K theory of finite fields (mod l homology)

Recently I have been reading about K theory of finite fields. Here I write some summaries about this calculation and calculate the homology ring of F\psi ^q which is one step in the computation of finite fields, see section 4, {Mitchell}. This post may be continued.

This post has been continued on A Summary of Quillen’s K-theory of Finite Fields (Oct 16, 2021).

Thoughts on Physics, Maths and Reality (Part II)

This is a rethink of the same topic and I realised that at this point of my life I have corrected my definitions and philosophical system ever since I’m concerned about the issue of reality. I remember that when I wrote down these thoughts in part I, I used lots of parenthesises to indicate that these words are not well-defined and the discussion is loose and casual. What I’m trying to convey here is my current view on this topic and my personal systemising.

Given the advance of physical sciences, the temptation is to downgrade the subjectivity of mental entity. Yet subjectivity is an undeniable part of the definition of reality. By saying that it’s undeniable I mean incorporating the subjectivity of mental state into the definition of reality is an inevitable part of developing a more consistent philosophical theory. Some attempts to define reality without account for subjectivity of mental entity include:

1. defining reality as concrete existence that is consistent for everyone.

2. defining reality as an existence of something independent of observations, the so called “outside world” as in part I.

The first one is a raw definition and has already been refined or abandoned by most people gradually in their cognitive development. Mathematical theory, together with concrete existences (experiments, observations etc.), are intrinsically accepted by most people as reality. People accept a later established definition and build their theories upon it until at some points they realised this contradicts their raw definition of reality and posted the question what reality is. When this contradiction arises, it’s the raw definition to be abandoned.

The second one is a very common one among physicists: Reality is something existing independent of observation which reflects its existence on our perception. Such existence is a very fascinating part of our worldview that motivates discovery and exploration . This is a very sound standpoint and would not contradict anything that I try to establish. To resolve some unsolvable problems like whether multiple universes exist, whether god exists, we want to incorporate subjectivity of mental entity in such way that this questions become either solvable or turned into a systematic investigation of our own mental states. Take thinking bear as an example. For me, my thinking bear is a soulmate since they fit into the several defining properties of a soulmate. Physically, most of my bear’s body is a bunch of cotton and fabrics, eyes are made of plastic. These materials are arranged in such a way that the materials in bear’s facial area stimulate the part of my brain responsible for facial recognition which in turns activate a series of mental activities and other brain activities. Bear’s fur casting shadow on their facial area results in different perceived facial expressions under different lighting and viewing angles. This gives an impression of changing moods. Bear’s long limbs have greater velocity than other parts of the body when the whole bear is given a small amount of momentum. Such movements of the limbs are perceived as activities of the bear. So far this can be summarised as personification based on concrete characters of bear’s appearance. To get a complete figure, my imagination fills in the nonconcrete parts based on my preference. Bear’s personality, interests and emotional states are created by me. All these imaginations together with concrete characters created a figure matching the defining properties of a soulmate. There can also be such an argument in terms of whether god exists. I would need to define god as I did not get a predefined definition of god from the community where I grew up.

One thing that I need to clarify at this point is when I say a word is not defined, it doesn’t mean it’s not defined in dictionary, in society or in general discussion. I mean this word is not incorporated into my philosophical system. Indeed any generally well defined word could end up having various definitions for different people. Most of the time when we make arguments, we could work with others to try to agree on the objective aspects of the definitions but what’re mostly neglected are the definition on the aspect of mental state. For example, people who doubt whether god exists are likely those that define god based on concrete properties. Yet for those who believe in god, the spiritual existence of god is in the definition hence their answers would be positive. When an atheist try to argue with an theist that god doesn’t exist. Almost both sides try to explicitly define god by concrete properties and based on their arguments from there. But what make their arguments diverge is the difference of the mental aspects of the definition of god. Hence in most cases, to resolve such a problem is to analyse the mental aspect of a definition. Such analysis is usually not easy, it can be personal, requires some kind of self-awareness and systemising. But by inspecting and systemising the mental aspect of our definitions, we incorporate subjectivity of mental entity in such way that many potentially disturbing questions become either solvable or turned into a systematic investigation of our own mental states.

Masculinity, femininity and the inequality

This post was inspired by an interview with Beauvoir 25 years after the publication of The Second Sex. Here is the link to the interview: The Second Sex 25 years later: Interview with Simone de Beauvoir

What Beauvoir said is classic. For women who can live independently and don’t feel many obstacles in their life, it’s easy to forget that gender inequality is real. You may be a privileged woman because you’re an intellectual, or you have more masculinity and suppressed femininity (a so-called “class collaborationist”) and benefit from the male-oriented society. You might tend to think that other women have choices too, but they choose to be in a lower position because of their suppressed nature. No, most of them don’t. The inequality is real and we need to understand the struggle of femininity. 

Here I say the struggle of femininity instead of women. Since women are not the sole targets being discriminated. Discrimination is not targeted mainly on biological sex. Feminism has greatly promoted rights for women. But what still need to be changed is discrimination against femininity. Most people don’t consciously discriminate against certain biological gender, which is a sign of an improvement in the history of gender inequality. But unconscious bias always exists and, without correction, creates discrimination. In some communities, there is certain expectation of manhood in men while feminine expression in men is discouraged or even despised. This is also a case of discrimination against femininity.

Historical inequality in biological gender resulted in a society strongly valuing masculinity. Traditionally masculine personalities are often more highly regarded than feminine ones. Besides historical reasons, people tend to relate masculine traits to success, which is just another stereotype.

Moreover, social expectations on how gendered personalities match biological genders create gender norms. Under such gender norms, regardless of biological gender, everyone can become a victim of gender inequality. 

The specific form of gender norms also varies among different communities. In some cultures, the same feminine traits in men and women often result in more serious discrimination against men. Some society accepts and welcomes masculine expression from females but discriminates against feminine expression from males.

From a more psychological point of view, as complete human beings, we have both masculine and feminine sides and integrating them is part of the individualization process. As Jung puts it in his Red book

What about masculinity? Do you know how much femininity man lacks for completeness? Do you know how much masculinity woman lacks for completeness? You seek the feminine in women and the masculine in men. And thus there are always only men and women. But where are people? You, man, should not seek the feminine in women, but seek and recognize it in yourself as you /possess it from the beginning. It pleases you, however, to play at manliness, because it travels on a well-worn track. You, woman, should not seek the masculine in men, but assume the masculine in yourself since you possess it from the beginning. But it amuses you and is easy to play at femininity; consequently man
despises you because he despises his femininity. But humankind is masculine and feminine, not just man or woman. You can hardly say of your soul what sex it is. But if you pay close attention, you will see that the most masculine man has a feminine soul, and the most feminine woman has a masculine soul. The more manly you are, the more remote from you is what woman really is, since the feminine in yourself is alien and contemptuous. *

The Red Book, P.263

* In 1921 in Psychological Types, Jung wrote: ”A very feminine woman has a masculine soul, and a very masculine man has a feminine soul. The contrast is due to the fact that for example a man is not in all things wholly masculine, but also normally has certain feminine traits. The more masculine his outer attitude is, the more his
feminine traits are obliterated: instead, they appear in the unconscious” (CW 6; §804). He designated the man’s feminine soul as the anima, and the woman’s masculine soul as the animus, and described how individuals projected their soul images onto members of the opposite sex (§805).

Lastly, here are some of my favourite excerpts from the interview with Beauvoir.

“In writing The Second Sex I became aware, for the first time, that I myself was leading a false life, or rather, that I was profiting from this male-oriented society without even knowing it. What had happened is that quite early in my life I had accepted the male values, and was living accordingly. Of course, I was quite successful, and that reinforced in me the belief that man and woman could be equal if the woman wanted such equality. In other words, I was an intellectual. I had the luck to come from a sector of society, the bourgeoisie, which could afford not only to send me to the best schools but also to allow me to play leisurely with ideas. Because of that I managed to enter the man’s world without too much difficulty. I showed that I could discuss philosophy, art, literature, etc., on “man’s level.” I kept whatever was particular to womanhood to myself. I was then reinforced by my success to continue. As I did, I saw I could earn as good a living as any male intellectual and that I was taken as seriously as any of my male peers. “

“Each stage fortified my sense of independence and equality. It became, therefore, very easy for me to forget that a secretary could in no way enjoy the same privileges. She could not sit in a café and read a book without being molested. She was rarely invited to parties for “her mind.” She could not establish credit or own property. I could. More importantly still, I tended to scorn the kind of woman who felt incapable, financially or spiritually, to show her independence from men. In effect, I was thinking, without even saying it to myself, “if I can, so can they.” In researching and writing The Second Sex I did come to realize that my privileges were the result of my having abdicated, in some crucial respects at least, my womanhood. If we put it in class economic terms, you would understand it easily: I had become a class collaborationist. Well, I was sort of the equivalent in terms of the sex struggle. Through The Second Sex I became aware of the struggle needed. I understood that the vast majority of women simply did not have the choices that I had had, that women are, in fact, defined and treated as a second sex by a male-oriented society whose structure would totally collapse if that orientation was genuinely destroyed. But like economically and politically dominated peoples anywhere, it is very hard and very slow for rebellion to develop. “


Serre-Swan Theorem and some K groups

This is an overview on Serre-Swan theorem and some ideas on the construction of K-groups for a Banach category. Serre-Swan theorem establishes equivalences between the categories of topological vector bundles over a compact Hausdorff space X, the category of finitely generated projective C(X)-modules and the categories of algebraic vector bundles of finite rank over
the affine scheme \mathrm{Spec}C(X). This theorem connects different objects of interest in K-theory.
It also introduces some ideas on the construction of K-groups for a Banach category and
in particular for compact topological spaces and Banach algebras.

Calculate (co)limits as (co)equalisers (two examples)

There is a general formulation for constructing limits as equalisers: see Theorem 1 in Section V.2, Maclane. For the dual version, see Theorem A.2.1 in Appendix A written by me.

The constructions look like these (see the links above for details):


But in practice, these diagrams may not be helpful to see what the equalisers should be. Now I give proofs for the (co)equalisers in two examples: the connected component of a simplicial set and the sheaf condition.

The connected components [Background]

For definitions and other backgrounds, see Subsection 00G5. For the record, see [P12, DLOR07] for the cosimplicial identities and Tag 000G for simplicial identities. (These identities are used in my proofs.)

Two examples of (co)limits as (co)equalisers

Pdf here: Two examples of (co)limits as (co)equalisers


[Short Notes] Non-compactness of the closed unit ball in an infinite-dimensional Banach space

This is about an exercise in [Bass]:

Exercise 19.5. Prove that if H is infinte-dimensional, that is, it has no finite basis, then the closed unit ball in H is not compact.

Proof. Choose an orthonormal basis \{x_i\}, then ||x_i-x_j||^2=||x_i||^2+||x_j||^2=2. This means the sequence is not Cauchy hence has no convergent subsequence.

For a Banach space, by Riesz’s lemma to find a non-Cauchy sequence.


[Bass] Bass, R. F. (2013). Real analysis for graduate students. Createspace Ind Pub.

Krull’s Principal Ideal Theorem in Dimension Theory and Regularity

This post is about some applications of Krull’s Principal Ideal Theorem and regular local rings in dimension theory and regularity of schemes [Part IV, Vakil], with the aim of connecting the 2018-2019 Warwick course MA4H8 Ring Theory with algebraic geometry. The lecture notes/algebraic references are here:  2018-2019 Ring Theory.  Note that the algebraic results included here follow the notes. Alternatively, one can also find them in [Vakil] either as exercises or proved results for which I have included the references.

Besides including results in both their geometric and algebraic statements, I have given proofs to a selection of exercises in Part IV, [Vakil] to illustrate more applications and other connections to the contents in the Ring Theory course. The indexes for exercises follow those in [Vakil].

See here for the full post: Application of Krull’s Principal Ideal Theorem

Please also let me know if you find any errors or have suggestions on any of my posts.