Blog: Articles filed under NSERC 2010

This Friday, Saturday, and Sunday I attended the eighth annual Combinatorial Algebra meets Algebraic Combinatorics Conference. No, I didn’t record awesome video diaries as I did when I attended the 2010 Canadian Undergraduate Mathematics Conference. I did meet many experts in these fields, listened to interesting talks that I didn’t really understand, and gave a talk of my own!

Combinatorial algebra and algebraic combinatorics are, as the conference’s title and purpose expresses, two sides of the same mathematical coin. They are areas of mathematics that combine techniques from combinatorics and abstract algebra (notably, commutative algebra) to solve a variety of problems in algebra, combinatorics, and even algebraic geometry. Now, these fields are specialized. I got the impression that even among the thirty or so graduate students, postdocs, and professors in attendance, many of them were struggling to keep up with some of the talks, because the topics in this area, as with any specialized field, can get pretty esoteric. One fellow gave a talk on cluster algebras, and the room was rather silent when it came time for questions.

Still, it was exciting to attend the conference even though I, as an undergraduate student with only two courses of basic abstract algebra under my belt, understood very little of any of the talks. I was invited to speak at the conference by Adam Van Tuyl, chair of our mathematics department and one of the conference organizers. He supervised my summer NSERC USRA. I previously gave a talk about that research in the fall, and he felt it would be a good fit for the conference. I was a little sceptical, not to mention a little intimidated by the notion of talking in front of all these learned academics. Nevertheless, I acquiesced—I mean, that opportunity might not come again. I‘m getting a lot of mileage out of this talk.

If you are interested, I’ve set up a page explaining my research on the spreading and covering numbers. Unless you are familiar with abstract algebra or graph theory, most of it will sound like gibberish, but check it out any way. You can also download a copy of the talk I gave, as well as the Macaulay2 code I wrote.

Giving my talk, which was well-received, was one of the high points of the conference, of course. For one thing, I‘m pretty sure everyone there followed what I was talking about, since I was presenting it on a more elementary level than a postdoc or professor would. And that’s fine. More importantly, a few of the attendees had some interesting ideas that might help me in the future. I am currently applying for another NSERC grant to continue working on this project this summer; hopefully I’ll get the grant and be able to put some of those ideas into practice. If anything, going to the conference has made me more excited about working on this problem again.

Another high point was meeting Tony Geramita. He co-authored the paper that introduces the spreading and covering numbers, essentially making him the originator of what I studied. And he knows his stuff; he seemed to switch gears effortlessly between each talk and ask intelligent questions (or at least, from my limited understanding of the topics, they seemed intelligent) whenever he needed clarification. So meeting him, and giving a talk about these spreading and covering numbers in front of him, was kind of a big deal. Plus, my natural tendency toward introversion means it takes me a while to warm up to new people, especially ones whom I meet in an artificial, arranged way like this.

So imagine my surprise and amusement when, at lunch, I brought out my copy of Forest Mage, and he said, “Ah, you’re reading Robin Hobb.” From there we conversed about our mutual love of science fiction and fantasy. Later, we started talking about eBooks, and he spontaneously asked if I had a thumb drive on me so he could give me a 1 GB library of eBooks he has on his computer. I was somewhat taken aback by this random and generous windfall. (I used my phone, since it had 11 GB free on its internal SD card. I should probably get an external one too.) This unforeseen icebreaker made it easier for me to think of him as a person, not just a Smart Math Individual, and much easier to give my talk.

Saturday night, after the conference, we went to the Masala Grille for dinner. Although my dad and I have ordered takeout from this Indian restaurant in the past, I had never actually been there to eat, so that was an interesting new experience. We had the upper room to ourselves, and the food was good (although I made the mistake of putting too much sauce on my plate). I had some interesting conversation with the people at my table about a variety of things, mathematics and non-mathematics alike, including an opportunity to talk to an Iranian fellow who is at Dalhousie for the summer. This was his first trip outside of Iran, and it was cool to hear about the situation in that country from someone who has grown up and lived there.

All in all, I have to admit the conference was a great experience, even though it did have people at it and did not in fact consist of me sitting in a chair reading a book all weekend. Sacrifices had to be made, and they were worth it! But don’t think this means I’m going to grad school just yet, despite the fact that more-than-hints have started to drop! But that is another topic for another blog post. Now I have to concentrate on finishing the rough draft of my honours thesis, for it is due on Thursday.

Hello September. I have missed you. You might be my favourite among all months, but don’t tell the others. And no, it’s not because my birthday is in September (although that helps). Nor is it because September signals the start of fall television, with new episodes of Castle, Chuck, House, Stargate Universe, etc. More than any other month, even that notorious January, September is a month of changes and new beginnings. For those of us biased in our perceptions by our position in the northern hemisphere, summer will soon be a memory; the leaves will change colour; and I’ll be back in school, where I belong.

I spent this summer doing research and quite enjoyed it. We didn’t make as much progress toward a solution as I had hoped, but I learned a lot, both about mathematics and research in general. I’m comfortable using LaTeX (which is sexy) and have had some experience with Macaulay2 (also pretty hot). I even went to a conference, something that surprised me.

With my research finished, I have these two weeks off before school begins on September 13. Next week I return to work at the art gallery. I don’t look forward to returning to the job that much; my relative solitude of this summer has left me even less eager to interact with people in a customer-service-based position. But I do miss my coworkers, my fellow front desk attendants, so I look forward to returning to them.

I anticipate another great year of school as well. This is my honours year for my math degree, and the Honours Seminar will consist of a sort of research-based project supervised by a prof. We’ll have to write a math paper and give a talk. This is a nice departure from lecture-based courses (I don’t much care for lectures); also, having done research, read papers, and written up results for the past four months, I feel somewhat prepared.

And with summer endings and fall beginnings come changes. My site last had a major redesign over two years ago. I’m still happy with the design in general; however, there have always been certain rough edges I wanted to correct. Now I‘ve done so. A few weeks ago, I rolled out tweaks to the design and significant changes to the backend.

I’ve reorganized the content on the home page. It’s my portal on the Web, something that lets people access my content whether it’s on this site or elsewhere. I‘ve tried to lay it out so that everything is on offer.

You’ll also notice that I have a new background image. Now that is definitely tea. The other image was tea, but ambiguously so, and the berries were an odd addition—it was a very Christmas-like cup of tea. It was the best photo I could find at the time. This new photo is exactly what I envisioned when I originally decided to use a cup of tea as my background image, and I‘m very happy with it.

For a long time, the only real content on this site has been my blog and the About section. Everything else consists of links and a little aggregated content. I have plans to change that soon and add more pages dedicated to original content (or specific aggregated content). For example, you’ll notice that my home page no longer displays my most recent book review from Goodreads. I want to keep my home page compact, and you can easily access my 15 most recent reviews from the books on the sidebar. Instead, I intend to create a new section of the site devoted to my reading habits—not just reviews, but top 10 lists, statistics, etc.

This sort of flexibility is thanks to the new backend. I’ve finally gone over to the dark side and started using a CMS—but not just any CMS. It’s Symphony, an XSLT-based CMS that is both minimalist and developer-friendly. The custom-coded backend I was using was rubbish, and I don’t need anything as powerful as an entire framework. Symphony is exactly what I need, and I highly recommend it.

It is Saturday, but it doesn’t feel like Saturday, mostly because I’m … at school. This is the last day of the CUMC. I’m in the last talk of the day, having chosen to attend “Perfect Matchings and Shuffling.” Afterward, there is the final keynote, which Ram Murty will deliver on the Riemann hypothesis.

Yesterday I went to a talk on fractal image compression. The talk itself was not stellar, but there were some good questions on the applications of this type of lossy compression, and the speaker addressed those well.

In the afternoon Aaron, Rachael, and I took a bus—yes, a bus—down to King St. This was my first time riding public transit, and it wasn’t in my own city! Aaron wanted to visit a small record store, Orange Monkey Records, and then i checked out a used bookstore known as Old Goat Books. I bought more books than I should have, considering they need to fit in my sparse luggage—but I couldn’t resist.

The final keynote of the day was delivered by Greg Brill, of Infusion. Although titled “The Evolution of Technology,” Brill’s talk was not what I expected. He has a Masters in computer science (after coming from a liberal arts background!) but talks like a showman rather than an academic or a businessman. He discussed how mathematics—and hence, mathematicians—are essential to the development of technology, particular business products. For example, he mentioned how his company had been working with motion-sensing technology similar to Kinect, and that the main problem was not a lack of technology but a lack of the mathematics necessary to achieve what we want in that area. Brill is very keen on the idea that we are moving from an idiomatic society to an idiom-less one and is convinced that mathematicians will help make that happen.

Dinner came in banquet form, and while the food was OK, the dancing was better. That’s right: dancing. I love to dance, and I had a great deal of fun on the dance floor for about an hour or so before calling it a night. I’m not as young as I used to be.

Tonight we fly home, and tomorrow I go to my nephew’s first birthday party (no weekend recovery for me). Then it is back to math research: reading papers, re-reading papers, writing algorithms, and making tea. CUMC has been fun, but I will be glad to be home.

It is Thursday, July 8.

After the first talk this morning—on set theory, particularly ZFC—I spent time caressing the lovely wireless network by way of uploading some photos to Flickr. When attempting to geotag them, however, I ran into the slight problem, in that typing “University of Waterloo” into the Flickr map’s location finder produced no results.

So, Yahoo!, in case you are wondering why people drool over Google and its products, here is a hint: we are lazy. When I type in the name of a major university, your map should be able to find it for me. I should not have to go find a postal code on my own, enter that, and wind up in the general vicinity of the campus. (I used Google Maps to find the postal code too, which just seems wrong). It is not that I am a Google fanboy, Yahoo!—they just do it so much better.

At lunch, I did something completely out of character and chose to be adventurous, purchasing bubble tea for the first time. My less adventurous self was soon vindicated. We went to a fast food place called “The Grill” for food. I attempted to poke my straw through the seal placed over my cup—urged on by Rachael’s encouragements of, “Just do it!”—and after one mighty stab, the straw went through … and the bubble tea exploded. A plume escaped from the top, but the cup also developed a leak in the bottom somehow, and it spilt all over the table and down onto the floor. We don’t cry over spilt milk, but what about spilt bubble tea?

I also decided to be adventurous when it came to food. The menu had a “lamb burger” on it. I have had lamb before, but never in burger form, so I ordered one of those. Its taste was similar to a regular hamburger, which disappointed me.

For some reason, I was lethargic after lunch and greatly desired a nap. I blame the heat. I struggled to stay awake and pay attention to the afternoon’s talks—first one on computability theory, and then another on universal algebra. After that, we had a little break before going for dinner. Rachael and I ordered some chicken fried rice from a Chinese place, while Aaron opted for shrimp wonton soup. The price was right and the portions huge—I could not finish mine, although I came close, while Rachael ate a lot and left even more.

The morning keynote speaker was Michele Mosca, from the University of Waterloo. He talked to us about quantum computing, with a particular focus on quantum cryptography. The talk was more about mathematics than of mathematics, with only a little actual math involved. I quite enjoyed the subject. Quantum computing is a concept that sounds like science fiction, but it is real; we have quantum computers—albeit primitive ones—right now! The future is here.

It is Wednesday, July 7. The CUMC talks began today.

I went to four talks today. Rather than summarize them all—I enjoyed them all—I’ll mention some highlights. The first talk of the afternoon was both my least favourite and most favourite talk. Entitled “The Ontology of Mathematics: Do Numbers Exist?,” the presenter read from dense slides, which did not make for the most riveting experience. There was some lively discussion among the audience, however, and I enjoy talks like that.

Comparing CUMC to the Combinatorics & Optimization workshop that preceded it, I prefer the student talks of the former. The topics are so varied—there is so much choice within each time slot, that it is difficult to decide which talks to attend. The atmosphere is less intimidating, because it’s undergraduates talking to undergraduates. I almost regret not giving a talk myself—almost, for it would involve public speaking, and long gone are the days when classes made that mandatory.

There were two keynote speakers, one at lunch and one at the end of the day. First, Frank Morgan, from Williams College, gave a talk on densities and the Poincaré conjecture. As I have never studied differential geometry, most of the mathematics went over my head. The audience in general got into it, however, asked great questions, and we all tried answering the questions Morgan asked of us. In the end, I learned from the talk, which is all one can ask, right? The second talk was easier for me to understand, because it involved matrices and metric spaces. I love metric spaces! Carsten Thomassen, visiting from the Technical University of Denmark, was the speaker; he also gave two talks at the Combinatorics & Optimization workshop.

After the last keynote, Aaron, Rachael, and I walked down to the campus plaza, which has a cornucopia of restaurants. We elected to share a pizza, placed an order, and then took it back to the air-conditioned environment of another building. The Waterloo campus is beautiful, but the heat makes any sort of lengthy walk unattractive. Waterloo campus is also big—compared to Lakehead’s, at least—so every walk is lengthy.

The pizza proved a good choice, as it was tasty and filling. We walked back to the residence where Aaron and Rachael stayed, and then Rachael and I listened to Aaron’s talk, which he is presenting tomorrow afternoon (it concerns the classical Cantor set). Tomorrow I plan to attend talks on set theory, computability theory, universal algebra, and perhaps one on range-sum queries.

I’ve uploaded some photos from my trip so far. They are all accessible in this Flickr set, and new ones will be added there as well.

It is Tuesday, July 6.

Today’s four talks began with electrical networks and random walks. That is, suppose you have a graph that describes a network through which electricity flows. Starting at a vertex x, what is the probability that, when walking at random along the graph, we will arrive at a vertex s instead of a vertex t? This talk was very easy to follow (for which I am thankful), even though I don’t have any engineering or physics background with which to understand the electrical current aspects (like voltage law).

Unfortunately, the second talk involved probability. Probability is great, but I find it very difficult, so this talk was hard to follow. The third talk was about embedding locally-compact metric spaces on surfaces (it is not as scary as it sounds). Finally, the fourth talk was about matching polynomials. The speaker went rather briskly, so it was difficult to take detailed notes, but I enjoyed the subject. Before this summer, I had no idea that polynomials and graphs went so well together. Now it seems like they’re inseparable.

And that concludes the Combinatorial and Optimization workshop. There was a banquet for CUMC at the Huether Hotel, and it was not what I was expecting—very crowded, although the food was good.

Prior to the banquet, Phelim P. Boyle delivered the first keynote speech for CUMC. Boyle is a mathematician of finance, he is interested in the recent financial crisis. He discussed option pricing and the Black-Scholes equation. As with probability, finance is an area of mathematics I avoid, because of its strong dependency on number. Nevertheless, I enjoyed the talk.

I now have access to reliable wireless on campus, although such a phenomenon continues to elude me at my grandparents’ house. Never has my dependency on the Internet been so apparent.

I wrote this last night at my grandparents’ house, which has no Internet connection I can feasibly use (dial-up does not count), so I had to wait until today to post it from the University of Waterloo campus. All references to “today” refer to Monday, July 5.

This week, Rachael, Aaron, and I have travelled to Waterloo, Ontario for two math conferences. The first is the Combinatorics & Optimization Summer School, a two-day event consisting of several talks and, yes, food! The second is the Canadian Undergraduate Math Conference, which also entails much talking and eating. I was reluctant to attend at first, because I dislike travelling. However, my grandparents live in Waterloo, so this was a convenient way to visit them for a week while still getting paid. With that incentive, I managed to convince myself that these conferences would be interesting and probably even useful to my research. This was only the first day, but so far I remain convinced in those respects.

I’ve been up since 4:30 in the morning. Let me take a moment to reflect on the fact that we flew from Thunder Bay to Toronto in an hour and a half, traversing—or rather, bypassing—the largest freshwater lake in the world. And we did it in a metal behemoth that harnesses complex physics and engineering to work miracles.

Flight is awesome.

OK, science-geeky moment over: back to math.

Today there were four talks. We arrived late to the first talk, by about fifteen minutes, but it was still very interesting. It concerned the colouring of graphs on surfaces.

Following a short break, the second talk discussed the Borsuk conjecture, which asks a question about the existence of a certain partition of any set in d dimensions. This was my favourite talk of the day, for several reasons. Firstly, I learned a lot about the diameter of sets, a topic with which I was not familiar. Topology involves a lot of geometry, something for which I lack proper intuition. Yet still it interests me, probably because of its ability to formalize that geometry. I like abstraction. Secondly, the presenter told the story of how Kahn and Kalai proved the Borsuk conjecture false. They took a problem that had been open for nearly seventy years, solved it in a week, and wrote a short, about one-paragraph proof. It’s a wonderful example of how unpredictable and exciting mathematics can be: sure, sometimes math research involves long, boring days reading papers and staring at a problem on a chalkboard. Sometimes, just sometimes, it leads to the most interesting results.

After lunch, we listened to a talk about cutting cake—specifically, how to divide a cake into sections such that no one person would complain that he or she received a worse section. It was by the far the most accessible of the three talks, and the presenter had a very engaging manner. Unfortunately, my fatigue caught up with me during this talk, and I found myself nodding off during the most interesting parts. We learned a little about hypergraphs, which, as the name implies, are like regular graphs but on crack.

The last talk was on symmetric groups and their combinatoric properties. Last week, my prof showed me how we may be able to make use of the symmetric groups to solve the problem on which I‘m working this summer. The talk was more of a review of things I had already learned in group theory two years ago, which was still useful considering the gap in time.

The day began winding down as we went to a pub-like house for dinner. Then we trekked across campus to the residence where Rachael and Aaron are staying. We got lost in the process, of course, but eventually found our way thanks to a map and, moreso, a helpful student.

More to come on Tuesday’s schenanigans tonight or tomorrow morning!

Last week, I discussed how maths is hard, but I spent plenty of time solving a Rubik’s cube anyway. At this rate, you are going to get the idea that I don’t do any work at all. Nevertheless, a desire for accuracy and lulz requires me to remain truthful regarding how I spent this week in the office.

We made a piñata.

We named him Stanley the Resurrection Pig.

I don’t recall who came up with the initial idea. As with all good, crazy plots, it starts off as an innocuous hypothetical scenario: piñatas equal fun, fun equal good, we could make a piñata! This is the last week all four of us will be in the office together—Aaron, Rachael, and I are going to Waterloo next week for a conference, and Jessica is off to Ireland, returning only after Aaron and Rachael’s contracts are finished. So if ever there was a time to set aside the math papers and construct a papier-mâché animal, then savagely beat it to a pulp, this was that time.

None of us are piñata-making experts, and that was probably for the best. Rachael had some experience with papier-mâché—also for the best—so we made her foreman and gave her a silly newspaper hat to go with the title. In remarkably little time, we gathered together the hodge-podge of materials required to manufacture a piñata. We decided on a simple shape, assembled the skeletal structure from balloons, and mixed up a batch of goo to begin the work of creating Stanley.

Over three days, Stanley emerged from a series of colour balloons. He grew stubby legs, ears, and a snout. We named him Stanley because none of us knew anyone named Stanley, and it sounded like a good name for something we would beat to death. (I apologize to all those named Stanley reading this.) Jessica, in particular, was quite bloodthirsty about the whole project. By Friday, however, as we stuffed Stanley full of candy and trussed him in string, we were all savouring the anticipation of Resurrection-Pigpocalpyse.

Stanley met his demise rather quickly. We took him outside, where it was the warmest it has been all summer so far, and suspended him upon a suitable tree branch. Jessica, as the aforementioned most eager participant in this piñata-bashing, got the first swing. I had brought a thin, metal beam that had been propped up in one corner of the hallway outside our office with other thin, metal beams, but we started with a stick to maximize Stanley’s torment. After a few swings from Jessica, however, the stick broke in two. Stanley one, us zero.

So we switched to the metal beam, and Stanley’s death came swift. Jessica pretty much decapitated him with a single, fearsome blow. Aaron, Rachael, and I quickly followed, each of us contributing to his destruction in our own way, until finally he lay on the ground, battered and broken, a shell of his former self.

Stanley was no more. But in his death, he gave us one final gift: lots and lots of candy. Oh, and math riddles. But moreso candy. Really, way too much candy. We had all brought candy, and even though much of the chocolate melted from the heat, there was more than we wanted to take home with us. There is still some of it languishing in the office despite our forthcoming week-long absence.

I could talk about what I‘ve been researching this week, how my supervising prof was in town only for the two days we were dunking our hands in flour-water to make a piñata in the office. I could mention that I’ve started running programs on SHARCNET and it’s awesome. Really, all of these things pale in comparison to spending a week making, and breaking, a piñata.

This was the eighth week of my research. I’m now halfway through my summer job, and it feels like I’ve barely begun. Wow.

Farewell, Stanley the Resurrection Pig. You served but a brief, miserable existence, but you served it well. So long, and thanks for all the fish—er, candy.

I like to joke with my friends about how easy I have it this summer. I‘m sitting in a cozy little office with a fan, proximity to a kettle, and a high-speed Internet connection. Unlike a summer research student in, say, chemistry or biology, I don’t have to manipulate lab equipment or sex fruit flies (Cassie :P). The extent of my experimentation will involve uploading programs to a high-powered computing network and asking it kindly to compute a few more numbers for me. I Google math papers relevant to my problem, try to understand what they say, and see if I can come up with my own ideas. One thing I love about math research, especially in my area of interest, is how much it’s thought. All I really need is a blackboard and chalk, or pencil and paper. (That being said, the high-powered computing network does help when I get to the computation step!)

Of course, it’s not all fun and games (even though I did learn how to solve a Rubik’s cube last week). Maths is hard! And right now, even though I’ve been in university for three years, I feel like an amateur groping around an unsolved problem. I know that research can be like that in general, and I’m still having lots of fun—and learning a lot. Nevertheless, sometimes I feel like a poser. And nothing is worse than a math poser!

I was all excited, two weeks ago, because I had almost finished an algorithm to compute the spreading number recursively. I was tackling the problem as one of finding a maximum independent set. The spreading number is, among other things, the cardinality of the maximum independent set of a certain type of graph. (The covering number is an analogous clique cover cardinality). The general problem of finding a maximum independent set is NP-hard. This means that there likely isn’t a very efficient algorithm for solving the problem (if there were, then P=NP, and that’s way above my pay grade). The best I could hope for was a good algorithm for my specific case; indeed, that was my hope for this algorithm.

After returning from the weekend, I finished the algorithm and happily set Macaulay2 to work, asking it to compute the spreading numbers and compare it with the values we already know. Alas, there were discrepancies, and I quickly understood why: I had made a fundamentally flawed assumption in constructing the algorithm. So while the algorithm did exactly what I wanted it to do, it turns out that what I wanted would not give me the graph’s maximum independent set.

Back to square one!

Frustrated but not very surprised am I. The problem is non-trivial, so I did not really expect such a simple solution. And I have plenty of summer left in which to try new ideas. Right now I am looking at Hilbert series. Most computer algebra systems, including Macaulay2, use Hilbert series to compute the dimension of rings (and this is how my professor’s orginal algorithm computes the spreading number). For larger rings, this computation takes up too much memory.

The easiest solution is, of course, to throw more memory at the problem. We had hoped my computer would be able to compute at least another two or three of the numbers, but this was not to be. Even without any refinements to the algorithm, however, SHARCNET should blow my computer out of the water. This week, I am looking at ways of breaking the computation of the Hilbert series into independent tasks so I can make use of throughput computing.

Oh, and I did learn how to solve a Rubik’s cube. I obtained one in my young adolescent days, but because I have poor spatial skills, I was never able to solve it on my own. Last week I observed Rachael manipulating her cube like a pro. I expressed my admiration and awe, and she just shrugged and mentioned that it was a matter of using certain algorithms (which makes sense). I was doubtful of my ability to learn the necessary algorithms; fortunately, I think I understand enough now to solve the cube reliably. I doubt I’ll ever be a speedcuber, but that is one puzzle down.

Now back to my shiny infinite polynomial series.

Yes, yes, I know. At this rate, my weekly recap will become bi-weekly. I didn’t do a lot the week before last, owing to Victoria Day making for a shortened week. So rather than two very short blog posts, I decided to forbear and write one short blog post instead.

The last two weeks have been more reading, more learning, and a little thinking. I hesitate to ascribe a label like “productive,” since it’s hard to quantify. I think I understand my problem now, but there remains a lot for me to learn in order to start trying solutions.

I tried running the original algorithm for computing the spreading number, which was written in CoCoA, on my computer. I had hoped that my 2 GB of RAM and 1.83 GHz processor would have enough memory to compute some additional numbers. Alas, CoCoA stubbornly crashed (after several long hours) each time I instructed it to do so.

So I ported the code to Macaulay2. It’s even slower, which makes me suspicious that I’m missing something—after all, I am learning both languages, so I‘m sure that in transliterating the code I managed to miss an obvious way to make it more efficient. Still, it looks like the original algorithm won’t produce many more useful results, at least not until I stick it on SHARCNET.

My supervising prof pointed me to a series of lectures he gave on combinatoric commutative algebra. Last week I started working through those, and I’ll continue doing so this week. He’s given me several promising “leads,” I suppose you‘d call them, but at this point, I have to start exploring avenues of interest and seeing if they produce any interesting results. I’ve already toyed with some alternative approaches in Macaulay2, familiarizing myself more with the language, but I think I need more experience with the mathematics first.

Probably the most significant news of the past two weeks would be my decision to attend the Canadian Undergraduate Mathematics Conference at the University of Waterloo and the Combinatorics & Optimization Summer School preceding it. Initially I was reluctant to go, because I don’t like to travel, but Aaron and (maybe) Rachael are going, so I won’t be alone. Plus, I’ll get to visit my grandparents. That’s July 5-10, a few more weeks away. Until then … time for more learning.

Shorter entry this week, as I didn’t do much new and exciting in week 2 of my research project. I‘m still having fun, but because it’s so early in the summer, that fun mostly takes the form of reading.

As tweeted earlier, the secret to reading (and understanding) math papers is simple. First, always read it twice. Then read it again. But to make sure you really understand, you need to take notes. Write down what’s implicit in the paper, the steps the author leaves out because “it is obvious” or “it is clear to the reader” or, even worse, “this has been left has an exercise for the reader.” Once you‘ve done that, the final step is to read the paper again.

I spent all week reading two papers, one of which expands on the findings of the other. The first investigates the spreading and covering numbers in relation to the ideal generation conjecture. Much of the paper goes over my head. Nevertheless, there were some very useful figures, and the use of graph theory in one paper and set theory in another helped improve my comprehension of what these numbers are. The second paper, in particular, was devoted to finding explicit values and bounds for the covering number using a combinatorial/set theory approach.

One of my goals is to improve, if I can, upon the bounds found in these papers. The actual values computed by my supervising prof suggest that there’s room for improvement. I’m a little daunted by this prospect. I feel like I understand the proofs present in these two papers regarding the bounds for the covering number … but I‘m not so sure I understand the procedures well enough to build upon them. Granted, I’ve only been doing this for two weeks. As the summer progresses, I’ll learn more and become more confident. For now, however, I’m just a wee bit intimidated by what I will try to accomplish.

Don’t mistake trepidation for discontent. The best is yet to come! Soon I’ll be playing with CoCoA and Macaulay2. This week, I‘m learning about resolution, which leads to a generalizatio of dimension from ordinary vector spaces to modules. Oh, and I’m having a lot of fun learning how to typeset my proofs in LaTeX. Math is totally the language of the universe, and LaTeX is its markup.

I am now into the second week of my NSERC summer research project. So far, I’m having a lot of fun. The subject of my research is interesting and exactly the type of mathematics that I want to study. The “daily grind,” such as it is, does not grind at all—it helps that there are three other undergraduate students doing research this summer, and we all share the sessional lecturer office. We can distract each other, when needed, and pick each other’s brains for help with particularly puzzling proofs.

So what exactly am I doing? Well, it’s esoteric even for those who enjoyed math up until the first years of university. I‘m going to drop some math jargon in the next few paragraphs, so don’t worry if your eyes start to glaze over. Photos and hilarious video will follow!

Since my prof was leaving town at the end of the week, we met several times so he could give me some lectures and we could discuss my project. The work I’m doing relates to ring theory, which is a course I took nearly two years ago, so I have a lot of review to do. Most of the week, like the next few weeks will, involved preliminaries. I found all of the references my prof recommended to me. I began reading the three textbooks among those references, learning about monomial ideals and simplicial complexes.

These, however, are but means to an end. After I have mastered the secrets of these wonderful algebraic concepts, I can use them toward the eventual goal of finding better algorithms for calculating the spreading and covering numbers. These relate to the maximum and minimum dimension, respectively, of a monomial subspace of a vector space over all polynomials of a given degree such that the subspace fulfils two respective properties.

On Thursday, my prof went over what’s changed since he and his colleagues wrote the paper from which my research project comes. In particular, they’ve learned about a connection between edge ideals and the Stanley-Reisner ideal. In the paper, they showed that calculating the dimension of the Stanley-Reisner ring is sufficient to find the spreading number. (A similiar result makes calculating the covering number possible.)

This connection is really cool for two reasons. Firstly, it makes the connection to graph theory stronger, which gives us another avenue for exploring the problem. Secondly, it might provide an alternative way ofcalculating spreading numbers (graph theory is also useful in this respect). The algorithm in the paper finds the Stanley-Reisner ring and then uses a computer algebra system to find the dimension of the ring. They did this on a Pentium II, so they could only find a few of the numbers before the calculations became impractical given the available computer memory. Computing power has improved considerably since then, so my first step will be to see how my little laptop compares against their Pentium II using the algorithm in the paper. Later in the summer, I’ll be creating alternative (hopefully more efficient) algorithms in Macaulay2 and running them on SHARCNET.

Of the three other students sharing the office with me this summer, Aaron is in the same year as me, and Jessica and Rachael are a year behind us. Aaron and Rachel are working on the same project, which involves fractals and Cantor sets. Jessica is also working on something related to commutative algebra (affine varieties and Gröbner bases). So not only do I get to learn about simplicial complexes and monomial ideals, but I’ll be learning about affine spaces and some more real analysis as well.

And for those of you who wonder exactly what math research looks like, I can attest that it’s pretty much like this clip from The Big Bang Theory. Aaron and I spent a good deal of Friday afternoon staring at my faulty proof regarding prime ideals on the chalk board. I did manage to figure it out eventually, but imagine if we had had a montage!

This January, I applied for a summer Undergraduate Student Research Award (USRA) from the Natural Sciences & Engineering Research Council (NSERC). Lakehead University has 20 such awards to give to applicants this year, and on Monday, I learned that I am the recipient of one!

I was (still am) a mixture of elation and trepidation. Part of me is still in a state of shock and can’t quite believe that this is real. I spend a good half hour after learning I got the grant just trying to calm down so I would not run up to everyone I encountered and yell, “I GOT A GRANT!” Another part of me is saying, “What do you think you‘re doing, Ben? You don’t even understand what it is you’re going to be researching!” As anyone who has ever looked at a higher math textbook knows, the language is just scary sometimes.

I applied for the NSERC grant for two reasons. Firstly, it’s a different summer employment opportunity than my default, which is the art gallery. Don’t get me wrong: I love working at the gallery. You can’t beat the hours, and I have an awesome boss—she took the news that I wouldn’t be working there over the summer much easier than I thought she would. Nevertheless, I’ve worked there for four consecutive summers. I‘m not averse to trying something new, particularly something related to my area of interest.

Secondly, since this is a research position, I’ll get a chance to experience exactly what “math research” is all about. Sometimes people will ask me why I’m becoming a high school teacher instead of going on to graduate school and becoming a professor; usually my answer is somewhere along the lines that I‘m not sure I’d like doing “math research” and writing “math papers.” I‘m more in it for the teaching. This grant is a perfect way to see if, in fact, I like or dislike doing research, without committing to something like graduate school first.

So I’m excited about this change, but also just a little bit anxious—it is a big change in how I’ll be spending my summer, and a different responsibility. After four years at the gallery, I’m so used to doing the same thing every summer that it’s hard imagining myself doing anything else.

The position itself is a full-time for 16 weeks. My area of interest in mathematics lies in commutative algebra, so Dr. Adam Van Tuyl has agreed to be my supervisor. He’s come up with a neat project for me, and I’ll try to explain some of it. I don’t fully understand what I’m doing yet myself; for the first few weeks I’ll need to review my ring theory from last year and then work to learn new concepts we didn’t even cover in that class.

Ultimately I’ll be continuing work that Dr. Van Tuyl did on computing spreading and covering numbers for monomial ideals. One of the issues he and his colleagues encountered when they first worked on this problem was a lack of computational power for calculating values for these numbers. Later in the project, I’m going to be writing my own algorithms for calculating these numbers, and I should be able to run them SHARCNET, a network of high performance computers maintained by several academic institutions in Ontario.

I plan to blog about the project as the summer goes on. I start working on May 10, so I probably won’t have much to say on the subject until then. For now I need to focus on finishing the school year!