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2013 Class Schedule

Link to Lecture Notes

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Colloquium Speaker

Erdős Magic - Joel Spencer, Courant Institute

Teacher info: Joel Spencer is a professor of mathematics and computer science at the Courant Institute. Professor Spencer has been a member of the Courant family for over 20 years and was a chair of an American Mathematical Society committee that gives grants to High School Math Camps. As such, he loves cSplash: so much so that he agreed to come and give a talk on his birthday! Did I mention that he was once the winner of the Putnam competition, the college equivalent of USAMO? That he has an Erdős number of 1? --Mark Kim

Abstract: Paul Erdos was a giant of twentieth century mathematics. We celebrate the centennial of his birth this year. Among his many contributions was Erdos Magic. This technique allows us to prove that something (perhaps a graph or a coloring or most anything) exists without actually exhibiting it. The notion is to envision a random algorithm whose goal is to create the object. If one can show that the random algorithm has a positive probability of succeeding then success absolutely must exist. Paul Erdos had a unique personality and spirit and the speaker, who worked with Erdos for many decades, will sprinkle the talk with anecdotes about him.

Notes: courant13.pdf

Period 1

Bitcoin: The math in the money

Bitcoin is a new experimental digital currency that is rapidly gaining popularity. Bitcoins can be transferred between users worldwide over the internet using a computer (or smart phone) without any intermediary financial institution (such as your bank, Visa etc) - and with no or tiny fees. You can use it to buy things online as well as in physical stores, some of which are located on Manhattan. Bitcoin is a so-called peer-to-peer currency which requires no central bank to issue money or ensure the trust in the system. Instead, the security of the system relies on math in the form of cryptography. In one year the dollar value of a bit coin has risen from about 10$/bitcoin to 200$/bitcoin. This, combined with the fact that no one knows the true identity of the inventor Satoshi Nakamoto, has led some to speculate that bitcoins is some kind of scam or a new type of Ponzi scheme. In this class, we will go behind the buzz and take a look at how the bitcoin protocol works. The class will also include some practical demonstration of bitcoin transfers.

Length: 1 Hour

Prerequisites: A curious mind and a willingness to exercise the brain cells.

Teacher info: Jens Jørgensen (jcbjorgensen [at] gmail [dot] com), Courant Math, 5th year

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 some...unexpected conclusions.

Length: 1 Hour

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, First-year Ph.D.

Surprising Probabilities

Problems involving randomness can lead to situations that seem counter intuitive. One thing that is often surprising is how probabilities can change when you are given some extra information. In this talk we will develop the tools we need to handle this and we will look at some examples where the results are surprising. The Monty Hall Problem is a famous example of the type of problem we will look at.

Length: 1 Hour

Prerequisites: Very basic knowledge of probability. If you have seen the formula: "Probability = # of favorable outcome divided by total # of outcomes", then you are all set! (I plan on reviewing this during the talk)

Teacher info: Mihai Nica (nica [at] cims.nyu [dot] edu), Courant Math, 2nd Year PhD

Puzzles that Feel Magical

Sometimes puzzles are so surprising that they appear to be magical. I will pose and, if asked, solve a few. Then there are those for which I know of no solution...

Length: 1 Hour

Prerequisites: High school

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

Wireless Communications - why your cellphone works (and doesn't)

Ever wonder how your cellphone works, and why you are able to text and call? Do you contemplate the future of wireless technology, as the computer and cellphone continue to morph into personal devices that will continue to impact your life? This course introduces the field of wireless communciations, and shows how the vast networks of cell towers and femtocells provide you with wireless connectivity. We mathematically derive how radio signals fade, thus causing service outages, and how engineers are solving these problems with technology.

Length: 1 Hour

Prerequisites: Some trigonometry and geometry is helpful. Physics also useful.

Teacher info: Ted Rappaport (ted.rappaport [at] nyu [dot] edu), NYU-Poly Electrical Engineering; Courant Computer Science, Lee/Weber Chaired Professor

Law of Cosines

What we can do when we are in the top of mountains and try to see where the fire is? Using Law of Cosines, we will focus on real-life applications of law of cosines.

Length: 1 Hour

Prerequisites: Basic fact of trigonometry

Teacher info: Jong Woo (John) Yoon (jyoon0529 [at] gmail [dot] com), Courant Math, 4th year undergrad

Modeling an Epidemic

In this talk, we will walk through how to model a disease infecting a population. We will start with a basic set of assumptions about the disease (What does it mean to be sick? To get better? How quickly do people get infected?) then build a working MATLAB model. We will see if flu shots actually make a difference. If so, how many people should get one? We will also discuss what makes a "good" epidemic – it shouldn't be too easy or too hard to get someone sick.

Length: 1 Hour

Prerequisites: Familiarity with calculus would be helpful, but it's not absolutely necessary. We will be using differential equations, but not actually solving them.

Teacher info: Tyler Palsulich (tpalsulich [at] nyu [dot] edu), Courant Math and Computer Science, 2nd Year Undergrad

Software Synthesis

Back in the old days an educated person was expected to speak Latin. In the recent short film "What Most Schools Don't Teach" Bill Gates, Mark Zuckerberg, will.i.am and others called coding "the Latin of the 21st century". Writing programs can be fun and an exciting experience, but sometimes it also involves routine tasks that are a burden for programmers. Let us ask a higher level question: can we teach computers to write programs? Can we tell a computer WHAT needs to be done without instructing HOW it should be done. This class will introduce software synthesis, a technique for automatically generating code. To understand better how software synthesis works, you will get a glimpse into the world of automated theorem proving, where mathematical theorems are proved by computers.

Length: 1 Hour

Prerequisites: basic algebra

Teacher info: Ruzica Piskac (piskac [at] mpi-sws [dot] org), Max Planck Institute for Software Systems, Germany, Faculty (academic visitor at Courant Computer Science)

Schrodinger's Cat in Equations; the mathematics of Quantum Mechanics

This talk will cover introductory quantum physics by solving Schrodinger's wave equation using operators and calculus. We will also derive Heisenberg's Uncertainty Principle and quantum tunneling. Bring a question about physics (classical or quantum) or astrophysics to class!

Length: 1 Hour

Prerequisites: Advanced and differentiable calculus, operator algebra, and trigonometry. We will develop an understanding of eigenfunctions and eigenvalues. An open mind!

Teacher info: Gladys Velez Caicedo (gbv2105 [at] columbia [dot] edu), Columbia University, Department of Astronomy & Astrophysics, 2nd Year Undergraduate

Probability and the Birthday problem

Imagine that you are sitting in a class with 22 other students. Guess the probability of at least two of you having the same birthday. When the number of student becomes 57, the probability reaches an even higher number that you would not believe! This course will take you to explore the world of probability and how it can be applied to many interesting real life problems such as the Birthday problem. After the class, students will understand more deeply that in the universe of probability, they should always rely on numbers and actual calculations instead of intuitions.

Length: 1 Hour

Prerequisites: Basic probability knowledge

Teacher info: Mengmei Chen (mc4522 [at] nyu [dot] edu), Courant Math, 1st Year Undergrad

Period 2

A Taste of Celestial Navigation

It is fascinating that by looking at the sky, you can figure out your latitude and longitude on Earth. That’s the task of celestial navigation, used for centuries to navigate the open oceans. It relies on spherical trigonometry, a beautiful branch of mathematics, which was important throughout history until the advent of GPS hid it from public view. In this brief tour, we will talk about finding one’s position on Earth by observing the Sun and the stars. We’ll try our hand at using a sextant, learn about distances and coordinates on a sphere, and discuss the mathematics of converting a measurement into a location. Some fun problems and puzzles related to the mathematics of the Earth will be provided.

Length: 1 Hour

Prerequisites: Geometry; trigonometry helpful but not required.

Teacher info: Dmitry Sagalovskiy (dsagal [at] nymathcircle [dot] org), New York Math Circle

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!

Length: 1 Hour

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

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

Machine Learning

Machine learning lies at the heart of many of the latest technologies: Microsoft's Kinect, Apple's Siri, Google's self-driving cars, face recognition, Amazon.com's recommendations. Come and discover what it takes to create an algorithm that learns from data.

Length: 1 Hour

Prerequisites: None

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

Method of Proving Trigonometry Identities

There are so many trigonometry identities that students can prove. I will first teach the method of proving trigonometry identities. Then, I will introduce basic trigonometry concepts and show the example. Then, I will tell student to try one and present on the board. Then, I will give practice problems.

Length: 1 Hour

Prerequisites: Trigonometry

Teacher info: Jong Woo (John) Yoon (jyoon0529 [at] gmail [dot] com), Courant Math, 4th year

Quantum Cryptography: The Physics of Keeping Secrets

Our understanding of quantum mechanics now makes it possible to create codes which achieve the ultimate level of security: breaking them is literally against the laws of physics. This course will give a very brief overview of traditional cryptography and the relevant concepts from quantum mechanics, and then describe the unique way these ideas can be combined and exploited to keep information secret and secure. We will culminate with the presentation of an encryption scheme currently being adopted by corporations and organizations with the highest level of security needs, as well as a conceptually satisfying proof of its ultimate security.

Length: 1 Hour

Prerequisites: Basic probability is all that is required. Matrix and vector operations are a plus, as is some basic exposure to quantum mechanics, but the key ideas of both will be reviewed as part of the natural progression of the material.

Teacher info: Colin G. West (colin.west [at] stonybrook [dot] edu), Stony Brook University, Dept. of Physics and Astronomy and Yang Institute for Theoretical Physics, 3rd year PhD

Merger Math: The Numbers Behind Corporate Acquisitions

We will cover the math behind mergers and acquisitions, including concepts such as discounted cash flows, relative valuation, synergy analysis, and accretion / dilution.

Length: 1 Hour

Prerequisites: Algebra

Teacher info: Annie Wei (annie.wei [at] carlyle [dot] com), University of Pennsylvania - Finance, Alumna

Paradoxes of the Continuum

The continuum refers to any space that is continuous. Like the Real Numbers. But does Cauchy's epsilon-delta characterization cover all of continuum? There is more than meets the eye! Continuum mechanics is the study of fluids and material. Fluids and material are atomistic in nature. Surely "continuum" is an oxymoron?

Length: 1 Hour

Prerequisites: calculus

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

Robotic Navigators and Cartographers

What do Phoenician merchants, Claudius Ptolemy, Christopher Columbus, certain brands of vacuum cleaners and the self-driving Google car have in common? All need to navigate under uncertainty or to map the unknown. They guess their next location and correct their predictions thanks to (always noisy) observations of some kind, be it from the North Star, Jupiter's satellites, a compass, a Kinect camera or a laser LIDAR system. They define coordinate systems and piece together small maps. We will see how robots can do this.

Length: 1 Hour

Prerequisites: Trigonometry, probabilities.

Teacher info: Piotr Mirowski (piotr.mirowski [at] computer [dot] org), ex-Courant CS (now at Bell Labs), Research Scientist

Water waves

Consider the following thought experiment. Suppose you have an ocean that is at rest, and then you perturb it just a little (imagine putting your foot in and stirring it up a bit). What happens? Do the ripples you just created die off, or do they persist forever? If you don't kick too hard, does the surface of the ocean remain smooth, or does it create waves that overturn? The answer to these questions is, more-or-less, that we don't know. This is strange. The modern theory of fluids dates back to the mid-18th century, and water has always been a subject of intense interest. Certainly it's reasonable to expect that we know all there is to know about water. Unfortunately, that's not even remotely true. While we understand a lot, many, many fundamental questions about how water behaves (like the one above) remain unanswered. Starting from the absolute basics, in this course we will look at the equations that govern water waves, and try to get a sense of what types of problems contemporary mathematicians and physicist are trying to answer.

Length: 1 Hour

Prerequisites: Students should be familiar with calculus and basic physics

Teacher info: Samuel Walsh (walsh [at] cims.nyu [dot] edu), Courant Math, Courant Instructor

Period 3a

The Whispers of Stars: Information from Stellar Objects

Stars are more than pretty objects in the sky-- they reveal invaluable information to scientists. Redshifts in stars uncover the true age of the universe. Binary systems are used to identify elusive black holes. And formations of galaxies indicate the behavior of the early universe. In this course we will explore the various techniques used to extract as much information from stars as possible. We will explore locations of super massive black holes, the looming star Nemesis that caused the extinction of the dinosaurs, and how every element around us came into being.

Length: 1 Hour

Prerequisites: It is recommended students have taken a prior physics course dealing in mechanics and basic Newtonian physics. Students should also have taken a course in chemistry (should be familiar with subatomic particles).

Teacher info: Isabel Baransky (isabelbaransky [at] yahoo [dot] com), Columbia University Applied Physics, 2nd Year Sophomore, Undergraduate

Programming with Alligators

What have alligators and their offspring to do with programming? We will start by learning an idealized programming language called Alligators and Eggs. Surprisingly, every computer program can be translated into this very simple language. To prove this, we will study Church encodings and fixpoint combinators. Along the way, we will touch on fundamental questions in computer science such as "What is computable?". I will conclude by giving you a glimpse of a real programming language. You will observe that many modern programming languages are actually Alligators and Eggs in disguise.

Length: 1 Hour

Prerequisites: Basic algebra, recursive equations

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

Math from the Movies

Are you a movie buff who is also a budding mathematician? Did you watch "Mean Girls" and wonder if that limit really DOESN'T exist? Would you like to be smarter than the MIT professors in "Good Will Hunting"? In this class, we'll be taking a look at a handful of math problems and concepts that are presented in movies to not only understand what they were really talking about, but also to get a feel of different areas of maths.

Length: 1 Hour

Prerequisites: Basic Calculus is a plus--taking limits, taking derivatives. We will also be working with infinite series and products. Apart from this, anyone willing to learn and participate is welcome!

Teacher info: Sam Jeralds (sjj280 [at] nyu [dot] edu), Courant Math, 2nd Year Undergrad

Ergodic Theory and Applications

Ergodic theory originated as a mathematical method to provide rigorous justification for the thermodynamical features of very large systems of molecules. The implications of this theory are surprisingly varied, having implications ranging from physics to the decimal expansions of real numbers! Using two key results, the Poincare Recurrence Theorem and the Birkhoff Ergodic Theorem, we'll explore three implications: the (apparent) breakdown of the second law of thermodynamics, a result on the decimal expansions of real numbers (i.e. how frequently does a 7 appear in the decimal expansion of "most" real numbers?), and Bedford's Law (a method in statistics to find fraud in large sets of data) applied to the first digits of the powers of 2.

Length: 1 Hour

Prerequisites: experience dealing with probability on finite event spaces.

Teacher info: Alex Blumenthal (alex.blumenthal [at] gmail [dot] com), Courant Math, 2nd Year PhD

Two Approaches to an Olympiad Problem

Along the way, participants will learn a few techniques for summing a series and a proof of a famous combinatoric identity, and will get a glimpse of mathematical elegance.

Length: 1 Hour

Prerequisites: Higher level problem solving.

Teacher info: David Hankin (hankin207 [at] aol [dot] com), New York Math Circle, Senior Instructor

Period 3b

The World on the Back of an Envelope: How to Solve Any Problem Without Google or Calculators and While Standing on One Foot

Of course the title is a bit of an exaggeration. But not completely. Enrico Fermi was famous for asking Ph.D. students during their final orals at the University of Chicago, "How many piano tuners are there in Chicago?" The idea is to develop the skill of approaching a problem out of left field with a certain amount of fearlessness and creativity, learning to make reasonable estimates, and gaining a handle on the solution quickly and with minimal effort. This is not only important and useful in your own work, but also enables you to answer anyone who comes up to you and asks you how many atoms that once resided in the body of Julius Caesar you take in with each breath. (Useful for betting also.)

Length: 1 Hour

Prerequisites: High school math is sufficient

Teacher info: Daniel Stein (daniel.stein [at] nyu [dot] edu), Courant Math/FAS Physics, Professor

Mathematics as a Language for Music

When people describe music, they often use a very traditional language and notation. They assign letters to the pitch of sound, sometimes with an additional sign (e.g. "A#"), they talk about chords, modes and the key of a musical piece... It usually takes a while to get used to this language, and even then it can be hard to explain what the concepts actually mean. One of the major drawbacks of the traditional language is that it hides the underlying structure of the music. We will talk about using a more mathematical language, so that musical patterns become more clearly visible. This not only helps with understanding music (theory) better, but also with playing music, improvising and playing by ear.

Length: 1 Hour

Prerequisites: Just some basic algebra is all you will need to understand the talk. In particular, knowledge about music theory is absolutely not required.

Teacher info: Jim Portegies (jim [at] cims.nyu [dot] edu), Courant Math, 4th Year PhD

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.

Length: 1 Hour

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

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

Mathematics of Evolution

Does evolution reward generosity, or does it pay to cheat? Nature is full of examples that might lead you to either conclusion. For instance, worker bees are prepared to sacrifice themselves for the sake of the hive and the queen bee. On the other hand, some cuckoos notoriously lay their eggs in other species' nests at the expense of their unsuspecting hosts. Although these two examples may appear unrelated, both cases share a type of evolutionary stability that we will seek to understand using mathematics. Connections will be drawn with concepts from game theory, such as the famous prisoner's dilemma, and from modeling with differential equations, such as phase plane analysis. At the core of this discussion is that the behavior of all living things depends on the code contained in their DNA. This course is inspired by Richard Dawkins' seminal book "The Selfish Gene."

Length: 1 Hour

Prerequisites: You should have some experience with derivatives, but prior knowledge of differential equations is not necessary. Those with a basic understanding of DNA and meiosis (it's only a few wikipedia pages away!) may find the course more enjoyable.

Teacher info: Thomas Fai (tfai [at] cims.nyu [dot] edu), Courant Math, 5th Year PhD

Difficult Probability Problems

The real world is full of examples of people making decisions based on probability, but probability is often counterintuitive and people frequently draw incorrect conclusions. In this class, we’ll discuss a few difficult probability problems and concepts. Focus will be on problems with counterintuitive results and real-life applications.

Length: 1 Hour

Prerequisites: Algebra and exposure to probability (expected value). Calculus will be helpful but not required.

Teacher info: David Vincent (d.w.vincent [at] gmail [dot] com), MIT class of 2007 / Jane Street Capital

Period 4

Computer Arithmetic

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.

Length: 1 Hour

Prerequisites: None

Teacher info: Michael Overton (overton [at] cs.nyu [dot] edu), Courant CS, Professor

Mystic Magic Squares

Magic Square is an interesting mathematical object that involves basic algebra--addition--to interpret the result but attracts many people's interest. I will mainly focus on the Magic Squares of order 5 while briefly touching orders of 3 and 4, and will talk about how to construct such Magic Squares both in mathematical and "informal" ways.

Length: 1 Hour

Prerequisites: basic algebra (addition)

Teacher info: Henry Lim (uny0426 [at] gmail [dot] com), Courant Math, 1st year MS in math

An Introduction to Particle Physics

The universe is more complicated than the simple formula of protons, neutrons, and electrons. Quarks, muons, and antimatter all exist, swirling around us. Neutrinos constantly change mass, yet particles such as the photon have no mass at all. In this course we will learn about an array of subatomic particles and their applications in the real world (and throw in a bit of dark matter just for the fun of it).

Length: 1 Hour

Prerequisites: An understanding of basic chemistry and physics is highly recommended.

Teacher info: Isabel baransky (isabelbaransky [at] yahoo [dot] com), Columbia University Applied Physics, 2nd Year Sophomore, Undergraduate

Understanding fish swimming and bird flight

Often we wonder how fish swim and how birds fly. In an era humans can send probes to Mars and can detect the earliest galaxies across the universe, you might think that the relevant scientific questions regarding biological locomotion (swimming, flying, walking and crawling) have been addressed long time back. The fact, however, is that we still don't fully understand how birds take off and land and how fish detect the surrounding flows and use it to move about. And, these aren't the only pending questions. In this talk I will walk you through a few experiments that have been conducted in the Courant Institute's Applied Math Lab, will introduce the ideas on how to approach the questions in biolocomotion with some simple steps. I will also demonstrate how experimenters excise scientific common sense with solid measurement as they conduct research activities.

Length: 1 Hour

Prerequisites: Nothing specific required. This talk is for general public who have the basic curiosity in nature and in science.

Teacher info: Jun Zhang (jun [at] cims.nyu [dot] edu), Dept of Physics, and Courant Institute, Professor

Creating order from disorder

What does it mean for a particle to travel randomly in a liquid or gas? How come "a drunk man will find his way home, but a drunk bird may get lost forever." (-Shizuo Kakutani) ? What are the underlying mathematical processes creating this picture? I will provide some answers to these questions, and introduce you to the notion of a random walk, a mathematical process consisting of a sequence of random steps, and self-organized criticality, the idea that order can arise in a system without outside intervention. I will also show one of the most beautiful theorems in mathematics, the approximation of random walk by Brownian motion (the random movement of particles in a fluid).

Length: 1 Hour

Prerequisites: Some calculus and curiosity

Teacher info: Laura Florescu (florescu [at] cims.nyu [dot] edu), Courant CS, 1st year PhD

An Ideal Gas Monte Carlo

In statistical mechanics, physicists develop equations describing the macroscopic behavior of complicated systems based on the dynamics of their microscopic constituents. But what if we want to study a system in which these microscopic dynamics are too complicated to lead to nice, useful macroscopic equations? Of course, computers will come to the rescue, but how? One possible answer is a numerical analysis technique called Monte Carlo simulation, in which a large number of simple interactions are simulated and their results aggregated and/or averaged. In this course, we will briefly introduce some of the basic concepts behind Monte Carlo simulations, and then explore a simple ideal gas simulator implemented in Python.

Length: 1 Hour

Prerequisites: Physics and chemistry (mechanics, Newton’s laws, electrostatic forces, ideal gas behavior). Calculus will be kept to a minimum. Some programming knowledge will be helpful.

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

One, Two - One, Two, Three, Fourier!

Have you ever wondered why an F# played on a guitar sounds so different than an F# played on a saxophone? Or how we can use computer programs to manipulate the frequencies of sounds? Well, it turns out that math and music are closely related! In this class, we will discuss the basics of Fourier Series and see how they can be used to analyze and model music. We will also learn about and use a useful tool called the Fast Fourier Transform, which has exciting applications in music.

Length: 1 Hour

Prerequisites: You should be familiar with the behavior of the sine and cosine functions. Knowledge of infinite series and integration will help your understanding of Fourier Series, but we will go over the basics together. An interest in music will certainly make the class more fun, but knowledge of formal music theory is not necessary.

Teacher info: Olivia Chu (olivia.chu [at] nyu [dot] edu), Courant Math, 2nd Year Undergrad

How Hard Is a Problem--Complexity theory

I will start out with the question "How do we measure how hard a math problem is?" For example, how hard is it to multiply 2,345 * 6,789? I'll make some attempts at a simple definition, and, after those fail spectacularly, I'll introduce the notion of complexity theory--Computer science's answer to my question. I'll present the basic concepts behind complexity theory, explain why it's defined in a way that might seem a bit strange at first, and provide some examples of how it's used in practice.

Length: 1 Hour

Prerequisites: All that's needed is knowledge of basic algebra, but the talk will move fast and assume an audience that is comfortable with math.

Teacher info: Noah Stephens-Davidowitz (noahsd [at] gmail [dot] com), Courant Computer Science, 1st Year PhD

Period 5

How does a Web Search Engine Work?

In a fraction of a second, Google and Bing can find and rank the web pages that match your search query, from among the billions of pages on the Web. Find out how it all works!

Length: 1 Hour

Prerequisites: None.

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

Circuit Analysis Basic Techniques

So you have played around with Arduino or Raspberry Pi, or fancying the thought of getting your hands dirty with them. This workshop will focus on introducing you to basic techniques of circuit analysis. Some of the topics we'll cover include Thevenin and Norton Theorems, Mesh and Node Analysis, Laplace Transforms, Voltage and Current Divider Rules, Transfer Function, Root Mean Square, etc. This should be a session not to miss for anyone interested in applying basic logic for understanding the flow of current, and electrical potential. The techniques learned will help you better debug your hardware with some logical thinking under your belt!

Length: 1 Hour

Prerequisites: Algebra

Teacher info: ZEESHAN MUGHAL (zmughal89 [at] gmail [dot] com), Department of Electrical & Computer Engineering, Stony Brook University, Graduate

Finding your roots: an introduction to population genetics

You might already know that your genes decide a lot about you: your height, your eye color, and whether or not you can eat that scoop of lactose-filled ice cream without being stuck in the bathroom for hours. But what can we learn about your past using just your DNA sequence? As it turns out, a lot! Learn how quantitative genetic analysis can be used to trace your ancestry back thousands of years, estimate the time of the first human migration out of Africa, and tell you approximately how much of your DNA you share with a Neanderthal. Come find out what your genes REALLY say about you!

Length: 1 Hour

Prerequisites: Excitement! And some (really) basic probability.

Teacher info: Jasmine Nirody (jnirody [at] berkeley [dot] edu), Berkeley Biophysics, 1st year PhD

Connecting the Dots: Using Simple Visual Models to Explore Complex Relationships

In a social circle of three hundred individuals, how can the most tightly-knit group of friends be identified? If two strangers share one friend in common, are they likely to become friends? Questions like this can be answered using graphs: mathematical structures that illustrate relationships between pairs of individuals. In fact, many problems in science can also be solved in a similar manner, when searching for meaningful relationships in large sets of data. An overview of different types of graphs and their features will be introduced, and example applications in social science, biology and the physical sciences will be discussed to demonstrate the many contexts in which graph theory is useful.

Length: 1 Hour

Prerequisites: None.

Teacher info: Loretta Au (loretta.au [at] stonybrook [dot] edu), Stony Brook University, Dept. of Applied Mathematics, 5th year PhD student

Special Relativity

This talk will be an introduction to Einstein's Special Theory of Relativity. Topics covered will include time dilation, length contraction, Lorentz transformations and more. We will talk about how two observers can disagree on the simultaneity of events. We will also cover all of the classic paradoxes of special relativity. If we have time, we will also cover the implications for astronomy.

Length: 1 Hour

Prerequisites: Basic trigonometry and algebra.

Teacher info: David Mykytyn (dwm261 [at] nyu [dot] edu), NYU Physics, 3rd Year Undergrad

Math Tricks, Teasers, and their Purposes

For those of you who thought that math could not be interesting and fun, this is the class to take. Over the years, numerous mathematicians have derived tricks and tips to simplify mathematical problems. This class will teach and analyze some of these tools and their significance to the field of mathematics. From the Fibonacci Sequence to decimal tricks to the Gauss trick and many more, this class will enlighten students in using the tools that we often take for granted.

Length: 1 Hour

Prerequisites: Basic algebra, rational thinking skills

Teacher info: Kristina Atalla (kla311 [at] nyu [dot] edu), NYU College of Arts and Sciences Mathematics , 1st year undergraduate

My Life as a Higgs Boson

The latter half of the twentieth century radically altered our perception of classical physics and the world around us. After decades of research, the confirmation of the existence of the Higgs boson at CERN last summer was the final fundamental piece needed to complete the Standard Model of physics. But what exactly does that even mean? What is the Higgs boson? Why do we need it? What comes next? This lecture will discuss these most troubling questions in an attempt to demystify the elusive Higgs boson. If we have time we'll derive neutrino oscillations probability using simple trig and linear algebra.

Length: 1 Hour

Prerequisites: An open mind! Bring one question about Standard Model/particle physics to class!

Teacher info: Gladys Vélez Caicedo (gbv2105 [at] columbia [dot] edu), Columbia University, Department of Astronomy and Astrophysics, 2nd Year Undergraduate

Fair Divisions and Allocations

You likely know how to divide a cake fairly between two people: one person cuts, the other chooses. But how would you do this for three people, or more? Another approach is to have an arbitrator divide property, e.g. in a divorce, or in setting a boundary. The first step is to agree on what it means to be fair. Answers go back at least to antiquity. Finally, if time allows, we will look at how resident slots in hospitals are allocated. This is often called the Stable Marriage Problem.

Length: 1 Hour

Prerequisites: Basic algebra would be helpful.

Teacher info: Richard Cole (cole [at] cs.nyu [dot] edu), Courant/CS, Professor

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!

Table of Course Notes:

Talk Name Teacher Link to Notes
Killing the Hydra Mark Kim http://markhkim.com/2013/10/killing-the-hydra/
Modeling an Epidemic Tyler Palsulich 2013-2ad61970a7efb68a8a5fbb0b60babb4f.pdf
Software Synthesis Ruzica Piskac 2013-15042bba959e35980286e2b218a55cda.pptx
A Taste of Celestial Navigation Dmitry Sagalovskiy 2013-43c254802d2e7481903a68bbb33f2575.pdf
Paradoxes of the Continuum Chee Yap 2013-10b1d0a2ccb9fa9965ad8901bf2157a5.pdf
Robotic Navigators and Cartographers Piotr Mirowski 2013-9de6e5a749818edb4df4f8d4ecfc3518.ppt
The Whispers of Stars: Information from Stellar Objects Isabel Baransky 2013-20b5efac26e19cf0f382e7df4f8f01d3.pptx,2013-20b5efac26e19cf0f382e7df4f8f01d3-1.pptx
MapReduce, the Big Data Workhorse Vyassa Baratham 2013-0b713788b8d02c38ac27f5d9ee4e8967.pptx
An Introduction to Particle Physics Isabel baransky 2013-8a91a1395dd0207d7d839da8120da20b.pptx
Creating order from disorder Laura Florescu 2013-6de148ce6c942e2ffde81655a8e7af4d.pdf
An Ideal Gas Monte Carlo Vyassa Baratham 2013-71f3774c4476e6ae2e22088da3c135b9.pptx
How Hard Is a Problem--Complexity theory Noah Stephens-Davidowitz 2013-f305fe3545faaf71d1241c8f6412659a.key
How does a Web Search Engine Work? Ernest Davis http://www.cs.nyu.edu/faculty/davise/presentations/webCsplash.html
Finding your roots: an introduction to population genetics Jasmine Nirody 2013-7d3734b9eb255800cc3d39e8acd3f392.pdf ,2013-7d3734b9eb255800cc3d39e8acd3f392-1.pdf
Fair Divisions and Allocations Richard Cole 2013-444a7f6e5dda1860b58ca6c7619898da.pptx