MathPath Breakout Catalog 2015

V5.0 7/18/15

Week 1, Morning    

Number Theory II, Mira Bernstein

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. 2 stars

 

Number Theory I, David Grant

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. 1 star

 

Kendoku, Mr L

Kendoku is a combination of KenKen and Sudoku. Ken means clever or wisdom in Japanese. So KenKen, a puzzle invented by Tetsuya Miyamoto in 2004, means wisdom squared. As in Sudoku, the goal of each puzzle is to fill a grid with digits –– 1 through 4 for a 4x4 grid, 1 through 5 for a

5x5, etc. –– so that no digit appears more than once in any row or any column. Additionally, KenKen grids are divided into heavily outlined groups of cells –– often called “cages” –– and the numbers in the cells of each cage must produce a certain “target” number when combined using a specified mathematical operation. Numbers CAN be repeated in a cage.

Take the basic KenKen grid and add the Sudoku constraint of the Sudoku boxes. Kendoku puzzles can only be 6x6 or 9x9. That is because of the addition of 6 Sudoku boxes each 2x3 in the 6x6 puzzle and the standard 9 Sudoku boxes each 3x3 in the 9x9 puzzle. .

Deductive reasoning, extensive use of logic AND the ability to persevere in the face of frustration are the only prerequisites for this course. Guessing will not be tolerated. 2 stars

AMC8 Mr M (Steve Maurer) This course is for young students who have yet to take any AMC exams or any students who will be taking the AMC8 at least one more time and have not yet scored close to a perfect 25. It is also not for students who have already worked through all the past AMC8 exams, since that's where we will get our problems. On two or three days we will take a whole test (maybe from about 10 years ago) under test conditions. On the other days we will analyze particular types of problems and problem-solving techniques that either seem particularly valuable for the most recent AMC8's or particularly valuable considering what you got wrong on our testing days. This is the one contest-oriented breakout in Week1. There will be one or two other contest-oriented breakouts per week in later weeks, aimed at more advanced contests. Students who have so far mostly approached math through contests, and are worried that maybe they are not ready for the different and more advanced approach to math in most MathPath courses, may wish to take this course as a way to have something in their comfort zone while adjusting to MathPath. 1 star Integer Partitions, sVsen-san (Tom Roby) How many ways can one you write 4 as a sum of positive integers? If order matters, so that we consider 3+1 and 1+3 to be different, then the answer (8) is an easy application of very basic combinatorics. But if we only care about the (unordered) *set* of addends, the answer is 5. This simple question leads to a rich, beautiful, and surprisingly challenging mathematical area on the intersection of combinatorics and number theory. We will scratch a few of its many surfaces, including some beautiful bijections, clever counting, revealing recursions, delicious diagram dissections, and playful polynomials. For this breakout, some previous experience with elementary combinatorics (sum rule, product rule, permutations, combinations, the binomial theorem) and facility with manipulation of polynomials and geometric series will be helpful. 3-4 stars    

Analytical Geometry, Dr T (George Thomas)

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 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 a unique conic passing through them. Equations of tangents, normals, subtangents, and 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 an ellipse is transformed in this manner to another ellipse and not, for instance, a hyperbola, there is a transformation, more general, that takes any conic to any other conic. It is the “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

Week 1, Afternoon

Number Theory III (Sums of Squares), David Grant

Sums of (perfect) squares show up in all sorts of interesting number theory contexts. When a square is a sum of two squares precisely characterizes when a right triangle has sides of integer length (and when they do, we call the lengths a "Pythagorean Triple"). In addition, is a surprising theorem of Fermat that every prime number congruent to 1 modulo 4 is a sum of two squares. It's even more surprising theorem of Lagrange that every positive integer is the sum of at most four squares. In this breakout, we'll show how to find all Pythagorean Triples, and prove Fermat's result, which will bring us into the realm of the Gaussian integers (complex numbers which behave like integers!) This will even give us a glimpse into why Lagrange's theorem is true. 3-4 stars

Cutting Cakes, and other Fair Division Questions, Kathyrn Nyman Whether dealing with property, inheritance, revenue, or taxes, the question of how to divide “goods” (or "bads") fairly among a group of people is a common problem. One delicious way to model "goods" is as a cake. At first glance, it seems easy to divide a cake fairly among n people: just cut it into n equal-sized

pieces. But, while this might work for a homogeneous chocolate cake (without frosting??), consider a cake that is half chocolate and half vanilla, with sprinkles in one corner; that is, a cake where each person may value different parts of the cake differently. Now making sure each person gets a piece they are happy with becomes a curiously complicated problem. We will talk about what it means to be fair, and how mathematical algorithms can help ensure an equitable division of tasty treats. There are no prerequisites except the ability to reason carefully and think logically about algorithms (sequence of steps). 2 stars Basic Counting (Combinatorics), sVsen-san (Tom Roby)

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”, sometimes called proof by story. Time permitting, we will look at inclusion-exclusion, and pigeon-hole arguments. This course is a good prerequisite for various later courses that involve counting, such as Proof by Story and The Poisson Distribution. 2 stars

Heavenly Mathematics, Glen Van Brummelen

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. 2 stars

Geometric Inequalities, Dr V (Sam Vandervelde)

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. 3 stars

Density, Paul Zeitz

The density of a subset of the natural numbers is a measure of its relative frequency. A trivial example: the density of the even numbers is 0.5. In this breakout, we will explore the concept of density to look at much more interesting sets. We will use number theory, probability, and counting arguments. Expect visits by some of your favorite numbers, including pi, the golden ratio, and others. One of our triumphs will be to determine the probability that two “randomly selected” positive integers are relatively prime. 4 stars

Week 2, Morning

Cryptology, Dr C (David Clark)

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. Knowledge of modular arithmetic, for instance from Number Theory I or the equivalent, will be used throughout the week. Additional number theory will be introduced as needed, so prior knowledge of additional number theory is helpful but not necessary. 2 stars

Infinity, Prof D (Matt DeLong)

The concept of infinity has long fascinated mathematicians, philosophers, theologians, and artists. The paradox and power of the infinite inspires wonder and fascination. In this breakout we will wrestle with some of the paradoxes that arise when working with the infinite, as well as see the power that results from incorporating the infinite into our mathematics. We will follow the ideas of Georg Cantor, whose creative ideas systematized the study of the infinite in a way that was profoundly brilliant and (at the time) controversial. We will see how Cantor’s ideas allow us to understand the nature of the real numbers more deeply, as well as to answer questions like, “Are there different sizes of infinity?” and “Is there a largest infinite set?” We will also relate the infinite to different areas of mathematics as well as to art, music, and literature. 2 stars

Induction, Mr M (Steve Maurer)

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.

The point: Even if you have done induction before, you don’t really know induction, because it can be used in so many ways in so many parts of mathematics. Every mathematician will probably do 1000 inductions in his/her life. Get 50 under your belt in this course. 2 stars

National MATHCOUNTS, Ms O’Neill

We will study topics from number theory, algebra, geometry, logic, and combinatorics as well as learn problem-solving techniques by working on problems from old MATHCOUNTS tests. The intention is to limit ourselves to problems from old national MATHCOUNTS tests, not chapter and state. Most days will include a brief Countdown Round practice. Prior contest math experience will be helpful. 2 stars

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 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

Distance in different flavors, Prof Cornelia Van Cott

We usually define the distance between two points to be the length of the straight-line segment connecting the points. This definition of distance – which we call Euclidean distance – is a reasonable way to measure distance if, say, you’re a crow or you own a helicopter. But there are other ways to measure distance – infinitely many ways, in fact!

We will start by investigating the most familiar alternative to Euclidean distance – taxicab distance, and then we will investigate a variety of other options, as well. With each of these new perspectives on distance, we will discover a new geometry on the plane. Circles change shape, pi changes value, and so on. These geometries which come from different distance functions are called Minkowski geometries on the plane, and they give a colorful contrast to the familiar world of Euclidean geometry.

Prerequisites: full familiarity with the Cartesian plane and basic Euclidean geometry (Euclidean distance, circles, ellipses, squares, regular hexagons, and equations of lines/circles, and so on.) 2-3 stars

Week 2, Afternoon

Mathematica I, Dr C (David Clark)

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.) 1 star

Note: If you hope to take this course, it would be very helpful if you bring to MathPath a flashdrive with capacity at least 1GB. Lewis & Clark is not allowing us to store work on their computers or network from session to session, so we need our own storage devices. We will have some flashdrives for students who do not. Also, if your parents have authorized you on our electronics form to bring a laptop or pad computer to camp, and it has Mathematica on it, please inform Dr C. This may make doing homework easier, as there is currently no computer lab in our dorm.

Knot Theory, Prof D (Matt DeLong)

Knots have fascinated farmers, sailors, artists, and myth-makers since prehistoric times, both for their practical usefulness and for their aesthetic symbolism. For over 200 years, mathematicians have intensely studied knots, both for their own intrinsic mathematical interest and for their potential application to chemistry, physics and biology. Today, Knot Theory is an active field of mathematical research with many important applications. It is visual, computational and hands- on, and there are many easily stated open problems.

This class will give an introduction to the fundamental questions of Knot Theory. It will take students from simple activities with strings to open problems in the field. It will answer such questions as, “what is the difference between a knot in the mathematical sense and a knot in the everyday sense?,” “how do I tell two knots apart?,” “how can I tell whether a knot can be untangled?,” and “how many different knots are there?” 2-stars

AMC 12, Ms O’Neill

This course is for students planning to take the AMC 12 for the first time next spring, and for students who have taken it before but want to improve their score, perhaps enough to qualify for

the AIME or an Olympiad. We will emphasize topics covered on the AMC 12 that are not covered on the AMC 10, namely, precalculus topics, such as functions, trigonometry, complex numbers, polynomials, and solid geometry. Some days we will do problems to illustrate one or two concepts or problem-solving techniques; some days we will take a test under test conditions. Prior contest math experience recommended 2-3 stars

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. 3 stars

Discovering n-flips, Prof Cornelia Van Cott

Start with the number 1089. Multiply by 9, and you get 9801. Notice that the digits of the answer are in reverse order from the number we started with. This means that 1089 is called a 9-flip. More generally, a number M is an n-flip if n*M equals the digits of M in reverse order.

Some families of integers require tough mathematical machinery to understand them deeply, but n-flips are not so high maintenance. We can learn a lot about them with basic tools from number theory and a good dose of curiosity. For example, we will discuss the following questions: How many n-flips are there for a particular value of n? Is it possible to describe all n-flips in a simple way? What happens if we work in other bases?

Prerequisites: induction, bases other than base 10, sequence notation, Number Theory I course at MathPath or equivalent. 3 stars

The Other Triangular Numbers, Dr V (Sam Vandervelde)

How many lattice points can be trapped within an equilateral triangle? This innocent question quickly leads to deep number theoretic waters. Our investigation will involve complex numbers (via cube roots of unity) and an elegant analogue to Gaussian integers. This breakout will move fast and assumes students have a firm grasp of both algebra and complex numbers. 4 stars

Week 3, Morning

Error-correcting codes, David Clark (Dr C)

If you've ever heard bad static in a phone call or seen strange blocks appear on your TV, then you needed an error-correcting code! 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. In this breakout, we'll transmit messages to each other and even have the chance to play the part of static: the evil corrupter of messages. We'll learn about vectors, matrices, and how to use them to help cell phones, computers, and even deep-space probes to overcome mistakes and make your messages understood. Number theory I recommended as a pre-requisite but not required. 2 stars

Number Theory II, Prof D (Matt DeLong)

See description under Week 1 morning.

Discrete Morse Theory, Prof K (Prof Dominic Klyve)

Discrete Morse Theory is a new field of mathematics -- it was created just 20 years ago, and it is still rapidly developing. A special case of the more general theory involves graphs, or networks. We can think of a network as a map, and describe the height of each node and of all the lines connecting them. Discrete Morse Theory involves trying to understand and describe what the map looks like when we only look at the parts below a certain height. Students will learn the basics of this exciting new field of mathematics, and even have the chance to look at problems that nobody currently knows how to solve. 2-3 stars 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 𝑎 + 𝑏 𝑆   ≠ 𝑎𝑆 + 𝑏𝑆.

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. 3-4 stars Proof by Story, Granya O’Neill

Many identities involving, say, binomial coefficients or Fibonacci numbers that are commonly proven by induction or manipulating algebraic expressions also have beautiful bijection proofs. Such bijection proofs are often called proof by story, since we come up with two different ways (“stories”) to count the same thing. These proofs often give insight into why the identity holds, and are so elegant that you will be convinced that there could not be a better proof of that particular theorem. Paul Erdos said that such proofs were from “The Book.” In this breakout, we will devise as many proofs by story for identities involving binomial coefficients and Fibonacci numbers as we can.  2-3 stars Exploding Dots, Prof S (Daniel Showalter) Mathematicians know the value of looking at a problem from several different perspectives. This course will introduce a perspective known as the “exploding dots machine,” which can be helpful in approaching various problems in algebra, calculus, and number theory. Consider a row of adjacent boxes with a nonnegative integer number of dots in each box. In a “3 to 1” exploding dots machine, dots follow these two rules: Any group of 3 dots in a box can explode into a single dot in the box to the left, and any single dot can explode into 3 dots when moving to the right. Think about it, we have a representation of the base 3 system. If we replace the 3 with a variable, our machine can now be used to study polynomials. But what happens when we allow for negative dots? What would a “3 to 2” machine look like in terms of base systems? Does each integer have a unique representation in a given machine? How could we represent non-integers? Exploding dots can be used to pose and approach complex mathematical problems in a simple visual format. For instance, exploding dots can be used to solve the question about your polynomial-picking enemy from this year’s MathPath Qualifying Test. (Hint: Think about what information a “1 to 1” machine could provide.) This course will begin with an introduction to exploding dots and move through polynomial manipulations to advanced mathematical questions that are still unsolved. 1 star  

Week 3, Afternoon

Mathematica II, Prof 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. 2-3 stars

Note: If you hope to take this course, it would be very helpful if you bring to MathPath a flashdrive with capacity at least 1GB. Lewis & Clark is not allowing us to store work on their computers or network from session to session, so we need our own storage devices. We will have some flashdrives for students who do not. Also, if your parents have authorized you on our electronics form to bring a laptop or pad computer to camp, and it has Mathematica on it, please inform Dr C. This may make doing homework easier, as there is currently no computer lab in our dorm.

Elementary Logic, Prof D (Matt DeLong)

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. Also, there are paradoxes which seem to defy common sense. We shall see how putting some paradoxical examples into the regimented forms demanded by logic causes the paradoxical element to disappear. 1-2 stars

AMC 10, Ms O’Neill

Working on problems from old AMC-10 exams, we will study the many ways in which the traditional topics covered in early high school math classes can provide the basis for more challenging questions. Students will become familiar with many problem-solving techniques as well as interesting applications of familiar concepts. 2 stars

Abelian Sandpile Models, David Perkinson

The Abelian sandpile model (ASM) is a game played on a graph. It has surprising and wonderful connections with a wide range of modern mathematics: combinatorics, probability theory, algebraic geometry, number theory, and pattern formation. The ASM provides a great pathway into these subjects. One goal of the course could be the recent graphical version of the Riemann-Roch theorem. 2-3 stars

Probability through Games, Prof S (Daniel Showalter) In this course, we will explore some concepts of probability underlying two rather simple games, one involving dice and the other involving playing cards. For the dice game, you and your opponent each have k tokens that you must place on the integers 2 to 12. You then take turns rolling a pair of dice. If the sum of the dice matches a number that you have at least one token on, you remove one token from that stack. The winner is the first player to remove all their tokens. We will use probability to figure out the optimal placement of k tokens, assuming that you have no information about your opponent’s placement. We will then tackle more difficult problems such as how the optimal strategy might vary based on information about your opponent’s placement or different types of costs and rewards for playing the game. Studying the probability behind these two games more in depth will (hopefully) help prompt similar types of questions and explorations related to your favorite games and more general situations involving probability. A basic understanding of probability will be helpful, but not necessary. 2 stars

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 the sphere in building the model of elliptic Geometry. 3 stars

Week 4, Morning

The Binomial and Poisson Distributions, Prof B (Owen Byer) What is the probability that three different calls are made to an Emergency Response Unit in a given hour? This is a typical Poisson Distribution question. In this breakout, we will derive the formulas for the Binomial and Poisson Distributions from beginning principles, using the notion of a limit to see how the number e appears in the Poisson formula. We will see many applications of the two distributions, both whimsical and practical. Students taking this course should be familiar with basic probability and familiarity with combinations and permutations will be very helpful. 3 stars

Ramsey Theory, Ben DeMeo

Ramsey theory challenges the notion of randomness by showing that in certain cases, order must appear. For example, within any group of six people, there will always be either a group of three mutual friends or a group of three mutual strangers. This is the most intuitive statement of a fundamental graph-theoretic concept: if we two-color the edges of a complete graph on six vertices, we are guaranteed a monochromatic triangle. Indeed, Given enough vertices, we can guarantee complete monochromatic subgraphs of arbitrary size! Other surprising results abound. We will show, for example, that for K large enough any two-coloring of the integers 1 through K contains a monochromatic arithmetic progression, and that any sufficiently large collection of points in the plane contains a convex n-gon. There will be some overlap with the advanced graph theory course, but this course will focus more exclusively on Ramsey theory and cover it in more depth. Background in graph theory and combinatorics (as would be offered by basic courses here or on AoPS) is required. 3 stars

AIME Problems, Sinan Kanbir

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. 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.

Finite State Machines, David Mende

A finite-state machine is an abstract mathematical model of a computer with a finite amount of memory. A deterministic finite automaton (DFA) consists of a finite alphabet of symbols, a set of labeled states (including start and end), and a transition function for each state that reads in a symbol and gives the next state. One way to describe a DFA is by the set of strings that it accepts, when the last transition is to an end state. Here are some of the questions we will explore:

Given a set of strings, is there a unique DFA that accepts it?

If not, how can we tell if two DFAs are equivalent, that is, accept exactly the same set of strings?

If they're equivalent, how do we find the one with the fewest number of states?

Can we find ways to combine multiple DFAs into a single one with the same functionality?

Are there any sets of strings for which it is impossible to build a DFA that accepts only those strings? Can we characterize these?

What happens if we allow non-determinism by allowing multiple transitions after reading in a symbol?

What if we allow the machine to have unbounded memory or read the input more than once? In analyzing these questions, we will discover many interesting results together. If you want to eventually understand Turing machines, the Halting problem, or the P vs. NP problem, you need to start with finite-state machines!

Cryptology, Jon Rogness

See the description in Week 2 morning.

Topics in Advanced Graph Theory, Kip

We will study Ramsey Numbers, from the basic idea through to an exploration of how the search continues for larger ones, and we’ll prove an important theorem that helps narrow that search. There will be colouring tasks which underpin a greater understanding of the concept. We will also study two theorems associated with Hamiltonian Cycles in graphs, one providing a sufficient condition and the other a necessary condition for the existence of a HC in a graph. Time permitting, we will also look at algorithms associated with the Chinese Postman Problem, and the achieving of upper and lower bounds for the Travelling Salesman Problem. 2-3 stars. Prerequisite: some previous experience with graph theory, such as from taking the Elementary Graph Theory Breakout.

Week 4, Afternoon

Boolean Algebra, Quarternions and Other Strange Creatures, Dr. Janet (Prof Janet Barnett) Alternate titles for this course might be “What (good) is British Symbolic Algebra?”or “Network circuit design made easy through algebra”. As important as these titles, which will make more sense afterwards, is the method. Namely, we will follow the advice of celebrated mathematician Niels Abel and actually “Study the Masters!” That is, we will read original papers written by the individuals who were responsible for major mathematical breakthroughs in the creation and use of a new kind of algebra. Actually, we will complete a set of short “primary source projects” which contain part or all of the original papers, along with commentary and exercises that will guide us to learn the subject. Dr. Janet is an expert on writing and using such projects. Our focus will be on learning about the structure of a Boolean algebra through the project "Applications of Boolean Algebra: Claude Shannon and Circuit Design”. But first we will briefly explore the quarternion group, an algebraic system discovered by William Rowan Hamilton in 1843 which remains important in both mathematics and quantum physics today. Time permitting, we will also look at how Hamilton connected an even more complicated algebraic structure with the study of graph theory in the project “Hamiltonian Circuits and the Icosian Game.” 2-3 stars Polygonal Areas, Prof B (Owen Byer) We all know the simple formula for finding the area of a triangle, but there are many other more interesting formulas, some of which extend to finding the areas of quadrilaterals and general polygons. We will learn lots of geometry while exploring these formulas, including the fact that if every other vertex of a polygon is translated by a fixed vector, the areas of the original and resulting polygon are equal. 2 stars

Olympiad Problems, Sinan Kanbir

Practice with Olympiad problems drawing on the USAMO and possibly other national Olympiads, including the Turkish Olympiad. Prerequisite: you have already taken the AIME. 4 stars

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. We'll start by learning a few basic rules in topology. Then we'll look at how to construct the torous and 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. 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.) 2 stars

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

Elementary Graph Theory, April (April Verser)

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. 1 star

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