Breakout Descriptions Last updated 7/19/13

Week 1

Analytical Geometry, Dr T Francois Vieta’s introduction of literal symbols for representing a class of numerical values was followed in 1637 by Rene Descartes’ application of it in Geometry to represent a point and for the representation of lines and curves by equations. He set up an oblique coordinate system and then the rectangular coordinate system. A first degree equation in x and y gives a straight line in this system, and conversely, a straight line has a first degree equation. The discussion goes over very briefly over the various forms of the equation of a straight line, and an example of how algebra is used to “analyze” geometric construction problems. The area of a triangle is obtained in two ways: In terms of the coordinates of the vertices and in terms of the equations of the three lines constituting the sides. Equations of the various conics are obtained from the Homogeneous General Equation of the Second Degree. The method of the rotation of axes is used to determine the conic an equation of the second degree represents. It is shown that given any five points in the plane there is unique conic passing through them. Equations of tangents, normals, subtangents,a nd subnormals are obtained for the circle first and then for the other conics. Poles and Polars, and the “correlation” between pole and its polar for a conic are shown. The general second degree equation in two variables is shown with its nine affine forms of which ellipse, parabola, and hyperbola are just three. The method of transformation of coordinates to convert a second degree equation to its most simplified form is illustrated. The simplified form is related – “affinis” – to the original and this leads in to a discussion of affine transformations and affine geometry. Whereas as an ellipse is transformed in this manner to another ellipse and not, for instance, a hyperbola, students are shown that there is a transformation, more general, that takes any conic to any other conic. Students are shown such a (projective) transformation – all conics are circles when viewed from the vertex of an infinite cone. This example is used to point out the role played by Analytic Geometry in the mathematical theory of perspective for curves: when a curve is specified an equation, the equation of the perspective view is obtainable by suitably transforming x and y. Pre-requisites: Algebra 2 and Trigonometry Difficulty – 3 stars Continued Fractions, Dr T Every representation of Real Numbers is in terms of positive integers and 0. One instance is the decimal representation. Another is the binary representation. Continued Fractions is another. But an important aspect of Continued Fraction representation of real numbers is that they are in 1 to 1 correspondence with the real numbers, unlike decimal (.5 = .49999...) or rationals (2/3 = 6/9). Continued Fractions reveal the nature of real numbers. To begin with we will find that rational numbers are finite Continued Fractions and quadratic irrationals are periodic continued Fractions. Irrational numbers will be found to correspond uniquely to infinite sequences of integers.

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One way of visualizing an irrational number is to find a rational approximation of it. Rational approximation theory begins in the Continued Fractions representation of the irrational number, namely, that there is an infinite number of rationals p/q so that the irrational is closer to p/q than 1/(q^2). We also touch on the important use of Continued Fractions in the solutions of Linear Diophantine equations and Pell’s equations.   Taming the Torus, Dr V A torus is the mathematical term for a shape such as a donut or bagel. We will examine the geometry of the torus, followed by a discussion of its topological properties, including triangulation, graphs, and games on the surface of a torus. Heavenly Mathematics, Glen How were the ancient astronomers able to find their way around the heavens without anything even as sophisticated as a telescope? With some clever observations and a little math, it’s amazing how much you can infer. Following the footsteps of the ancient Greeks, we will eventually determine the distance from the Earth to the Moon…using only our brains and a meter stick. Along the way, we will develop the fundamentals of the subject the Greeks invented for this purpose, now a part of the school mathematics curriculum: trigonometry. Scientific calculators are encouraged. Cryptology, Jon Rogness Cryptology is the art of creating – and breaking! – ciphers which take information and make it secret. We'll begin by looking at basic codes, and using modular arithmetic to explain how certain ciphers can be described mathematically. We'll also examine the Vigenere Cipher, which was described as "unbreakable" just 100 years ago, but is nearly as easy for mathematicians to break as a simple "Cryptoquip" in the newspaper. By the end of the week we'll cover modern cryptosystems, including methods used by websites like Facebook and Amazon to protect passwords and credit card numbers. You should have taken one of the number theory courses this year or a previous year at MathPath. In particular we will use modular arithmetic throughout the week. The Shape of Space, Jon Rogness This breakout will explore ideas in the area of math known as topology, where a donut (torus) and coffee cup are equivalent objects. It will provide a nice followup for any students who have done the previous breakouts this year which covered the torus, although those courses are not prerequisites for this one. We'll start by learning a few basic rules in topology. Then we'll look at the torus, either as a review or brief introduction depending on whether students were in those previous breakouts. Then we'll look at how to construct other two dimensional surfaces, like the Mobius strip and Klein Bottle. As it turns out, in one of the triumphs of mathematics, we can describe how to build all of the so-called "closed surfaces" by sewing together spheres, Moebius strips and donuts.

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After we deal with surfaces, we'll have the necessary skills to move on to three dimensions. That means we can think about different possible shapes for the universe -- i.e. the Shape of Space -- and explain why the picture below with the dodecahedra might be a model of how our universe is put together. There won't be much in the way of computations or algebra in this breakout, but you should like thinking about three dimensional shapes. (For example, if you like looking at nets and figuring out what the resulting shape is, we'll be doing more complicated versions of that process.) Induction, Mr M If you line up infinitely many dominoes on their ends, with each one close enough to the previous one, and knock over the first, then all infinitely many fall down. This is the essence of mathematical induction, the main proof technique when you have infinitely many statements to prove indexed by the integers, such as 1-ring Towers of Hanoi can be won. 2-ring Towers of Hanoi can be won. . . . 487-ring Towers of Hanoi can be won. . . . Every budding mathematician needs to know mathematical induction, and it's a great proof technique to learn first, because it has a standard template (unlike most proof techniques) and yet leaves room for an infinite amount of variety and ingenuity. Thus this course is one of MathPath's foundation courses. But don’t take my word for it. Here is what a MathPath student wrote on an AoPS MathPath forum on June 24, 2010: I took induction last year, and I knew induction before the class. But it was very well taught, and I learned how to write proofs by induction, which was very valuable on the USAJMO. I'd recommend this class to anyone who wants to learn about induction, whether you know it at the beginning or not.

Mathematica I, Ms C Mathematica is a popular computer software program used by mathematicians and scientists. We will learn how to use Mathematica to perform basic mathematical operations, solve equations, create lists, and generate 2D graphics. No computer programming experience is necessary. (At the end of camp, each MathPath participant will receive a free copy of Mathematica.) Regional MATHCOUNTS, Ms C

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We will use old chapter and state MATHCOUNTS problems to discuss basic problem solving topics including number theory, combinatorics and geometry. Each day will include a brief Countdown Round practice. No prior MATHCOUNTS experience is necessary. Fast Mental Arithmetic, Prof Su In this breakout we will learn how to do arithmetic quickly in our heads. Learn to add and subtract the way the experts do it--- from left to right. Learn how to multiply 2-digit numbers (and maybe even 3-digit numbers) in your head, and square numbers quickly! Impress your friends! This class is for beginners, not for those who are already good at fast mental arithmetic. No background needed, but you must be willing to memorize some key numerical facts. Number Theory I, Sunil Number Theory is the study of the positive integers. How challenging can it be to understand such simple numbers? The wonderful answer is: plenty. The objects are so simple, yet the mathematics is deep. The main topics for NT1 are: Divisibility GCD/LCM, relatively prime integers, the Euclidean algorithm, primes and composites, and an introduction to modular arithmetic. We will prove that there are infinitely many primes, and that every positive integer > 1 has a prime factorization, though the proof that this factorization is unique awaits a later course. If you have seen most of these topics, you are probably ready to begin in NT2. Number Theory II, Sunil Suppose a fruit stand owner wants to arrange oranges neatly. If the oranges are arranged in rows of 5 then there are 2 left over, if arranged in rows of 6 then there is 1 left over, and if arranged in rows of 7 there are 3 left over. How many oranges could the owner have? Is there a unique solution? Answering this question requires a deeper understanding of modular arithmetic than we achieve in NT1. So in NT2 we discuss inverses mod m, the Chinese Remainder Theorem, Fermat's Little Theorem, Wilson's Theorem, the Euler phi-function and Euler's extension of Fermat's Little Theorem. These results will allow us to answer other questions as well, such as: How many ways can 1 be factored? What happens when one repeatedly multiplies a number by itself? And because the answers to such questions are so interesting, we can all enjoy quick divisibility tricks, and other applications.

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Week 2 AMC 10, Adam Learn about Algebra, Combinatorics, Geometry, and other contest math topics using problems from AMC10 tests. Previous contest math experience would be useful, but is not required. Graph Theory, April Most people think of a "graph" as a visual representation of data - a function, assorted information, etc. In the study of graph theory, we define a "graph" to be a set of points, called "vertices", and a set of lines connecting two of those "vertices", called "edges". Graphs of this sort can be used to diagram, understand, and solve many mathematical problems; some of which may surprise you! In this breakout, we will be investigating the fundamentals of graph theory, and discovering problems that can be solved by being diagrammed as graphs. Coding Theory, Dr C If u cn rd ths, u r wlcme n ths brkout! A code is not a secret -- at least not the kind of code that we'll be playing with. (For that sort of coding, take the Cryptology course.) Our codes are exactly the opposite: A way of writing down a message which makes it *easier* to understand and *protects* it from mistakes or errors. The first sentence shows how the English language is filled with extra information which helps us make sense of even horribly mangled messages. Codes are used by computers to do the same thing, and they are essential in everyday communications: If you've ever sent an email, used a cell phone, watched TV, or even saved a file on your computer, you've used a code. If you've heard bad static in a phone call or seen strange blocks appear on your TV, you've learned about the limitations of codes. In this breakout, we'll answer the questions: How far can you shorten a sentence before it loses its meaning? How can you rewrite it to protect against mistakes? Can you do both at the same time? We'll transmit messages to each other and even have the chance to play the part of static: the evil corrupter of messages. But most of all, we'll see how some clever mathematics makes it possible for cell phones, computers, and even deep-space probes to communicate with each other. Hyperbolic Geometry, Dr T

This is an ideal course for the future mathematician in that it combines history and the axiomatic method. The course begins with a discussion of Neutral Geometry – axioms, the Exterior Angle Theorem, Alternate Interior Angle theorem and the Saccheri-Legendre Theorem. Next: Euclid’s Fifth postulate (“Fifth”) is added; its equivalence with the Euclidean Parallel Postulate and the Converse of the Alternate Interior Angle Theorem is shown. The efforts of two millennia to prove the Fifth from the Neutral Geometry postulates and the introduction of the hyperbolic postulate. H. Liebmann’s proof: The area of a singly asymptotic triangle is finite. The finiteness of triangular area and Gauss’s proof of the area of a triangle in terms of angular defect. The Bolya formula connecting the angle that a perpendicular of given

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length to a given line makes with a parallel to the given line is derived. Bolya’s construction of ultraparallels. The Klein, Poincare and Beltrami models.   Pre-­‐requisites:  Euclidean  Geometry,  Trigonometry      Difficulty  –  3  stars       Fabulous Fibonacci Array, Dr V We will discover, formulate, and prove several Brickbreaker theorems. Each student will write and revise a proof of one such theorem over the course of the week. Then we will segue into a remarkable arrays of integers that catalogs all nontrivial Fibonacci sequences. Geometric Inequalities, Dr V Beginning with classic questions, such as minimizing the length of a path that must cross a road perpendicularly, we will develop an arsenal of techniques for handling optimization problems involving length and area, including Ptolemy's inequality and the Isoperimetric Problem. Spherical Trigonometry, Glen To find your way through the heavens, along the earth, or across the oceans, you need mathematics. But the math you learn in school mostly takes place on a flat surface --- not the sphere of the heavens, or the earth. To navigate properly, we develop a completely new trigonometry that allows us to find our way around a sphere. We shall discover surprising symmetries and a rich world of theorems, many of which are beautiful and unexpected extensions of some of the most familiar geometric theorems we have learned in school. Scientific calculators are encouraged. Cryptology, Jon Rogness Cryptology is the art of creating – and breaking! – ciphers which take information and make it secret. We'll begin by looking at basic codes, and using modular arithmetic to explain how certain ciphers can be described mathematically. We'll also examine the Vigenere Cipher, which was described as "unbreakable" just 100 years ago, but is nearly as easy for mathematicians to break as a simple "Cryptoquip" in the newspaper. By the end of the week we'll cover modern cryptosystems, including methods used by websites like Facebook and Amazon to protect passwords and credit card numbers. You should have taken one of the number theory courses this year or a previous year at MathPath. In particular we will use modular arithmetic throughout the week. Origami, Jon Rogness Most of us are familiar with origami, the traditional Japanese art of paper folding, but you might be surprised to know that there are strong connections between origami and mathematics. During this breakout session we'll explore a number of mathematical ideas with folding projects. In some cases we'll create geometric objects (such as dodecahedra) which illustrate mathematical ideas. In other cases, paper folding can be used to solve a mathematical problem.

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If you like folding animals and other objects out of origami, this could be a fun and interesting breakout for you. However, previous experience or skill with origami is not required to take this breakout – just a willingness to learn and attention to detail in your folding. Combinatorics, Mr M For students who know a little about how to count, but want to know more and get better. We start with the basics: sum rule, product rule, permutations, combinations, the binomial theorem. Then we learn about “combinatorial arguments” – a good warmup for Dr V’s Proof by Story in Week 3. Time permitting, we will look at inclusion-exclusion, or pigeon-hole arguments, or maybe even get to the powerful theory of linear difference equations. This course is a good prerequisite for the various counting courses in later weeks, such as Proof by Story, Catalan Numbers, and Probability. Mathematica II, Ms C This course will expand on the topics covered in Mathematica I. We will learn how to define our own functions, how to write loops, how to create animations using the Manipulate command, and how to generate 3D graphics. The exercises will be more challenging than the ones in Mathematica I. National MATHCOUNTS, Ms C We will work on a selection of old national MATHCOUNTS problems, including questions from the 2013 contest. Each day will include a brief Countdown Round practice. Prior contest math experience will be helpful.

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Week 3

AMC 12, Adam This class will cover contest math topics at a slightly more advanced level than the AMC10 course. Prior contest math experience recommended.

Symbolic Logic, Brenna and Seth

If you attend MathPath, you cannot be a cow. Bessie is a cow. Most of us would correctly deduce that Bessie cannot attend MathPath, but how do we know this, and how can we prove it? Logic serves as the foundation for our reasoning, setting out formally the rules we understand intuitively. Much as algebra allows us to generalize relationships among numbers by using variables to represent quantities, we will use symbols to represent propositions, demonstrating the form or structure of an argument (premises leading to a conclusion). Then we will formally prove these arguments using rules such as Modus Ponendo Ponens, Modus Tollendo Tollens, and Reductio Ad Absurdum. We will explore truth tables, semantic trees, symbolic logic proofs, and potentially predicate logic. Polynomials, Coach D Polynomials crop up all over mathematics, and there are interesting things to say about them that aren’t limited to solving equations with polynomials in them. We’ll start by looking at the well-known binomial theorem and generalizations of that. For example, what happens if one increases the number of terms that are being raised to a power? What happens if the power becomes something other than a positive integer? We’ll also look at what happens when one transforms the variable in a polynomial by a linear transformation. This leads to the subject of invariants, which were the hottest topic in late nineteenth-century algebra. While they may not quite be at the top of the charts these days, finding invariants for polynomial expressions offers insight into the interplay between algebra and geometry. We’ll also talk some about how to arrange the terms of a polynomial in more than one variable, which turns out to be more than just a matter of lexicography. Of course, we are interested in trying to solve equations with polynomials in them. We’ll build on what we did in the plenary on the cubic and quartic equation to see what we can say about the solutions of equations, even if we can’t find them exactly. We’ll also explain what the fundamental theorem of algebra does for us and mention some of the ways in which Gauss and others tried to prove it. We’ll end by talking about polynomial equations in more than one variable. In particular, we’ll consider Diophantine equations, where we’re limiting the possible solutions to integers. There are some famous polynomial equations of this sort, some of which don’t have any solutions. We’ll end by looking at the question of whether we can find a method of figuring out whether a Diophantine equation has a solution in general. The answer has surprising consequences.

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Being Precise about Randomness, Dr Gray How do you say something mathematically precise about a situation that is completely uncertain and unpredictable? This question frustrated mathematicians for hundreds of years, in spite of the fact that randomness and uncertainty are unavoidable in every part of life. In this course, we will play with several mathematical methods and ideas, such as chip-firing and directed graphs, that allow us to make accurate statements about a variety of interesting and amusing scenarios involving randomness. We will also see how much can be accomplished just from the simple idea of averaging. As a result, we will be able to uncover the causes of amazing coincidences, learn how to solve perplexing probability puzzles, and make infallible predictions about hopelessly unreliable objects. Elliptic Geometry, Dr T In this course we explain why there are exactly three 3-dimensional constant-curvature geometries – Euclidean, hyperbolic and elliptic. Whereas the Euclidean and hyperbolic geometries are neutral geometries, we show that Elliptic Geometry is not a Neutral Geometry. However, Elliptic Geometry complements the Euclidean and Hyperbolic geometries in terms of the number of parallels, namely none. We discuss Elliptic Geometry using two models – Spherical Geometry and the Klein Circle model – to show also that it is a projective space with an elliptic metric. We also prove some fundamental properties of a sphere including that a geodesic is a great circle on the sphere. Pre-requisite: Trigonometry. Those who took Spherical Geometry will find some overlap due to the use of sphere in building the model of elliptic Geometry.   Proof by Story, Dr V This breakout introduces the technique of combinatorial proof for establishing various identities. We will focus on identities involving binomial coefficients for the first half, followed by those for Fibonacci numbers. I will challenge the class as a group to complete and present a ``Story Challenge.'' Morley's Theorem, Dr V Following a brief review of complex (imaginary) numbers we will quickly develop their application to plane geometry, use them to prove a number of tricky little results, then finally apply our knowledge to establish a legendary and difficult theorem involving trisection of angles and equilateral triangles.  Algebraic  Inequalities,  Dr  W    Often  questions  in  Olympiad-­‐type  mathematics  competitions  require  you  to  compare  two  expressions  and  show  that  one  is  larger  than  the  other.  Inequalities  are  the  standard  tools  that  enable  one  to  do  so.    Starting  with  the  "trivial"  inequality  x^2  >  0  whenever  x  is  real  and  nonzero,  we  will  discuss  both  the  AM-­‐GM  inequality  and  the  Cauchy-­‐Schwarz  inequality  along  with  various  ways  these  can  be  used.  

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 Number Theory II, Dr W Suppose a fruit stand owner wants to arrange oranges neatly. If the oranges are arranged in rows of 5 then there are 2 left over, if arranged in rows of 6 then there is 1 left over, and if arranged in rows of 7 there are 3 left over. How many oranges could the owner have? Is there a unique solution? Answering this question requires a deeper understanding of modular arithmetic than we achieve in NT1. So in NT2 we discuss inverses mod m, the Chinese Remainder Theorem, Fermat's Little Theorem, Wilson's Theorem, the Euler phi-function and Euler's extension of Fermat's Little Theorem. These results will allow us to answer other questions as well, such as: How many ways can 1 be factored? What happens when one repeatedly multiplies a number by itself? And because the answers to such questions are so interesting, we can all enjoy quick divisibility tricks, and other applications. Adventures in Group Theory, Mr H In this course, we will explore patterns and properties of * permutations like 123456 --> 523461, and * symmetries of triangles, squares, cubes, tetrahedrons, icosahedrons, graphs, and molecules. Together we will discover the algebraic properties that these things share, and we will learn to express our ideas using group theory, the language of symmetry. Many hands on calculations and experiments will help us develop our abstract reasoning in this beautiful world of mathematics. Pólya's  Enumeration  Theorem,  Max  3  stars    How  many  different  (cubical)  dice  can  you  design  using  the  numbers  1-­‐6?    What  if  you  can  repeat  numbers?    These  questions  beg  for  a  better  understanding  of  symmetry.    In  this  breakout,  we  will  introduce  the  language  of  groups,  cycle  notation,  and  group  actions  so  that  we  can  effectively  talk  about  symmetry.    Symmetry  language  in  hand,  we  will  tackle  pieces  of  the  same  dice  question,  for  some...different  dice.    By  counting  the  same  things  in  different  ways,  we  will  prove  Pólya's  Enumeration  Theorem  and  apply  it  to  the  cube,  maybe  even  other  solids!    

Week  3  courses  continued  on  next  page                      

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Linear  Set  Geometry  I,  Mr  M    You  know  about  number  algebra  and  perhaps  vector  algebra.  These  are  both  examples  of  linear  algebras,  because  there  is  both  addition  and  scalar  multiplication.  In  the  plane,  or  in  space,  there  is  also  linear  point  algebra  and  linear  set  algebra.  We  can  take  scalar  multiples  of  sets  and  the  sum  of  sets  –  this  is  not  the  union  of  sets.  So  for  instance,  if  the  red  set  below  is  added  to  itself  you  get  the  blue  set.    What?      Point  and  set  linear  algebra,  combined  with  the  traditional  set  algebra  of  subsets,  unions  and  intersections,  allow  us  to  treat  certain  important  geometric  ideas,  such  as  symmetry  and  convexity,  with  powerful  algebraic  tools.    In  Week  1,  we  most  get  used  to  point  and  set  algebra.    Set  algebra  takes  some  getting  used  to,  because  several  “obvious”  rules  are  false.    For  instance,  in  general  

.  We  first  develop  a  feel  for  the  algebra  and  then  prove  what  we  discover.    Finally,  we  learn  how  to  express  convexity  and  symmetry  using  point  and  set  algebra,  and  begin  to  prove  geometry  facts,  say,  about  the  sum  of  convex  sets.  

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Week 4 Team Math Contests and Puzzles, Adam, 2-3 stars Don't want to spend 9 hours working on 6 problems? We'll do some math contests with more interaction and a faster format than Olympiads, including ARML Team Rounds, ARML Relays, and HMMT Guts Rounds. We may also try mathematical puzzles such as Sudoku and Kenken. Infinity, Coach D There’s a great deal to say about infinity, and some of it has been said by authors from Norton Juster (The Phantom Tollbooth) to Douglas Adams (The Hitch-Hiker’s Guide to the Galaxy). Philosophers continue to argue over whether there are contradictions at the heart of the idea of infinity. In the meantime, mathematicians have been able to proceed to work with and around infinity. The breakouts will start with getting everyone to say something about infinity (but there’s only a finite amount of time to say it). We shall compare our views of the subject with what has been said in the past. We shall talk about Zeno’s paradoxes and the problems that arise from treating infinity as a number. After all, if we can talk about a circle as a polygon with an infinite number of sides, how does that help us to understand either circles or polygons? One of the best ways to deal with infinity is to translate sentences that talk about ‘infinity’ into sentences from which the word has been removed. We can talk about an ‘infinity’ of primes, but we don’t need the idea of infinity. There are plenty of places where infinity seems to show up where a good translation avoids some of the problems that the idea creates. In fact, we’ll even be able to show how having ‘infinity’ in a problem makes it easier to solve, whether it involves repeating decimals, geometric series, continued fractions, or nested radicals. One of the real accomplishments of the nineteenth century was figuring out how to talk about calculus without involving the notion of ‘infinite’ or ‘infintely small’. If we want to allow the notion of infinity into our mathematics, we can follow the ideas of Georg Cantor, who built up a whole hierarchy of infinities. We can come up with (not surprisingly) paradoxical results about how big the sets of whole numbers, rational numbers, and algebraic numbers are. Then we can take a jump to the real numbers (and perhaps to Professor Conway’s ‘surreal numbers’). One of the issues that mathematicians continue to debate is the relationship between the number of whole numbers and the number of reals. Can we draw a picture of infinity? After all, artists might want to draw a picture of railroad tracks, which could be a challenge if the canvas isn’t infinitely big. The invention of perspective had an influence on projective geometry, and the influence of mathematics in art has been equally conspicuous. The Dutch artist M.C. Escher was not formally trained in mathematics, but some of his later work was inspired by mathematicians and their work in hyperbolic geometry. We shall end by tying Escher’s art work into music and even logic.

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p-adics, Deven, 3 stars What is a number? It is very difficult to give a concrete definition of such a thing. Almost everyone will agree that 1/2 and 75/40 are numbers. However, once we leave the world of fractions the answer is no longer so clear. For instance is 2 a number? What about something like

nn=1

!

" ! = 1! + 2! + 3! + 4! + ... ?

Most of us would probably answer yes, no respectively. The reason for this is that we are so used to Archimedean distance. In this course we will examine what happens when we start with fractions and change our notion of distance so that numbers are really close to 0 if they are divisible by lots of powers of p (for a given prime p). This creates a new number system called the p-adic numbers, denoted ! p . We will explore the p-adic universe; contemplating the question "what is a number" (for instance, the sum above is definitely a number but the square root of two might not be!) and examining the shape of the p-adic universe itself (for example, the integers are all contained in the unit interval and all triangles are isosceles!). Prerequisites. Because we talk about numbers being close when they have lots of powers of prime p, it would be nice to know a bit of modular arithmetic. However, the only real prerequisite is to know what a prime is, and to be mathematically mature and willing to explore exotic ideas. AIME, Dr P Practice with lots of AIME problems. For students who have taken the AIME but not done well yet, and students who hope to qualify next year. Olympiads, Dr P Practice with Olympiad problems drawing on the vast untranslated olympiad training literature from the Former Soviet Union. Dr P (Prof Pasha Pylyavskyy of U Minnesota) was a 2-time IMO gold medalist for Ukraine and will translate some of this literature for you. He also plans to run a team competition about submitting the best solutions for homework. Prerequisite: you have already taken the AIME or you did well in the Algebraic Inequalities breakout. Great Expectations, Dr V Expected value is a tool that can be used to analyze who has the advantage in mathematical games of chance, and how much they stand to gain. We will examine a variety of classic such problems, then develop our own games of chance in teams and hold a probability fair.

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Projective Geometry, Dr T

Euclidean Geometry is the geometry of lines and circles: its tools are the straight edge (unmarked ruler) and compass. There is a geometry where its constructions need only a straight edge. In this geometry a straight line joins two points and two lines never fail to meet. It turns out to be simpler than Euclid’s but not too simple to be interesting. This is Projective Geometry which turns out to be the mother of all geometries, for Euclidean, Elliptic, Hyperbolic and Minkowskian Geometries are but special cases of it. The course begins with Pappus’s ancient theorem. The discussion then proceeds as follows: The new Geometry’s axioms, the principle of duality, the concept of projectivity, Harmonic sets, The Fundamental Theorem of Projective Geometry, and Desargue’s Theorem. Pre-requisites: None Difficulty – 3 stars    Linear Set Geometry II, Mr M We continue studying the effect of set sums and scalar multiples on convex, symmetric, and also balanced sets. In addition, we introduce hulls and kernels. For instance, the convex hull of S is the smallest convex superset of S, and it turns out every set has a unique convex hull. We learn to answer such questions as Is the sum of the hulls the hull of the sum? This is very much a proof course. Prerequisite: Linear Set Geometry I during some summer at MathPath.  OR    Recurrences,  Mr  M    The  usual  definition  of  the  Fibonacci  numbers,    

fn+1 = fn + fn!1, f0 = f1 = 1 ,    

is  a  recurrence,  that  is,  an  expression  which  doesn’t  tell  you  the  value  of  any  terms  in  the  sequence  directly,  but  tells  you  how  to  find  them    in  terms  of  previous  values.    This  contrasts  with  a  closed  form  or  explicit  formula,  like   an = 2

n ,  where  you  can  compute  values  directly.    It  turns  out  that  there  are  many  problems  where  finding  a  closed  form  is  very  hard,  but  with  practice  finding  a  recurrence  is  doable.    From  there,  sometimes  one  can  find  a  closed  form,  and  even  if  one  can’t,  one  can  compute  many  terms  and  prove  theorems  about  the  behavior  of  the  terms.    For  instance,  consider  the  question:  How  many  binary  sequences  of  length  n  are  there  in  which  no  two  0’s  are  consecutive?    If  we  let   an  be  that  number,  then  it  turns  out  that    

an+1 = an + an!1, a1 = 2, a2 = 3 ,    

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from  which  one  proves  that   an = fn+1 ,  so  if  we  know  a  lot  about  the  fibonacci’s  (either  by  formula  or  by  other  means),  then  we  know  a  lot  about  these  binary  sequences.  Furthermore,  it  is  not  hard  to  discover  and  justify  the  recurrence  above.    In  this  breakout,  if  offered,  we  will  spend  about  half  the  time  practicing  finding  recurrences  for  various  problems  and  about  half  the  time  leaning  how  to  find  closed  forms  from  recurrences  or  prove  other  results  from  the  recurrences.  For  instance,  in  turns  out  that  any  constant  coefficient  linear  recurrence  (CCLR)  can  be  solved  in  closed  form,  and  you  will  learn  how.    Since  the  Fibonacci  recurrence  is  a  CCLR,  you  will  learn  where  the  closed  form  for  the  fibonacci’s,  with  all  those   5 ’s,  comes  from.          Imaginary Roots and Cubics, Mr R Though most students first encounter imaginary numbers in connection with their study of the quadratic equation, that is not the context in which such numbers became respectable members in the world of numbers. For reasons to be made clear in this class, imaginary numbers were welcomed into the company of numbers important to know about when a method for solving cubic equations was being studied. Clearly, to understand this development, we will have to learn how to solve cubic equations. We will not only learn how to solve them, but we will take a look at some of the chicanery involved among those involved in discovering a method for solving the cubic equation. Inevitably, our discussions will take us on sidetrips: synthetic division, the relations between roots of polynomials and their coefficients, efforts to find ways to solve polynomials of higher degree, other places in which imaginary numbers have proved their value. We will see that because mathematics is a human activity, human foibles enter into its development.

Sequences  and  Series,  Mr  R  

Sequences  are  simply  unending  strings  of  numbers.    Some,  like  

       1,  1,  2,  3,  5,  8,  13,  .  .  .  are  famous.      

Some,  like    

      3,  3.1,  3.14,  3.141,  3.145,    .  .  .    

are  used  to  define  a  single  number.  

Interesting  questions  can  be  asked.    Can  a  sequence  continually  increase  (that  is,  every  number  is  greater  than  the  one  in  front  of  it),  yet  never  exceed  some  fixed  number,  such  as  2?    While  pondering    that  question,  you  might  want  to  keep  in  mind  that  the  sequence  an    defined  by    

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a1 = 1

a2 = 1+12= 1.5

a3 = 1+12+ 13= 1.8333…

!

a10 = 1+12+ 13+ 14+ 15+ 16+ 17+ 18+ 19+ 110

= 2.928968…

!

   

will  eventually  exceed  1000.    In  fact,  it  will  eventually  exceed  any  positive  integer  you  care  to  name.  

Other  questions  abound.    What  is  the  meaning,  exactly,  of  the  dots  .  .  .  used  so  generously  above?    Why  do  they  sometimes  indicate  a  repeating  pattern,  as  in    

     

113

= .076923076923076923…    

and  sometimes  indicate  a  pattern  that  will  never  repeat,  as  in  

      2 = 1.414213562…    

Does  it  bother  you  when  you  read  that  the  decimal  representation  of   2  will  never  repeat?    How  can  one  be  sure  that  if  some  energetic  soul  works  long  enough,  the  pattern  won't  begin  to  repeat?    Isn't  this  an  example  of  mathematical  arrogance  to  say  to  someone,  "You  can  stop  looking,  because  you  will  never  find  a  repeating  pattern?  

Can  you  tell  which  numbers  will  be  represented  by  a  repeating  decimal,  and  which  ones  will  have  decimal  representations  that  will  never  repeat?  

Is  .99999999  .  .  .  just  a  tiny  bit  smaller  than  1?  

I'm  not  out  of  questions,  but  I'm  out  of  space  in  which  I'm  allowed  to  ask  them.  

 The Catalan Numbers, Prof B The Catalan numbers might be the second most famous sequence in combinatorics. The sequence 1,1,2,5,14,42,142,... counts many cool structures, from mountain ranges to handshakes to trees. In this breakout, we will use the Catalan numbers to explore various counting methods, including recurrence relations, counting proofs, and generating functions. We will look at lots of different examples of Catalan problems and find some lovely and surprising correspondences between these families of structures.

Graph Ramsey Games, Prof Butterfield

It would be surprising if there were ten students in my class who had the same birthday. It would not be surprising if there were ten students in Minnesota who had the same birthday. That is because when things are large we expect patterns to show up. Ramsey Theory asks the question “how big does the group have to be for a particular pattern to always happen?” In the context of graph theory, we might ask how many edges a graph can have before it has to contain a clique of

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size three. Questions like this form the basis of Graph Ramsey Theory. In this breakout session, we will answer some Graph Ramsey questions, and learn just how hard the unanswered ones are.

Then we will spend the rest of our time playing games--I mean, “considering Graph Ramsey Theory from a competitive perspective”. Some of these games will be between two intelligent players, and some will be between an intelligent player and a random adversary. We will show how these games are related, and try to decide who wins them. This area is still very recent, and so many easy-to-ask questions are still unanswered (and are possibly very difficult).

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