Mathcamp 2000 -- Schedule for Week 2 -------------------------------------- Tuesday, July 18th ----------------------- 9:00 - 9:55 [160] Topology Track **** [166] Number Theory Track ** [256] Problem Solving * (Chris) [260] Calculus 5 ** (Madeeha) 10:00 - 11:55 [166] Colloquium: Folding Polygons into Polyhedra Anna Lubiw 12:00 - 1:25 Lunch 1:30 - 2:25 [160] Discrete Math Track * [166] Algebra Track *** [256] Problem Solving *-**** (Mark) [260] Geometric Inversion 1 ** (Aytek) 2:30 - 3:25 PAUSE TO THINK 3:30 - 4:25 [160] Adding Things Up 1 * (Julian) [166] Epsilon and Deltas 1 ** (Evelyne) [256] Weird numbers 1 (indigo, Mira) [260] Olympiad Topics and Problems **** (Kiran Kedlaya) 4:30 - 5:30 [160] Proofs and Reasoning 4 * (Meep) [166] Cauchy's Rigidity Theorem ** (Anna Lubiw) [256] Trees and Ps and Things that Sneeze 1 ** (Chris) [260] Generating Functions and Partitions 1 *** (Mark, Julian) Wednesday, July 19th ----------------------- 9:00 - 9:55 [160] Discrete Math Track * [166] Algebra Track *** [256] Problem Solving *** (Aytek) 10:00 - 11:55 [166] Colloquium: TBA (Kiran Kedlaya) 12:00 - 1:25 Lunch 1:30 - 2:25 [160] Topology Track **** [166] Number Theory Track ** [256] Problem Solving *-*** (Julian) [260] False Proofs ** (Meep) 2:30 - 3:25 PAUSE TO THINK 3:30 - 4:25 [160] Igor's Mathematical Medley **-*** (Igor Pavlovsky, new mentor) [166] Advanced Number Theory 3 *** (Dave) [256] Non-Euclidean Geometry 4 * (Dr. Thomas) [260] Linear Algebra 5 ** (Mira, Aytek) 4:30 - 5:30 [160] More on Mathematics and the Brain (Josh Tenenbaum) [166] Folding + 1 Cut = Everything * (Anna Lubiw) [256] Reconstructing the Reals *** (Chris, indigo) [260] Lagrangian Mechanics 1 **** (Noah; requires calculus!) Thursday, July 20th ----------------------- 9:00 - 9:55 [160] Topology Track **** [166] Number Theory Track ** [256] Problem Solving *-**** (Mark) [260] Calculus 6 ** (Madeeha) 10:00 - 11:55 [166] Colloquium: Digital Image Compression Hany Farid 12:00 - 1:25 Lunch 1:30 - 2:25 [160] Discrete Math Track * [166] Algebra Track *** [256] Problem Solving *-*** (Chris) [260] Geometric Inversion 2 ** (Aytek) 2:30 - 3:25 PAUSE TO THINK 3:30 - 4:25 [160] Adding Things Up 2 * (Julian) [166] Digital Image Morphing ** (Hany Farid) [256] Weird Numbers 2 *** (Mira, indigo) [260] Olympiad Topics and Problems **** (Kiran Kedlaya) 4:30 - 5:30 [160] Advanced Number Theory 4 *** (Dave) [166] Efficient Algorithms and the P=NP Question ** (Anna Lubiw) [256] Non-Euclidean Geometry 5 * (Dr. Thomas) [260] Linear Algebra 6 ** (Aytek, Mira) 3:30 - 5:30 [Shuswap, near Suite 213] Zometool Geometry Workshop 1 (George Hart) Friday, July 21st ----------------------- 9:00 - 9:55 [160] Discrete Math Track * [166] Algebra Track *** [256] Problem Solving *-**** (Mark) [260] Olympiad Topics and Problems **** (Kiran Kedlaya) 10:00 - 11:55 [166] Colloquium: Geometry and Art George Hart 12:00 - 1:25 Lunch 1:30 - 2:25 [160] Topology Track **** [166] Number Theory Track ** [256] Problem Solving (Igor) [260] False Proofs 2 ** (Meep) 2:30 - 3:25 PAUSE TO THINK 3:30 - 4:25 [160] Epsilons and Deltas 2 ** (Evelyne) [166] Huffman Coding ** (Hany Farid) [256] Weird Numbers 3 *** (Mira, indigo) [260] Lagrangian Mechanics 1 **** (Noah; requires calculus!) 4:30 - 5:30 [160] Proofs and Reasoning 5 * (Meep) [166] Igor's Mathematical Medley **-*** (Igor) [256] Trees and Ps and Things that Sneeze 2 ** (Chris) [260] Generating Functions and Partitions 2 *** (Mark, Julian) 3:30 - 5:30 [Shuswap, near Suite 213] Zometool Geometry Workshop 2 (George Hart) Saturday, July 22nd ----------------------- 9:00 - 9:55 [166] CAMP ASSEMBLY -- REQUIRED! 10:00 - 11:55 [166] Colloquium: Quantum Computation Jonathan Tannenhauser 11:00 - 11:55 [160] The Basics of Complex Numbers (Mark) 12:00 - 1:25 Lunch 1:30 - 2:25 [160] Advanced Number Theory 5 *** (Dave) [166] Digital Image Enhancement ** (Hany Farid) [256] Non-Euclidean Geometry 6 * (Dr. Thomas) [260] Linear Algebra 7 ** (Mira, Aytek) 2:30 - 3:30 [160] Calculus 7 ** (Madeeha) [166] More about Mathematics and the Mind ** (Josh Tenenbaum) [256] Igor's Mathematical Medley **-*** (Igor) [260] Lagrangian Mechanics 2 **** (Noah; requires calculus!) 3:30 - 4:25 [Shuswap, near Suite 213] Zometool geometry Workshop 3 (George Hart) [Shuswap, lounge] Relays! --------------------------- More Information on Classes and Talks: -------------------------------------- Anna Lubiw: Folding Polygons into Polyhedra. This talk is about polyhedra, which have fascinated people for centuries. One way to build models of polyhedra is to start with a piece of paper, cut it out to the right shape, and fold it up, gluing the edges together. For example, you can make a tetrahedron from a triangle by folding up the three corners and gluing them together. In this lecture I will talk about this process. In one direction, we want to slice along the edges of a polyhedron and unfold it to get a polygon. Can this always be done? In the other direction, we start with a polygon and want to glue up its perimeter to form a polyhedron. Can this always be done? What different polyhedra can be made from one given polygon? The talk will include basic material on polyhedra, discussion of a powerful theorem of Aleksandrov, some new research and algorithms, and some hands-on folding of polyhedra. Cauchy's Rigidity Theorem. Though Cauchy is most famous for his results in analysis, early in his career he proved an important result in geometry. The result is about the rigidity of convex polyhedra. It says that if you specify the geometry of each face of a convex polyhedron, and specify which faces attach to which other faces, then the whole polyhedron is uniquely determined. To put it a little more physically, a cardboard model of a convex polyhedron is rigid -- it cannot be flexed into different polyhedra. In this talk I will prove Cauchy's theorem. Though the result is deep, the talk will be elementary. Folding + 1 Cut = Everything. This lecture assumes very little background: just some Euclidean geometry, which I will review. Suppose you take a piece of paper, fold it up to something flat, and chop it with a paper-cutter. What can you get? The surprising result is that you can get anything -- for any collection of straight line segments on the paper, there is a way to fold the paper and chop once so that you cut exactly the given line segments. In this talk I will show many examples of this, invite you to discover how to get any triangle, and begin to prove the main result. Efficient Algorithms and the P=NP Question. One of the most fundamental open problems in mathematics is actually a problem from computer science, the P=NP question. I will give an elementary introduction to this topic. The talk will not assume any particular programming or computer experience. Hany Farid: Digital Image Processing. The colloquium talk will be called "Digital Image Compression". The core mathematical idea here is the notion of a linear basis, with an emphasis on the Fourier basis and its use in image compression. The math is not too complicated, and there will be lots of demos. I will also show a simple way to do digital morphing using the Fourier basis. The second talk of that same day will be titled "Digital Image Morphing". Here we will discuss in more detail how to do Fourier-based and geometric-based morphing. I am hoping to take pictures of the campers and do some live demos. The third talk will be titled "Huffman Coding". This is a simple technique for coding and underlies most image compression schemes. It is more of a computer science talk, as it relies on some simple abstract data types. George Hart: Geometry and Art. Sculptor/mathematician George Hart will present some interesting connections between geometry and art, both historically and in his own work. This includes slides of some of his sculpture made of metal, wood, acrylic, paper, CD-ROMs, floppy disks, forks, knives, spoons, etc. Each of Hart's works is founded on a mathematical structure of some type. The underlying geometry of several pieces will be explained. For examples of his work, see http://www.georgehart.com/ Kiran Kedlaya: Olympiad Topics and Problems. I will focus on specific topics the way I do at the US Math Olympiad Program: e.g., a lecture and problem set on number theory, one on combinatorics, etc. Regular Staff: Trees and Ps and things that sneeze (Chris). If you wanted to reconstruct your family tree, you'd probably start with birth and death records. But what would you do if you wanted to construct your species's family tree? I'll talk about some of the mathematics used in constructing evolutionary trees from data such as DNA sequences. (Note: there will be a colloquium talk on a similar topic next week.) Epsilons and Deltas (Evelyne). Dissatisfied with the sloppy, handwaving definitions of limits and continuity given in your high-school Calculus class? Join us for a two-hour discussion of the "real stuff": the so-called epsilon-delta definitions. Generating functions and partitions (Mark and Julian). Generating functions are a powerful tool for analyzing sequences, and sometimes allow us to get explicit formulas that are hard to get in any other way. After a general introduction, we will apply generating function techniques to the sequence p(n) giving the number of partitions of a positive integer n. A beautiful combinatorial proof due to Franklin will lead to Euler's pentagonal number theorem, which yields a recurrence relation for p(n). (note: This course will probably run over into next week.) Reconstructing the reals (Chris and indigo). Last week they were taken apart, this week we'll put them back together! We'll aim to emerge with a good working definition of what a real number is. Weird numbers (indigo and Mira). Every day, starting from Day 0, new "numbers" are created according to the following rules: Rule 1: Every number corresponds to two sets of previously created numbers, such that no member of the left set is greater than or equal to any member of the right set. Rule 2: One number is less than or equal to another number if and only if no member of the first number's left set is greater than or equal to the second number, and no member of the second number's right set is less than or equal to the first number. Sounds confusing? Sounds weird? You bet! We'll see what we can derive from the two rules for these "weird numbers". We will be working from scratch, so at first we will be showing things like "if a