Archive for the 'mathematics' Category



Doing A Graduate Degree In Taiwan

NTNU Main building on Heping Dong Rd.

I just read Fili’s most recent post about his stint at a Taiwanese university (I couldn’t tell which from the post). Having been an international student at NTNU, in the graduate program of Mathematics since ’09, I have different things to report. Granted, I am studying in sciences at the graduate level, so the classes are ultimately very small. I have 4 students in one class and 2 in another.

Still, there are some important points to remember when you think about doing a graduate program in Taiwan.

Continue reading ‘Doing A Graduate Degree In Taiwan’

Mathematics Graduate Class Lecture Format

Atiyah's Introduction to Commutative Algebra

This year, the graduate class format changed dramatically for me. I went from a normal class, filled with students, to classes with at the most 4 students and a professor. Actually, my Complex Analysis II class has only another student enrolled. As such, the format has changed. The professors no longer give 3h-lectures, the students do, each in turn.

Basically, each graduate student will prepare a 3h-lecture¹. In one class, that means that I lecture every 4 weeks. In another, it’s every other week². Preparing the lecture involves going over the textbook and the proofs. Depending on how detailed the proofs are, you’ll need to flesh them out further, and make them understandable, citing the right theorems, propositions, etc. Depending on what book/resources you are using, this might take quite a few hours. It also depends on the overall complexity of the class and the overall sparseness of the authors of the book. Atiyah’s books is very sparse. The proofs are sometimes quite short and they need to be expanded significantly.

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Algebra and Scaffolding

Giving Lectures & Presentations in Graduate Classes (Commutative Algebra)†

Introduction to Commutative Algebra

Last Thursday, I gave my first lecture in a graduate  class of mathematics. There were three other students in that class, and the professor. All of the students were graduate students in Algebra. I was the sole person in Analysis. At first, this intimate setting was pretty daunting. I hadn’t taken the class last semester and the prof obviously didn’t like me being part of it¹.

Continue reading ‘Giving Lectures & Presentations in Graduate Classes (Commutative Algebra)†’

The Asthray By Errol Morris Part 3

Le penseur par Rodin, via Wikipedia

The third installment of The Ashtray is online over at the NYT. It’s the longest of the series, until now.

Commensurability or incommensurability is a concept in the philosophy of science. Scientific theories are described as commensurable if one can compare them to determine which is more accurate; if theories are incommensurable, there is no way in which one can compare them to each other in order to determine which is more accurate.

Pythagoras and the Pythagoreans were devoted to a higher spookiness. It is their distinction. With his vein-ruined hands describing circles in the smoky air, Pythagoras has come to believe in numbers, their unearthly harmonies and strange symmetries. ‘Number is the first principle,’ he affirmed, ‘a thing which is undefined, incomprehensible, having in itself all numbers…’ Half-mad, I suppose, and ecstatic, Pythagorean thought offers us the chance to peer downward into the deep unconscious place where mathematics has its origins, the natural numbers seen as they must have been seen for the very first time, and that is as some powerful erotic aspect of creation itself…
David Berlinski, “Infinite Ascent”

There is an anomaly — an inability to find a rational fraction that measures the diagonal of a unit-square. This is followed by a mathematical proof that shows conclusively, irrefutably that there is, that there can be, no such fraction.

It’s a very nice essay, presented in an almost academic fashion. A must for any mathematician.

Grigory Perelman’s Perfect Rigor

The NY Review of Books reviews Masha Gessen’s Perfect Rigor, a fascinating biography of Grigory (Grisha) Perelman, the Russian mathematician who proved the Poincaré Conjecture. What I find surprising is that she wrote this biography without talking with Perelman or his mother.

010110

Dang, while working on some posts I just noticed that the date today is all in binary!

If you didn’t know, that’s 22 in decimal. So today’s 22!

22

Igon Value: Pinker on Gladwell’s What The Dog Saw

I’m kind of surprised at the ignorance of some writers, especially when it comes to mathematical terms, such as Igon Value. For those of you who don’t know, it’s not igon value, but eigenvalue. Here’s what eigenvalues are all about. The name come from the German word eigen, which means “self”.

Eigenvalues, eigenvectors and eigenspaces are properties of a matrix. They are computed by a method described below, give important information about the matrix, and can be used in matrix factorization. They have applications in areas of applied mathematics as diverse as finance and quantum mechanics.

 

An eclectic essayist is necessarily a dilettante, which is not in itself a bad thing. But Gladwell frequently holds forth about statistics and psychology, and his lack of technical grounding in these subjects can be jarring. He provides misleading definitions of “homology,” “sagittal plane” and “power law” and quotes an expert speaking about an “igon value” (that’s eigenvalue, a basic concept in linear algebra). In the spirit of Gladwell, who likes to give portentous names to his aperçus, I will call this the Igon Value Problem: when a writer’s education on a topic consists in interviewing an expert, he is apt to offer generalizations that are banal, obtuse or flat wrong.

 

Paracompact Spaces

Toroid, via Wikivisual

I

It’s strange that I was actually researching paracompact topological spaces on Sunday and that we are seeing those types of spaces in my complex analysis class. We just started the Berenstein & Gay Complex Variables¹ book and things are pretty interesting. I actually deduced that we were heading there because of some of the concepts that we are seeing.

A paracompact space is a topological space in which every open cover admits an open locally finite refinement.

Continue reading ‘Paracompact Spaces’

Concretizable Categories

As some topological spaces can be metrizable, the same can be said of categories. Certain categories can be concretizable, if they are isomorphic to a concrete category. I must be making progress as I understood almost everything in that post from David Speyer.


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