Articles of soft question

Beginner material for mathematical logic

I am looking for study and beginner material to study mathematical logic. I understand that it is a very broad topic but I would like to know what the best path there is to learning mathematical logic. Where should one start? What are the best resources? If someone could paint a time line of events […]

Is the infinitesimal generator for Lie groups the same as the infinitesimal generator of a Markov semigroup?

Is the infinitesimal generator for Lie groups related to the infinitesimal generator of a Markov semigroup? Or are they totally different concepts? https://en.wikipedia.org/wiki/Lie_group#The_exponential_map https://en.wikipedia.org/wiki/Infinitesimal_generator_(stochastic_processes) Both seem to be related to the exponential map. The connection would also explain why so many people, when discussing infinitesimal generators of a Markov process, seem to have such a […]

Can I skip the first chapter in Rudin's Principles of Mathematical Analysis?

I am a statistician who wishes to learn real analysis in order to better understand the foundations of statistics. With that aim in mind I plan to go through Rudin’s classic on “Principles of Mathematical Analysis”. Given the above context can I skip chapter 1? It seems to me that the material in chapter 1 […]

Lie algebra-like structure corresponding to noncrystallographic root systems

In the classification of Coxeter groups, or equivalently root systems: $$A_n, B_n/C_n, D_n, E_6, E_7, E_8, F_4, G_2, H_2, H_3, H_4, I_2(p)$$ with $p \geq 7$, the last four fail to generate any simple finite dimensional Lie algebras over fields of characteristic zero because of the crystallographic restriction theorem. However, I know that there exist […]

Why are Lie Groups so “rigid”?

This is probably a naive question, but here goes. To motivate my question, I’ll consider a unit circle in $\mathbb C$ or $\mathbb R^2$. This is a compact Lie group equipped with the usual exponential map. However, any deformation no matter how smooth of the unit circle makes it lose the group closure property (say […]

Learning differential calculus through infinitesimals

In class, we’ve studied differential calculus and integral calculus through limits. We reconstructed the concepts from scratch beginning by the definition of limits, licit operations, derivatives and then integrals. But the teacher really did everything to avoid talking about infinitessimals. For instance when we talked about variable changes we had to swallow that for a […]

Can I get a PhD in Stochastic Analysis given this limited background?

General advice on PhD apps welcome Given my limited background in stochastic analysis and other information (below), can I apply for a PhD with stochastic analysis for my dissertation topic? 1/4 I am currently a masteral student of mathematical finance, expecting to graduate sometime this year. I am not particularly interested in mathematical finance anymore […]

Mnemonics for linear algebra

Sometimes formulas in linear algebra are not easy to remember. Some usefulness for the process of remembering can provide application of mnemonics. Do you know some useful mnemonics for this purpose? I’ll give two examples: For the process of finding the inverse of matrix it could be used mnemonic Detminstra what can be translated as […]

What do we mean by an “Elegant Proof”?

What do we mean when we say that a mathematical proof is elegant? Of course one can say that the proof is beautiful, but what do we precisely mean when we say that a proof is beautiful ? Is there a precise way to measure the elegance of a mathematical proof ? I have thought […]

Hairy Ball theorem and its applications

While searching a question about fibre bundles, which was asked here, i got directed to Vector bundles. I noticed this word “Hairy Ball” which sounded eccentric and made a search at Wikipedia. How is the hairy ball theorem related to this statement: You can’t comb the hair on a coconut.