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Geometric justification for the prime spectrum and “generic points”

I realize there have been plently of discussions about this, but most of them are over my head and I never understand the geometric intuition behind them. I’m trying to make a big list of “justifications” for using the prime spectrum of a ring. So far, I have two fairly pleasing formal ones: Preimages of […]

Homological algebra in PDE

I have been fascinated by the power and wide applicability of homological methods in algebra and topology. Because I am also interested in PDE, there arises a natural question for me. What is known about applications of methods from homological algebra to the analysis of solutions of PDE on domains in $\mathbb{R}^n$?

Recommendations for an “illuminating” (explained in the post) group theory/abstract algebra resource?

I recently asked a question regarding why homomorphisms and isomorphisms are important. The best answer to that question was actually a comment, which referred me to Brian M. Scott’s answer here: That answer was mind blowingly insightful for me. I finally began to understand why someone would care about homomorphisms, and why the “kernel” […]

What is elliptic bootstrapping?

While reading about elliptic differential operators, I have seen the phrase elliptic bootstrapping in several places, but none of them explain exactly what it means. I know it has something to do with the regularity of elliptic differential equations. Does elliptic bootstrapping refer to a collection of theorems, a method/technique, or just a vague concept? […]

How is the concept of the limit the foundation of calculus?

My casual study of mathematics and calculus introduced me to the notion that calculus didn’t find a firm foundation until Cauchy, Weierstrauss (et al) developed set theory some ~100 years after Newton and Leibniz. The concept of limits (and their epsilon-delta proofs) was what allowed calculus to get past the shaky logic of infinitesimals. (The […]

Why are modular lattices important?

A lattice $(L,\leq)$ is said to be modular when $$ (\forall x,a,b\in L)\quad x \leq b \implies x \vee (a \wedge b) = (x \vee a) \wedge b, $$ where $\vee$ is the join operation, and $\wedge$ is the meet operation. (Join and meet.) The ideals of a ring form a modular lattice. So do […]

Do groups, rings and fields have practical applications in CS? If so, what are some?

This is ONE thing about my undergraduate studies in computer science that I haven’t been able to ‘link’ in my real life (academic and professional). Almost everything I studied I’ve observed be applied (directly or indirectly) or has given me Aha! moments understanding the principles behind the applications. Groups, Rings and Fields have always eluded […]

Understanding the Laplace operator conceptually

The Laplace operator: those of you who now understand it, how would you explain what it “does” conceptually? How do you wish you had been taught it? Any good essays (combining both history and conceptual understanding) on the Laplace operator, and its subsequent variations (e.g. Laplace-Bertrami) that you would highly recommend?

Why are groups more important than semigroups?

This is an open-ended question, as is probably obvious from the title. I understand that it may not be appreciated and I will try not to ask too many such questions. But this one has been bothering me for quite some time and I’m not entirely certain that it’s completely worthless, which is why I’ve […]

Why are all the interesting constants so small?

A quick look at the wikipedia entry on mathematical constants suggests that the most important fundamental constants all live in the immediate neighborhood of the first few positive integers. Is there some kind of normalization going on, or some other reasonable explanation for why we have only identified interesting small constants? EDIT: I may have […]