# Talks

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 2012 Class Schedule Link to Lecture Notes Icons indicate difficulty rating, in ascending order. Description of the icon system.
 Period 3a Mathematics in Finance and Economics The class will begin with simple ideas and concepts in Economics, and we will quickly go into the concept of indifference curves and budget constraints. Then we will teach the idea of optimization and the use of Lagrangian. After this, we try to drive towards optimizing across two time periods. This is the idea of the discount factor. After this, depending on class performance, we can derive the Sharpe Ratio and Beta, or we can explain the Time Value of Money. Length: 1 HourPrerequisites: Calculus and basic differentials. The Lagrangian concept is supposed to be Calculus III but can be introduced during the class. Teacher info: Jeremiah Leong (ghl230 [at] stern.nyu [dot] edu), Stern and Courant Math, JuniorCourse notes available here Mathematics: Its Past, Present, and Future Have you ever been sitting in math class and wondered "When am I ever going to use this?" or "How did we ever come up with this stuff?" Well then, this class will at least try to answer those questions. From likes of Archimedes to recent mathematicians like Mandelbrot and Courant, this class gives a bird's eye view of the context within which modern day mathematics operates and influences the world. Philosophers often place ideas within the context of a period of time, so why shouldn’t we, as mathematicians, do the same? Just as human thought flowed from Platonic philosophy to existentialism to post-modernism, so too did mathematics from Euclid to Newton’s Calculus to modern day Chaos Theory. The placement of ideas within this historical context helps to illuminate why these mathematical methods were invented in the first place, and what they can help us with on a day to day basis, whether pondering the philosophical truths of this universe or building an econometric model. In essence, uncovering our mathematical past will frame our modern day condition and adjust our expectations for the future of mathematics. Length: 1 HourPrerequisites: One should be familiar and comfortable with mathematical reason. High school algebra and pre-calculus would be a big plus, but we will be brushing through everything from basic trigonometry to more complex topics. Really all that is required is a logical mind and willingness to think. Teacher info: Ador Michael Cristofi (amc738 [at] nyu [dot] edu), Stern Finance and Courant Math, 2nd Year Undergrad, Double Major in Math and Finance 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 we will see how to give explicit descriptions of optimal strategies for some very special games. Length: 1 HourPrerequisites: noneTeacher info: Wesley Pegden (pegden [at] math.nyu [dot] edu), Courant Math, NSF Postdoctoral FellowCourse notes available here Intro to Programming (II) A gentle, fun introduction to programming. We will be working in the computer lab to design a simple graphics application in the Python programming language. Length: 1 HourPrerequisites: No programming experience necessary. Algebra, trigonometry, and analytic geometry (i.e., precalculus) will be useful. Teacher info: Paul Gazzillo (pcg234 [at] nyu [dot] edu), Courant Computer Science, 2nd Year PhD Counting with Algebra I will present a way that lets you use tricks from algebra to let you count things. This idea is known as "generating functions". I will also develop an explicit formula for the n-th Fibonacci number.Length: 1 HourPrerequisites: Familiarity manipulating algebraic equations; know how to sum geometric sequencesTeacher info: MihaI Nica (nica [at] cims.nyu [dot] edu), Courant Math, 1st Year PhDCourse notes available here
 Period 3b Math made Difficult: Equivalence Relations It's the next best thing to equality - it's equivalence! Join us for the formal mathematical definition of an equivalence relation, and then test out your new knowledge by discerning some motivating examples from a few non-motivating non-examples.Length: 1 HourPrerequisites: The willingness to learn from examples, and the ability to make great mistakes. Material will be pulled from many levels of mathematics.Teacher info: Japheth Wood (japheth [at] nymathcircle [dot] org), New York Math Circle, PresidentCourse notes available here From Leibniz to a Little Big Planet (TM): Using First-Principles Physics to Simulate Complex Motion Have you ever wondered how character movements in animated movies or video games look so realistic? What concepts in physics, mathematics and computer science are engineers using to create these graphics? It turns out that basic laws of physics, such as velocity and acceleration, can be integrated with computational methods to generate these results. We will begin with a discussion on what derivatives and differential equations are, and how they can be used in modeling the dynamics of movement. Commonly used computational algorithms for solving these equations will be introduced, and we will define areas where these approaches have proven useful, such as providing entertainment and even solving scientific problems.Length: 1 HourPrerequisites: None.Teacher info: Loretta Au (lau [at] ams.sunysb [dot] edu), Stony Brook University, Dept. of Applied Mathematics, 4th year PhD student A Few Great Problems Classes at the New York Math Circle are problem solving sessions. During this talk I'll discuss a small number of my favorite problems, some from combinatorics, some from probability, and some from geometry. Hopefully we'll solve them all!Length: 1 HourPrerequisites: Geometry, basic combinatorics and probabilityTeacher info: David Gomprecht (davidgomprecht [at] gmail [dot] com), New York Math Circle and the Dalton School, Math Teacher 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. A little mobile robot will make a guest appearance.Length: 1 HourPrerequisites: Trigonometry, probabilities.Teacher info: Piotr Mirowski (mirowski [at] cs.nyu [dot] edu), Bell Labs, Statistics and Learning group, Research ScientistCourse notes available here
 Period 5 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 HourPrerequisites: Numbers. Maybe fractions?Teacher info: Ken Ho (ho [at] courant.nyu [dot] edu), Courant, Computational Biology, 5th year PhDCourse notes available here Why the bell curve? The bell curve occurs frequently in nature. Heights, weights, IQs and many other values have a bell shaped distribution. In this class, we will do a hands-on activity and discuss the mathematics behind this surprisingly common phenomenon. Length: 1 HourPrerequisites: NoneTeacher info: Meredith Burr (mburr [at] ric [dot] edu), Rhode Island College Math Department, Professor Stirling Numbers A teacher wants to divide four students into groups. How many such divisions are possible? We will start with this simple counting question, and explore some of its ramifications. Along the way, we will solve more challenging problems and develop a few fantastic formulas.Length: 1 HourPrerequisites: Preferably students should be familiar with combinations, the Inclusion-Exclusion Principle and recursive equations.Teacher info: David Hankin (oana [at] nymathcircle [dot] org), New York Math Circle, Mathematics Teacher An Introduction to Memory Corruption and Exploit Development Computer security is always big news, but it's hard to get started without a proper introduction. This talk will be a brief immersion into one of the most technical areas in information security. We'll quickly cover the basics of C and x86 and talk in depth about memory corruption vulnerabilities on modern platforms. We will analyze several real vulnerabilities and write exploits for them. Many topics will be quickly covered as we discuss arbitrary code execution including: compilers, optimization, reverse engineering, smashing the stack, ASLR, NX, ROP, clowns and sorcery.Length: 1 HourPrerequisites: Computer programming experience, lower level languages are better.Teacher info: Julian Cohen (hockeyinjune [at] isis.poly [dot] edu), NYU Poly, ISIS Lab, Junior UndergraduateCourse notes available here Exploring Infinity (and Beyond?) I want to give a short talk about the cardinality of the natural numbers and the real numbers, and show the students the diagonal argument. When I was first shown this proof it "blew my mind" and really sparked a long interest in Math, and I hope if I showed it to others, it could inspire them as well. If there is time, I would also go into the density of the rationals and their cardinality as well.Length: 1 HourPrerequisites: A suggested prerequisite of PrecalculusTeacher info: Patrick Song (patsong [at] nyu [dot] edu), Courant Math, 4th year undergraduate Fractals In the Limit! Exploring Fractals Using Iterated Function Systems. Fractals are amazing forms that appear throughout the natural and mathematical worlds. How can we understand, define, and create them? In this class we’ll look at one way of understanding fractals, as the result of iterated function systems. This will allow us to explore some of the defining features of fractals in more depth. We’ll dive into the math behind why Iterated Function systems produce fractals. We’ll end up talking about things like the “distance between two pictures”, and what a “sequence of pictures” might be. Then, we’ll see how Iterated Function Systems produce sequences of pictures that converge to fractals! Awesome.Length: 1 HourPrerequisites: We’ll be thinking about things fairly abstractly. Familiarity with sequences and limits is a plus. But we’ll also be seeing a lot of neat fractals that require no prerequisites to appreciate.Teacher info: Arjun Kataria and Ezra Winston (ezrawinston [at] gmail [dot] com)

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 Take a Tour with Euler Amro Mosaad ../data/notes/2012-c1f00cafc35c2d3dd2cf7b61b39948dc.ppt Fundamental Identities in Trigonometry Jong Woo (John) Yoon ../data/notes/2012-4217166843abce0efd9b7bc17da31f89.pdf Mathematics in Finance and Economics Jeremiah Leong ../data/notes/2012-a1e3f7493e5075bc748f7a94e03a82f1.pdf The Mathematics of Games Wesley Pegden ../data/notes/2012-ccde8fecb96346287f6117dc846546bf.pdf Counting with Algebra MihaI Nica ../data/notes/2012-ce1f8e642e1cb77717d74f0327c01df0.pdf Math made Difficult: Equivalence Relations Japheth Wood ../data/notes/2012-f47b216e046fb2358959aea30c9f1fcd.pdf Robotic Navigators and Cartographers Piotr Mirowski ../data/notes/2012-5c0e7715225a95a4ae4d538e2b92fae4.pdf Stars and their Physical Properties Isabel baransky ../data/notes/2012-d748d01390b738fd5a96b00d517ad11a.pptx Biomechanics: consequences of size in nature Ken Ho ../data/notes/2012-7c20a4aee3804d65691d314fc4022a32.pdf Matrix Partitions and Extensions Daniel Zhou ../data/notes/2012-273b19a852b65d38aba48035283163f0.doc Fibonacci Numbers in Nature Ken Ho ../data/notes/2012-1a40fcb167b4d0cc5402025e6b4dfff6.pdf An Introduction to Memory Corruption and Exploit Development Julian Cohen ../data/notes/2012-de3aaf02f77214f5085a2847efa7899b.pdf