple's Siri, Google's self-driving cars, Facebook's face recognition, Amazon.com's recommendations. Come and discover what it takes to create an algorithm that learns from data.

Prerequisites: No prerequisites.

Teacher info: David Sontag (dsontag [at] cs.nyu [dot] edu), Courant Computer Science, Professor

Class Size: 70 (enrolled: 34)


A probability paradox

Throw a stick at a circle. What is the length of the inscribed chord, on average? There are three ways to answer this (at least), and all of them give a *different* answer. Yet, we could also just do the experiment, and obtain only one answer. Or would we? In this class we'll talk about this paradox, do an experiment, and learn a little about the philosophy of probability theory.

Prerequisites: A bit of exposure to probability will help, but is not necessary. Enthusiasm to do an experiment is required.

Teacher info: Miranda Holmes-Cerfon (holmes [at] cims.nyu [dot] edu), Courant, Math, Assistant professor

Class Size: 15 (enrolled: 15)


The cSplash! Calculus Crash Course: An Introduction to Optimization

In this course, we will explore the basic yet ubiquitous idea of optimization. Starting with unconstrained problems in a single variable, we will discuss local and global optima before expanding into functions of two variables (twice the fun!). Then we will dive into the subtle area of constrained optimization, where we will introduce ideas including duality and sensitivity analysis. Finally, we will touch upon the numerical algorithms used for optimization in practice.

Prerequisites: familiarity with basic functions, algebra, and pre-calculus. calculus itself is not assumed and is not be required.

Teacher info: Scott Yang (yangs [at] cims.nyu [dot] edu), Courant Math, 2nd Year PhD

Class Size: 10 (enrolled: 9)


Git into GitHub

Knowing how to use Git is essential to being a modern software developer. Git is a form of version control, a way of managing changes to source code. GitHub hosts some of the largest open source projects in the world using Git -- from Linux to Rails to Bootstrap. In this talk, we will cover the basics of Git, GitHub, and why they should matter to you.

Prerequisites: Basic programming ability would be very helpful.

Teacher info: Tyler Palsulich (tpalsulich [at] nyu [dot] edu), Courant Computer Science, First Year Master's

Class Size: 35 (enrolled: 3)


Hey! Are We There Yet? Strategies for Finding the Most Efficient Route between Two Locations

How does Google Maps plan your road trip? The shortest distance between any two points is, theoretically, a straight line. In practice, however, we must carefully consider how roads are already laid out to account for rivers, mountains and other changes in terrain, like towns and cities. A number of well-established search algorithms are available to help evaluate the costs and benefits of choosing different possible paths. We will discuss how uninformed searches can be improved to assess alternative courses in addressing a route-finding problem, and the effectiveness of different algorithms will be compared. Additional applications ranging from robotics to biology will also be introduced.

Prerequisites: None.

Teacher info: Loretta Au (loretta.au [at] stonybrook [dot] edu), Stony Brook University, Department of Applied Mathematics and Statistics, 6th year PhD

Class Size: 50 (enrolled: 26)


Pearls in Graph Theory

Graph theory is an important branch of Mathematics today. Euler wrote a paper in 1736 on the Seven Bridges of Königsberg and the theory of Graphs was born. What is a graph? Why are they so important? We will use Euler's paper as a starting point and see what we can get from there. Topics should include(but are not limited to): definition of a graph and related terminologies, planar graphs, Kuratowski's theorem, Euler's formula, vertex coloring, six/five color theorems, and a brief discussion of the famous four color theorem. Note: the title is dedicated to N. Hartsfield and G. Ringel's book of the same name.

Prerequisites: We won't be using anything beyond high school algebra, but some mathematical maturity is required. Familiarity with proofs and mathematical induction is recommended. We will be moving rather quickly because we want to eventually discuss the four color theorem!

Teacher info: Junichi Koganemaru (jk3967 [at] nyu [dot] edu), Courant Math, 2nd Year Undergrad

Class Size: 35 (enrolled: 13)


Edge Detection: How Computers See

Edge detection is a basic yet fundamental problem in image processing and computer vision. In this course, we will talk about some basic tools and concepts in image processing, then use this foundation to study the Canny Edge Detection algorithm and some of its optimizations. It is my goal for everyone in the class to be understand the Canny edge detector well enough to implement it on their own, although we will not do this during the class. If time permits, we will also take a brief high level look at some more modern approaches to edge detection, especially those involving machine learning.

Prerequisites: If you know what a derivative is and have some programming knowledge (any language), you should understand everything in this course. Experience with MATLAB is a plus.

Teacher info: Vyassa Baratham (vyassa.baratham [at] stonybrook [dot] edu), Stony Brook University Physics, 2nd year undergrad

Class Size: 35 (enrolled: 11)


Period 2

Cryptography: Real World Magic

It may have started with sending secret instructions to Generals across enemy territory during the expansion of the Roman Empire, but Cryptography is now far more than just keeping sensitive messages from prying eyes. How do you play a multi-player version of Where is Waldo? How do you get Amazon servers to compute functions on your DNA without compromising your privacy? How do you ensure fairness in a network-play version of Rock-Paper-Scissors? Believe it or not, with cryptography.

Prerequisites: I'll try to make it as self-contained as possible - and make it more of a qualitative and anecdotal talk than a technical one.

Teacher info: Siddharth Krishna (siddharth [at] cs.nyu [dot] edu), Courant Computer Science, 1st Year PhD

Class Size: 35 (enrolled: 35)


The cSplash! Calculus Crash Course: Biology and Medicine

In this course, we will explore some fundamental concepts in single-variable calculus by considering real-life problems in biology and medicine. As we investigate these biological problems, we will introduce and analyze relevant topics in calculus and see how they can help us model and solve the problems.

Prerequisites: Familiarity with basic functions. Precalculus would be helpful. Knowledge of calculus is not required; we will build the necessary machinery together.

Teacher info: Olivia Chu (olivia.chu [at] nyu [dot] edu), Courant Math, 3rd Year Undergrad

Class Size: 10 (enrolled: 10)


Practical Tractor Beams

Beams of light can exert forces on illuminated objects. How to control those forces has been worked out only recently. After introducing the theory of photokinetic effects, we will put it into practice with a new experimental technique called holographic optical trapping, in which computer-generated holograms are used to exert total control over the dynamics of micrometer-scale objects. This interplay of theory and technology has yielded the first experimental demonstration of a knotted force field and, more recently, the first practical realization of a tractor beam, a propagation-invariant traveling wave that can transport illuminated objects back to its source.

Prerequisites: High-school algebra will be very helpful for understanding the theory. Any level of mathematical preparation is fine for appreciating the experiments, which are carried out with holograms and are visually appealing.

Teacher info: David Grier (david.grier [at] nyu [dot] edu), FAS Physics, Professor

Class Size: 50 (enrolled: 12)


MapReduce, The Big Data Workhorse

An Intel Core i7 980 XE processor can run 100 billion floating point operations every second. But some data processing jobs require astronomically huge computing resources, which require tasks to be distributed over several machines. Often, this means using an algorithm called MapReduce, which deals with the fact that two pieces of data sent to two different machines may, in fact, depend on each other. In this course, we will explore some basics of distributed computing, and then talk about the MapReduce algorithm conceptually, before seeing a basic example and discussing some practical aspects of the algorithm and its open source implementation, Hadoop, and Amazon's MapReduce service, EMR.

Prerequisites: Some programming knowledge (small amounts of pseudocode will be shown, so any language should be fine).

Teacher info: Vyassa Baratham (vyassa.baratham [at] stonybrook [dot] edu), Stony Brook Physics, 2nd year undergrad

Class Size: 35 (enrolled: 4)


The Fibonacci Numbers: Counting with Algebra

The Fibonacci numbers are a famous sequence of numbers where each number is given by the sum of the previous two numbers in the sequence. Because of this recursive definition, it is not clear how to calculate the 1000th Fibonacci number without first calculating the first 999. In this talk, we will use algebra to tackle this problem and learn more about the Fibonacci numbers. We will also explain a neat trick that lets one convert miles to kilometers using only the Fibonacci numbers.

Prerequisites: Basic algebra and manipulating algebraic expressions. Knowing how to sum an infinite geometric series is a bonus, but is not required and will be explained in the talk!

Teacher info: Mihai Nica (nica [at] cims.nyu [dot] edu), Courant Math, 3rd year PhD

Class Size: 70 (enrolled: 21)


Information Theory in Biology

Time is money, but information is priceless! Or is it? In this course, we'll discuss quantitative measures of information and mutual information, discovered by Claude Shannon, and consider several examples to build intuition. Information theory clarifies what it means to communicate efficiently and relates to biology via DNA, which contains instructions for assembling the proteins required to sustain life. We will compare the efficiencies of DNA coding for different organisms, and also apply information theory to study transcription factor binding, which is thought to explain the majority of variations between species. As we know from everyday experience, messages can become distorted as they travel from sender to recipient, and we will consider such noisy channels as well. Time permitting, fundamental connections with the concept of entropy in physics will be explored. Though widely accessible, this course will require deep thinking. Welcome to the world of information!

Prerequisites: Familiarity with logarithms and basic probability

Teacher info: Thomas Fai (tfai [at] cims.nyu [dot] edu), Courant Math, 6th Year PhD

Class Size: 50 (enrolled: 23)


Quantum Information: The Surprisingly Simple Math of the Impossible

New quantum technologies are bringing us face to face with seemingly impossible realizations about the world. This talk will include some examples of real-life paradoxes that happen in the lab all the time, as well as one or two of the (simplest) algorithms that are in use in quantum technologies today and in the near future.

Prerequisites: algebra, basic probability

Teacher info: Seth Cottrell (cottrell [at] cims.nyu [dot] edu), Courant, venerable PhD student

Class Size: 35 (enrolled: 30)


Period 3a

What gives the sky its colors

I will derive and solve the equations pertaining to driven simple harmonic oscillators and resonance, and demonstrate using the example of a pendulum. I will follow with a simple description of the phenomenology of electromagnetic radiation, connect this description to resonant systems, and derive a simple model for Rayleigh scattering. Rayleigh scattering explains the color of the sky.

Prerequisites: single variable calculus, basic knowledge of E&M.

Teacher info: Arya Tafvizi (arya [at] math.nyu [dot] edu), Courant Math, 1st year PhD

Class Size: 50 (enrolled: 9)


Programming in an Eggshell

In this lecture, you will learn a simple puzzle game called Alligators and Eggs. What does this have to do with programming, you might ask? Quite a lot! Surprisingly, every computer program can be translated into this very simple game. To understand how this works, we will study Church encodings (no praying involved - I promise) and fixpoint combinators. Along the way, we will touch on fundamental questions in computer science such as "What is computable?". I will end by giving you a glimpse of a real programming language. You will see that programming is actually like playing alligators and eggs in disguise.

Prerequisites: basic algebra; induction

Teacher info: Thomas Wies (wies [at] cs.nyu [dot] edu), Courant Computer Science, Assistant Professor

Class Size: 70 (enrolled: 16)


All kinds of pi

You've learned by now that the length of the circumference of a circle is 2 pi times the radius. You've also learned that pi is some fixed constant, a fundamental property of the university, and certainly never changing. But is it? In this talk we will see how the traditional value of pi=3.14... is intricately linked with how we measure distance; and that once we decide to measure distance differently, the idea of pi as a static constant disappears.

Prerequisites: High-school geometry and trigonometry.

Teacher info: James Fennell (fennell [at] cims.nyu [dot] edu), Courant Mathematics, 1st year PhD

Class Size: 35 (enrolled: 3)


Projective Geometry

The projective plane is an amazing place. Every pair of lines meets in a point. Circles, ellipses, parabolas, and hyperbolas are all the same shape; and all quadrilaterals are the same shape. You can turn all the points into lines and all the lines into points, and nobody will notice. The price you pay is that you can't compare two distances or two angles, and, if you have three points on a line, you can't say which one is in the middle. This lecture will give an introduction to projective geometry, one of the most important examples of a non-Euclidean geometry. We will explain how projective geometry relates to perspective in drawing and images and how you can use projective geometry to prove the astonishing and important Pappus' theorem. Pappus' theorem: Draw two lines L and L'. Choose three points A, B, C on L and D, E, F on L'. Let X be the intersection of AE with BD; let Y be the intersection of AF with CD; and let Z be the intersection of BF with CE. Then X, Y, and Z are collinear.

Prerequisites: Geometry and algebra

Teacher info: Ernest Davis (davise [at] cs.nyu [dot] edu), Courant Computer Science, Professor

Class Size: 50 (enrolled: 12)


Randomness and Computation

It turns out that if we allow computers to have access to random numbers, we can come up with faster and more elegant algorithms for some problems. However, there are some algorithms for which we can eliminate the need for access to random numbers. In this class we will see a couple of my favorite results that explore the relationship between randomness and computation.

Prerequisites: Basic probability. Experience with algorithms, graph theory, and modular arithmetic would be helpful, but not necessary.

Teacher info: Shravas Rao (rao [at] cs.nyu [dot] edu), Courant, Computer Science, 1st Year PhD

Class Size: 35 (enrolled: 9)


Period 3b

Vision by Humans and by Machines

We will introduce Vision, the amazing abilities needed to process images. We will analyze a few optical illusions. We will discuss why mathematics and computers are need to model it. Then, we will illustrate such modeling with an introduction to Haar Wavelets, a multiscale representation of images (and the base of JPEG 2000 compression.) We will show the link of wavelets to neuroscience.

Prerequisites: basic algebra

Teacher info: Davi Geiger (geiger [at] cims.nyu [dot] edu), Computer Science and Neural Science, professor

Class Size: 50 (enrolled: 15)


The Mathematics of Pleasure and Reward

I will discuss how neurons in the dopaminergic system in the brain fire as a function of presentation of unexpected reward. A relatively simple equation can capture the properties of the firing of these dopaminergic cells with profound accuracy. The insights gleaned from this simple relationship can describe how we learn associations between stimuli and reward and how we reinforce behaviors that are pleasurable to us.

Prerequisites: Basic algebra

Teacher info: Stephen Keeley (stephenlkeeley [at] gmail [dot] com), Computational Biology at Courant, Home Dept: Center for Neural Science , 3rd Year PhD Student

Class Size: 70 (enrolled: 18)


Electronic Privacy and Security: Introductory Cryptography

In a world where the scope of electronic surveillance is becoming more known, questions about just what security is and how it works are becoming increasingly prevalent. In this talk, we will discuss the basic tenants of cryptography and what it means for something to be “secret” or “secure”. We will look at some of the cryptographic methods used from ancient times up to the present and we will consider various cryptographic attacks.

Prerequisites: Standard algebra is necessary. Basic knowledge of probability and binary are highly suggested.

Teacher info: George Wong (gwong [at] nyu [dot] edu), Courant Mathematics, 3rd Year Undergraduate

Class Size: 35 (enrolled: 6)


Your Brain on Bayes: Understanding How Your Brain Uses Bayesian Inference to Influence Perception and Guide Action

Bayesian Inference is a statistical method that optimally combines prior knowledge with current evidence. Behavioral studies have shown that many human actions can be fit with Bayesian models. Recent work suggests that neurons in the brain can compute the components of these Bayesian models, and thus provide the mechanism for Bayesian behavior. In this class, we will go over basic notions of probability and how probability distributions can be updated by Bayes' theorem. Then, we will look at how Bayesian inference can explain several behavioral phenomenon (e.g., optical illusions).

Prerequisites: Knowledge of basic probability (probability spaces, distribution functions, etc.).

Teacher info: Ryan Shewcraft (ryan.shewcraft [at] nyu [dot] edu), Center for Neural Science, 3rd Year PhD

Class Size: 35 (enrolled: 14)


This Machine Kills Cancer

Cancer is a terrible disease in which the cells of your body rebel against their normal function and multiply out of control. Despite decades of research on cancer treatments, most cancers are still treated with medieval methods: cutting and poison. There has been a promising new trend toward analyzing each cancer's DNA in order to make personalized treatments for every patient. How do you sequence DNA? How do you look at DNA on a computer? What mathematical models help you build a personalized cancer treatment from a DNA sequence? All these questions, and more, will be answered!

Prerequisites: Biology (should know what DNA is) A little bit of experience with programming

Teacher info: Alex Rubinsteyn (alex.rubinsteyn [at] gmail [dot] com), Mount Sinai School of Medicine, Bioinformatics Researcher (just finished my CS PhD @ NYU)

Class Size: 50 (enrolled: 20)


Period 4

The M Stands for Magic

The M in mathematics stands for magic. Join us for a brief survey of the most fantastic results in mathematics. From Euler's formula to Euclid's proof, to the mathematical algorithms that hold internet banking together, math can go from the theoretically beautiful to the surprisingly practical. We will go through results in geometry, calculus, and number theory. And if time allows, cryptography and group theory.

Prerequisites: Some familiarity in calculus and trigonometry would be helpful.

Teacher info: Imran Qureshi (iq214 [at] nyu [dot] edu), Courant Math, Bsc. Alumni

Class Size: 35 (enrolled: 25)


Killing the Hydra

In Greek mythology, Hercules was ordered to kill the Lernaean Hydra as a part of his Twelve Labors. Many years have passed since then, and the modern-day Hydra that we now face has evolved to be much more vicious. In this mini-course, we will model the kill-the-Hydra game by a simple numeric problem. We will then tackle the problem via the technique of transfinite induction, which will produce unexpected conclusions.

Prerequisites: It helps to know what it means to compute the limit of a sequence, but this isn't strictly necessary.

Teacher info: Mark Kim (markhkim [at] math.nyu [dot] edu), Courant Math, 2nd Year PhD

Class Size: 70 (enrolled: 14)


Hyperplanes!

Hyperplanes, unfortunately, don't describe aircraft that are super excited and giddy. But they do describe spaces that are of the utmost importance in linear algebra, multivariable calculus, and even convex analysis. In this talk, I will develop an elementary introduction to the geometry of hyperplanes and explain some different types (e.g. supporting, separating). Then I'll jump into a description of the applications, which span from linear approximations and tangent hyperplanes to describing convex sets.

Prerequisites: Basic geometry, basic set theory, some exposure to calculus could be helpful

Teacher info: Mohammad Manzari (mtm430 [at] cims.nyu [dot] edu), Courant Math, 1st Year Masters

Class Size: 35 (enrolled: 3)


Paradoxes of the Continuum

The real numbers is the main example of a continuum. Intuitively, it is a continuous medium with no gaps any where. For instance, the integers clearly has big gaps, but even the rational numbers has many gaps (like square root of 2). In Calculus, you learn about continuous functions: this is defined using Cauchy's idea of epsilon-delta continuity. But the concept of a continuum is rife with paradoxes. Does Cauchy's idea capture all the properties of the real continuum? It turns out that we can introduce new kinds of numbers (infinitesimal) into the real line! We live in the space-time continuum. So is the physical world a continuum? Newtonian physics assumes so, but quantum physics denies it. Continuum mechanics is a field that studies the ``mechanical properties'' of fluids and continuous material. But surely, the name is an oxymoron? Come join us as we explore this paradox.

Prerequisites: A bit of calculus.

Teacher info: Chee Yap (yap [at] cs.nyu [dot] edu), Courant Computer Sciene Dept, Professor

Class Size: 50 (enrolled: 8)


Hilbert's Hotel

The story of Hilbert's hotel, a hotel with an infinite number of rooms, reveals some unintuitive facts about infinity. The hotel is full, and yet it has vacancies. We'll explore the example of Hilbert's hotel and some of the mathematical ideas behind it. We'll discuss the notion of countably infinite, and learn how to determine if an infinite set is countable. Finally, we'll see an example of an infinite set that is not countable, and if time allows we will discuss how to prove it is not countable.

Prerequisites: Familiarity with sets will be helpful.

Teacher info: Steven Delong (delong [at] cims.nyu [dot] edu), Courant Math, Fourth-year PhD

Class Size: 35 (enrolled: 11)


Large-Scale Structure of the Universe

Over the last few decades cosmology has developed a very detailed picture of how the large-scale structure we see in the universe (galaxies, clusters of galaxies, and so on) arose from very small ripples in the distribution of matter and energy at very early times (as early as 10^-34 seconds after the Big Bang). I will discuss how small fluctuations in the early universe are generated and then amplified as they evolve until the present time and how cosmologists tests these ideas with telescopes and satellite observations.

Prerequisites: calculus for the most technical bit, but for the most part algebra and trigonometry is enough.

Teacher info: Roman Scoccimarro (rs123 [at] nyu [dot] edu), Center for Cosmology and Particle Physics, NYU Physics, Professor

Class Size: 50 (enrolled: 11)


The Halting Problem, Incompleteness, and the Limits of Mathematics

Anyone who's spent much time programming knows that it's surprisingly easy (and very frustrating!) to accidentally write programs that never stop running. To save ourselves some time and embarrassment, we'll ask a very simple question: Given some computer program, how can we tell whether it will run forever? Unfortunately, this basic question will prove to be maddeningly difficult to answer. We'll quickly find that computer programs can go into infinite loops in dizzyingly complex ways. Without intending to do so, we'll somehow find ourselves talking about some seemingly unrelated problems in mathematics that have stumped mathematicians for centuries. We'll actually learn that we were trying to solve them without realizing it. Oops... This will lead us, by way of Alan Turing's famous Halting Problem, to a simple proof of one of the most profound and humbling results in the philosophy of mathematics: Kurt Godel's First Incompleteness Theorem. Namely, we will show that, in a sense, NO formal system can encompass all of mathematics. We'll ponder the philosophical meaning of this statement and, if we have time, conclude by considering the statement "Nothing that is interesting can be defined unambiguously."

Prerequisites: Students should have a very basic understanding of computer programming: If you know what an If statement is, then you're fine. Other than that, this talk will be completely self-contained. However, it will be advanced and move relatively quickly, so students should be very comfortable with mathematics in general. Students who have already heard about the halting problem or Godel's Incompleteness Theorem(s) are encouraged to attend, as this presentation will likely be quite different from what they've seen.

Teacher info: Noah Stephens-Davidowitz (noahsd [at] gmail [dot] com), Courant Computer Science, 2nd Year PhD Student

Class Size: 35 (enrolled: 17)


Introduction to Quantum Mechanics

Although quantum mechanics is fundamental to our understanding of the universe, conceptually, it is entirely foreign to our everyday experience. We will use mathematics to develop some of the framework for quantum mechanics (in one dimension). Along the way, we will consider and discuss particle-wave duality, wavefunctions, the Schrödinger Equation, the square well and harmonic oscillator potentials, and quantum tunneling. Questions are welcome and expected and will hopefully play an important part in the discussion.

Prerequisites: Knowledge of graphs and certain types of functions (trigonometric and piecewise) is necessary. A general understanding of linear algebra, complex numbers, and calculus (differentiation / integration in one variable) is highly suggested but not necessary. We will functionally develop necessary concepts over the course of the lecture.

Teacher info: George Wong (gwong [at] nyu [dot] edu), NYU Physics, 3rd Year Undergraduate

Class Size: 35 (enrolled: 18)


Period 5

The Hardware of a Computer

Ever wonder how a computer works? This talk will cover the basics of the internal structure and design of a computer. It will cover the basics including CPUs, the memory hierarchy, pipelining and multicore processors.

Prerequisites: Curiosity

Teacher info: Mike Jaoudi (mjaoudi [at] nyu [dot] edu), Courant Computer Science, 3rd Year Undergrad

Class Size: 35 (enrolled: 35)


Nuclear fusion: the fuel of stars, and energy for the future?

Using elementary math tools, we will look at the physics of nuclear fusion. We will explain how the Sun has been able to shine and heat us for so long, and talk about the progress we are making in bringing nuclear fusion on Earth for a carbon-free, quasi-infinite energy source. In this class, you will see a few cool formulas, curves, and pictures, and no long boring derivations. We will also make sure to keep plenty of time for all the questions you want to ask!

Prerequisites: Elementary algebra and the ability to read/interpret graphs!

Teacher info: Antoine Cerfon (cerfon [at] cims.nyu [dot] edu), Courant Math, Assistant Professor of Mathematics

Class Size: 50 (enrolled: 28)


The cSplash! Calculus Crash Course: Length, Area, and Volume

A basic question in mathematics is to compute the length of a curve. In this course, we will learn how to compute the lengths, areas, and volumes of various objects by the ideas of calculus. Some examples will include a surface with infinite area but which encloses a finite volume, along with a formula for pi involving only the number 2.

Prerequisites: knowledge of calculus is not required

Teacher info: Ian Tobasco (tobasco [at] cims.nyu [dot] edu), Courant Math, 3rd Year PhD

Class Size: 35 (enrolled: 25)


Puzzles and Invention

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. Sometimes entire new fields ensue. We present and challenge students to solve several such puzzles having inspired advances in cryptography, graph theory, and even diplomacy.

Prerequisites: Basic algebra

Teacher info: Dennis Shasha (shasha [at] cims.nyu [dot] edu), Courant Computer Science, Prof

Class Size: 70 (enrolled: 15)


Coloring planar graphs

The four-color theorem states that any map in a plane can be colored using four colors in such a way that no regions sharing a common boundary have the same color. This theorem is very well known because it was the first major result to require a computer program to solve it, and there was controversy in the mathematical community on whether such a proof should be accepted. In this talk we will discover that with a little bit of combinatorics, we can prove the six-color theorem. In the process we will derive the beautiful Euler's formula. If there is enough time, we will also prove the five-color theorem as a bonus.

Prerequisites: Students should be comfortable with combinatorial reasoning and also should know how to color!

Teacher info: Igor Balla (balla [at] cims.nyu [dot] edu), Courant Math, 2nd Year PhD

Class Size: 35 (enrolled: 2)


The 15-puzzle, bubble sort, and symmetric groups

The 15-puzzle is a famous 4-by-4 sliding puzzle from the 19th century, consisting of 15 numbered tiles in a 4-by-4 grid. In this mini-course, we will tackle Sam Loyd's one-thousand-dollar problem of swapping 14 and 15 while keeping all the other tiles in place. Along the way, we will devise a simple sorting algorithm and use it to study an abstract mathematical object called the symmetric group. This mini-course will provide you with sufficient mathematical background to study the mathematics of Rubik's cube, which we unfortunately will not have time to discuss.

Prerequisites: Some familiarity with mathematical proofs. For example, a course in Euclidean geometry is sufficient (though we won't use any geometric reasoning).

Teacher info: Mark Kim (markhkim [at] math.nyu [dot] edu), Courant Math, 2nd Year PhD

Class Size: 35 (enrolled: 14)


An easy introduction to difficult probability

This class will introduce the concepts of probability limit theorems and their applications through an interesting example. We will analyze a random experiment that can estimate the value of pi to any desired accuracy! In order to analyze this convergence result I will introduce powerful probability tools such as Hoeffding's concentration inequality as well as the law of iterated logarithm and the central limit theorem.

Prerequisites: Multivariable calculus, basic notions of probability such as independence, joint probability distributions, expectation ...

Teacher info: Andres Munoz Medina (amunoz88 [at] gmail [dot] com), Courant Math, 4th year PhD

Class Size: 50 (enrolled: 13)


About the difficulty icons:

We have developed 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 somewhere-in-between points 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!