Geometrica

x üzeri n

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İnsan (bu insan) önem verdiği şeylere çok zor başlıyor. Daha önemsiz bulduklarını kasmadan, rahat rahat yapıyor, güzel şeyler oluyor. Önem verdiklerini elleyemiyor bile bir türlü.

Bikaç bilimsel site planı/projesi var yıllardır kafamda  döndürdüğüm. Baktım “istediğim gibi” yapabilme kaygısıyla (ve “Her projeye ayrı alan adı mı alsam?”, “Önce içerik mi oluştursam?”, “Alt yapıyı şöyle mi yapsam?”, “Kendi adımla mı yazsam?” gibi sorular arasında attığım turları sayarak) ömür geçiyor, bodoslama girmeye karar verdim. Buyrun.

Bir eski ev arkadaşımla (B.) beyaz tahtamız vardı evimizde. Arada geçer başına, kafamıza takılan şeyleri anlamaya çalışırdık. Sınavlar, ödevler, işler, güçler beklerken, kaç gece alakasız bir şeylere takılıp kendimizi kaybettik, uykusuz kaldık, uğraştık durduk. Kimi zaman çözdük uğraştığımız şeyi, kimi zaman çözemedik. Belki biraz da alanlarımız farklı olduğu için, öyle çok uçuk, ileri seviye şeyler de konuşmadık pek. Ta lisede, hatta ortaokulda öğretilen, çoktan anlamış olmamız beklenecek, ama anlamadığımızı fark ettiğimiz şeylerle uğraştık. Bilmem kaç yüzyıl önce Antik Yunan’da anlaşılmış da olsa uğraştığımız şey, çözünce/anlayınca yeni bulmuş gibi sevindik, suratımızda bir sırıtışla gittik uyumaya.

İşte o kafa kafaya verip uğraştığımız sorulardan biri, bu “bodoslama giriş”te sormak istediğim:

x^n‘in integrali, neden x^{n+1} ile orantılı?

Biraz garip bir soru, bir bakıma. Türev/integral öğretilirken ilk anlatılan şeylerden biri aslında, x^n‘in integrali. İspatı da (bir yere kadar) lisede bile veriliyor. Herhangi bir analiz (calculus) kitabını açıp okumak mümkün. Bir şeyin doğruluğunu adım adım ispatlayınca, önceden bilinen şeylere dayandırınca, o şeyin niye doğru olduğunu göstermiş de olmuyor muyuz? Elimizde ispat varken, bir de “neden?” diye sormanın anlamı var mı? Bir önermenin muhtelif ispatlarından bazılarının, onun neden doğru olduğunu açıklamaya daha çok yaklaştığını söylemek, anlamlı mı?  Eğer bazı ispatların ayrıcalıklı oluğunu söyleyeceksek, “insanın bilişsel mekanizmasına hitap etme”nin ötesinde bir ayrıcalıktan bahsedilebilir mi?

Zor, derin, güzel mevzular. Ama genel (ve belki kolayca afaki hale gelebilecek) bir tartışmaya girmektense, yukarıdaki probleme dönelim diyorum. Oradan öğrendiklerimiz ne kadar yardımcı olacak, bakalım.

Birkaç arkadaşıma sordum bu soruyu yıllar içinde. İkisi benim cevabımdan başka cevaplar buldular, ufkumu genişlettiler. Bildiğim cevapları bir sonraki yazıda anlatacağım, ama gene bilmediğim, düşünmediğim bir şey çıkar, yeni pencereler açılır belki diye bir umutla: Bu soruya aradığım türden bir cevap bulana, benden kitap. Eğer bildiğim cevaplardan biri gelirse, böyle sorular üzerine kafa yormayı seven birinin seveceğini umduğum türden bir e-kitap. Yeni bir cevap gelirse, kağıttan kitap. (Kurallar: Bir dahaki yazıya kadar, internet’ten bakmak yok. Bir yerlerde okunmuş, önceden duyulmuş bir cevabı vermek yok—kendiniz düşünün. Bir de, “Çünkü integral türevin tersi, ve x^{n+1}‘in türevi, x^n ile orantılı (gerekirse limit alıp ispatlayabilirim)” gibi cevaplar vermek yok.)

Girin bakalım, bodoslama.

x üzeri 0, 1, 2, 3

Written by mkz

July 4, 2011 at 9:24 pm

Posted in Uncategorized

Language vs. music

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I’d like to think a little about this, before starting to read.

Papers, books, talks:

“The aim of the present study was the investigation of neural correlates of music processing with fMRI. Chord sequences were presented to the participants, infrequently containing unexpected musical events. […] [T]he cortical network comprising all these structures has up to now been thought to be domain-specific for language processing. To what extent this network might also be activated by the processing of non-linguistic information has remained unknown. The present fMRI-data reveal that the human brain employs this neuronal network also for the processing of musical information”

“To appreciate how our species makes sense of sound we must study the brain’s response to a wide
variety of music, languages and musical languages, urges Aniruddh D. Patel.”

“This review focuses on syntax, using recent neuroimaging data and cognitive theory to propose a specific point of convergence between syntactic processing in language and music. This leads to testable predictions, including the prediction that that syntactic comprehension problems in Broca’s aphasia are not selective to language but influence music perception as well.” (Not surprising? –mkz)

“Semantics is a key feature of language, but whether or not music can activate brain mechanisms related to the processing of semantic meaning is not known. […] Our results indicate that both music and language can prime the meaning of a word, and that music can, as language, determine physiological indices of semantic processing.”

“Research on how the brain processes music is emerging as a rich and stimulating area of investigation of perception, memory, emotion, and performance. Results emanating from both lesion studies and neuroimaging techniques are reviewed and integrated for each of these musical functions. […] Unfortunately, due to scarcity of research on the macrostructure of music organization and on cultural differences, the musical material under focus is at the level of the musical phrase, as typically used in Western popular music.”

Places, people:

Related:

Thanks to Emre Sevinc for the reference to the Peretz-Zattore book.

Stefan Koelschb, a, 1, Thomas C. Guntera, D. Yves v. Cramona, Stefan Zysseta, Gabriele Lohmanna and Angela D. Friedericia

Written by mkz

May 1, 2010 at 10:51 pm

Posted in Uncategorized

Discrete differential geometry

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Various communities of researchers have thought about this class of problems; there are results scattered through journals on image processing, geometric mechanics, numerical analysis, graph theory, machine learning, general relativity (and more?). Besides its use in various domains of application, discrete geometry also provides fresh insights into classical, continuous geometry. (See, e.g., Regge’s discussion of Bianchi identities in his original paper on Regge calculus.)

Discrete exterior calculus, cohomology, etc.

Discrete Riemannian geometry, connections, etc.

Discrete spectral geometry

  • References on graph laplacians, etc.

Convergence of discrete geometric quantities to their continuous counterparts.

  • Add references.

Discrete geometric mechanics, discrete field theory

General links

People

Written by mkz

March 29, 2010 at 10:04 pm

Posted in Uncategorized

Advice to a Ph.D. student

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This is from an email I wrote to a student a few years ago. I’ve only scratched the surface, but it is a start…

Hi X,

This won’t be a coherent essay, I’ll just “core-dump” various things that have been floating in my mind. I would have found these things useful when I was an undergrad/grad student, I hope some of it will be useful to you, too. Part of what I’ll say will be intended for a theorist like myself (I don’t have much insight about experimental work), and part of it will be just general stuff. People sometimes function in very different ways, keep in mind that some of the things I’ll mention are probably related to my personal quirks. Thanks for making me think about this stuff, I think it is quite important to pass on these sorts of things. Hopefully, someday I’ll come back to this for a second draft.

– Learning/understanding things –

In grad school, learning things was quite difficult at times. This was partly due to the complexity of the things I was trying to learn, and partly due to the sheer volume of the new stuff. Later, I was able to understand some of these things in simple terms, but understanding simple things can be difficult, too, especially if you are facing many of them at the same time. And not being able to understand something that you feel is simple can be quite discouraging.

A couple of points:

1) Be intellectually honest to *yourself*.

There will be *many* things you don’t understand, some old, some new, some simple, some complex. Some people will seem to have no problems with the things you find confusing. You will often feel stupid. You will have an adviser/professor that you need to impress. It will be difficult to admit/imply to other people that you don’t understand some of the things you “should have understood a long time ago”. Under these pressures, one may end up pretending to understand. Even worse, one may end up pretending this to *oneself*, instead of admitting confusion, subconsciously equating it with “stupidity”.

I suggest that you do away with this last tendency, and be honest to yourself about what you understand. When you are alone, when there is no one around to judge you, you *can* tell yourself, openly, what you understand, and what you don’t. Don’t fear you will come off as stupid to yourself. Very clearly, and openly identify things that you don’t understand. These may well be simple things, all the better. If you state your confusions explicitly, you will have a well-defined battle to fight, and well-defined battles are much easier to win.

Let me state this explicitly:

2) Make mental notes (better yet, physical notes) of things you don’t understand.

This not only helps you define your battles, but it also allows you to see your progress. When I find my notes from grad school about things that seem trivial now, but seemed very confusing then, I realize that I did, in fact, progress over time. I also realize that what seems confusing now will, too, probably, become clear one day.

Another reason that I think it is useful to face one’s confusions is that the sense of confusion creates a tension that tends to stick in one’s mind. So,

3) Utilize the tension of confusion.

When I see something that confuses the hell out of me, it makes a strong impression in my mind. “How could this thing be like this?”, “what is the ‘real’ reason for this strange phenomenon predicted by the equations?”, “what is the meaning behind this complicated mathematical structure that I can barely grasp?”, “I have a hard time even following it from a book, how can somebody come up with an idea like this?” etc., are questions that create this sort of tension. And this lasting tension makes you recognize the answer, once you see it. So don’t shy away from this tension, face it, and utilize it. Let it make its impression in your mind. The sense of relief when these tensions get resolved is so strong that the relevant ideas/solutions tend to get implanted in your mind almost permanently.

4) Persevere.

This is almost cliche, but very true. As a child with scientific inclinations, you may have gotten used to understanding some things the first time you see them, and things no longer work that way in research (unless you are lucky); most scientists need to face this fact at some point. You will be seeing many things that are new, difficult, not explained clearly, etc. One pass will not do it. Finish reading that paper, even if you get lost at some point. Go read some other stuff, and get back to it again. Then again. This is the way to understand hard/new stuff.

Once again, make notes of things you don’t understand in a paper/book etc. These may be simple leaps of logic, or even some words—I remember making a *long* list of the technical terms that I didn’t understand in the first paper I read as a grad student. Again, these notes will define many small battles to win, and you *will* progress this way.

5) Give simple things the importance they deserve.

One sometimes thinks that complicated things are the important ones, and one should spend most of one’s time trying to understand those. In fact, your understanding will always be based on the simpler stuff you were able to nail down. One builds more complicated things out of simple building blocks, and at any level, there will be just a few building blocks. A professor of mine once said that the people we call geniuses are those that are able to understand really simple things, really well, and I think there is some truth to this.

So spend time on simple things, whenever you have the opportunity. I don’t mean to tell you to leave what you are doing and open your freshman calculus book, but when, during your research, you face some basic thing that you think you don’t quite have a nice mental picture of, spend a little extra time trying to build that picture. Even if you don’t succeed, the tension that this creates will help you in the future, as I mentioned above.

As you spend more time using the elementary tools, you will unconsciously develop intuition about them, and this will help you come up with such nice mental pictures, but you need to make some conscious effort for this.

Next, some more general comments:

– Keeping your curiosities/interests alive –

During grad school, I ended up working on a problem that did not really interest me that much. There were interesting parts and I tried to learn more about them (which improved my mental condition quite a bit) but all in all, it definitely wouldn’t be my first choice as a project. Worse, it took a lot of time, so there wasn’t much time left to continue to learn and think about the things that really interested me. But I did, every now and then, audit a course unrelated to my thesis, discuss with friends who worked in areas that I found interesting, etc.

It so happened that many of these things that I had an honest interest in, and spent time on, helped me enormously later on. I am currently working on two research directions that have almost nothing to do with with my Ph.D. thesis, but are related to things I was curious about, and spent some time to think about in an honest way. So I suggest that if you end up working on something that is not very exciting to you, keep your curiosities alive, spend time on things that interest you—and these things may not even be scientific! Who knows where life will take you in 10, 15 years? I have an ex-mathematician friend who is an artist now, an ex-engineer friend who is a musician, another ex-engineer friend who is a sociologist. These people are much happier now than they were in their “previous lives”. It is definitely worth spending some time and effort on keeping an open mind and looking for things that you are genuinely interested in.

I don’t know if you knew XX when you were at YY—he was a prof in ZZ. I used to hang out with him every now and then, and one day I asked him what he would have done differently when he was younger, if he knew what he knows today. He gave a very simple answer: “I would have spent more time trying to find things that interest me.”

(There is a delicate issue here; in many types of research, one tends to deal with technical difficulties and routine work most of the time, rather than cool, deep ideas, and sometimes another field seems cool and exciting because from a distance all you see are the cool, exciting things. On the other hand, what one person sees as boring routine work in a field may actually be fun for another. Let me leave these sorts of things to your judgement.)

Curiosity and honest interest are very strong feelings—driven by them, one can easily spend a whole night reading/thinking about/working on something that he/she has a genuine interest in, even if there are no obligations to do so. This is something “pure work” is incapable of (at least for me). If you are working on something you are curious about, that is wonderful—work hard to keep that curiosity alive, and don’t let the technical difficulties make you forget the big picture. And if the big picture starts to seem incapable of justifying the boring stuff, wrap up what you are doing, and move on to something you are more interested in. And keep your eyes open, so that you can recognize such an opportunity to move on, when you see it.

Written by mkz

December 9, 2009 at 10:48 am

Posted in Uncategorized