From the teaching frontAdded September 21, 2010: One of the teachers who took the class has used Minkowski theory in her own lesson and reported that the lesson went well. The students turned out to be inquisitive and discovered by themselves that there are symmetrical convex figures with area greater than four not intersecting the lattice at all if the figure is not centered on the origin. They asked: "Why did Minkowski center the figure on the origin?"Such questions are hard to answer because the obvious answer "without that assumption there is no theorem" sounds cheap. It also provokes questions like whether there is a related theorem (with additional assumption) in which only the symmetry assumption with respect to an other point is assumed. GraduationAdded June 2, 2010: graduation of Kalyan Mekala and Anselm Snodgrass (both students in Math E320). Click for a larger photo.DynamicsWhile working on the Collatz problem, Amselm brought up the Palyndrome 196 problem. Let S(x) be the number x written reversed. Like S(1262) = 2621. Define T(x) = x+S(x). A number x is called a Lychrel Number, if the orbit x_{n} of x never hits the palindrome set P = { x = S(x) } . The following Mathematica code (written during class) does itF[n_]:=Module[{},s=IntegerDigits[n];m=Sum[s[[k]]10^(k1),{k,Length[s]}];If[m==n,Print[n]];m+n] ListPlot[Log[NestList[F,696,1000]]]See p196.org. Apropos Collatz: see this. Cryptology 
Set theory 
CalculusNick sent this link about Ed Burger (who appeared in the presentation) winning an award for great teaching.When looking at Poincaré's counter intuitive examples sum 1000^{n}/n! sum n!/1000^{n}(the first sum diverges experimentally and the second one converges experimentally), we have all seen pretty quickly that the left hand side is just e^{1000} and convergent. To see that the right hand side is divergent, we had to improvise a bit but we found out with the help of Sterlings formula that already for n=3000, the terms n!/1000^{n} are larger than 1. Here are some additional and still sketchy remarks [PDF] complementing the summary handout. Kalyan mentioned the Indian mathematician Madhava who has developed some series in the 14th century. Added, April 3rd: Take it to the limit. A nice presentation of Archimedes argument on how to derive the area of a disc. First get the circumference, then the Area. Note that pictures can be misleading as the Schwarz paradox of 1890 shows and which I learned as a first year calculus student in Christian Blatter's book "Analysis I,II". one can triangulate a cylinder surface S of radius 1 and height into finer and finer pieces and take the limit of these areas and get infinity. Make a subdivision into n^{4} squares by subdividing the height into n^{3} and the angles into n pieces. We get a triangulation into 4 n^{4} triangles if we draw the diagonals in each square. The area of each triangle is large than 2 sin(pi/n) sin^2 (pi/2n) > 4/n^{3}. The sum of the areas goes to infinity like n. Added, April 18, 2010: Strogatz column in the NYT on the fundamental theorem of calculus. 
Group TheoryMany puzzles are too difficult to cover in one hour. But there are some which work. The pyramorphix puzzle has 24 group elements. An other group accessible is part of the edge corner group of the rubik cube, which has only 36 elements. One can easily check this with GAP.gap> Size(Group(( 1,2,3,4)(5,6), ( 1,2)(3,4,5,6))); 36Here is the size computation of the Rubik cube: rubik := Group( ( 1, 3, 8, 6)( 2, 5, 7, 4)( 9,33,25,17)(10,34,26,18)(11,35,27,19), ( 9,11,16,14)(10,13,15,12)( 1,17,41,40)( 4,20,44,37)( 6,22,46,35), (17,19,24,22)(18,21,23,20)( 6,25,43,16)( 7,28,42,13)( 8,30,41,11), (25,27,32,30)(26,29,31,28)( 3,38,43,19)( 5,36,45,21)( 8,33,48,24), (33,35,40,38)(34,37,39,36)( 3, 9,46,32)( 2,12,47,29)( 1,14,48,27), (41,43,48,46)(42,45,47,44)(14,22,30,38)(15,23,31,39)(16,24,32,40) ); Size(rubik); 43252003274489856000Online solver for rubic puzzles. Added March 8: An article of Melanie Bayley in the New York times takes about the Math in "Alice in Wonderland". Some quotes:

Forth lectureThere are two difficulties to understand fully the proof of Wilson's theorem:1) If p is a prime, then for any a, there is a pair b, such that a*b = 1 mod p has exactly one solution 2) If p is a prime, then the equation x^{2}=1 mod p has two solutions x=1 and x=1. To 1) One could see this with Fermat's theorem: just take b = a^{p2} to get one solution. It can not be that we have two solutions b,c because a*b = 1 mod p and a*c = 1 mod p implies a(bc) = 0 mod p which means that p either divides a or (bc). It can not divide a because a is smaller than p. It must divide (bc) therefore, which means that b and c differ by a multiple of p. To 2) x=1 and x=1 both are solutions. Assume there is an other solution y, then y^{2}1 is divisible by p. Write y^{2} 1 = (y1) (y+1). Therefore, p divides (y1) or (y+1) which means that y differs from 1 by a multiple of p or differs from 1 by a multiple of p. As Brahim has pointed out, it is not obvious to see from the expression / n \ n (n1) (n2) .... (nm+1) ( ) =  \ m / m (m1) (m2) ... 2 1is an integer. It follows from the fact that these terms appear when expanding (x+1)^{n} and from combinatorial consideration as the number of possibilities to chose m elements from n. We have seen that this Binomial coefficient is divisible by n, if n is prime and if m is positive and smaller than m because there is nothing in the denominator which could cancel n. It is actually true that all these terms are all divisible by n if and only if n is prime. During the presentation part, I mentioned that the coincidence that the perfect number 28 is the length of the month might have contributed to the mystical attraction to primes. Eliza has pointed out that the month is not 28 days but 29. Indeed, there are many different months: on a lighter note, we can add that the average of the month lengths {27.554549878, 27.321661547, 27.321582241, 27.212220817, 29.530588853 } is 27.7881, pretty close to 28 .... Aliza also pointed out that the bible reference (given in Dickson) to the perfect number 6 is in the source actually referring to 56. While this confirms how crazy any numerology is, it is interesting to put oneselves into the mind of the spiritual past. It has influenced, how mathematics has developed thousands of years ago and why certain questions were persued more vigorously. Spiritual considerations certainly did influence the Pythagoreans. Added February 24: we have seen in class how many open problems there are in number theory. Elementary number theory is attractive for amateur mathematicians because one can do so many experiments. As an example, I go an email tonight from a number theory enthusiast Bill McEachen about a conjecture he calls the primorial conjecture which is of a similar spirit than Goldbach. It is a claim about the primorial numbers pm(n) = p_1 p_2 ... p_n, the products of the first primes. The primorial numbers PM={2,6,30,210,...} grow much faster than the factorials {1,2,6,24,120,...}. If P={2,3,5,7,...} is the set of primes and Q={...7,5,3,2,1,1,2,3,5,7,...}, the claim is that PM+Q contains all prime numbers. It is hard to get a grip on this numerically because primorials grow so fast. (I myself do not know much about this conjecture and only checked it for the first few thousand primes). Goldbach's claim is that P+P contains (2N+2), if N={1,2,3,4,...} is the set of natural numbers. The difficulty of these problems is that the addition and multiplication in N are so incompatible. In general, it is hard to make statements about the addition A+B={a+b  a in A, b in B} of sets. The Sieve of Erasthostenes with which our lecture started, is the statement that the complement of M*M is P, where M={2,3,4,5,6,...} and A*B = { a*b a in A and b in B }. The complement of the primes P is a union of regular arithmetic progressions like 7M. But P is very irregular. Erastothenes insight is to note that M*M = (2M v 3M v 4M v ...) where v is the union and to see that this is an effective method to compute the primes. Third lectureAs expected, we did not have time for all handout topics, but we did work out Pythagoras, Thales and Hypochrates in detailed and discussed in the presentation part the 4 special points in the triangle. Some interesting points:
Added March 15, 2010: The new Strogatz article in the NYT deals with geometry. 
Second lectureWe have seen two great moments in mathematics: the introduction of counting, and the realization that there are numbers beyond fractions. A few important things:
Here are some additional remarks about arithmetics [PDF] complementing the summary handout. Added February 14: Strogatz column about negative numbers explains why (1) x (1) = 1. Already the title of the column does. But the article explains well that such computations are on a different level than 3*4 = 12 which have a concrete realization as taking 3 groups of 4 objects. The computations for negative numbers follow rules so that an algebraic structure prevails and for example the arithmetic progression 2*(1) = 2, 1*(1) = 1, 0*(1) = 0, (1) * (1) = 1 survives. The multiplication 3*(4) = 12 still has an interpretation like 3 times borrowing 4 objects but (4)x3 does not have it so easy any more. Today, Mathematicians rely on algebraic structures like "any Abelian group is also a Z module" and feel at home, but  as Strogatz points out  there can still be discomfort when learning or teaching how to work with negative numbers. One of the readers of the columns recalls a joke illustrating the mystery of negative numbers: A mathematician, a biologist and a physicist are sitting in a street cafe watching people going in and coming out of the house on the other side of the street. First they see two people going into the house. Time passes. After a while they notice three persons coming out of the house. The physicist: "The measurement wasn't accurate." The biologist: "They have reproduced." The mathematician: "If now exactly one person enters the house then it will be empty again."A math teacher explains it as follows: When good things happen to good people, it's good. (+)(+)=(+) When bad things happen to good people, it's bad. ()(+)=() When good things happen to bad people, it's bad. (+)()=() When bad things happen to bad people, it's good. ()()=(+)Added March 4, 2010: The earliest writings and maybe also computations seem older than thought. Eggshells used as containers were labeled (with numbers telling how much they contained?) Added April 22, 2010: the following story shows that fractions can be tricky. Without a calculator it is obvious that 136 votes against 70 votes is less than 2/3 because 140/70 would be a 2/3 vote. I think the sin to use 0.66 for 2/3 is a typical spreadsheet sin. Using inaccurate numbers in spreadsheets combined with rounding errors make spreadsheet errors so abundant. 
First lecture
Added February 23, 2010: the Notices of March 2010 contain an interview with Mikhail Gromov. He says: "Mathematics not only deals with what you see with your eye but what you see in the structure of things, at a more fundamental level, I would say." An other citation is by David Ruelle writes in his book "The Mathematician's Brain": "From what we have seen, mathematics appears to have a dual nature. On the one hand, it can be developed using a formal language, strict rules of deduction and a system or axioms. All the theorems can then be obtained and checked mechanically. We may call this the formal aspect of mathematics. On the other hand, the practice of mathematics is based on ideas, like Klein's idea of different geometries. This may be called the conceptual or structural aspect." These thoughts fit well with the definition attempt "Mathematics is the science of structure" given at the beginning of this course. 