What is mathematical research ?

What is mathematical research? And how is it done? Questions that have been on my mind for very long.

We know what researchers do:
- answering questions asked in the literature;
- discovering a new theorem;
- publishing an article in a journal ( i.e. Journal for Number Theory );
- writing a book;
- lecturing about their work and giving talks;
- being part of a research community.

Researchers are either employed by a university or by a corporation. University researchers have a commitment to teach and corporate researchers aren't free to chose their topics. Both are pressed to publish often in the best journals possible.

But -HOW- to they do it? What makes them successful in their field? Questions, I can't answer. Let's go and search for answers elsewhere. Starting with Manning, perhaps.



and this one:



Enjoy.

Open University M336 video lectures

#M336 #geometry #openuniversity

The Open University M336 course comes with 7 lectures on one DVD of about half an hour each, or almost four hours of lectures. The lectures are titled:
- Living with patterns
- Friezes
- Counting with groups
- Incidence symbols
- Lattices and wallpaper patterns
- Regular solids
- Octet for truss and comb

These lectures are additions to the booklets and excercises and are not meant to learn new material from, instead they reinforce what has been learned before.

So, although there is no entire lecture series covering the M336 materials you could easily create one by cherry picking lectures from the internet. For Group Theory you can use the first half of the Harvard Abstract Algebra course which covers group theory upto the Sylow Theorems.

For the geometry part you could use the MIT Course 'An Introduction to Crystallography'. This course contains 41 video lectures of which lectures 5 to 27 cover the material of the Geometry track in M336.

Link: Symmetry, Structure, and Tensor Properties of Materials MIT OpenCourseWare

Escher's imaginery workplace

#mathematics #art #Open University #m336 #Escher

The Scream by Edvard Munch was sold for USD 120 million. I didn't like it yesterday and I don't like it now that I know it's perceived value. My favourite artists are Escher, Kandinsky and Dali, their work inspires me, and I am truly impressed by what they have created, art needs beauty. M336 brings group theory and geometry together through visual symmetry, or the symmetry Escher used in a lot of his work. I browse a lot through work of Escher as a result of M336 studies. Recently I came across this sensational video. A must see, really.


( A short movie inspired on Escher's works and a free vision on how it could be his workplace. )

This is another video by Eterea.

( A short movie about numbers and geometry. )

M336 - Group Theory - Fundamental Theorem of Abelian Groups

#openuniversity #m336 #video

One of the theorems that is discussed in the group theory track in the Open University Course 'M336 Groups and Geometry' is the Fundamental Theorem of Abelian Groups. Early on in Group Theory it becomes clear that there is a connection between group theory and number theory in Langrange's theorem and the Sylow Theorems ( also part of M336 ) but only after studying the Fundamental Theorem of Abelian Groups you'll get a notion of the depth of the connection between Group Theory and Number Theory.

MathDoctorBob ( his YouTube alias ) made a short video lecture on the topic. Precise as always.

Japanese Precision

I don't know how this art is called in Japan, but it is awesome. It is not dance, it is not mathematics, it is both!

Abstract Algebra E-222 video 26 Rings 2

#maths

Watched lecture 26 of the Harvard Abstract Algebra series. - What can you say about the complex number $z$ if $(2+i)z$ must be an integer?

Prof. Gross ... "ideals in the Gaussian integers $\mathbf{Z}\left[i\right]$ of type $\mathbf{Z}/p\mathbf{Z}$".

These lectures were recorded in 2003 and are basically saved for all generations to come. Imagine that the lectures of Gauss were recorded on video! Euler and Gauss will be remembered forever by their name and picture, but the great mathematicians of today and tomorrow will be remembered by their video lectures.

Abstract Algebra E-222 video 25 Rings 2

Every word counts in mathematics.

Every non zero single-variable polynomial with complex coefficients has exactly as many complex roots as its degree, if each root is counted up to its multiplicity. ( Fundamental theorem of algebra, Wikipedia )

The polynomial $x^2-1=0$ has ( thus ) two roots: $(1, -1)$. However, if we consider the coefficients of the polynomial as elements of the ring $\mathbf{Z/8Z}$ then the polynomial has four roots: $(1, -1, 3, -3)$.

$x^2-1$ has $4$ roots...

In lecture 25 professor Gross explains how the division and Euclidean Algorithm can be applied to polynomials in Polynomial Rings over a field.

Abstract Algebra E-222 video 24 Rings 1

#M336

Just watched a video where Benedict Gross introduces Ring Theory. I don't think you can learn Ring Theory ( or any mathematics for that matter ) by just watching a video.

In the business of commercial education, in programming for example, teachers are often confronted with students ( sent by their employers ) who expect to leave as a qualified programmer just by hanging in their chairs during the course. Needless to say they leave as empty headed as they came in.

But if you watch prepared you can pick up a lot from this professor. In this first lecture he explains why there is such a field as Ring Theory in the first place. Where did it come from? And most of all: what are the important topics we have to watch in this field? ( I.e. Ideals and Unit Groups ). You may wonder why I gave this post the M336 ( Groups and Geometry ) hash-tag, it is because Rings and abelian Groups ( and Number Theory ) are intimately connected and one of the objectives of M336 is the classification of all abelian groups. - By the way, the word is abelian group and not Abelian group despite the fact that the word abelian comes from Niels Abel. Writing a name lowercase is the highest possible honor in mathematics. ( So I have been told... ).

$(\mathbf{Z/nZ})^{\times}$ has $\phi(n)$ elements

At the end of this lecture he mentions that Group Theory is a really hard subject and all that. The thing with Group Theory is that it has to sink in quite a while before it clicks and opens up to you.

Riemann Hypothesis ( Video )

Very well done video about the history, and the implications of the RH.

[Video] Deriving Binet's formula for the Fibonacci numbers

One of those formulas every mathematician loves ( I think ):

$$F_n = \frac{1}{\sqrt{5}}( \phi^n - (1-\phi)^n )$$

Here's how MathDoctorBob explains it.



Although I like The Doctor's videos ( I wished the real doctor would show up accusing me for abusing his name but taking me for a ride in his phone box anyway ), I wouldn't like to have Doctor Bob as a tutor in class, I simply wouldn't be able to catch up and I am not the audible type anyway. I like to read a bit, play and think a bit, read a bit, and so on. Every person has its own unique style of learning that works for him. Part of studying is discovering your own learning style.

Oh, and I think this formula beats the one of Binet ( although strictly speaking not in closed form ), because it fascinates me that the Fibonacci numbers are actually -in- the triangle of Pascal.

$$F_{n+1} = \sum_{k=0}^{n} {n-k \choose k}$$

Example of a Galois Group of order 8 ( Introducing Math Doctor Bob )

Regular readers must have noticed my interest in Abstract Algebra, of which I am currently studying, in different ways, the topic of Galois Theory. If you have chosen a different route in mathematics ( computation, statistics, and so forth ) or if you are at the early undergraduate level you may have difficulty picturing what Galois Theory is -all about-. I am trying to communicate that idea by summarizing the popular introduction to the field 'Fearless Symmetry' which basically introduces Galois Theory to the general ( but educated ) public. ( Currently working on part 7 out of 23). But as they say, one picture says more than a thousand words. For those that want to get an idea, fast and easy, and *now*, I recommend the following video ( mini lecture ). Don't expect you can master the subject by watching a ten minute video but the ten minutes are well worth it. The video lecturer is 'Math Doctor Bob', who uploaded about 600 mini lectures on various mathematical topics.



See also:
- Fearless Symmetry

More Alan Turing

( Although I have read more Fearless Symmetry haven't done a new summary yet. )

Anyway, I have selected two Alan Turing videos. First a clip from Nottingham Trent University in the series FavScientist.



Then the great Derek Jacobi as Turing in a clip from the docu drama Breaking the Code.

Peter Neumann speaks about Galois.

Found this video more or less by accident on YouTube.

Mathemusician puts Pi to music.

I -love- creative people.

Video lectures on Theory Of Automata, Formal Languages and Computation

NPTEL released a new series of video lectures featuring Prof.Kamala Krithivasan from the Department of Computer Science and Engineering IIT in Madras on  the Theory of Automata, Formal Languages and Computation. The first lecture in the series is called GRAMMARS AND NATURAL LANGUAGE PROCESSING.

Goedel, Escher, Bach - Lecture 4(1)

My brain ran those neural network algorithms.

Justin Curry

The reading assignment was chapter 6 'The location of meaning'. It is basically about coding and decoding. Of course Hofstadter mentioned the Rosetta Stone in this context, the key to ancient Egypt. It contained a parallel text in three languages and was deciphered in 1821 by Champollion.

A recent example in the context of chapter 6 is space archeology. Archeologists and Egyptologists were able to interpret satellite pictures of Egypt which lead to the sensational discovery of new pyramids.

Let's go to the lecture. ( This is a 1h46m lecture and will be discussed in two posts. )

Curry talked about Goedel numbering again, a method Goedel used to code strings in formal number theory to numbers. What Curry said about coding a string in formal number theory, playing with it and then code it back is only true in theory. Simply because Goedel numbers become -extremely large-. Think of numbers built from pages full of digits. ( Would Curry ever have calculated a Goedel number? This reminds me of a DBA course I attended once. The trainer talked about all the beautiful properties of the then new RMAN from Oracle as if backups could be recovered in an instant. It turned out that he never worked in the trenches of 7 x 24 administration of large databases. )

Dialog "Contracrostipunctus" on page 75 of GEB is discussed. How this dialog has meaning on several levels. The dialog refers to itself that it contains a hidden message. The concept of 'Self' is introduced here.

Starting with what does "Snow is white" mean? he builds an argument that there is an isomorphism between electrical activity in the brain and the interpretation of symbols. ( Thought reading might be possible after all, one day. Isn't it true that man can create everything he is able to envision? )

Adam and Eve

There are at least two phases in the proces of assigning meaning two a string. The first is parsing the string, the second is the interpretation of the parsed words. Interpretation depends on the context of the interpreter.

Message in a bottle

He introduces the concept of information. For example how physicists reduce complex physical behaviour to a small sequence of symbols. Like for example how a pendulum works.

Pendulum ?

Goedel, Escher, Bach - Lecture 3

A guy named Euclid.

Justin Curry

Curry briefly explaines the concepts:
- consistency
- completeness
- and geometry.

A consistent system leads to conclusions that are not contradictory in any sense. A statement is either true or false, and never both true and false.
A system is complete if everything that is true in the context of that system can be derived from the axioms.
Regarding geometry he mentioned that there are Euclidean non-Euclidean geometries.

Then he attempts to explain Goedel's Incompleteness Theorems.
1. Any system as powerful as number theory which can prove its own consistency is necessarily inconsistent.
2. Any system as powerful as number theory is necessarily incomplete.
He explains that Goedel managed to transform the idea of provability to a property of numbers by introducing his Goedel numbers.
He says that students should now have a notion of the Goedel theorems and promises that this is just a first glance at Goedel's theorem. ( Not sure if he meant he would come back at Goedel in this lecture series. )

Trying to explain Goedel

He then talks about Euclid and his postulates.
(1) Any straight line segment can be drawn joining any two points.
(2) Any straight line segment can be extended indefinitely in a straight line
(3) Given any straight line, a circle can be drawn having the segment as radius and the
(4) All right angles are congruent.
===
(5) If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles

He explains that the 5th postulate is consistent in Euclidean Geometry but not in spherical and hyperbolic geometry.

Hofstadter Dialog - Little Harmonic Labyrinth is removed from the video due to copyright concerns. It is part of what makes GEB such a difficult book. Here is part of it.

The Tortoise and Achilles are spending a day at Coney Island. After buying a couple of cotton candies, they decide to take a ride on the Ferris wheel.
Tortoise: This is my favorite ride. One seems to move so far, and yet in
reality one gets nowhere.
Achilles: I can see why it would appeal to you. Are you all strapped in?
Tortoise: Yes, I think I've got this buckle done. Well, here we go. Whee!
Achilles: You certainly are exuberant today.
Tortoise: I have good reason to be. My aunt, who is a fortune-teller, told me that a stroke of Good Fortune would befall me today. So I am tingling with anticipation.
Achilles: Don't tell me you believe in fortune-telling!
Tortoise: No . . . but they say it works even if you don't believe in it.

About 1% of the Little Harmonic Labyrinth dialog.

He explains the cardinal arithmetic, the arithmetic of infinities.
An interesting definition of infinity is that a set can be mapped to a subset of itself. I.e. the natural numbers can be bijectively mapped to the even numbers. The points on the real line can be bijectively mapped to the points on the line between 0 and 1.

Non-Euclidean geometries

Goedel, Escher, Bach - Lecture 2

It will get a little bit mathy, but that's ok.

Curran Kelleher

Lecture 2 is given by Curran Kelleher. This lecture is all about recursion and ends with a nice explanation of the Mandelbrot set.

He starts out with the traditional examples factorial:

factorial[0]:=1;
factorial[n_]:=factorial[n-1]*n; 

and fibonacci sequence:

fib[1]:=1;
fib[2]:=1;
fib[n_]:=fib[n-1]+fib[n-2];

Kelleher's factorial program

Kelleher's hand-out ( pdf ) contains examples of Java code for drawings of the Koch curve and Sierpinski triangle. Although I fast-forwarded through this part of the lecture, it may be very interesting for non-programmers.

Explaining the Fern algorithm

Complex number implemented as a class in Groovy 

At around 1:00 he starts with the topic of the Mandelbrot set. Starting with f(z) = z^2 + c he manages to give a nice explanation of the Mandelbrot set. The color of a point in the Mandelbrot set is based on the number of iterations it took f to 'escape' a circle. Since this is done on a pixel by pixel basis and one pixel may generate not one but several iterations this explains the long time it takes to generate a Mandelbrot set.

P.S.
GEB does not seem 'outdated' at all although it was written in the late seventies. A time when there were no mobile phones, no PCs, let alone laptops and the internet was still in its toddler phase.

Goedel, Escher, Bach - Lecture 1

'Understanding Goedel' is one of the major goals I set for myself.

This final unit brings together all the ideas introduced in the course. These ideas constitute the technical machinery that enables us to prove some very important theorems which answer what we have called Leibniz's and Hilbert's Questions. These theorems, Goedel's Incompleteness Theorems, are among the most profound intellectual discoveries of the the twentieth century. Thus you should not be surprised if you find this unit hard going in places.

M381 - Unit 8.

In Goedel, Escher, Bach (GEB) Hofstadter asks the question: what happens when 'things' start referencing themselves? ( Like people do who are in essence not more than a set of linked molecules. )

At last I took the time to watch video 1 of the GEB series.

Justin Curry

The teacher is Justin Curry. He started by telling that most undergraduates don't get through GEB in less than 13 weeks and that it took him seven years to get through the book. I am not sure but I think my first attempt in reading GEB was in 2007 or 2008. It took me almost six months to get through it. Which I thought was really bad. When I finished the book and still didn't understand what he was talking about I started to seriously doubt my learning abilities. I have to admit that I still don't get it but I made progress. And I am getting closer, thanks to M381 Mathematical Logic ( read: Nigel Cutland ).

Anyway, to the point: the lecture.

He starts with the concept of isomorphism. In GEB, Hofstadter explains isomorphism as a map between structures that maps parts with similar purpose to similar purpose ( my words ). This is different than the mathematical definition which states that an isomorphic map is both surjective and injective. Hofstadters definition can be understood immediately, whereas the mathematical definition needs understanding of layer upon layer upon layer. Since what is a map in mathematical sense? What does surjective mean? What does injective mean? Analyzing a mathematical sentence always creates a ( large ) tree structure.

Recursion. The concept of recursive definition. A fascinating concept which I use a lot, since I am a programmer by profession. Curry uses the example of the Fibonacci sequence 1,1,2,3,5,8,13,... and translates it to f(n) = f(n-1) + f(n-2) and the Sierpinski triangle ( fractal ).

Drawing the Sierpinski triangle

( To be continued in the next post. ) Edit: Nope. I'll make a last note about lecture 1 here and continue with lecture 2 next time.

Some remarks, tips for if you want to give it a try ( like myself ). I was not in continuous awe while watching this lecture. You know when like you are watching the latest BBC Horizon or similar. It's not like that. I don't have the feeling as if I have wasted my time, not at all. I am going to watch lecture 2 soon.

- You definitely need the 720+ pages ( 20 chapters ) book. ( Details on the course site. )
- You need to be ( somewhat ) familiar with Bach's music, or at least -know- someone who is. ( What are forums for anyway? ) To fully grasp the genius of Hofstadter's work.
- If you are a religuous person than GEB might not be for you.

There is an audio set in the lecture room. Near the end of the lecture a piece of Bach is played. Students familiar with that music could elaborate on it. Since I am ignorant to most classical music I must have missed a lot of what Hofstadter said. It might be an opportunity to start listening to some Bach, who knows what happens.,

So far for lecture 1,

M381 mathematical logic - or Escher, Goedel, Bach - Revisited

Although I believe the academic world is at least as closed as it always has been, the access to ( mathematical knowledge ) is ( almost ) a level playing field for those in and those outside the thick walls of academia. All thanks to the development of Internet. Some people say we live in a short period of a truly free Internet that once the technology stabilizes ( whenever that will be ) more and more sites will become accessible to the elite only. A bit like today's access to scientific journals for example, which give professors a six month to a year lead.

To the point: I still haven't watched the MIT video-series 'Gödel, Escher, Bach: A Mental Space Odyssey' because it is stored in Real format and I don't want to install Real with it's bogus Adware. But these files play well in VLC Media Player which is my favorite media player. Now is the time to watch these videos, I suppose. In the middle of doing M381 mathematical logic. Now that I have two ( real ) books on the subject the Open University booklets seem so much more accessible. In retrospect I think that M208 is an extremely difficult course IF you go by the booklets alone. Enough said over that subject.

I noticed that understanding a mathematical topic is something continuous it's not a discrete 'I get it' versus 'I don't get it'. The Jigsaw pieces are slowly getting on their right place and the image becomes clearer every day, Goedel's Incompleteness Theorems. I remember that when I read Hofstadter's book that I thought that he was an absolute genius ( he might be ) and that it was impossible for me to =ever= understand Goedel at a mathematical level. And now, years later, I am getting closer to that every day. I still have this idea about unreachable highs of mathematical knowledge represented by mountains disappearing in the clouds. I am far from the mountains on low ground. As long as I keep walking I must get there one day.

MIT video, six lectures on GEB.