I love what I do for living, but not gonna lie. Sometimes being a mathematical-physicist it's kinda depressing. You study the most part of the time, you fail some exams, you keep studying and then you realize you're into an SU(n) so you can't escape lmao.
Anyways. I love quantum mechanics, numerical methods but I fckn' hate theoretical mechanics bc of newton's formulations. I got an 8 (I think it's like B+ out there) in Lagrange equations this week.
Once I finish this degree, gotta forget about physics and get a math major. You really don't know how beautiful maths are beyond physics derivations. I do really love working with groups, symmetries, categories, manifolds and all that kinky stuff.
Yeh, I know. Some people might ask: why didn't you study maths?
Well, I came here bc I thought physics was beautiful and had a lot of interesting math. I would never have imagined that physical "assumptions" were so ridiculous when solving problems.
Anyways. I really hope ur having a nice week. See ya soon, buddies 👍
Man, this week was kinda weird. I got a 0 in theoretical mechanics (f u Newton), I took some really interesting classes on quantum computing, cosmology, math-phys and I really enjoyed QM classes. I feel kinda lost, I was alone this week but this time... I couldn't find myself. Usually I get up and take responsibility for my decisions, now I feel tired and kinda depressed bc of some consequences... Idk. I came back to my road but feeling a lil empty, damaged.
Pd. I really enjoyed the Klein-Gordon equation derivation (and it is not part of my current classes lmao).
Anyways. I hope to see some light at the end of the road (semester or theoretical mechanics course). See you soon, guys 🫂
Man, what a week. After I had a reunion with ma bae, I got sick, I finished my numerical methods hw, I did some theoretical mechanics exercises and took a QM class, THEN I assisted to the Geometry in Quantum Mechanics Advanced School. I understood some complicated concepts I was not used to like entanglement, Hilbert projective space, group representation and a MASSIVE etcetera. Furthermore, I found a new branch that made me smile for several minutes and gonna change my thesis topic once again hehe. I mean, I don't want to be the confused kiddo who's not able to make a decision, but I want to be sure that my topic is gonna make me feel comfortable every time I have to study something related to it.
I won't say either which is it, but I'll show some stuff at the end of the semester. Dude, am very excited about some decisions I made this week.
Hope u got a good week, good time, good life. See u soon 🙂↕️
What a week. I mean, good scientific stuff, I've seen three QM Pilar's (Harmonic oscillator was there), started numerical lineal algebra, tried some rigid body problems and I think picked a satisfying thesis topic related to IBM (I won't put the meaning of that 'cause am jealous of my kinky topic). Over a different section of my life, I came back to gymnastics and I think I solved some personal troubles I was dealing with. Anyways, interesting scientific/social environment this week.
Well, numerical methods exam, continuum qm and more theoretical physics this week. Math and physical interpretation is getting a little tough these days.
Anyways, I keep smiling every time I see phys-math connections in my lectures.
I mean. After taking 4 calculus, some complex variable and linear algebra; in the nuclear engineering branch I had to take 2 math tools (numerical methods and probability), I took probability in the previous course , so am taking numerical methods. It is not my favorite one, but I consider the subject "interesting". Also I decided to add theoretical physics I and quantum mechanics I. I'll catch you up about my progress. See ya soon, buddies.
Pd. Gotta be honest, I'd prefer differential geometry, modern algebra or analysis, but they're not available in my nuclear branch; I'll have to give them a chance as a listener.
P.Pd. Yeh. I attended a SUSY QM conference and I've earned this cup. I love it heh
Well, maybe nuclear engineering isn't that bad. I mean, I hated chemical engineering, but nuclear totally changes my perspective. I'll keep on mathematical-physics anyway lmao.
I wanna be a mathematical physicist, I'm not sure about taking a branch, but I'm pretty sure that my next "grad" will be related to Operator Theory. I might get crazy, but man, just imagine it.
Introduction to Homological Algebra 1: Chain Complexes and Exact Sequences
At its heart, homological algebra is the study of chain complexes, exact sequences and measuring how chain complexes fail to be exact. In this post we shall introduce the first two concepts and talk about how to measure the failure of exactness (homology) in the next post. Later, we will also want to study how well functors preserve exactness and this will amount to measuring how certain complexes fail to be exact.
I will be assuming some familiarity with category theory and module theory. I have a post about category theory here and I may write a primer post on module theory in the future. I will recall the definition of a module over a ring before we proceed!
R-modules:
Definition 1.1:
Let R be a ring. A (left) R-module is an abelian group (M,+) with a scalar mulitplication R×M->M such that
If R=ℤ, ℤ-modules are precisely abelian groups. If R=k is a field, k-modules are vector spaces over k. If R is not commutative, we also get a distinct notion of right R-modules where scalar multiplication is on the right. The only axiom which changes is (4) which becomes (ms)r=m(sr). When R is commutative, sr=rs so the objects are the same.
Definition 1.2: A function f:M->N between left R-modules is said to be an R-module homomorphism (or an R-linear map) if it is a group homomorphism, i.e. f(m+m')=f(m)+f(m'), such that f(rm)=rf(m) for all m∈M and r∈R. If instead M and N are right R-modules, we require f(mr)=f(m)r.
With this definition, one can show that the identity map and the composition of R-module homs are both R-module homs. Hence we have a category of (left) R-modules and R-module homs which we denote R-Mod. We denote the category of right R-modules and R-module homes Mod-R.
In what follows, R-module will always mean left R-module unless stated otherwise.
Chain Complexes:
Definition 1.3:
Examples 1.4:
We may also consider complexes where the operators increase in degree instead and we call these cochain complexes. Mathematically, these are the same up to relabelling but it is often useful to treat them as distinct as we shall see later on.
We get the following immediate result.
Lemma 1.5:
One might care about complexes where this is actually an equality and indeed this is our definition of an exact sequence!
Definition 1.6:
From this definition, we see that any exact sequence is also a chain complex. We distiguish short exact sequences for two reasons. One is that in a sense (see the next example) short exact sequences are the shortest "interesting" exact sequences. The second is that they turn out to be very important in homological algebra. Also note that to show a sequence is short exact, we need only check that φ is injective, imφ=kerψ and that ψ is surjective.
Examples 1.7:
Comparing examples 1 and 7, we see that short exact given R-modules A and C, we don't necessarily have that all short exact sequences 0->A->B->C->0 must have B≅A⊕C. Now comparing with examples 3 and 4, we see that short exact sequences are indeed the shortest exact sequences where something more interesting can happen. However do not discount the utility of examples 3 and 4!
Definition 1.8:
One may now wonder whether we can tell when a short exact sequence is split and indeed the next result gives a full classification!
Proposition 1.9 (Splitting Lemma):
Chain Maps:
As is common in maths, now that we have seen some objects we should should talk about maps between these objects. This leads us to talking about chain maps.
Definition 1.10:
As one might expect, chain complexes over R with chain maps forms a category, which we denote by Ch. This is the result on the next lemma.
Lemma 1.11:
Note that the composition of chain maps is also associative. This follows from the associativity of the composition of set functions.
This allows us to talk of isomorphisms of chain complexes. A chain isomorphism is an isomorphism in Ch. Equivalently, a chain isomorphism is a chain map such that each map in the sequence is an isomorphism.
We conlude with a useful result involving chain maps between exact sequences.
Theorem 1.12 (The Five Lemma):
The first of these results is sometimes referred to as the Short Five Lemma.
Note that in the proof of the splitting lemma, in particular in (ii)⇒(i), we need not have checked that f is an isomorphism or that B=φ(A)⊕η(C). If, after constructing f as we did, we check that the following diagram commutes, then the short five lemma implies that f is an isomorphism:
In the next post, we shall discussion homology (and cohomology) and how it measures the failure of exactness as well as a few very important results!
