Matroids Wk 3 (Happy New Year)

Happy New Year

• I’m so happy to be home this year. It’s been awesome and so relaxing (I legit had a (real; not that fake overpriced nonsense in the US grocery stores!) mango this week, and conchs!! And I saw some parang!) , and I have actually been doing work that doesn’t feel like work at all, in the late night to early mornings. There are some friends I’m catching up with in the New Year, but to be honest, I really just want to chill out at home for the most part. Being home reminds me of how awesome my parents are, and it’s been a lot of fun. Our whole family is. They’re hilarious and we really enjoy watching British Detective movies together. I’ve been hooked on Silent Witness, which they introduced me to this year (they also like Wycliffe.) The deal is that my dad is the one who likes blood splashing everywhere in movies, and my mom likes movies with people who are already dead, with things like maggots and worms crawling about. She is totally into autopsies with scalpels and bloated bodies with worms crawling out and knows some of the antibiotics and chemical reactions and compounds of things just randomly because she’s just that kind of person. My dad is more into the ten-people’s-heads-being-cut-off-with-a-sword and blood gushing everywhere movies. They’re both very clinical, analytical people, so there is another layer of hilarity by their casualness about these things, that would make most ordinary people squeamish. And I love it. They also LOVE numbers (my dad, as it relates to people and systems, as he is trained in economics, and will talk your ears off about stats and databases, and my mom just generally, in terms of mathematical puzzles; she’s really good at them and totally unapologetic about it!
• Interestingly, there was a time in my home country where they would not hire women in managerial positions at the banks (they were for mostly white expat men, and the programme apparently was literally called “Boys in Training”, which makes me want to throw up in my mouth. I did not know this before this Christmas season, btw. Both mom and dad told me about this, as they’ve been together forever practically (like, since high school), so my dad knows a fair amount about the kind of stuff she had to put up with early on, and vice versa). And even then, when she worked in one briefly, they had her in jobs that required acute mathematical ability, particularly when there was a customer rush. She later quit when she realized how stupid the rules of the banks were regarding management, and shortly thereafter, the Black Power movement took place, which resulted in the banks changing their policies. I mean, imagine this happening in a predominantly black country, too? Wth.). Anyways, it’s nice to be in a place where I can be myself, where I am assumed to be intelligent, and laugh heartily about macabre things without getting stares. It reminds me that I fit here.

Anyways

• I started digging into the Oxley book this week. I just completed Chapter 1, and it’s a lot. I have a separate book with hand-drawn notes and attempts of proofs and stuff like that, but they won’t live here. Just general topics / notes and readings will.

Topics

• Chapter 1:
  • Circuit elimination axiom
  • E is usually called the ground set in the book (also denoted E(M) for ground set of the matroid
  • A matroid that is isomorphic to the cycle matroid of a graph is called graphic
  • A matroid that is representable over some field (F) is called representable, coordinatizable or linear or metric
  • Not every matroid is graphic or representable
  • An element e is a loop of an arbitrary matroid M if {e} is a circuit of M.
  • A parallel class of M is a maximal subset X of E(M) st. any two distinct members of X are parallel and no member of X is a loop. It is trivial if it contains one element.
  • If M has no loops and no non-trivial parallel classes, it is called simple. For Simple Matroid (Macaulay has good definition): one that is obtained from M by deleting all loops, and all but one element from each parallel class. Lattice of flats has the empty set as minimal set, and all atoms are singletons. See link
  • A maximal independent set of M is called a basis or base of M.
  • Uniform matroid of rank U_{m, n}. Unique matroid on the empty set is called empty.
  • Paving : if a matroid has no circuits of size less than r(M) where r(M) is rank.
  • See Welsh (1976) for elongation of M to rank k
  • Closure operator: function of M satisfying properties (see CL1 through CL4)
  • A flat or closed set of M (see def)
  • Spanning sets of M. See rules for minimal spanning set, etc.
  • Strong circuit elimination
  • Binary matroids
  • Geometric matroids of small rank (1.5)
  • Fano Matroid (F_7): Covered in Federico lectures. It is the matroid that corresponds to the 7-point projective plane.
  • Escher Matroid is a thing (Brylawski and Kelly, 1980).
  • Binary affine cube as it relates to the six faces of the cube (two twisted planes). Also real affine cube.
  • The relaxation of M, where each matroid is obtained by relaxing one of the second circuit-hyperplanes in F_7.
  • Pappus and non-Pappus Matroids (relates to Pappus configuration in projective geometry). See Pappus Hexagon Theorem
  • Transversals and partial transversals (we use the idea of matchings to check these)
  • Simplification of a matroid
  • Chains and the Jordan-Dedekind chain condition
  • Atom (as relating to cover of posets)
  • Semimodularity : satisfies the Jordan-Dedekind chain condition and satisfies for every pair (x,y) of elements of (finite) lattice L (compare with geometric lattice of finite semimodularity). The partition lattice (K_n)
  • Kruskal’s Algorithm (1956)

Readings

• Welsh, 1976 (Elongation of M to rank k)
• Brylawski, 1986a (Higgs Lift)
• Brylawski and Kelly, 1980: Escher Matroid

Other:

• Read through Quantum Chapter 2 over the course of a day this week. I’m probably going to do another read through before the next gathering, and I began making my way through the proofs (things like, prove that these are conjugate linear, working with Hermitian operators, etc). I do have one more week to do another read through, but exercises are a lot like Spectral so they definitely are doable.

For the New Year

• I don’t know. I’m so focused and being home has been such a joy that I just want to continue on that vibe of not being bothered by annoyingly toxic people. I joked that this year was that of “Bye Felicia”. I really like that for the most part, my family and that of my peers in Pure Maths in grad school understand when you need to focus, and give you room, although we do have social events and that sort of thing. They’re incredibly positive and I feel supported in that space, like I am not being undermined or anything like that (I didn’t find any knives in my back recently!). And I’ve already felt like I’ve grown so much, and want to live up to the standard at which my peers in that broader community do research and publish (it lines up with my desires for my personal standards, too!). I also feel like I’m ready to move on from some of the other communities that were kind of just exploitive of me earlier in my degree, and haven’t really given back or supported me much, IMO. Going back would feel like trying to force a conversation with an acquaintance with whom I don’t have much in common anymore. It is what it is and I really like the direction I pivoted to, and I want to stick with that. And I don’t need to explain it to anyone, or justify anything, or bring anyone along. This is my journey. And I’m pretty psyched. It will continue to be challenging, but I want it, in a way that I was lukewarm about other things in the past a couple years ago. I also just don’t feel like I have time for the other nonsense distractions. There is so much stupid nonsense, especially at schools where there aren’t a lot of people who look like you. None of that is my job and some tenured person with a fancy salary and actual proper health insurance can totally do that stuff (ha). So I think I’m in a good place, and just want that to continue in 2023. I’m super thankful for my Pure Maths friends in 2022, and for my family (and the dogs!). That’s really all I need.
• Also, there are things I am excited about. I will try to blog about them after they happen, I think (a consolidated post).
• I should really get to bed; my mom is probably going to yell at me to head to sleep, because we shared a pot of tea (we drink a lot of tea; former British colony and all that!) and had Chamomile together, but apparently it worked on her and not on me or something.

And that’s it!