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2011 Class Schedule

Link to Lecture Notes

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

Colloquium Speaker

Teacher info: Mehryar Mohri (cs prof)

Notes: csplash_2011.pdf"

Period 1

OddzRGood

Let's play a prototype of a game in which we explore probability and statistics, interactively. How well do you know your theoretical probabilities? How well do you know the probability of events occurring in the world? How well do you know facts about the world that are based on statistics?

Length: 1 Hour

Prerequisites: Multiplication, division, addition, subtraction, fractions.

Teacher info: Josephine Giaimo (josephinegiaimo [at] gmail [dot] com), Masters of Science


Vedic Math - Magic of Mental Math

Are you stuck in class doing calculations, and algebraic problems slowly? Or do you want to sharpen your algebraic skills to the point that multiplication and division of any two numbers is done mentally? Welcome to the wonderful world of Vedic Mathematics, a system far simpler and enjoyable than modern math. Vedic Mathematics is the ancient system of mathematics developed in India. According to the vedas, all of mathematics is based on sixteen sutras or formulae. These formulae describe the way the mind naturally works and are therefore a great help in directing the student to the appropriate method of solution. The simplicity of Vedic Mathematics means that calculations can be carried out mentally (though the methods can also be written down). This leads to more creative, interested and intelligent pupils. Even complex problems involving a good number of mathematical operations, the time taken by the Vedic method will be a third, a fourth, a tenth or even a much smaller fraction of the time required using modern methods. Come join us and get acquainted with the wonderful technique of doing complex math in your mind!

Length: 1 Hour 15 minutes

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

Teacher info: Ankit Ashok Parekh (aparek01 [at] students.poly [dot] edu), NYU-Poly Math, 1st Year, Masters Applied Math


Feynmanomena

Feynman wasn't your typical physicist - renowned for his quirky sense of humor and remarkable discoveries in quantum electrodynamics, he's right up there with the likes of Maxwell and Newton. We'll discuss his Caltech years, ogle the sweet shuttlecraft named after him and look at a few of his published papers. An extensive knowledge in physics isn't required but you must have a good sense of humor!

Length: 1 Hour 15 minutes

Prerequisites: N/A

Teacher info: Amber Bennoui (amberb [at] mit [dot] edu), Simmons College, Physics and Math, 2nd year undergrad


So You Think You Can Count?

It's easy enough to count like Count von Count from Sesame Street, but how about counting large? How long it would take to crack a safe by trying all combinations? Or how about the number of ways to arrange the books on your bookshelf or distribute candy to groups of kids? Seating arrangements at a dinner party? Now you too can learn some tricks for those times when you get so bored that you start counting sheep.

Length: 1 Hour

Prerequisites: None

Teacher info: Evan Chou (chou [at] cims.nyu [dot] edu), Courant Math, 3rd Year PhD

Course notes available here


The Mathematics of Games

Combinatorial game theory studies games. Which games have winning strategies? Can we describe them? Do games like chess and checkers have optimal strategies? In this class, we will see how to *prove* that games have optimal strategies, and how to give explicit descriptions of optimal strategies for some very special games.

Length: 1 Hour 15 minutes

Prerequisites: no prerequisites.

Teacher info: Wesley Pegden (pegden [at] math.nyu [dot] edu), Courant Math, NSF Postdoctoral Fellow


Algebraic Geometry

Algebraic geometry is one of the oldest mathematical subjects, it has attracted research over thousands of years, and it is still very hot! We will do some time traveling; we will start off with curves in the plane, find out what polynomials give certain shapes (for example a cardioid) and end by constructing a space in which all lines intersect (even parallels).

Length: 1 Hour 15 minutes

Prerequisites: Geometric intuition is strongly required, everyone attending should have a good understanding of plane geometry (i.e. what are parallel lines, and how to check this property, how to compute the intersection of lines?). Some knowledge of algebra is useful, though not necessary.

Teacher info: Lukas Koehler (lmk368 [at] nyu [dot] edu), NYU Courant, 2nd Year PhD


Scattering Process: Rainbow, DNA, Visual Cells, Wavelets

Many phenomena in nature such as rainbows and blue skies come from scattering process: the interaction between waves and particles. Nobel Prize winners F. Crick, J. Watson, and M. Wilkins discovered DNA based on an X-ray scattering pattern. Interestingly, recent advances in harmonic analysis connect this natural process to Nobel Prize winners D. Hubel, T. Wiesel's work to explain how information is processed in the brain.

Length: 1 Hour

Prerequisites: Curiosity

Teacher info: Sixin Zhang (zsx [at] cs.courant.nyu [dot] edu)

Course notes available here


Philosophies of Mathematics

So what is mathematics anyway? This may seem like quite an easy question to answer. You might go up to the chalkboard and write 2x + 1 = 3, solve the equation, and rightly say this is mathematics. Or maybe you will claim that the sum of the angles of a triangle equals to two right angles, and that this is mathematics. After all, mathematics is just mathematics. However, if we start asking more penetrating questions this apparently easy explanation seems to dissolve. Are mathematicians like geologists, surveying an already existing mathematical landscape? Or are they like artists, thinking up and creating mathematical theorems out of thin air? Would an alien race have the same mathematics as we do? What is the foundation of mathematics? Why is mathematics so useful? Why does nature conform to mathematical laws? This class will not give definitive answers to any of these questions (as they are still highly debated today), but we will instead raise them. We’ll talk about different philosophical views of mathematics like Platonism, empiricism, logicism and intuitionalism, and also discuss the historical development of mathematics’ foundations. The goal of this class is to get you thinking about mathematics in a different way, and give you an appreciation for its complexity.

Length: 1 Hour 15 minutes

Prerequisites: Familiarity with calculus and infinitesimals will be helpful but not required. However, we will be discussing some very abstract concepts, so an ability to think abstractly and “outside the box” will be crucial.

Teacher info: Josh Fry (jpfry [at] uchicago [dot] edu), University of Chicago, Undergraduate


Bézier Curves: A geometrically-intuitive polynomial representation

The classic power-series polynomial representation taught in Algebra hides wonderful geometric properties of polynomials. On the other hand, the Bernstein/Bezier representation, used frequently in drawing programs, was developed to design the surfaces of cars. We will see how the geometric properties of this representation suggest an iterative corner-cutting procedure to carve the top of a triangular block of wood into a stylish polynomial curve. We will see how undesired artifacts of polynomial interpolation can be avoided by approximating using the Bezier representation. We will see how to generalize this representation to define surface patches, which can be tiled to create large, complex surfaces like those often used in computer-animated movies.

Length: 1 Hour

Prerequisites: Trigonometry (for the sake of exposure); Calculus is a plus.

Teacher info: Ashish Myles (amyles [at] cs.nyu [dot] edu), Courant Computer Science, Postdoc

Course notes available here


Algorithms and complexity

An algorithm is the "how" of a computation. For example, one algorithm to multiply binomials is the "first, outside, inside, last" method. Algorithms tell a computer how to perform complicated operations (like factorization of an integer) in terms of simple functions (like division) that it already knows. Some algorithms take longer than others to execute. A number of fascinating and counterintuitive results pertain to the time that an algorithm takes to run. We will consider some of the most interesting algorithms used in modern computer science.

Length: 1 Hour 15 minutes

Prerequisites: Some programming experience would be helpful. Mathematically, you should be familiar with logarithms and other functions.

Teacher info: David Lawrence (dlaw [at] mit [dot] edu), MIT computer science, First year undergraduate

Course notes available here


Period 2

The Mathematics of Juggling

What on earth does juggling have to do with mathematics?! "Siteswap notation" is a means of expressing juggling patterns through strings of numbers. After introducing siteswap notation, we will use it to observe (and then prove!) some interesting and unexpected mathematical properties of juggling. Mathematically, we will reason using methods from combinatorics, and we will then draw connections to topics as seemingly unrelated as the limits of infinite series and the topology of braids. Even if they've never juggled in their lives, students will learn to visualize juggling patterns they've never seen before, and even invent new tricks. The teacher is himself a juggler, and he will juggle many of the patterns that are discussed.

Length: 1 Hour

Prerequisites: None.

Teacher info: Jeremy Kuhn (jdk360 [at] nyu [dot] edu), Linguistics, 1st Year PhD


A tour of math in the movies

Many live action and animated movies today use computer generated special effects to create more realistic animations of physical phenomena such as fire, water, explosions, fracture, and others. In this talk I'll describe some of the physics and mathematics used to create such special effects, giving a high level tour of some of the mathematics used in the movies.

Length: 1 Hour

Prerequisites: none

Teacher info: Tamar Shinar (ttshinar [at] gmail [dot] com), NYU, postdoc


Mathematics of Voting Systems

What does it mean for a voting method to be fair? Can we find a fair voting system? Is Democracy fair? We will look at what it means for an election to be fair and how the mathematics used to determine the winner affects the outcome.

Length: 1 Hour 15 minutes

Prerequisites: None

Teacher info: Meredith Burr (mburr [at] ric [dot] edu), Rhode Island College, Math & Computer Science, Assistant Professor


Dimension X: Fractals

Have you ever wondered what a four-dimensional object would "look" like? In this session, we'll do more than tackle this question. It turns out that our common sense notions about what the word "dimension" means aren't enough to describe some things. These complex objects are called "fractals," and as the name suggests, they aren't at all like your garden variety lines or planes from geometry. As we'll see, it's disturbingly easy to construct something that has fractional, or non-integer, dimension. A recurring theme in this talk will be how complexity (even chaos) can arise from iterating simple processes. In addition to meaningful explanations for all the colorful images we shall encounter, we will discuss fractals in the real world. Rather than just telling you about them, I'll show you how to actually make one using some holiday decorations and light bulbs. (Live demonstration included.)

Length: 1 Hour

Prerequisites: Complex numbers and logarithms will make brief appearances, but familiarity with them is not at all essential. We won't use much aside from very basic algebra.

Teacher info: Jarrett Lancaster (jll419 [at] nyu [dot] edu), NYU Physics, 4th year PhD


Problem Solving Through Contest Problems

We will be discussing interesting topics in number theory, algebra, and geometry, and address subtle mathematical ideas, including the nature and construction of proofs. Some material will come in the form of intriguing problems from contests such as AMC 10, AMC 12 and AIME. This talk will be led by a veteran teacher, former AIME Chair and instructor at the New York Math Circle (nymathcircle.org.) NYMC is a non-profit organization formed by several former math team teachers and coaches offering year round math enrichment programs at Courant, for high school students.

Length: 1 Hour 15 minutes

Prerequisites: You should be enthusiastic about the study of mathematics, and very comfortable with regular high school material

Teacher info: David Hankin (oana [at] nymathcircle [dot] org)


Linguistics and Computational Linguistics

The North American Computational Linguistics Olympiad (www.naclo.cs.cmu.edu) is in its fifth year now. It is a contest for high school students interested in languages, logic, and computation. I am the coach of the US national team - the team has won numerous gold medals at the International Linguistics Olympiad since 2007. http://en.wikipedia.org/wiki/North_American_Computational_Linguistics_Olympiad http://ioling.org The lecture will include several puzzles in linguistics and computational linguistics.

Length: 1 Hour 15 minutes

Prerequisites: nothing

Teacher info: Dragomir R. Radev (radev [at] umich [dot] edu), CSE, U. Michigan, Professor

Course notes available here


Introduction to logic

Spock:"Really, Dr. McCoy. You must learn to govern your passions; they will be your undoing. Logic suggests..." McCoy:"Logic? Why, the man's talking about logic; we're talking about universal Armageddon! You green-blooded, inhuman..." (Star Trek VI: The undiscovered Country) Even outside Spock and the Star Trek universe, we often say, "So and So was not being very logical" or "This was the most logical choice". But what do we mean exactly when we say something is logical? What criteria should we use to make such an assertion? This class will explore these questions and develop a potential solution, Propositional Logic, from the ground up. We will cover rules of inferences and how to use these rules to prove logical statements. After discussing the completeness of our logical system, we will briefly consider the relationship between logic and mathematics. Embrace your inner Vulcan!

Length: 1 Hour 15 minutes

Prerequisites: There are no formal requirements, but a knack for abstraction and a good intuition of everyday logic will really help!

Teacher info: Josh Fry (jpfry [at] uchicago [dot] edu), University of Chicago, Undergraduate


Circles, parallels and points at infinity

What would happen if we bent geometry a little and instead of having only one parallel to a line through a given point you would have infinitely many? What does reflecting through a circle means? How are these two questions related at all? Find out when you learn about hyperbolic geometry: how it started, how you model it, and how you relate it to the geometry you normally see.

Length: 1 Hour

Prerequisites: A basic knowledge of Euclidean geometry, similarities and triangle geometry

Teacher info: Andres Muñoz (munoz [at] cims.nyu [dot] edu), Courant Math, 1st year PhD


Extinction in Branching Processes

In this talk, I will introduce the probability generating function (pgf) and I will discuss how to use pgf to compute the moments of probability distributions and tackle certain probability problems. Once the students are comfortable with the idea of pgf, I will apply it to study extinction problems in branching processes.

Length: 1 Hour 15 minutes

Prerequisites: Some knowledge of calculus and probability would be helpful.

Teacher info: Ling Jiong Zhu (ling [at] cims.nyu [dot] edu), Courant Institute, NYU, 3rd Year PhD

Course notes available here


How Programming Works

Digital computers are fast, powerful machines that are used to predict the weather, display 3D scenes, and play movies and music. But how do programmers instruct these machines to perform such complicated tasks, when most processors only understand simple commands, such as adding two numbers and checking whether a number is zero? We will learn how programming languages bridge human expression and the machine. We will program together in a computer lab an arithmetic expression evaluator that can calculate the result of expressions such as (8+8/2)*(2*8-4). Arithmetic expressions are natural to humans, but computers cannot understand them without our help.

Length: 1 Hour 15 minutes

Prerequisites: Students must have programmed before. Python and Ruby templates will be provided for the expression evaluator, but students are free to use any language available on the provided workstations (perl, java, etc). Feel free to email the teacher to find out which.

Teacher info: Paul Gazzillo (pcg234 [at] nyu [dot] edu), Courant Computer Science, 1st Year PhD


Meta-Mathematics

Through a series of examples in various fields of mathematics we are going to look at: 1) The structure of mathematics. 2) What do mathematicians do? 3) How does one succeed as a mathematician? 4) Application of mathematics to science and other fields. 5) Applications of mathematics to life.

Length: 1 Hour 15 minutes

Prerequisites: No prerequisites are necessary but it would be good to read (wiki is enough) on axiomatic systems, Zermelo-Fraenkel axioms, Gödel's incompleteness. The course will be challenging not in terms of the prerequisite material but in speed and abstraction.

Teacher info: Dmytro Karabash (karabash [at] cims.nyu [dot] edu), Courant Math, 3rd Year PhD

Course notes available here


Period 3

Mathematics in Cryptography

We will discuss the Vigenère cipher, cryptanalysis through statistical techniques, and how the one time pad overcomes the weaknesses in the Vigenère Cipher. Afterwards, students should be able to encrypt, decrypt, and crack the Vigenère Cipher, as well as see how mathematics is used in cryptography.

Length: 1 Hour 15 minutes

Prerequisites: Understadning of modular arithmetic and some basic statistics will be preferable.

Teacher info: Harold Metz (Ham298 [at] nyu [dot] edu), Courant Math, 1st year masters


The Special Theory of Relativity

In the year 1905 Albert Einstein published a paper that revolutionized our understanding of space and time. The theory he proposed, nowadays known as Special Relativity, shows us that when traveling very fast, the world looks very different from what our intuition leads us to expect. Space and time, although different in nature, are put on an equal footing. This class will teach us how a few simple but deep modifications of our picture of the world naturally lead to consequences which may seem outlandish: time dilates, space contracts and not everyone agrees on whether or not two events took place simultaneously.

Length: 1 Hour 15 minutes

Prerequisites: Basic Newtonian kinematics and trigonometry

Teacher info: Klaus Widmayer (klaus.widmayer [at] cims.nyu [dot] edu), ETH Zurich, Courant / Math, M.S. Mathematics


Game Theory: The Mathematics of Decison-Making

Game theory seeks to mathematically model people's behavior in strategic situations, or games, where the players' outcomes depend on the decisions of the other players. This class will be an informal introduction to the subject, where you will be introduced to the basic concepts, such as Nash equilibrium and extensive and normal form representation. We will then use these concepts to analyze some famous mind-bending game theory problems! Additionally, this class will explore the many real-life situations where game theory comes into play, such as decision-making, auctions, and popular games of strategy. By the end of class, you will have learned some new strategies to help you make decisions and obtain the best outcomes for yourself. We will also play some games during the class to gain insight on how people think strategically!

Length: 1 Hour

Prerequisites: Basic algebra and probability, and an inclination to think rationally!

Teacher info: Victoria Gregory (vg559 [at] nyu [dot] edu), Mathematics and Economics, Undergraduate Junior

Course notes available here


Problem Solving Through Contest Problems (Advanced)

We will be discussing interesting topics in number theory, algebra, and geometry, and address subtle mathematical ideas, including the nature and construction of proofs. Much of the material will come from contests such as USAMO and International Math Olympiad (IMO). New York Math Circle (nymathcircle.org) is a non-profit organization formed by several former math team teachers and coaches offering year round math enrichment programs at Courant, for high school students.

Length: 1 Hour

Prerequisites: You should be enthusiastic about the study of mathematics, and able to do the first few questions on AIME contest.

Teacher info: David Hankin (oana [at] nymathcircle [dot] org), New York Math Circle, Mathematics Teacher


Intro to Cryptography: History, Math and Techniques behind making and breaking codes

Since the times of ancient civilizations, making and breaking codes has been of paramount importance. Throughout history, the balance between the two has won wars, saved or doomed lives, helped unravel ancient scriptures, and ultimately allowed communication and business in this Information Age. These dual disciplines have both required a great deal of art, craft and science, including the study of linguistics, mathematics and computing. You will have a chance to take a peek into the history of making and breaking codes, and you will learn the basic methods to encrypt and decrypt messages. Who knows, maybe you will come up with a new one!

Length: 1 Hour 15 minutes

Prerequisites: No math prerequisites. Some knowledge of permutations or modular arithmetic might help, but is not necessary.

Teacher info: Eduardo Corona (corona [at] cims.nyu [dot] edu), Courant Math, 3rd Year PhD

Course notes available here


Time travel and the Universe

Ever wondered if you could travel in time? The short answer is yes. You could potentially travel to the year 3000 in a span of one year or even one month, if you had enough money. According to known laws of physics (Einstein’s theory of relativity) it’s not only possible to travel into the future, it’s been done already. How about traveling to the past? We cannot say decisively no. There are respectable scientists with theories that make it look as a possibility, but there is no evidence of it being done yet. If you come to this lecture, you’ll gain a basic understanding of relativity and the nature of time, speed and the universe we live in; this could be your first step to building yourself a time machine. This topic, being much used in Science fiction, may soon become as scientific and as doable as traveling to the moon.

Length: 1 Hour 15 minutes

Prerequisites: Basic arithmetics including square root and exponents

Teacher info: Jack Shutzman (js3104 [at] nyu [dot] edu), Courant CS, 2010 Alumnus M.S.

Course notes available here


Introduction to group theory

Many of the properties of the integers -- specifically, the existence of a unique additive identity (zero), the existence of unique additive inverses, and the associativity of addition -- can be generalized so that they apply to objects other than numbers. Any collection of objects that satisfies these properties is called a "group". Group theory is an extraordinarily rich and beautiful branch of mathematics concerned with the study of groups. We'll examine some of the most interesting and surprising parts of group theory!

Length: 1 Hour

Prerequisites: Experince doing algebra with variables.

Teacher info: David Lawrence (dlaw [at] mit [dot] edu), MIT computer science, First year undergraduate


Machine Learning

Fifteen years ago, the content of this class would have been considered science fiction. Cameras that detect and recognize human faces. Self-driving robots finding their way in a forest. Telephones that seem to understand spoken language. How was that made possible? Well, one uses “intelligent” algorithms that can learn from examples, and of course, lots of data. From Google search to washing machines, machine learning is now all the rage. By the way, did you know that information about who you are “friends” with on a social network could be used for advertising? We will learn about “features”, “linear classifiers”, “neural networks”, and applications of the Bayes’ Theorem. We will see how it all works on a relatively simple problem: spam filtering. Then, we will take a peek at more difficult problems like computer vision (e.g. avoiding obstacles using a Kinect sensor).

Length: 1 Hour 15 minutes

Prerequisites: vectors, equations and basic probabilities

Teacher info: Piotr Mirowski (mirowski [at] cs.nyu [dot] edu), Courant CS alumnus, Courant alumnus, research scientist at Bell Labs

Course notes available here


(Algorithmic) Game Theory

We will start by describing simple games that students can relate to (e.g. rock, paper, scissors) and gradually explain the abstract formal definition of a game in strategic normal form. We will explore notions such as the Nash Equilibrium and the Price of Anarchy of a Game. We will also be playing some fun games in class that will help students understand how players think and which strategies are good. Putting yourself into other people's shoes is the first thing that you want to do here before you choose your course of action.

Length: 1 Hour 15 minutes

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

Teacher info: Vasilis Gkatzelis (gkatz [at] cims.nyu [dot] edu), Courant Computer Science, 3rd Year PhD


Ring Theory

Take addition and multiplication. Figure out the properties they observe. What things that aren't numbers might you be able to add and multiply? Well, there are lots of things: matrices, polynomials, functions, and much more. When you take these properties you get what mathematicians call a "ring", a collection of elements with rules for addition and multiplication. Find out how to dramatically generalize things you've known since elementary school.

Length: 1 Hour 15 minutes

Prerequisites: You should enjoy a very abstract conversation!

Teacher info: Daniel Zaharopol (dan [at] learningu [dot] org), CEO, Learning Unlimited


Getting Computers To Learn

Machine learning is the branch of artificial intelligence which concerns itself with getting computers to learn automatically from data. Since 1985, this field went from having few commercial applications, to having literally thousands. I'll cover a few of the big ideas from the field, as well as some interesting applications.

Length: 1 Hour

Prerequisites: Some simple calculus (especially taking derivatives), and a basic understanding of matrices (e.g. operations like matrix multiplication) would be useful, though these are not essential to understand most of the material.

Teacher info: Spencer Greenberg (willfind [at] gmail [dot] com), Courant Math, 5th year PhD


Period 4

How Does a Search Engine Work?

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

Length: 1 Hour 15 minutes

Prerequisites: None

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

Course notes available here


Predators and prey, rabbits and sheep, love affairs, can you predict the outcome?

And who wouldn't want to know the answer to these tough questions! But sometimes we can model simple scenarios when two objects interact and use Math to gauge the final state. For example foxes hunt rabbits, but if they out-hunt them, they may run out of their food source and die out themselves. If rabbits and sheep compete for the same amount of grass, which species will survive a drought? Romeo is in love with Juliet, and he believes she likes him, but she runs away and hides every time he tries to ask her out. Do they have a chance of living happily ever after? You will learn how to introduce two variables, write the dynamics of the scenarios above in terms of simple equations, and determine the final state of a system through math analysis (with the help of the computer software "pplane").

Length: 1 Hour 15 minutes

Prerequisites: None, but the class will be much more enjoyable if you know how to solve a quadratic equation and know what a vector is.

Teacher info: Enkeleida Lushi (lushi [at] cims.nyu [dot] edu), Courant Math, PhD student


The Courant Code

Did you know that the 60 foot mural across the front of the Courant Institute contains dozens of hidden clues? In this class, we will unravel the Courant Code and take a journey through the secret history of Mathematics itself.

Length: 1 Hour

Prerequisites: None!

Teacher info: Shane Keating (skeating [at] cims.nyu [dot] edu), Center for Atmosphere-Ocean Science, Postdoctoral Researcher


Sports and Probability

Will Lebron win a championship with the Heat? Is Michael Phelps an aquatic anomaly or would some other lucky person have won his collection of gold medals? Even the most exciting and complicated real life situations can be modeled using basic concepts in probability. This talk will discuss some of those possible models.

Length: 1 Hour

Prerequisites: The class will include basic probability, so no prior knowledge is necessary.

Teacher info: Ima Narvell Abia (ina216 [at] nyu [dot] edu), Courant Math, 1st Year Masters


Can you count to Infinity?

In this talk we explore the nature of numbers and the eluding concept of infinity. On the journey we will visit the merchants of Babylonia, Arkimedes in Greece, Dedekin in Germany and spend a few nights at Hilbert's Hotel. In their company we will meet the natural, rational, trancendental and real numbers and revisit some of the questions that have occupied mathematicians for centuries.

Length: 1 Hour 15 minutes

Prerequisites: The talk presumes little more than familiarity with rational numbers except for mathematical courage to dive into problems that will set your head spinning.

Teacher info: Jens (jcbjorgensen [at] gmail [dot] com), Courant Math, 3rd year


Knot just a Knot...

Did you know knots are also mathematical objects?? How do you know if 2 knots are the same or not? I'll try to answer these questions and introduce you to a special class of "tangles," the rational ones, and then play an interesting game with them.

Length: 1 Hour

Prerequisites: 9th grade math should be enough

Teacher info: Domingos Lopes (domingos [at] cims.nyu [dot] edu), Courant Math, 2nd year PhD


What is Geometry? The Computational Clue

Geometry has been a wellspring of profound ideas in every branch of mathematics, from number theory to algebra, to analysis, to combinatorics, and even to computation. I will talk about this last connection. Many books, articles and talks have been entitled ``What is Geometry?'' One is reminded of the parable of the Elephant and the Blind Men. In this lecture, the Computer Scientist joins the Blind Men to probe this Geometric Elephant.

Length: 1 Hour 15 minutes

Prerequisites: High School Geometry. Nice to have some programming experience.

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


Game Theory in Biology and Models of Social Interaction

Ever wondered how altruistic behaviors can exist in a world of competition for food, space, and resources? Or why there are about as many males as females in (most) animal species? Surprisingly some relatively simple mathematics, combined with clever and subtle arguments from game theory, can shed a great deal of light on these dilemmas from biology. This course will give a very short introduction to game theory, a field which you may have heard of through such "artificial games" as Prisoner's Dilemma and their Nash Equilibria. We will then discuss how biological scientists found examples of situations in their field work that closely matched these seemingly simplistic models. Finally, we will explain how these game theoretic models lead to a solution to some puzzles in the biological and social sciences, and survey some big questions that still remain.

Length: 1 Hour

Prerequisites: Only high school algebra is required for most of the talk, though some familiarity with calculus will be helpful for understanding certain derivations.

Teacher info: Eliot Brenner (epb254 [at] cs.nyu [dot] edu), Courant Scientific Computing


Introduction to neuronal modeling

Neurons, the excitable cells of the nervous system, are often called the brain's basic computational units. Each neuron communicates with many others via electrical and chemical signals, allowing animals to create an internal representation of their surroundings. To understand how brains work, we often build simplified mathematical models and implement them on a computer, stringing together individual neurons into networks that perform rudimentary brain-like functions. As a first step, it makes sense to design an individual neuron. This already raises a whole series of questions. What does it mean to mathematically "model" a neuron, anyway? What assumptions can we make? How many annoying biological details can we throw away before we lose touch with reality? We'll start with a discussion of how real neurons communicate and what features we might want to build into a model neuron. Then we'll implement those features, using physical analogy, simple mathematical expressions, and computer code. If time permits, we will compare our model to others that are more sophisticated and biologically realistic, and discuss the advantages and disadvantages of each.

Length: 1 Hour 15 minutes

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

Teacher info: Robert Levy (rbl2 [at] nyu [dot] edu), NYU Center for Neural Science, postdoctoral fellow

Course notes available here


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 differential calculus and basic physics

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


A Taste of Computational Geometry

In this hands-on class, you'll have a chance to learn about a subject that mixes mathematics and computer science. Together, we'll look at a few problems dealing with points, triangles, and polygons and try to come up with algorithms to solve them on our own. Computational geometry is a subject which is studied with both mathematics and computer science. It can be thought of as teaching a computer how to work with simple geometric objects. One place where computational geometry shows up is in computer graphics, because graphics require keeping track of many different polygons so that objects look right (we won't talk about these applications in class though).

Length: 1 Hour 15 minutes

Prerequisites: High school geometry including triangle and circle theorems.

Teacher info: Michael Burr (mburr1 [at] fordham [dot] edu), Fordham University, Mathematics, Postdoc


Period 5

Mathematical Induction

Suppose one wants to prove a statement involving a parameter that is true for all natural numbers. One method of proof is mathematical induction, which proves a base case and then uses a boot-strapping step to extend the base case result to all numbers. We will give a quick introduction to mathematical induction and then use it to prove facts in various fields such as algebra, combinatorics, and geometry.

Length: 1 hour

Prerequisites: None

Teacher info: Sean Li (seanli [at] cims.nyu [dot] edu), NYU/Math, 3rd Year, Math PhD


3D-Stuff, Paraboloids and Solids of Revolution

Are you interested in vases? Are you interested in figuring out how much water a vase can hold? Are you interested in interesting 3-D Shapes (Like a 3-D Parabola) and how you can actually describe these shapes with MATH??? Well if this sounds like you on a Saturday night, sign up and learn!!

Length: 1 Hour

Prerequisites: Basic idea of Calculus I & Geometry should be more than sufficient.

Teacher info: Xiaowei Wang (xw379 [at] nyu [dot] edu), Stern, 2014 Undergraduate


Fibonacci numbers in nature

Most of you have probably heard about the Fibonacci numbers (1, 1, 2, 3, 5, 8, 13, ...) and how they are found practically everywhere in nature. But has anyone ever given you a convincing reason why? In this class, we will go through the history of the Fibonacci numbers and see for ourselves their allure through the ages, point out common misconceptions regarding their ubiquity, and finish up by giving a very "rational" reason for their emergence in the context of flower patterns and plant growth. No background necessary!--if you can reason about numbers, then you, too, can learn how nature "knows" math.

Length: 1 Hour

Prerequisites: Numbers. Maybe fractions?

Teacher info: Ken Ho (ho [at] courant.nyu [dot] edu), Courant, Computational Biology, 4th year PhD student

Course notes available here


How the Internet Works (in 75 Minutes)

When you fire up your web browser and visit a website, a staggering number of computers have to work in concert to get you the page you wanted. This talk will introduce you to what really happens when you browse the Internet, give you some historical context, and touch on some design issues that the Internet is currently facing.

Length: 1 Hour 15 minutes

Prerequisites: Helpful if students are already familiar with some general computing principles such as binary counting.

Teacher info: Guy Dickinson (guy [at] gdickinson.co [dot] uk), Courant CS, NYU ITS Technology Security Services, 2nd Year MS Student, also Network Security Analyst


Chaos and the Butterfly Effect

I am going to talk about some basic principles I am currently learning about in a math elective I'm taking called "Chaos and Dynamical Systems." I'm going to relate these math principles that explain a complex dynamical system to what we know as the "Butterfly Effect," which takes its root in chaos theory. This effect will then explain some real world phenomena such as weather. I will include simulations of the butterfly effect and talk about the 3D evolution of trajectories.

Length: 1 Hour

Prerequisites: calculus

Teacher info: Serena Sian Yuan (ssy248 [at] nyu [dot] edu), College of Arts and Science, Physics, Undergraduate, 2nd year

Course notes available here


Gyan from the Geek

Ruby: Are you a CS major? Geek: Yup. Ruby: Do you build cool websites? Geek: Nop. Ruby: Then what do you do? Geek: I solve problems using a computer. Ruby: I guess I need to attend "Gyan from the Geek". Let the Geek give you an insight as to what are the important concepts you learn as a computer science engineer. You will learn to solve puzzles in a really cool way, that you would have never thought of before. Also you will learn what computer science is really about. Basic Algorithms and data structures which are the foundations of computer science engineering will be taught in this class.

Length: 1 Hour 15 minutes

Prerequisites: Basic

Teacher info: Apoorva Vadi (av893 [at] nyu [dot] edu), Courant Computer Science Dept, 2nd year Masters student

Course notes available here


The Epidemics of Contagious Diseases

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

Length: 1 Hour 15 minutes

Prerequisites:

Teacher info: Enkeleida Lushi (lushi [at] cims.nyu [dot] edu), 5th year, Courant PhD student


The Fast and the Fourier

Whether you are whistling a tune, singing a song or jamming on your guitar, you are producing sound waves. The higher the frequency of the wave, the higher the pitch of the note. In music, usually various notes with different pitches sound at the same time. We will look at an amazing tool, called the Fast Fourier Tranform (FFT), that provides a way to find out which pitches are present in a sound wave. After discussing the mathematical theory, we will put it to the test. In a little experiment, we are going to let a computer recognize pitch by performing an FFT on the microphone signal of a standard laptop, and see if it can really distinguish musical notes.

Length: 1 Hour 15 minutes

Prerequisites: If you can graph the sine, cosine, exp, and log functions, you should be fine. Some interest in music is a plus, but knowledge of musical theory is not necessary.

Teacher info: Jim Portegies (jim [at] cims.nyu [dot] edu), Courant Math, 2nd Year PhD


Planes, Trains, and Automobiles: An Introduction to Graph Theory

Have you ever wondered what a traveling salesmen, Mapquest, and internet service providers have in common? They all want to move things (like cars or data) around quickly in some network of paths (like roads or cables), and graph theory, which concerns itself precisely with the properties of networks with pairwise relations, can tell them how to do it the quickest. In this talk we'll discuss the traveling salesman and shortest path problems, giving plenty of examples along the way. We'll also learn interesting properties of graphs, such as why it's impossible to do a tour of Königsberg, Germany crossing each bridge only once (due to Euler), and how results from graph theory can be applied to making maps.

Length: 1 Hour 15 minutes

Prerequisites: No previous knowledge is assumed, only an eagerness to learn!

Teacher info: Thomas Fai (tfai [at] cims.nyu [dot] edu), Courant Math, 3rd Year Ph.D.


Convergence: The simple, elegant idea underlying much of mathematics

There is a simple and elegant yet powerful idea known as convergence that allows virtually all of calculus to be done. In fact, a good proportion of higher-level mathematics makes use of this idea in some form. If you decide to major in Mathematics in college, you will see it quite often. In essence, the idea of convergence is that we might or might not be able to “get to a value,” but we can still get as close to that value as we want. I will illustrate this concept by detailing the convergence of sequences. I will then explain how that idea is used to conceptualize limits, derivatives, and integrals. I will also show how to calculate the exact value of certain simple infinite series.

Length: 1 Hour 15 minutes

Prerequisites: Good knowledge of elementary algebra, including inequalities and absolute value required. Some knowledge of sequences, series, derivatives, and integrals is helpful but not required. The class might be somewhat fast paced, hence the difficulty.

Teacher info: Zachary DeStefano (zrd202 [at] nyu [dot] edu), Courant Math, 4th Year Undergraduate


Where do numbers come from?

What are numbers, really? I mean, what are they? As children, we were taught how to count, as if numbers had always been there and were obvious. When you got to fractions, well, those were supposed to be clear too. And then real numbers? pi? It was always brushed under the rug... it's just some weird decimal that goes on forever, right? Well, you can't prove anything about numbers if you don't know what they really are. We're going to give you a taste for how you go about defining a mathematical object and start proving things about how they work.

Length: 1 Hour 15 minutes

Prerequisites: A willingness for extreme abstraction

Teacher info: Daniel Zaharopol (dan [at] learningu [dot] org), CEO, Learning unlimited


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!


Table of Course Notes:

Talk Name Teacher Link to Notes
Extinction in Branching Processes Ling Jiong Zhu notes/2011-24.pdf
So You Think You Can Count? Evan Chou notes/2011-16.pdf
Algorithms and complexity David Lawrence notes/2011-29.pdf
Meta-Mathematics Dmytro Karabash notes/2011-23.pdf
Linguistics and Computational Linguistics Dragomir R. Radev notes/2011-30.pdf
Introduction to neuronal modeling Robert Levy notes/2011-50.pdf
Scattering Process: Rainbow, DNA, Visual Cells, Wavelets Sixin Zhang notes/2011-18.pdf
Game Theory: The Mathematics of Decison-Making Victoria Gregory notes/2011-39.pdf
Time travel and the Universe Jack Shutzman http://www.nyu-phd-cs.com/school/Time_Travel_and_the_Universe.pdf
How Does a Search Engine Work? Ernest Davis http://www.cs.nyu.edu/faculty/davise/presentations/webCsplash.html
Gyan from the Geek Apoorva Vadi notes/2011-17.pptx
Fibonacci numbers in nature Ken Ho http://www.courant.nyu.edu/~ho/teaching/2011/csplash/fibonacci-numb-nat/
Machine Learning Piotr Mirowski notes/2011-03.pdf
Chaos and the Butterfly Effect Serena Sian Yuan notes/2011-36.odp
Bézier Curves: A geometrically-intuitive polynomial representation Ashish Myles http://www.ashishmyles.com/tutorials.shtml
Intro to Cryptography: History, Math and Techniques behind making and breaking codes Eduardo Corona notes/2011-48.pdf

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