Mathcamp 2000 -- Schedule for Week 4 -------------------------------------- Tuesday, Aug 1st ----------------------- 9:00 - 9:55 [160] Topology Track **** [166] Number Theory Track ** 10:00 - 11:55 [166] Colloquium: Fermat's Last Theorem I Ken Ribet 12:00 - 1:25 Lunch 1:30 - 2:25 [160] Advanced Number Theory **** (Ken Ribet) [166] More Monster Equations 1 ** (Igor) [256] Analytic geometry * (Dr. Thomas) [260] Functions of a complex variable 4 *** (Mark K, Chris) 2:30 - 3:25 [160] Determinants 1 ** (Mark K) [166] Mathematical genetics ** (David Reich) [256] p-adic numbers 1 **-*** (Cathy O'Neil) [260] Measure and category 1 **** (Aytek, Meep) 3:30 - 4:25 PAUSE TO THINK [256] Problem Solving *-**** (Mark S) 4:30 - 5:30 [160] Discrete Math Track * [166] Algebra Track *** [256] Combinatorics on Words Track **-*** (Mark S) [260] Problem Solving Wednesday, Aug 2nd ----------------------- 9:00 - 9:55 [160] Discrete Math Track * [166] Algebra Track *** [256] Combinatorics on Words Track **-*** (Mark S) 10:00 - 11:55 [166] Colloquium: Fermat's Last Theorem II Ken Ribet 12:00 - 1:25 Lunch 1:30 - 2:25 [160] Differential Equations 3 *** (Madeeha) [166] Cryptography 1 * (Julian) [256] Gorgeous geometry! ** (Igor) [260] Differential topology 3 **** (Noah) 2:30 - 3:25 [160] Probability paradoxes 1 * (Meep) [166] Projective and algebraic geometry 4 **-*** (Mira, Brenda) [256] Elliptic Curves 1 *** (Dave) [260] Special Relativity 1 ** (Noah) 3:30 - 4:25 PAUSE TO THINK [256] Problem Solving 4:30 - 5:30 [160] Topology Track **** [166] Number Theory Track ** [256] Problem Solving Thursday, Aug 3rd ----------------------- 9:00 - 9:55 [160] Topology Track **** [166] Number Theory Track ** 10:00 - 11:55 [166] Spreading out evenly: some problems Richard Anstee 12:00 - 1:25 Lunch 1:30 - 2:25 [160] Some really cool topic, TBA (Aaron Abrams) [166] More Monster Equations 2 ** (Igor) [256] Analytic geometry * (Dr. Thomas) [260] Functions of a complex variable 5 *** (Mark K, Chris) 2:30 - 3:25 [160] Determinants 2 ** (Mark K) [166] The Banach-Tarski Theorem 1 *** (Chris) [256] p-adic numbers 2 **-*** (Cathy O'Neill) [260] Measure and category 2 **** (Aytek, Meep) 3:30 - 4:25 PAUSE TO THINK [256] Problem Solving 4:30 - 5:30 [160] Discrete Math Track * [166] Algebra Track *** [256] Combinatorics on Words Track **-*** (Mark S) [260] Problem Solving Friday, Aug 4th ----------------------- 9:00 - 9:55 [160] Discrete Math Track * [166] Algebra Track *** [256] Combinatorics on Words Track **-*** (Mark S) 10:00 - 11:55 [166] Colloquium: Nim! How to beat your friends at simple games Cathy O'Neill 12:00 - 1:25 Lunch 1:30 - 2:25 [160] Differential Equations 4 *** (Madeeha) [166] Cryptography 2 * (Julian) [256] Young Diagrams ** (Igor) [260] Differential Topology 4 **** (Noah) 2:30 - 3:25 [160] Probability paradoxes 2 * (Meep) [166] Projective and algebraic geometry 5 **-*** (Mira, Brenda) [256] Elliptic Curves 2 *** (Dave) [260] Special Relativity 2 ** (Noah) 3:30 - 4:25 PAUSE TO THINK [256] Problem Solving 4:30 - 5:30 [160] Topology Track **** [166] Number Theory Track ** [256] Problem Solving Saturday, Aug 5th ----------------------- 9:00 - 9:55 [160] Some really cool topic, TBA (Aaron Abrams) [166] The Banach-Tarski Theorem 2 *** (Chris) [256] p-adic numbers 3 **-*** (Cathy O'Neill) [260] Measure and Category 3 **** (Aytek, Meep) 10:00 - 11:55 [166] Colloquium either Aaron Abrams or Mira, TBA 12:00 - 1:215 Lunch NOTE: Shorter than usual! 1:15 - 1:45 [166] CAMP ASSEMBLY -- REQUIRED! 1:45 - 2:40 [160] The 5 Color Theorem * (Chris) [166] Projective and algebraic geometry 6 **-*** (Mira, Brenda) [256] Elliptic Curves 3 *** (Dave) [260] Special Relativity 3 ** (Noah) 2:45 - 3:40 [160] A mathematical theory of drunkenness ** (Chris) [166] A friendly introduction to higher dimensions * (Mira) [260] Functions of a complex variable 6 *** (Mark K) 3:45 - 5:30 [Shuswap, lounge] Relays! ---------------------------- TRACKS Discrete Math * is doing game theory this week. If you want to join, you can -- we're starting from scratch! Algebra Track ***: In true Mathcamp spirit, the plan we told you about on Friday has been completely modified. There will still be only one algebra track (though Evelyne plans to teach representation theory in the evenings, if there's enough interest). It will probably not cover Galois theory. We'll talk some more about symmetry groups, permutation groups (with applications to the 15 puzzle), and then perhaps about finite fields. Combinatorics on words ***-**** (Mark Sapir). A really great class (all the mentors want to take it!), and still accepting new (advanced) students. VISITORS Ken Ribet, one of the key players on the proof of Fermat's Last Theorem, will be giving two colloquia about the history of the theorem. He will present an elementary outline of the proof, and talk about interesting questions that people have asked him about it over the years. He will also teach an advanced class, which should be accessible to everyone in the number theory track who knows about quadratic reciprocity. Cathy O'Neill, a number theorist from MIT, will be giving a colloquium on the game of Nim, a topic that should be accessible and fun for absolutely everyone. Don't miss Cathy, she's one of the coolest people I know! She is also teaching a class on p-adic numbers. Come find out about a strange (and extremely important!) universe where all triangles are isosceles and where 1+4+8+16+32+... is a convergent series. (Can you guess what it might converge to?) Aaron Abrams, a topologist from UC Berkeley, will be giving a couple of classes and possibly the colloquium on Saturday. Unfortunately, he hasn't told me anything about what he wants to teach -- so watch the bulletin board for announcements. (If he doesn't give Saturday's colloquium then it's all mine!) Richard Anstee, who does discrete math at UBC, will be giving a colloquium on "Spreading things out evenly". For example, can you sprinkle numbers from 1 to n into a rectangular grid in such a way that every rectangle has its fair share of every number? Can you arrange numbers from 1 to n in a circle so that any k of them will have the same average? Lots of nice problems will arise, and he is confident that you guys have s hot at solving some of them. NEW CLASSES Cryptography * (Julian). How can you flip a coin over the phone and both know that the other person didn't cheat? How can you hold a confidential conversation over the Internet? These problems can both be solved using cryptographic techniques. In the first session, we will look at some sorts of problems for which cryptography is useful and how we can use it. In the second we will look at two well-known encryption techniques: RSA and Diffie-Hellman. Prerequisites: the second session will make some use of Euler's theorem taught in the number theory track, but we'll go over the statement again. (If there is sufficient demand, this mini-course may extend into week 5.) Probability paradoxes * (Meep). Haven't had enough paradoxes yet? Add still more confusion to your life--this time about probability. The Qualifying Quiz project will probably be one of the topics discussed. Banach-Tarski Theorem -- or Paradox if you wish *** (Chris). So you're just about to take a big bite out of your apple when your friend comes up and asks to share it. Half an apple just won't be enough, but you don't want to refuse -- what to do? Is it a bird, is it a plane -- no, it's the Banach-Tarski theorem to your rescue! You can divide the apple into finitely many pieces, rearrange them using rotations and translations, and put them together to get two apples the same size as the first. Problem solved! With the help of group actions and the axiom of choice we'll prove this, and finish up by thinking about what goes wrong when you actually try to take the knife to the apple. Measure and Category **** (Aytek). After going over the basic definitions of measure, density, and nowhere density, we prove the Baire category theorem and start looking at some crazy-looking results. Clever constructions of bizarre sets will show how dramatically wrong intuition can be! Required background: notion of countability, being comfortable with epsilon-delta arguments. More Monster Equations** (Igor). We will dip further into the world of irrational equations and then explore techniques for solving trigonometric equations. We will even find pretty connections between the two worlds (Ramanujan's formulae). Attendance of last week's classes is not required. Gorgeous Geometry** (Igor). We will start off with a beautiful and elementary Steiner's Theorem (given a triangle, find the inscribed triangle with the smallest perimeter). Then we'll move into outer space and discover why exactly our solar day is slightly longer than our star day. Time permitting, we will derive a few analytical geometry results. Inequalities and Young's Diagrams ** (Igor). A general method for proving symmetric inequalities. Problem Solving Farewell Special ***-**** (Igor). This will be my last week of camp, so as a finale, I want to discuss the most interesting and difficult problems from the previous two weeks of problem sessions. Please pick up a copy of the problem sheet at the Mathcamp lounge in advance (will appear by Tuesday). If you'd like to present a solution, talk to me. The Five Colo(u)r Theorem * (Chris). How many colors do you need to color every country on a map so that no two adjacent countries are the same color? You've probably heard that you can do it with just four colors -- the famous four color theorem, proved in 1976 with the help of a computer by Appel and Haken, over one hundred years after the problem was first posed. We'll be a little less ambitious and prove the simpler "five color theorem" -- five colors are enough. The proof has most of the main ideas used in the proof of the four color theorem but is simple enough for a roomful of humans to carry out in an hour. Special Relativity ** (Noah). Title says it all: we'll learn about Einstein's special theory of relativity. I'll be using lots of pictures and geometric ideas to motivate the theory. Analytic Geometry * (Dr. Thomas). Analytic geometry is a method of solving geometric problems using the one-to-one correspondence between points on a line and the real numbers to establish a coordinate system for points in the plane. This allows the use of algebra and analysis to prove theorems in geometry. We shall discuss FIFTY GEMS from analytic geometry dealing with conditions and equations concerning points, lines, and conics. The classes are mostly independent from one another, so if you miss one or two, you can still come. Elliptic Curves *** (Dave). In last week's Diophantine Equations class, we studied primarily quadratic Diophantine equations, which had nontrivial solutions if and only if they had solutions (mod n) for all n. This week, we will discuss Diophantine equations called "elliptic curves", which are equations of the form y^2 = x^3 + ax + b. The main feature of these curves is that their points can be made into a group using geometric ideas, which allows fantastic insight into the arithmetic of these equations. You'll probably hear more about them at Ken Ribet's talk -- they're crucial to the proof of Fermat's Last Theorem. Determinants ** (Mark K). They come up absolutely everywhere (including next week's colloquium by Neil Sloane), they're incredibly important, and although they are part of linear algebra, they require almost no background. Come find out what they're about! A friendly into to higher dimensions * (Mira). Confused about what comes after 3? Come and practice looking at hypercubes and it'll make more sense! Lots of fun with patterns and numbers. EVENING CLASSES Yes, it's =that= time of camp again! The schedule is getting so full, and the demand is so high, that we are beginning to have evening lectures. Watch the calendar wall for announcements of dates, times, and places. Also, watch for math movies -- we have some really cool ones that everyone should see! Permutation Groups *** (Julian). We will explore some of the basic properties of groups of permutations in the first session, using them to derive some simple results from group theory. We will then apply these results in the second session to a more difficult situation, deriving some beautiful results called Sylow's theorems. Finally, in the third session, we will look at some algorithms for computing in permutation groups, and discover how we could use these results to solve problems such as Rubik's cube. Prerequisites: basic knowledge of group theory, up to and including normal subgroups, as has been taught in the Algebra track. Representation Theory *** (Evelyne). A beautiful area exploring the connection of group theory and linear algebra. Prerequisites: You should be pretty comfortable with the basics of both groups and linear algebra -- but it's worth it!