Just starting out this page. I want a log of the mathematics I think about on a daily basis, keeping track of the progress I have been making on my mathematical self improvement. On the 31st of December I was on a flight from Edinburgh. I spent that time listening to the original 'A Mathematician's Lament' by Paul Lockhart, and it inspired me to work with some ideas I had swimming in the back of my mind. I ended up playing with the concept of Fuzzy Graphs, and established the basic idea for a metric between vertices of this fuzzy graph. Would love to explain more with drawings, but there is more playing and some background reading to do.
I did some background reading on fuzzy sets, and realised that my formulation is slightly different from the usual one. My formulation of fuzzy logic uses multiplication as conjunction, but I now realise that this isn't standard. There are a lot of systems of fuzzy logic and picking one requires a pickyness I'm too stupid to have.
Also for my formulation of fuzzy graphs, my vertex set is a normal set, while the textbook I read uses fuzzy sets for the vertex and edge set of a graph. This is really weird to me, and I think I might stick to my formulation, adapting theorems to suit my formulation.
I want to think more about motivations for fuzzy graph theory. I think there are already tons of known results adapting classic graph theory results to a fuzzy context, but I believe there is more to this. Fuzzy Ramsey theory seems like an unexplored corner.
Watched a video about the construction of the Hyperreals. From what I understand it is a way to contruct a number system which allows for the existence of infinitisimals. It uses ring theory, quotient groups and an interesting theorem in first-order logic called the transfer principle. I cannot say that I understand it, but I find it fascinating. Will investigate further.
I haven't been keeping up this log because my every day mathematics at the university is too mundane to document. Although I will give a big picture overview of the stuff I've learned so far with some extra fun bits.
This semester I'm taking a course in Analysis, Groups, Automata theory, and Logic and Sets. All of these have been fascinating so far, although the course in Groups (my favourite in terms of topics) has the least new mathematics for me. I've learned about the theory of Ordinal numbers, formalisms of propositional logic, thought more about equivalence relatiions and group actions, and thought a lot about classes of languages. Its been a very discrete and very fun semester so far.
I also had the opportunity to hear the proof of a result about the Rado graph from Peter Cameron during a logic and sets tutorial. It was genuinely the coolest proof I've seen presented. "It is the most outrageous claim that I am fully convinced of." The idea of building a graph isomorphism by lining up countable vertex sets and going along them and connecting them is such a cute idea. Like two parallel train lines. I also think the result itself is mind bending.
The Rado graph haunts me every where I look. It turns out that the Rado graph peeks its head when you forget direction on the relationship graph of a model of ZFC. Potentially, very few axioms of ZFC are even required to produce the Rado graph. A friend of mine described it as the 'Pi of graphs' in the way it makes itself found in seemingly unrelated places.
I also have been reading about finite topologies and have done some exploration in the combinatorial aspect of finite topologies on a set. I have found with inspiration from Peter Cameron's book that there is an one to one correspondence between partial pre orders on a set and finite topologies. This is very cool, but counting these is very hard. The number grows very fast and there is no formula known to count these.
Do not get me started on the dutch baby pancakes.
A series of explorations led me to the partition numbers:
The story starts with finite abelian groups. How many finite abelian groups of order 64 are there? We know that each finite abelian group is uniquely (upto reordering) decomposible of cyclic groups of prime power order. So the problem becomes how many ways can we write 64 as a product of prime powers. 64 = 2^6, so this problem breaks down into writing the integer partitions of the number 6. This was the short more direct route to this conclusion, but the path I took was more perilous. I started off by considering multiset partitions of {1,1,1,1,1,1}, which led me to counting non-isomorphic trees of height 2 with n leaves, which was a bit of a road block. These numbers are equivalent to so many things its quite cool. Slightly trivially, these numbers also count the number of conjugacy classes of the symmetric group Sn, as each permutation can be written as a product of disjoint cycles, and must contain each number in exactly one cycle.
I've spent the past few thinking about computation. I read up on Hilbert's 10th problem, and how insane the Matiyasevich theorem is, A set in N is Diophantine iff. it is recursively enumerable. This solved Hilbert's 10th problem as a corollary. Hilbert's 10th problem: Asks about an algorithm for deciding whether a Diophantine equation has solutions. As there are are undecidable (semi-decidable) recursively enumerable sets, there are undecidable Diophantine sets.
I also spent a good while thinking about automatons and the Cook-Levin Theorem. The idea behind the Cook-Levin theorem reminds me of the DPLL algorithm for determining satisfiability. The 'computation tree' of a non-deterministic Turing machine seems like it has the same sort of structure as the tree obtained in the execution of that algorithm. It seems like the translation between non-deterministic Turing machine and a Boolean satisfiability problem could be more 'direct' (in a very imprecise sense), because all non-deterministic Turing machines can do is test whether a word 'satisfies' the conditions given by its automaton. What is SAT polynomially equivalent to?
TREE AUTOMATONS: I found out about this notion of automatons that operate on more arbitrary structures, like trees and finite graphs. This is fascinating but I am curious to know how they differ in expressiveness to word languages.
Orbits and Stabbing: I also spent a good amount of this week proving the first few basic theorems about group actions as a revision exercise for my groups module. It reminded me of the joy in group theory. This along with me working on my paper has got me pleasantly thinking about groups again.