Back in Toronto

Well.. I'm finally back in Toronto.. Didn't have so much time, nor the energy to do so much today. I went out walking around the area where the hostel is and through the nearby park. Here are some pictures.

Goodbye Fredericton!

After Summer school ended I spent the two next days sleeping...

And today Monday, was my last full day in Fredericton, so I spent it by walking around taking some pictures and looking at the city. It isn't as bad as I might have said it is. There wasn't very many things to do, and everyone left on Saturday, so I was all alone here.

Some of the buildings on campus was very impressive. There were parks, and departments I hadn't seen before.

A very nice last day.. Farewell Fredericton!

Roadtrip Part 3- Whale-watching

After our classes ended, me and two others began our drive down to St Andrews. We arrived just in time for the Whale watching, and headed out.

Some pictures of the whales we saw. We saw finback and minke whales, and I can't tell them apart on the pictures.

Of course there were lots of seagulls.

These are not dolphins, although they look a lot like them. They are porpoises (sv. "Tumlare") and they don't have the pointy "noses" that dolphins have. On this picture there is a mother and her calf (on the right). The porpoises that we saw were maybe 50 cm long, so they were small.

Here are some rocks that have appeared when the high tide was going out. On them, about 50 seals have decided to take a rest. After we got back, I sat down and did some Riemannian Geometry.

I part of me has always wondered how everything would have been if I had never lost my interest in whales.. Maybe I had become a Marine Biologist..? All I know is that Mathematics and Physics is way more AWESOME than some whales! :D My life had been so boring, just going around looking at some weird whales! Omg that would have been horrible.. I think that some things are better left as dreams..

Roadtrip Part 2 - Prince Edward Island

Day two of our travels, we went all across New Brunswick to get to Prince Edward Island. To get there we had to drive for several hours, and then cross an enormous bridge to get there.

In the background, the bridge to Prince Edward Island.

Prince Edward Island was very beautiful, everywhere you looked there were green fields, flowers and trees. After driving to across it, we went to a big beach area somewhere near Green Gables. If you have read/seen Anne of Green Gables, it is set in this area.

Prince Edward Island is very beautiful.

Even though this was the waters of the Northern Atlantic Ocean, it was really nice to swim, and there were some nice waves there. I really should learn how to surf.

After that, we drove to Charlottetown, where we had supper at an Italian Restaurant, before heading back. I might mention that all this driving didn't cost very much. The price for gasoline in New Brunswick is 1 CAD (about 6.8 SEK)/litre, for regular gas. And it is cheaper in other parts of Canada, and the gas prices in the US, is apparently way lower. Guess not everyone is having an oil crisis, eh?

After having dinner, we began our long drive back.. Another great day.

Roadtrip Part 1 - Saint John & St Andrews

Before the last weekend, a few of us had been talking about that we wanted to see more of the region, and not just Fredericton. I saw this as an excellent opportunity, because I had already decided that I wanted to go whale-watching, and buses were inconvenient and really expensive. So a couple of us, went to a car rental company and rented a car. It was supposed to be a compact car for 5 people, but when we went down to get it they didn't have one, so instead they gave us a minivan for 7 people for the same price. :)

We talked two more people into going and then we headed off, towards Saint John and then St Andrews.

Saint John is a nice, quite big port town, and is one of the oldest in New Brunswick. It's a very nice little town, with charming houses, parks and very friendly people.

Saint John's Saturday market is much more thrilling than Fredericton's. Here they have lots and lots of stuff, and even some live music. :)

What can you say..? It's Canada...

And here are the famous (haha) Reversing Falls. As the picture might tell, things start to get crazy when the laws of Physics break down. The Reversing falls is a river, that from high to low tide changes direction (because the water levels change). This creates some whirlpools and strong currents. Here the river is flowing upstream, up the river. (it just changed)

After a brief visit to Saint John, we (Michal) drove down along the coast line, almost to the border of Maine, to the small harbour town of St Andrews.

This is pretty much the shopping street, but it is a nice little town, so don't be fooled by its looks. Unfortunately, all the whale-watching tours were booked for the day (I didn't make a reservation, because I wasn't sure we were going to get there) Fortunately though, we had the car until Tuesday morning, because everything was closed on Monday, because it's New Brunswick Day! ;) But we made a reservation, to go whale-watching then after class..

The rest of the afternoon, we spent on a desolate beach.. After that we wnt back home.

Kings Landing

AARMS had planned a trip for us summer school participants; a day trip to Kings Landing Historical Settlement, which is basicly a couple of houses with some people showing how the first settlers in New Brunswick lived. For the people of NB, this might appear like a big part of their history, and very old, but being from Europe in general, this isn't very old. (I mean, Old Town in Stockholm is hundreds of years older than their country).

Anyhow, this was a nice trip.

A typical house.

The cows are of a different breed than the one we have in Sweden.

A saw mill, or rather the waterworks that power it.

Hm... Where have I seen these kinds of stoves before??? ;)

The weekend and Monday, Tuesday, Wednesday..

Well.. I didn't really feel that I have had time to post anything new on my blog during the last couple of days. I have been studying really hard.

My courses are really advanced, and all of a sudden we started using K-theory and Vector Bundles in Noncommutative Geometry. Taking three courses is a lot of work, but I think I'll manage...somehow. ;P Time to remind myself that I'm awesome. ;)

Friday evening, I needed a break from quivers, Lie groups, Gelfand transforms... So I went down to the common room and played my first game of Foosball. After an intense match I and my Italian friend stood victorious. =)

After that we started talking about getting the crew together and go to a pub or a club (assuming that Fredericton had one.. haha :P). So the "Math gang" headed downtown. Here are some pictures of some of us.

As you can see I am drinking a Gin and Tonic, for as I have said before, beer taste like piss.

After that we headed back, I taught them how to dance (Salsa, Swing...) in the common room, and then I went to sleep..

Next morning, some of us were going down to check out the "famous Saturday Market". It was supposedly a great place to go to, and they were supposed to have lots of good stuff.

As you might have guessed, this place was not my kind off place to be at... It was too crowded, small, sold junk, and even smelled bad. (Sorry Fredericton) :(

After that I headed back and studied some more.. Sunday was just really stressful, because there was a lot of things to finish for the Noncommutative Geometry lecture on Monday. So I sat down and proved Heisenberg's uncertainty principle (awesome proof by yours truly), or rather showed that there is no unital Banach Algebra that exist, such that the quotients cancel. =)

After that, I went out for an evening walk, cleared my mind and watched the stars. It was really nice to not think of things that are not Noncommutative (or at least not in an obvious way).

Monday, Tuesday and today have just been full of stuff to do, still I've managed to go for a walk every day. The only bad thing is that I am reminded as soon as I leave Campus, how polluted the air is.. :( You can really smell the cars, even in this small town. :/

A couple of my friends here and I were talking about renting a car and go down to Saint John. We haven't really decided if we are really going, but if we are, then maybe next weekend. I have also been looking into taking a two day trip from Toronto to Ottawa, so I get to see the capital, but haven't decided yet.. Figured I might be able to take two night buses, so I don't have to pay for a hostel two nights, then it could fit into my budget. But haven't decided yet..

Swimming

Today was yet another day here. It was basicly the same as other days, except from the fact that it was sunny. :) After class, I went with one of my friends to the swimming hall on campus, and swam for an hour. I got to use my special guest card from the AARMS. :)

Awesome! =) They've got several sports halls, which are always free for me with this card, so why not use it? My next project is to find someone to play badminton with..

After swimming and having supper, I got back to my room and continued with my studies..

I realized that I am kind of taking 3 courses, which is supposed to be an impossible amount of work as the director of the summer school said. But if I find NCAG too difficult, then I'll drop it. :P I might only get this chance once, so why not go for it?

Not much really...

Today was yet another day. It is basicly like most days are here; wake up, shower, breakfast, Riemannian Geometry, Noncommutative Geometry, lunch, power nap, read some, Noncommutative Algebraic Geometry, read some more, have supper, study and do homework, sleep etc... Because of this there isn't really that much to write about, so don't be surprised if I don't write that much while I'm here..

One interesting thing though, one of my Polish friends here and I were talking about going to swim, since there is a swimming hall on Campus, and we can go there for free through the university. Maybe tomorrow? They were open really strange hours..

Today in Noncommutative Geometry we introduced the Gelfand transform and showed that it has the same basic properties as the Fourier transform. This is nice, since I am quite familiar with the Fourier transform. :)

In NCAG (abbreviation for NonCommutative Algebraic Geometry from now on) we introduced quivers, which is really just Directed Graphs with a defined Algebra over it. For some reason, algebraists seem to like to take simple things, rename them, and then use them (I am very aware that there are purposes for this but still). Anyways, quivers will enable us to go from the Noncommutative Algebra to the Algebraic Geometry and vice versa, so that we will be able to use methods directly between them. The name quivers is quite ingenious though, because a quiver is the container that holds the arrows for an archer, and directed graphs basicly look like arrows, so a quiver becomes in the same sense the thing that holds the arrows.

I found out something interesting as well between classes, when I spoke briefly to our professor, Massoud Khalkhali. I asked him if we were going to cover the Non-Hausdorff spaces in the course, and his response was that we weren't. All that we will cover is the spaces that can fail to some extent to be Hausdorff (wikipedia it you are wondering what a Hausdorff space is). That is an interesting case, but I don't think that is where we will find the answers that explain the universe. I think that the Geometry and Algebras that will explain most of Physics and Mathematics are actually contained in the abstract spaces which are not Hausdorff at all. But as it seems, there isn't much progress in that field, because well it doesn't really make sense..

The first days.

Monday morning (yesterday) at 8.30 am we had our first lecture, in Riemannian Geometry. The first part of our course is devoted to developing some concepts from Differential Geometry (which is my major field) so it hasn't been so challenging. :)

The thing about his lectures is that he keeps saying these things, which made me think that Riemannian Geometry (or rather Differential Geometry) isn't his actual field. His approach towards proving properties of curves, by finding a frame, was purely analytical and, for me quite strange, because it got rid of almost all geometrical interpretations whatsoever. Curves are not really that interesting to study anyway, but then he went on and said something about that the real point of Differential Geometry was to use the properties of the embedding functions, which is not true at all. From what I know (I don't know anything really about anything, ask me in 10 years and I'll say the same thing) the point of Differential Geometry is just to make way for the Riemannian Geometry (which works for an arbitrary number of dimensions) unless you are in fact interested in studying curves and surfaces in 3 dimensional spaces. The biggest point of Differential Geometry, I would say, is to find that several properties are independent of the embedding in space (the opposite of what he said), like the Gaussian Curvature, and it is in fact these ideas that gave Riemann the idea of developing a Differential Geometry for arbitrary spaces which are purely based on these intrinsic properties that are invariant of the embeddings.

The second lecture I had that day was in Noncommutative Geometry. We started by showing the correspondences between Noncommutative Algebra and Noncommutative Geometry, and then stated that the Noncommutative Geometry is really what justifies the Noncommutative Algebras. The professor is an expert in his field, and seems very intelligent. What is even more impressive is that he doesn't use lecture notes, instead gives theorems and proofs of very abstract concepts just out of his head. That is basicly the level of understanding that I one day hope to achieve.. Even though the course seems very good, it is 2 or 3 years above my level of understanding. But I'm reading a book right now, on Algebras and Categories which might give me the knowledge I need. :)

Since I got the chance of coming here and I really want to learn, I am also sitting and listening to another course, called "Noncommutative Algebraic Geometry" by a professor from Oxford. That course uses a more algebraic approach toward Noncommutative Geometry, and even if it is a bit over my head, like the other course, I'm hoping that some things might stuck, even if the whole picture will elude me for some time in the future.

After that, me and some friends went to Walmart and I bought some Coke to drink while studying and some Shower Gel. After that and after we ate in the cafeteria, I locked myself in my room and started to read some Abstract Algebra.

This morning I woke up, showered, had breakfast and then went to class. It is amazing how much fried food they seem to eat here. Every meal here since Sunday morning has had something fried, eg. Potatoes, Chips, Fries, something that looks like the Swedish "Rösti" etc. Fortunately it is somewhat like a buffet, so there are plenty of different kinds of things to eat, so I have been able to avoid the fried things. Another thing that I noted is that there are always sweets to have as dessert with the meals. I have been avoiding it so far, because that amount of sugar is just not good for you. And most people drink soda with their meals (sometimes I drink the diet Pepsi), but that has to give them a lot of extra sugar.. Well what can you say? North Americans...

Apart from my classes today I have mostly isolated myself in my room to read. I hang out with the people the minutes before and after the lectures and in the dining hall, but apart from that I try to study as much as possible, because well... In a month, I'll most likely never see most of them again, while my knowledge is forever (or until I get senile or die ;) ).