Lecture Notes: DG Categories and Homotopy Category of a DG Category

Math Questions: Differentiability, DIfferentials, Derivatives and... Derivable?

Reginald Anderson

17 February 2018

Math Questions

(I'm even just talking about real differentiability here, for instance)

Derivatives send functions to functions.

Differentials send differential (n)-forms to differential (n+1)-forms.

Want to define: differential equation

Differential equations relate functions of higher derivatives to a function. Where is the “differential” in differential equations? In slope fields, and graphing? In statements about boundaries, divergences, and curls? Where is d◦d=0 here? That if you integrate over the boundary of a boundary, you don’t?

Why is a function called differentiable at the point x, when all of its partial derivatives exist in some open neighborhood U, of x? Shouldn’t this function be called derivable, instead? Or is this statement that the function can be made into a 1-form, by applying the differential.

Attempt at an answer: Differentiable means there’s an exact 1 form which integrates to said function. [I am assuming that integration takes a differential n-form as input and spits out an n-1 form.] (Yeah, I know the limit definition of differentiability. Yeah, I know the derivative is a linear operator that satisfies the Chain Rule [also the Leibniz rule], the best local linear approximation, etc.) The function is not closed unless its derivative is zero, in which case it is called constant. But this suggests that not every function satisfies d◦d=0, because not every function differentiates to a 1 form which is constant from the point of view of a 2-form... It should be the case that if you apply the differential twice, you get 0, and I’m not sure how that occurs here. (I think this is a statement about the de Rham complex, for example.)

Thanks,

One Confused Student.

2017 Progress

2017 Progress

1) This is the first ArXiv article that I would venture to say I can actually understand some of:

"From Apollonius to Zaremba: Local Global Phenomena in Thin Orbits"
Alex Kontorovich
(Submitted on 27 Aug 2012)

https://arxiv.org/abs/1208.5460

2) I fully understood one page of Discrete Groups, Expanding Graphs, and Invariant Measures by Alex Lubotzky, a book for "mature graduate students." Namely, the definition of a c-expander graph. I don't really know what they look like yet, though.

3) My goal for 2017 is to learn measure theory. By June 1, 2017 I will have a link to a PDF file describing what I understand about measure theory. This will include a brief description of the theory (how it generalizes the notion of a Riemann integral, and will mention how it is used in other fields). [Really, this just means that I want to know what R_{'little ell'}, L^2, and L^p spaces are. 

Update: Now that I've seen and heard it explained once, it seems trivial. I need more ambitious goals.
Update 2: By June 1, 2017, I have, in fact, finished reading Rudin's 'Principles' closely for the first time (includes studying, working problems from earlier sections -- my point is that I have finished reading closely the sections on Measure Theory, but have yet to solve problems there.) 

4) I'm redoing Rogawski's Calculus, and then doing Gilbert Strang's Linear Algebra after that...

5) Programmed first lines of code, in Python 1/17/2017

Answered a discussion question correctly in Engineering Differential Equations 1/18/2017

6) I can understand a fair amount of this: https://arxiv.org/pdf/math/0605327v1.pdf

7) and I am working through Rudin's Principles of Mathematical Analysis, slowly.

• Constructed the field R from sets of cuts
• Induction proofs involving triangle inequality

8) College math professor said I should want to get my PhD in math and teach college...
        (1/30/2017)

8.5) First footnote in a LaTeX article, 2/4/2017 (this is just cool to me, idk why)

9) First positive response from a grad program: PhD interview with Penn State for Philosophy, 2/3/2017

10) Sat in on math courses at UChicago, and some big names (!) let me stay.

11) Read my second full article on arXiv, this time on Category Theory. Summary: Leibniz algebras are a generalization of Lie Algebras. read 4/9/2017. https://arxiv.org/pdf/1703.10426.pdf

12) Answered my first discussion question in Manifolds on 5/3/17 (fundamental group of the 2-sphere is the identity!). Correctly solved first problem in manifolds 5/4/17 (p. 40 from Tu).

13) 9 months ago all I knew at this point was that I was going to run out of money, and that if I wanted to continue in academia, I had to get into something good -- and quick! So, I took French I and German I (at the former advice of my philosophy advisor), made sure that I had every back-up plan check done for Philosophy PhD's (I ended up getting into a couple...! :) ) , and then kept going in Math & Comp Sci. This semester (Spring 2017) I took my first Computer Science course (Introduction to Computer Science -- included some Python, assembly language, and examples of machine language), and sat in on more math courses. This April, I signed my first contract with a mathematics department to start my graduate studies. My long-term goal has and will be to continue working towards a PhD in Mathematics, with teaching college (or going into research, or industry!) as my end goals for now. I am feeling good about having a grasp again on Calculus 1-3 (Green's Theorem, Stokes' Theorem, mixed partials are everything!) (it's more fun to read these things from Rudin) and Linear Algebra (went back through Strang's Linear Algebra)-- examples include diagonalizing a matrix using eigenvector matrices and their inverses, spectral analysis of matrices using the size of their eigenvalues, and am getting to Matrices of Linear Transformations again); I'm PUMPED for round two on partial differential equations (I'll be a bit smarter about things this go-around), I'm learning and catching up in abstract algebra [I'm doing one day of algebra after one day of analysis] (groups, modules, PIDS, Integral Domains, Euclidean Domains, Rings, Fields, Galois Theory, and Canonical Forms), and am starting Topology on my own [topology day is after algebra, and before analysis] (Munkres' Topology, would like to also do Hatcher and Guillemin & Pollack, and am starting to understand some of Peter May's Algebraic Topology book and Allen Hatcher's "AT"). Considering starting PDE's again, numerical analysis, and trying to learn more Applied Math.

Right now I'm doing one day of Analysis (Real and Complex: this has included reviewing Rogawski's Calculus for Calculus 1-3 up to Green's Theorem, Stokes Theorem, Divergence, Curl, Jacobians of a linear and non-linear transformation; Rudin's Principles of Analysis [am starting to read some Adult Rudin], Ahlfor's and Freitag & Busam's Complex Analysis), one day of Algebra (Dummit & Foote, Lang's Algebra on Galois Theory, Hungerford's; Insel, Spence, and ____; Hoffman & Kunz\:{e}; etc.)


14) Strong A on midterm, perfect score on final exam in Foundations of Analysis, and fell in love with two theorems in particular which I can't mention here. The best thing, though, was that I just decided that this was going to happen! "Rarely do we accomplish goals that we do not set out for explicitly" - this quote was from an MIT prof on OpenCourseWare for I think a "Teaching Physics" course -- I should really look up the name to cite this correctly. Also, need to update/make a CV in LaTeX.

15) Ohmanohmanohmanohmanohman... Just about understood this one --- need some homotopy in my life.  https://arxiv.org/pdf/1705.09323.pdf

16) Understood homotopy for the first time yesterday: https://en.wikipedia.org/wiki/Homotopy
Starting this now: https://www.math.cornell.edu/~hatcher/AT/AT.pdf.

17) Passed my first Qualifying Exam for a Graduate Mathematics Program: June 2017


Moments I am most proud of in mathematics:

1) Realizing that the Laplace Transform of the sine function comes from Euler's identity from Complex Analysis.

2) Scoring enough points on the first midterm of Complex Analysis to stay in the course when I wasn't guaranteed a spot: hadn't taken the pre-requisites, was driving a half-hour to get to class, and had to score well to stay alive in mathematics. 


Running List of Questions in Mathematics:

1) How to turn the columns of a matrix into linear combinations of eigenvectors?

2) How are row operations and similarity related in matrices?

D&F: how is xhx^{-1} = 2Z [need to look up page number for this example]

3) [a big one] how are the eigenvalues of a matrix related to a matrix? What can the eigenvalues of a matrix tell us about a linear transformation? [I feel like I should write a "practice paper" on eigenvalues of linear transformations].

Skepticism Mentioned versus Skepticism Embraced

How interesting it is that the only things that can be certain are those which we decide in advance to delimit as “true” or “certain,” because math is (certainly) the only realm where certainty lies, and math is founded upon an initial prejudice that consistency should be valued over other things like "diversity” or "the pure joy of movement.” What I mean by this is that the foundations in math, the hand-waving and ad-hoc demonstrations that we take and run with for quantitative thinking, is relative.


Admittedly, consistency and tautology are good places to start for any formal deductive system: assume that something is not the case, and you get that it is --- that seems like exactly the type of thing that you would want grounding your argument. But what if things are not that binary in truth value? What if the way you frame your argument from the get-go is wrong, is question-begging... What if other values are better at grounding our logic (if "logic" doesn't exist), our thinking, and, therefore, our lives?


Just because our mind looks for something does not mean that it exists -- a lot of the nihilism and thoughts of meaningless in life simply come from ascribing a meaning to the question of whether or not a universal meaning to life could be won without ascribing what a satisfactory answer to this question would look like. It therefore becomes a meaningless and therefore "unscientific" (inconsistent) question to ask.


In lieu of the acknowledgment that math (of all things!) cannot be grounded upon logic, we are left with the hair-raising result that proofs of certainty, of truth, of meaning are fragile (and a matter of taste!). It is true that there is always a gap between the is and the ought, but it seems like it must be the case (all instances of necessity would now become musical, as it were) that we are to listen to mathematics as one listens to music. Because it (knowledge) is fragile. Because we (math) are fragile. Because we (human beings are, basically everything...) are fragile. I don't draw the lines you would expect here because how can they be wrong? What are we left with to search for if any time you try to make a point, it won't go over, every time you put out energy it consumes itself, and the faster that you spin your wheels, the faster you have to run to stand in place?


What I don't get is how other people force necessity into their words and actions (their language), if the only thing we must do (of necessity) is to stop thinking of necessity literally. I'm assuming they don't force it, because then it would just be force, and not necessity. Either a) other people are really good at faking it, and we are to become genuine fakes, or b) this hasn't the slightest relevance to things at all and I've totally missed the mark. Probably the latter. Isn't it wrong to fake it, especially if that's something your doing as a "noble lie"? It is all dancing -- but it's not dance, because dance "is founded" on the joy of pure movement -- it acknowledges line and shape and form in and of themselves, it values emotion, and color, and an emergence of meaning that isn't so bluntly literal. This is what I try and put in a math proof, and they'll say it's wrong or not this, or not that. Okay. Alright. Growing process, I'll get there. It all comes in time... and no one can tell you how to get there, or where it goes, or where you are...


If there's any truth to this, don't take it too seriously! You'll crush it... There are people who will disagree with you not because of the content of what you'll say, but from the sheer fact that you try to make points. Pointy people like points! Un-pointy people do not! And there isn't the slightest difference between them. I have no way of knowing how fast your wheels are spinning when you give a response, or whether or not this is something to be valued. I have no way. If you find something you like, launch it into the sun. Just don't stoop. Don't stoop in your thoughts or your language, or when the sun comes up (or doesn't). There is no good response to this! There is no science of meaning! I mean it!


What do you run from? (The cure). My point is that making sense is not literal. And that is therefore a very hard thing to explain from a standpoint of use: we can mention this all of the time. But what is the difference between mentioning, and using? I do not think that we have fully grasped this difference. What does it mean to actually use math in our lives? To be distracted? What does it actually mean to "do politics" in our lives? To be distracted? How many people am I stealing these arguments from? How do you know who they belong to? How can there be a process by which we claim to call some forms of ideas 'common knowledge' by which we can extradite some forms, some assumptions from people, without citing them? How can there not be? What does it mean to be a skeptic about meaning... The fact that we can say "skeptic about meaning" and have it register in the same way something registers when we hear "the car is red" means that skepticism isn't taken seriously. It is taken as an attitude, a scoff, a 'fly in the ointment.' Wouldn't it be wonderful (and terrible) if we needed that skepticism in our lives? When we say healthy skepticism, this is never what we mean. We mean: "take the points that are being given, and see whether they can be verified. Or falsified. And repeated." We do not mean: "Throw the whole thing out. Discard it wholesale! Launch it into the sun! See if it grows back. See if it does do what I say it does in our absence." And this is because, knowledge is fragile. It takes nourishment, and love, and warmth. It takes belonging, it takes holding your views up and suspending their judgment. So, if this is the case -- what is a debate? What is a "political fact"? Is it "politics"? Is it making points...


I actually believe some of what I'm writing, and therefore need to let it go. It doesn't belong to me anymore. (Because if the argument isn't already tautologous and grows back in its absence, why keep it? Why "hold on" to anything, if there is such a thing as logic whose essence implies its existence, who springs from the void into a simultaneously coherent and accurate picture of itself and, therefore, everything else).

The reason I am writing this is because if logic is a matter of taste, then academic math can't really be as rigid as it is believed to be in practice. Maybe a lot of other people have these same sentiments. Maybe the whole point is to acknowledge that and keep going because, as stated above, logic is a matter of taste, and the better taste loves working on in silence more than stopping and asking questions. 

All I can really say is that I love finals. I love the cramming, I love the tests, I love the homework, the studying... I love all of the things that I spend so much time memorizing only to never use again. I love the things that I spend so much time learning that I will use again, and everything in between. I can't believe that the gap between academia and everyday life is so big that the things that we learn in school have no bearing on our daily lives, but it's also a very privileged thing to say that I value education because it is competitive. Because it is a competition, because it is a race that we are always running whether we choose to or not, so why not run faster than everyone else so that you finally outrun the question of why we are running at all in the first place? On my best days, I completely sideskirt the question of "why?" I'm learning anything at all, because I basically function by bullying the questioning parts of my brain out of existence from 9AM-5PM. I can't say where it leads or where it's going...

The only sustainable answer I can give to "Why learn math at all?" is that, maybe we will find something that is thought of as a contradiction, and offer a new way to deal with it that revolutionizes the theory, because that's how science moves forward. But it's all just running, to keep running, for more running, so that we can keep running, and then run some more...

These are very basic questions in the philosophy of science, and I feel like they are important because there is an opportunity cost to studying, and part of my conscience always wants to know why I'm spending any time at all on math, especially if it's not the type of thing that brings me closer to the world ("...keep your two feet on the ground!"). It's also definitely not the case that there are mathematicians or professors out there just waiting for someone to come around and save them by solving a problem or offering them help.

Is everything just a competition? How do you get a stable mathematics, if the only reason it exists at the highest levels is to beat other people to a pulp? How can we get logic out of competition that doesn't bully itself out of coherence and existence? Most days I feel like my own mind would leave me behind at any chance it would get, and I'm just left trying to keep up...

This is the circularity that I see in things: 1) Math is not giving a framework of the world. 2) Math is giving a framework and a foundation for math. It is self-fulfilling 3) Each has their own subjective definition of mathematics. 4) "Math is different from a hallucination because _________"

How do I fill in that blank? Because it's math? Because it's logical...? (Then what isn't susceptible to this circularity?) The thing that I actually tell myself is that math is the sort of thing that is worth listening to more than the part of ourselves asking "Why are we doing this?" As though we could just outrun it. I guess you just have to be hungry for it, and navigating the world is more like aesthetics than we thought.

Formal Logic

17 Sep 2016

[I'll update this post as I find things that are interesting in logic]

I'm actually familiar with some of the things here: https://www.youtube.com/watch?v=tpeeTHNVnxQ and am watching the second part now: https://www.youtube.com/watch?v=VOhGLnxJt4E. (Joel David Thomkins 'The Continuum Hypothesis and Other Set-Theoretic Ideas for Non-Set-Theorists), which makes me glad that I have taken courses in analysis.

I've glanced at some literature on the algebra of cardinals at the University of Iowa library which I would like to understand better.

I tried reading Hermann Weyl's The Continuum Hypothesis last summer and a lot of it went over my head. Sitting in on a formal logic course on Irving Copi's Symbolic Logic (I got the 5th Edition for like $9 online) has been helpful and I'm hoping to get back to Weyl's book eventually.

I've taken two analysis courses and two abstract algebra courses and have done homework assignments on the Riemann Zeta function, Gamma function, and have heard about elliptic curves in a lecture from a course that I was taking for a grade. This is farther than I ever thought I would get in math, so I am happy about that. What I like about formal logic is that it can give me a way back into math, with stronger foundations: Irving Copi's formal logic goes into calculus and algebra by the end, so I would have a way to "stay alive" in mathematics. If I am not accepted to any graduate courses in logic then I would be able to say that I have given it a good run in math and would feel accomplished in my work.

I did touch on some Set Theoretical basics with my '14-'15 Algebra I students, so I'm glad that they'll have some access to Set Theory.

I learned that DeMorgan's Laws for Logic give an analog for the way that I need to remember to explain semantic incompleteness. I had previously noted:

Week of 9/12/16: It is the case that in any language, there is always something that cannot be expressed. It is not the case that there is some thing that cannot be expressed in any language; for, to name it would be to invalidate the reason we were searching for it. - (On the notion of semantic completeness)

This can be expressed in formal logic as: 

¬∃x∀yφ(x,y)  = ∀x∃y¬φ(x,y)

I am thinking of the left side as:                                       and the right side as:

"It is not the case that there is a subset of the                   "For every subset of the real line there

real line which is such that for every language,          =   exists a language such that it is not the

the specific subset of the real line is inexpressible"         case that the subset of the real line is 

                                                                                       inexpressible."

Now, the left side is what we were given from the notion of semantic completeness. It is interesting that this is logically equivalent to saying that for every subset of the real line, there is some language in which the subset can be expressed.

The error I was tending to make in natural language was to say the negative case of the Week of 9/12/16 Quote: "It is not the case that there is some thing that cannot be expressed in any language..." Using formal logic keeps this straight in my head.

Quote of the Week

Week of 9/5/16 “It is striking that we are less likely to criticize violinists, say, than political philosophers, for failing to provide justice-promoting guidance, as if being interested in identifying truths about justice meant that one was more rather than less culpable for failing to tell us how to bring it about… But I find it hard to feel more impatient with political philosophers than with those who show no interest in justice at all” Adam Swift's The Value of Philosophy in Nonideal Circumstances, 367.

Week of 9/12/16: It is the case that in any language, there is always something that cannot be expressed. It is not the case that there is some thing that cannot be expressed in any language; for, to name it would be to invalidate the reason we were searching for it. - (On the notion of semantic completeness)

Derrida's 'Of Grammatology'

Gayatri Spivak’s translation Of Grammatology (1974) of Derrida’s De la Grammatologie (1967)

Derrida is considering Saussure’s linguistics and says that Peirce had some things right in linguistics, then talks about Rousseau, Claude Lèvi-Strauss, presence, metaphysics, and morality. The people he’s writing about that stuck out to me are Claude Lévi-Strauss, Rousseau, Saussure, Husserl, Hegel, Diderot, and Bergson.

Main Idea: Derrida says that metaphysics of presence cannot account for the spacing within writing. That is, if I print a written document, the "actual meanings" of the words on the page do not account for how the shape of the page, formatting, type of ink and paper, and shape of paper affect the meaning of the document as a whole. This has implications for the relationship between speech and writing: while each has characteristics that are unique to it, there is no sharp divide between the two. How would our lives (or, more exactly, our writings) be different if, instead of writing on rectangles, we wrote on circles or S-shaped paper? How would our lives be different ("how would the meanings of our writings change...") if, instead of writing upright we always had to stand on our hands to write as a social norm? The "literal meaning" (even these will have to be discarded) of the word "blue" will still mean what "blue" means regardless of the shape of the paper it is written on, or whether it is written by someone who is upright or standing on their hands. However, since metaphysics of presence can't account for spacing (I want to use the word style), repetition does not produce the same meaning time and again, but rather, a simulacrum is produced. In this way, there is always a trace of ineffability within articulation; of absence within presence. So "blue" is not a standalone referent from the signifier in the process of evocation and how we write or say the word (upright? upside-down? Is it spray painted?)... how we talk about it is tied up with what it is.  

Why it's useful: This forces us to consider whether, in language, it is the ideas themselves, reduced to simple (logical) statements, that are most important in describing what is being told. In a way, Derrida is saying that if you only focus on logic, you drain speech and logic of the character, musicality, and the things that make language and communication worth actually hearing or listening to. This is called Derrida's critique of logocentrism. I like this for my own life because it forces me to consider that to tell a good story, it's not about focusing solely on translating the plot into formal logic, but rather, the ideas and logical connectives become emergent from within a discourse through the act of explaining and gesturing. I think that this holds true for what masters of a discipline or sport would tell you as well: if you focus only on the finish line, you probably won't run a very good race! If you focus only on the degree, you probably won't get a very good education. If you only think about getting the dive over with, ya might not look so graceful in the air...

Key Terms

Differance: Tied to the notion that to mean something, it must differ from other things and defer to what system we use to eff something effable (i.e., “able to be grokked”; it’s really hard to explain what the “eff” in ineffable means[1]). Both difference and the supplement are emergent from Derrida’s consideration of the relationship between speech and writing.

The supplement: tied to the notion of evil, imitation, presence nature, “the light of day,”, inside versus outside, our relationship to signs and referents, our believing that we have a lack, the possibility of a guardrail in our thinking, and getting ahead of ourselves in our thinking. Heidegger also said in Being and Time that we exist as fallen, and as such tend to get ahead of ourselves. I did like the quote “As soon as man comes to life, he is ready to die,” some of the things he says about fallenness, and the anticipation of death serving as an individualizing principle, but that’s about as much as I want to say about that man’s work. A tangential point is the relationship one has to the footnotes of a book gets better and better the more one reads the book because the index and footnotes are getting closer – the first full sentence I understood in German was from Being and Time! So that was cool. There’s also a German dictionary of relevant terms in the HarperCollins edition. In getting ahead of ourselves, we can come to care more about the signs for things than the things themselves, and I think that’s part of what Derrida is saying about perversion. Anways, supplementarity is the play between substance and absence that no metaphysical or ontological concept can comprehend (266). 

Key Quotes: I see p. 180 & 181 as the “heart” of the text

Auto-effection is a universal structure of experience. All living things have the power of an auto-affection. And only a being capable of symbolizing, that is to say of auto-effecting, may let itself be affected by the other in general. Auto-effection is the condition of an experience in general. This possibility – another name for “life” – is a general structure articulated by the history of life, and provides space for complex and hierarchical operations. Auto-affection, the as-for-itself or for-itself – subjectivity – gains in power and in its mastery of the other to the extent that its power of repetition idealizes itself. Here idealization is the movement by which sensory exteriority, that which affects me or serves me as signifier, submits itself to my power of repetition, to what thenceforward appears to me as my spontaneity and escapes me less and less… Conversation is, then, a communication between two absolute origins that, if one may risk this formula, auto-effect reciprocally, repeating as immediate echo the auto-affection produced by the other. Immediacy is here the myth of consciousness… As soon as nonpresence comes to be felt within the voice [vowel] itself – and is at least pre-felt from the very threshold of articulation and diacriticity – writing is in some way fissured in its value (translated in 1974 from Derrida 1967, 180 – 181, – italics added by me!).

Notable WTF Moments from Rousseau

But in the north, where the inhabitants consume a great deal off a barren soil, men, subject to so many needs, are easily irritated. Everything that happens around them disturbs them. As they subsist only through effort, the poorer they are the more firmly they hold to the little they have. To approach them is to make an attempt on their lives. This accounts for their irascible temper, so quick to turn in fury against everything that offends them (Derrida 1967 in Spivak 1974, 244).  

The country is not a matter of indifference in the education [culture] of man; it is only in temperate climes that he comes to be everything he can be. The disadvantages of extreme climes are easily seen. A man is not planted in one place like a tree, to stay there the rest of his life, and to pass from one extreme to another you must travel twice as far as he who starts half-way…. A Frenchman can live in New Guinea or in Lapland, but a negro cannot live in Tornea nor a Samoyed in Benin. It seems also as if the brain were less perfectly organized in the two extremes. Neither the Negroes nor the Laps have the sense of the Europeans. So if I want my pupil to be a citizen of the world I will choose him in the temperate zone, in France for example, rather than elsewhere. In the north with its barren soil men devour much food, in the fertile south they eat little. This produces another difference: the one is industrious, the other contemplative (p. 27; italics added by Derrida) (on p. 242 in Spivak’s translation of Derrida).

Hopefully we have learned by now that if you call another culture primitive or basic you are only expressing your own ignorance of something beyond yourself!

A good quote from Rousseau: “He who imagines nothing senses no-one but himself; he is alone in the midst of humankind” (this is Rousseau on page 203 in Spivak’s translation Of Grammatology).