Tracking a Mathematics Ph.D. Student

Week 2 Done!

2010-06-11T01:34:00.003-04:00

I'm officially exhausted. Teaching 4 days a week is rough. We're definitely making movement in the course, and I'm starting to feel the pressure of fitting an entire linear algebra course in just 5 weeks. I am covering 2 sections of text every day, which amounts to covering 2 chapters a week. Next week is their midterm, and that puts me at the next challenge. I do have the added experience of being an exam writing for a program run through the department, which I've done for the last 2 years. And I was also a worksheet writer during the past two semesters for a program which gave calculus students the opportunity to discuss harder calculus problems than we would cover within the course, so I have some idea of how I should be doing phrasings and even slightly on timing as well. It's just very hard to come up with clever ways to present problems that cover knowledge of the material in a way such that those that know the material will be able to do the problems without much difficulty but those that did not put the work will have a fair amount of difficulty. But then again, it's simply impossible for the students to have a full working knowledge and mastery of the subject material after these quick lectures and limited homework.

I guess we will see. For now, it is the weekend. The midterm exam is written, and I will tackle writing up lectures for Monday-Wednesday another day.

Week 1 Down

2010-06-04T10:07:00.002-04:00

The first week of the summer linear algebra course is over and appears to have gone quite well. With the exception of maybe 3 or 4 students (out of approximately 20), the dynamic has gotten quieter and quieter. I haven't caught anyone full out sleeping yet though, so that's a good sign I suppose. We've completely covered the first 2 chapters of the text we are using, and I've already gotten a number of comments that I should do a few more examples. Hopefully I will have the time to do this as we get to the more difficult material. There's just only so many examples you can do of determining whether a transformation is linear or not, or determining the geometric interpretation of a 2x2 matrix!

Next week we start covering chapter 3, which deals with subspaces and their dimensions. It seems that I should be able to do a bit more example wise with this material. We'll see.

Lecture #1

2010-06-01T15:45:00.000-04:00

Talk about a whirlwind!!! Turns out I had EXACTLY enough material to cover the full 2 and a half hours of my course today. Granted, it was the whole first chapter, but I think it went well!!!

It's a different experience lecturing, as opposed to leading a recitation. I'm used to the students interacting with me. Today, all 20 of them just sat there. With one or two questions thrown in. It was bizarre.

It was definitely a roller coaster though. I thought I was going to run out of material at the beginning when it only took me 10 minutes to go over the entire course in general. Then in the middle I thought I wouldn't finish, so I sped up a bit. Then back up to not having enough material. I guess I planned it out just right though! I had to have enough material to get me til 3:30, and I finished at 3:28. Not too shabby for a first timer!

As for my actual lecturing, I need to tighten things up. There were a few minor slip ups, but my students were actually paying attention and got the words out for me, so that was nice. I hope that means they were really thinking about the material while I was covering it! I get to do it again tomorrow. I'm exhausted after just one day!!

Planning Lectures

2010-05-22T18:56:00.003-04:00

Calling all teachers of mathematics!! I'm teaching a course this summer (starting in a little over a week) on Linear Algebra. The course will run for 5 weeks with 4 classes a week except for the first, where there will be three (due to the US holiday). I've got the material all planned out, and when homeworks/quizzes will be due and what they will all be. I'm at the stage where I need to start planning out the lectures. Any tips for a newbie on motivating especially the first lecture? Each lecture is 2 and a half hours long, so I need to keep them in tune with what's going on at the board - easier said than done, I know. Any advice would be greatly appreciated!

Recognition!!

2010-05-07T09:46:00.002-04:00

Finally, some solid real recognition from my department! Two pieces really. First, was a very nice goodbye from my advisor, who said something to the effect that I would have been successful in the PHD program if I had been allowed to stay, and then gave me some pointers on where to go next in my reading to keep me motivated on the Algebraic Topology route. The next text up? McCleary's A User's Guide to Spectral Sequences, since ... those who are comfortable with spectral sequences have a large advantage over those who don't. This should be interesting enough, since most if not all of my questions this semester have been about spectral sequences! A User's Guide would be nice.

The second form of recognition meant a GREAT deal to me. There are 2 teaching awards that are given out each year. One given out purely by the department (of which 2 are given out), and one given out by a family member of a previous member of the department. Well, I got one of the ones given out by the department!! What a way to go, right? :-) It's wonderful to be recognized for my hard work in my teaching. Thankfully, my students recognized it, since they are the ones who nominated me. But, this is what I want to do with my degree. I want to teach.

Now, let's hope I come through for them again, and give them the tools to get decent grades on their final exam!!

Question

2010-04-10T10:36:00.002-04:00

Hello All,

This post is actually a question. I realized throughout my course at JHU that I had failed to learn a good chunk of advanced undergraduate concepts (for example, Jordan Blocks and Diagonalization in Linear Algebra, and I have completely forgotten most common examples/methods in Calculus from Series in Calculus II into the entirety of Calculus III with the exception of basic operations of vectors (adding, cross product, dot product,etc.) and I never even learned Stokes and Green's Theorems. Now, here's my question. I just finished going through (reading and doing all exercises) in a Linear Algebra textbook. So, I've filled in those gaps as far as Jordan Blocks, Diagonalization, Projections, Orthogonality, etc. But, I don't know what the next step is. I'm tempted to go back to Calculus and really nail down all of that, but fear it will take too long, and it should be stuff that I can pick up as I need it by reading a few sections in a standard calculus textbook. It's only really methods and basic tools that I've lost, but upon reading it I should be able to get it back fairly easily. I taught Differential Equations last summer, so I'm fairly comfortable with that material, and the next logical thing would be to move towards an Abstract Algebra textbook (at least for me). I know that I did not pass the analysis qualifying exam, but I'm comfortable with undergraduate analysis (a lot due to the fact that I did most of an undergraduate textbook last summer, while studying for my analysis exam, thinking that I had missed something in undergrad, but I didn't). Now, if that is the next step, I'm not sure which book I should use. I have 3 or 4 at my disposal:

Lay's A First Course in Abstract Algebra
Hungerford's Abstract Algebra: An Introduction
Nicholson's Introduction to Abstract Algebra
Lanski's Concepts in Abstract Algebra

It is my knowledge that Lay's is the most elementary of the 4, and it is the one I used as an undergraduate. At Hopkins, we used Lang's Algebra, so you can see that there was quite a gap. I have heard that Hungerford is good, and JHU is using Nicholson for their undergraduate course right now. Wesleyan will use Dummit and Foote, which I also semi have access to, but not fully, so I would like to avoid using that one at the current moment.

My question ... Help! Which book do I use!? And is this even the right direction to go? Maybe I should go back to Calculus? Or, since I may be studying moving to study Graph Theory at Wesleyan, do I do my Graph Theory text?

Thanks all!

Next Chapter!

2010-03-27T17:23:00.002-04:00

Hello All!! I've officially accepted an offer at Wesleyan University. I am extremely excited to start classes and such this fall.

I still am finishing up classes at JHU this semester. And I'm teaching Linear Algebra (solo) this summer through JHU. Then I'll be heading home for a month or so, and then I'm getting married, moving to CT and starting at Wesleyan. It's a very very exciting time, and I'm confident that it will work out better this time. I'll be closer to family and friends, and closer to my undergrad where I still have a great relationship in tact. Yay for new beginnings!

Done with Applications!

2010-01-21T14:09:00.002-05:00

Phew! Finally! The New Year has brought some much needed relief. I am DONE with Graduate Applications. This time, it better be for good. They are so expensive! When all is said and done, I applied to 10 schools (first time around I applied to 9). People do say that it's not an excessive amount, but don't tell that to my bank account. It got hit hard.

I got my first acceptance the first time around on Valentine's Day, so I still have a few more weeks to wait it out now. I think I have fairly good shots at getting into all the places I applied. I didn't apply to any real reach schools, except maybe one or two. 4 out of the 10 places, I applied to and got into back when I applied two years ago, so logic would say that I am more experienced this time around so I should get in again. But, then again, it really is extremely relative since I am being judged against the other applicants.

In other news, I nearly have my schedule completed for this semester, my final semester at JHU. I am taking a reading course with my advisor. I will be reading/discussing the text by Hilton and Stammback, "A Course in Homological Algebra." I am also continuing with the second semester of the Number Theory course. Alongside those two courses, I will also be working through my undergraduate Linear Algebra text, as I took the course freshman year of undergrad and found it to be a large disadvantage to have gotten so rusty on the material. Seems like things are lining up because it turns out that I will be teaching Linear Algebra during the summer session come May, so having it fresh in my memory will be helpful. I will hopefully finish this text by the end of March/beginning of April, and then will have a month or two to fully plan out the course schedule, exams, quizzes, and all the other stuff that comes with being an instructor to a course. This semester however, I am following up with my Calc I students from last semester and TAing Calc II. It isn't really calculus though, they cover a few integration techniques, basic differential equations, and then a lot of probability and statistics. It's quite a strange sequences, which is why I want to follow it through. I think it is meant for those seeking to take the MCAT's, since JHU is a large medical school. And lastly, I'll be continuing to attend the Topology seminars, and I will be organizing the slow-pitch seminars (series aimed at first and second year graduate students), and a few other odds and ends around the department, as I am a go-to girl for some of the faculty ...

It will be a busy semester for sure, with a lot of personal events going on as well and planning a move out of Maryland, but I'm looking forward to it all. It's an exciting time, and I'm very happy to have all these opportunities.

Updates

2009-12-02T10:52:00.003-05:00

Several comments were made on my previous post, which I will now take the time to answer.

First off, I greatly appreciate your comments. I'm glad to know that my thoughts can ease someone else with the fact that it happens to other people. Whether struggling through a first year graduate program, or being told that you can't do something that you love and have been told all through the rest of your life that you were good at it. It is a disheartening time, for sure. But, we will rise above it.

I am now in the process of applying to new graduate schools. I will continue to strive for my PhD in Mathematics. I still want to do active research, but I am also looking for an institution that will aid me in developing my teaching abilities, as that is truly what I do want to do. Luckily, I leave JHU with a masters, so that alone should help me to get into many of the places I am applying.

As for the qualifying exams at JHU, yes, you are given two chances with an option for a free chance when you first enter (for those with masters and such to place out of the first year courses). My problem was that I did not demand the help that I needed, and my fellow students worked in a very different manner than I did. I prefer to have readings done before classes, and assignments done with at least a full day to spare, so I can look over it and catch any mistakes. My classmates were the exact opposite, which worked to my disadvantage only. They were able to do this and be successful, and I lost out on the valuable discussion that I desperately needed.

If given the chance to do it all over, yes I would take the opportunity. I know that I can succeed at JHU, I just needed there to be more support and willingness to help.

I will be staying at JHU for the spring semester. My current advisor would still be happy to work with me, and I will continue getting teaching experience. I will also be teaching my own course over the summer through JHU, so I'm not quite done with them yet. Luckily for me, there are some at JHU that do like me, and want me to succeed.

Thanks for all your support! I do greatly appreciate it.

Back to Square One

2009-09-18T09:34:00.004-04:00

Hi All,

Haven't updated in quite a while, but now feels like a time to do so.

My last post (back in June) mentioned my retaking the Analysis qualifying exam this fall. The text was last week, and I got the results this week. Turns out that I did not pass the exam, and, as a consequence, have been terminated from the program. There are many things going through my mind at this point about the turnout of the exam. For one, I received more than half the points, which is the general unspoken rule for passing. Also, I was planning on going into Algebraic Topology, so perhaps there should be a slight lenience on my Analysis exam? Obviously, I would never ask for a "free pass" but lenience might be nice.

However, it's a blessing in disguise. I will continue elsewhere to obtain my degree. I will be successful, and I will be happier. So, I will show those at JHU that I CAN do this and I WILL do this, and it's their lose for not having me in their program. Plus, not to toot my own horn, but I had several students stay after my section yesterday and tell me that I was the best TA, and this from students who are notorious for hating math.

I will be staying at JHU at least through the end of this semester, but beyond that I am not sure what I will be doing. I have the option of staying through the end of the academic year, should I choose to continue TAing, but I'll evaluate that decision later on.

I'll keep this thing updated on decisions and such.

Either Or

2009-06-09T21:15:00.002-04:00

So, turns out I had to pick either doing work/studying or updating my blog, so I stopped updating.

I basically only studied for quals and such for the past several months. Turns out I passed Algebra (much to my surprise), and failed Analysis, which I'm dealing with. I'm now currently studying and prepping for teaching Differential Equations online this summer. And also studying for the Analysis exam which I'll retake in September.

Next semester is going to be a bit crazy as far as classes, so I'll post again in the future on the definite courses and such.

Thanks to everyone who reads my blog!

Ahead of the Game

2009-02-09T09:46:00.002-05:00

Accomplished is one way to express how I'm feeling right now. I'm overjoyed as well! I set out on a new way of thinking this past week, more organization, a more solidified schedule of when I want tasks to be completed. Overdoing it? Maybe. It's not often you see people with 2 page long detailed to do lists, including within it everything from detailed homework assignments, going to church, cleaning various rooms of the apartment. The only thing not on my to-do list is sleep, eat, shower, and other basic necessities of life. Now, I've made to-do lists in the past, so what is different about this to do list? Well, for one, I'm actually getting the things on it done!! But, it's how I'm doing my to-do list that's really making the difference. In the past, I've written everything down and did things in a half-random order (only semi-prioritizing items on the list, assignments always first, and after that, a free for all!). I also only really used my to-do list on the weekend, and once it was into the week, it was forgotten about.

This specialized to-do list contains everything I had in it before with one major addition. Now, I have a set schedule on when I want certain items done by. Now, it would be physically impossible to get everything on my list done in one week, which is why I allow for the schedule of "Next Week" to appear in a few places. This is ONLY on stuff from last semester that I did not get around to finishing (reading and such), and readings that are not due in that upcoming week. Each day of the week has a color, and for each day I have a set of things that I wish to accomplish, marked by a dot of the color next to the assignment. For example, today is Monday, and is marked by Yellow. And today's tasks (besides my attending class obligations) include rewriting my complex variables notes from today's lecture, reading two sections in Lang (from last semester), reading one section in Hatcher (from last semester), and go food shopping. This seems like a light load, and to be perfectly honest, it is. I should get those items done with no problems, and if that's the case, depending on the time I finish, I'll either relax a bit (take some me time) or read one more section in Lang, or something like that. Always moving forward!! But, I really feel accomplished. Out of the 73 items on my list, I have finished 32 of them. And being that my list started on Friday, and it's Monday and 12 of the items are listed for next week, I'd say that's pretty good. I think this is just going to be what I have to do! Go figure.

As for my actual classes, I'm thoroughly enjoying all of them! Complex is actually fun, I can visualize the material, which always makes it easier and more enjoyable for me. Algebra is very hard, as I knew it would be. It is my weakest subject. And, Algebraic Topology is starting to really sink in. This idea of cohomology that we're working on entirely this semester is quite fascinating, and the beauty of it all working dually with homology. Most wouldn't understand the beauty of this, and I'm sure I don't fully understand it either, but at least I can see it.

One Week Down

2009-01-30T09:57:00.002-05:00

Well, I've survived the first week. Initial reaction for all my classes??

Complex Variables I think will be a very very good course. I'm really pushing in this one at the moment because in the grand scheme of things, I have a qualifying exam on this combined with Real Variables in May. And, I definitely slacked last semester (although others would argue that I definitely didn't, so I guess that's all debatable), so I need to kick butt in complex and really understand it so I don't fail that qualifying exam. I figure, if I can nail the complex stuff, and at least look like I know what I'm talking about in reals, then I'll have a shot at passing. Plus, the material is actually kind of interesting. We worked mainly with the Cauchy Integral Formula, and the Cauchy Integral Theorem the last class. It's pretty interesting, to be quite honest! And, what's even better, I have an assignment due on Monday, which is DONE! Go me!

Algebraic Topology is going well. There's really nothing new in this course, nothing special to report on. Same as last semester! We covered cohomology yesterday, at least the definition of. I imagine we'll do some examples and proofs with it in the coming weeks, or actually the entirety of the semester (we're only doing Hatcher's chapter (3) on Cohomology, with a few things added if time allows). So far so good though!

And Algebra. I cannot express enough, my disdain for Algebra! (Yet, here I am, thinking of going into Algebraic Topology, go figure.) My professor for this course is, well, interesting. I cannot understand/hear (it bounces between) him most of the class, and I'm not sure he understands this limitation, because unlike my professor last semester who did understand this limitation, he puts nothing really up on the board except for extremely abstract diagrams of field extensions. And, these diagrams, he reuses for other purposes a half hour later in the lecture. So, I'm going to have to read before and after this class, so that I can get an idea in my head for the lecture and then put the diagrams back into the text for a deeper understanding. We'll see how that goes. Algebra is what's on task for this weekend, with an assignment due on Tuesday.

So, there you have it. Much less stress so far this semester, so I hope that keeps up. On to tackle Galois Theory, straight from Lang. I know that all you mathematicians out there are jealous ... (sarcasm).

One Lecture (Per Class) Done!

2009-01-27T11:37:00.002-05:00

First off ... Greetings to all my new readers! I'm so glad this blog is bringing in some interest. The comments you all leave, while I do not always respond, help to keep me going. It is very encouraging to know that there is such support in the math community! Responding to one comment on my last post: Favorite Talks at the JMM this past month - I attended a few Algebraic Topology talks, none of which I understood, but there was such a different dynamic in that room, as compared to other rooms. A fellow graduate student made the observation that it is because everyone in that room is a member of the field and has done, in some cases more than others, vast research in the area. Other fun ones included a few graph theory talks, a matrix theory talk, and a few arts/music talks. My overall favorite, I think, had to be the matrix theory one talking about ranks of graphs. This is partly because I have done research in the area of minimal rankings of graphs, and some of the results were the same as results I had proved, but done in a different way. Sorry for the long winded answer!

Onto the new stuff...

And, here we go again! This semester started off much better than last semester! Complex Variables yesterday went very well, except when he started talking about conformal mapping, hopefully we don't use that too much. On task for that course is to get the Cauchy-Riemann Equations memorized and comfortable in use. And I need to read the first two chapters in our text, and do the assignment which is due on Monday. Yikes!

Algebraic Topology thismorning as actually quite enjoyable. I really, REALLY like the subject (I say this now...). We did mainly warm-up material today. I was the only student that showed, (there's only one other in the course) so it was kind of awkward at first until we got going. We got some snow today, which in Maryland is not something that people know how to deal with, especially while driving. Being from NY and going to undergrad in CT, while I hate doing it and avoid it when I can, it was no big deal today. No assignment for that course yet, but I'm working on catching up on reading I didn't get finished last semester, as well as a few pages that he assigned for reading today.

Algebra was directly after Alg. Top. and should be interesting. For sitting in the course for an hour or so, I have less than 2 pages of notes, which is unheard of for me. This is due partly to the fact that he didn't cover much (very introductory stuff to Galois Theory, and not even using advanced terminology), and the other partly due to the fact that what he did throw up on the board, it was all symbols and diagrams, very few words. He claims that if we pass the course, we will have no problem passing the qualifying exam in May. I'm not so sure I believe that, but you know, it happens I suppose.

Push comes to shove, I'll be working much harder this semester than last semester. I wrote 24 pages of notes yesterday (reviewing last semester's Algebraic Topology course), in addition to the 4 pages of notes I took during Complex Variables. Hopefully this hard work will pay off. I have just over 2 hours before I leave here for a few hours to tutor at an alternative high school up the road, so I should eat lunch and get some work done.

New Semester

2009-01-26T08:54:00.002-05:00

After a very long (very relaxing, and unfortunately not productive) winter break, classes start back up again today.

I did attend the JMM in DC in January, which were very exciting. I may have even run into a few of those who have been commenting on my blog for the past few months, and not even known it!! Perhaps next time, I'll be more aware of that. Although, I don't think my school will fund me going to San Fran, but who knows.

Just an overview of what's going on this semester, I'm taking three classes and doing help room. No teaching!!! Qualifiers are in May, two exams, one in Algebra and one in Complex/Real Variables (combined exam). The goal is to pass both of them in May, otherwise I take them in September again, and that's my last shot at them. Two and out, so they say!

The classes that I'm taking this semester are Complex Variables, Graduate Algebra II, and Algebraic Topology II, and I may be the only student enrolled in that Algebraic Topology II course. So, needless to say, it should be an interesting semester. I'm getting myself over-geared up for the topology course so I can be on the top of my game. Complex seems like it should be a good course, I'm excited for that one. And Algebra, well ... I just don't like it, but it's partly because it's just beyond my grasp right now. I have the potential to fully understand it, and it's frustrating because it's right there, I can touch it with my fingertips, but I can't get a full hand around it. Maybe this semester it will all come together.

Beyond that, it should be a good semester. I'm trying to be more productive in my office/at the library than I was last semester, and I'm really going to be diligent about reading before class and again after class, at least reviewing the material.

So, here's to Spring 2009! My second semester of graduate school!

Feeling Human!

2008-12-23T15:39:00.002-05:00

Well, now that all that's over with, I can feel human again. No insane stress levels, no more feeling like I have to be super woman with teaching and studying and assignments. It's so nice to be able to relax, even if it's only for a few days!

I passed all my classes, and even somehow managed an A in Algebra ... no idea how I could possibly do that. Obviously, I won't complain though.

So, I'm going to work through some Complex Analysis on my own over my break, and review Algebra a bit too. But other than that, no major math for me! (Well, except the Joint Meetings, ha!)

Thanks for all the support from my readers throughout this past semester!!

The End is Near!

2008-12-10T11:02:00.002-05:00

I've kind of neglected this thing - for which I apologize. Things are finally wrapping up here. Algebraic Topology has been over for almost a week, there is no final exam in that course. Real Variables ended on Monday with a pretty rough exam, not by fault of any one particular person either. I just focused my studying on the wrong material. I figured that since we spend a good 2 and a half to 3 weeks doing Lp classes, that it would be a good idea to know how to work with them, so I studied most of my time on that stuff, and just reviewed a little bit of Fourier transforms (which he said would likely not be on the exam), and the other stuff (absolute continuity, Fubini/Tonelli, Lebesgue's Differentiation Theorem, etc.) . Most of that later stuff was valid for the exam, so I did look at that a bit more than the Fourier stuff, but still. The exam had a Fourier transform question (which we did in class, but I didn't review because he said it wouldn't be on the exam...), one 'silly' question on Lp classes (showing that L-infinity([0,1]) is contained in but not equal to L2([0,1]) is contained in but not equal to L1([0,1])), and then a question on little l infinity of a certain Banach space, and whether it was separable or not (again, in the text, but didn't really look over it, since separability was covered in 10 minutes, maybe), and a question on absolute continuity of a function. So, it was a little frustrating, but now I know next time I can't rely on what the professor says, and I will just have to know everything, hands down.

But, that's done, and now I've got a full week and a half to study for Algebra, which I'm going to need that full time. I'm angry because I forgot a book I needed at home, and it would be silly for me to drive all the way back there and get it, so I'll have to make due with studying some other way today. Go figure.

But, the end is near. Next semester I'm taking Algebra II, Complex Variables, and Algebraic Topology II. I'm probably the only one in the topology course, since everyone else pretty much hated it, but I'm considering it for my field of research, so I figured I'd best take the second semester of the course. It should be interesting, but I think I've realized what I need to do and how to do it, in order to succeed, so we're well on our way! But, I can't help but say ... 9 days until some sort of sanity!!!

The Good With The Bad

2008-11-14T10:20:00.001-05:00

Well it was a good and a bad week all rolled in one. I was so ahead the last time I wrote, I was done with my Analysis homework on Sunday, and it wasn't due until Wednesday! It was amazing! Then I started working on Algebra, and I went to go to my professors office hours because I am 100% completely lost. We're working on modules right now, and Lang is useless to me. So I've been reading it alongside Dummit and Foote, which seems to be okay, but I'm still lost. It's just not clicking with me right now. Anyway, I went to my professors office hours, and his door was shut. I have no idea if he was on the other side of it, but his door was shut. During his office hours, when his door should be open. I was not very happy. I went back 2 more times after that too, and his door was not opened. And I even told him earlier that morning that I'd be stopping by, it's just so frustrating!! So, I had to force through yet another assignment, and I didn't answer (or rather wrote jibberish down for) half of the questions. I was only even moderately okay with the 3 that I did answer, and those were mainly diagram chasing questions, which we covered in Algebraic Topology 2 weeks ago, actually we even proved one of our homework problems, so that doesn't say much. But, I went to talk to someone today to figure out how to deal with all this, and they acknowledged that I'm in a rough spot and they are going to try and help me figure out how to handle all of this, while still being sure to take care of myself. And that's that! I have a visitor coming this weekend (so excited) so I have to get my Algebraic Topology homework at least started in the two hours that I have before I have to meet up with the girls for lunch, and then head to the airport to pick up my visitor. Enjoy your weekends!

Caught Up?

2008-11-08T10:47:00.002-05:00

Well, I know it's been a while, 2 weeks or so I believe. It's been an interesting time here in Baltimore.

I guess I'll just recap and extend on my ongoing stories, in a systematic way.

First, Analysis. After getting a 45 on my midterm, I was kind of freaking out as to what I was going to do. I'm pretty sure that I won't really fail, but I was still very concerned. This is what I am producing, after getting NOTHING but A's and grades of 90% on everything in math. It's bad when my professors at SHU were dropping quiz grades that were perfect scores! At any rate, I was very concerned about my progress in the course, and overall my projected progress in the program. If I was doing so poorly in this introductory course, how could I possibly move on? Well, I went to meet with my professor, finally, last week. I had been trying to meet with him since I got my exam back, but he was never in his office every time I ran up to see him, including his office hour, which is frustrating. But, I finally met with him and here's how the conversation ran;

Me: I was wondering if we could discuss my progress in the course for a few minutes.
CS: Sure, have a seat. What did you get on the midterm? I don't remember.
Me: A 45...
CS: Oh, that's not so bad. There were a bunch of you in that range, a bunch in the middle (i.e. 60's), and only one or two that did better than that.

I never thought that a 45 would be "not so bad," but apparently in grad school it is. I was definitely at the lower rungs of the class, but I was not doing so terribly that he was concerned. We went on to discuss my homeworks and how I probably needed to be doing them differently. I had been collaborating with my office mates, who are also in the course, and I was probably (aka definitely) getting more help from them than I was giving. So, as a result, it was more like them giving me the answers and me writing it up. Thus, now I'm not working with anyone on my homeworks. This may result in a drop in grade, but, I think I'll learn more this way. And I want to be the best, so this is what has to happen as a necessary step to get there (although, not sufficient...). Yesterday (Friday) was my Analysis day so to speak. And I tried to do some of the homework problems, there are 6 this week. And, I did 3 of them last night! I was not expecting to do that, and now I have half the problems done, and the assignment isn't due until Wednesday!!! So, already, this is looking up for me. Usually I do the assignment the day before it's due, because I have other stuff I need to be doing all the other days, but, I'm turning over a new leaf, again. I must have an n-dimensional leaf, because I keep turning it over and over and over! So, that's Analysis. Hopefully going to be going better. I actually do understand most of what's going on there. Currently we're working on differentiation of the Lebesgue integral, last class we studied absolutely continuous functions, and singular functions. We did a lot earlier last week with the various versions of Vitali's Covering Lemma. My professor hates covering lemma's, so he does them out on the board with a certain disdain, it's kind of amusing - with his making politically charged jokes while doing it; i.e. "We're making Al Gore very unhappy right now with all this chalk usage..."

Second is Algebra. My last post was on the day of, or the day after, my midterm exam, where I stated that I failed, which I did. Although, I did better than I was expecting, and even better than in Analysis! I think he wanted to have pity on me though. I had emailed him two or three days before the exam asking what would be the best way to prepare (i.e. do old qual problems, review homework, read and know everything in the text...) and he told me to review the homwork and notes from class, and then relax. Of course, I didn't relax, but that's okay. Anywho, I got a 50% on the exam, and when he handed them back he said, "If you got a 60% on the exam, you did good enough. If you got a 50% on the exam, you're still okay, but you need to work harder. A good place to start with this is to not be doing your homework on Wednesday afternoon, when it is due on Thursday." This last sentence was said while stairing directly at me. He had seen me doing my homework on Wednesdays several times, and we (my deskmate and I) often go up to his office on Wednesday afternoons with definitional problems (i.e. whether a ring has a multiplicative identity or not, since we have two conflicting definitions... stupid texts.), and when he asked if we had questions other than on a definition, we looked at each other and shook our heads in shame. So, I haven't started the assignment for the week for that quite yet, but I'm going to attempt the other 3 problems in analysis today, and if I get them done then I'll move to Algebra, and if I get stuck ... I'll move on to Algebra. So, either way, I should touch on some of the assignment today, it isn't due until Thursday. Unfortunately, I'm completely lost by everything that's going on in my class right now. We did diagram chasing last class, which I understood, but then we moved on to some iffy stuff with modules and vector spaces, and all that. I guess because my linear algebra is not that great, vector spaces and basis ideas are not very clear to me. I'll have to work on that. I plan to meet with my professor this week to have the same conversation I had with my Analysis professor.

Third, Algebraic Topology. By far, this is my most interesting class as far as material. As far as presentation, it's not the greatest. However, I may be taking the second semester of it anyway, because I've recently had callings into this branch of mathematics for my research. Again, this is all massively up in the air at this point. I'm doing okay in this class. I'm holding about a 70% average on my homeworks, and the course is determined by homeworks, no exams. So, we'll see. Each week I gain that little bit more of clarity in understanding. We just finished a series of "Homologies." We covered Simplicial Homology, Singular Homology, and Cellular Homology. Unfortunately, I'm not sure what the difference between any of them are, so I'll have to work on that. We covered at the end of class last week the idea of splitting short exact sequences, so we're booking right along there. Another assignment is due next week, so once I'm done with Analysis and Algebra, or I'm stuck on both of them, I can start this assignment. Again, I plan to meet with this professor as well.

Finally, my teaching. I'm the most frustrated on this right now. My students had an exam last week, and there were two questions that were a little, "iffy" to say the least. The first was a proof of a limit, not a big deal, right? Choose your N and be done with it (this is for limits to infinity). Well, instead of just having n < N, like every other homework problem, example done in class, etc., it was n
Wow, okay - I rambled and vented far too long on this post. This was supposed to be a quick part of my day, but this took way too much time. Until next time!

Algebra Midterm Exam and Other

2008-10-29T11:24:00.003-04:00

So, what to write. I've had another couple of very rough days, and today will be no exception. To put it lightly, I am VERY frustrated right now. And that's not just in the in the realm of my classes, it's the program in general.

But, first and foremost, I'm pretty sure I failed my Algebra midterm yesterday. So I'm 0/2 on midterms for passing here. I'd say that pretty much sucks.

Then, let's see. I guess my current tiff right now is with my TA assignment. I have to regrade homework assignments because my students don't know how to read directions, or if they did and didn't understand, they don't know how to ask for help understanding the directions. Go figure. So, I get to regrade 60 homeworks next week due to that, on top of the homework that I have to grade anyway.

Now, also ... my classes have their midterm on Friday, so I held a review session on Monday, which was fine. I was very content with how it went, and things seem to be going well. Now, after a midterm, we have a "grading party" where we order pizza or something along those lines and grade for hours and hours. I'm sure if you were ever a mathematics (or other discipline, I'm sure) TA, you've heard of these types of things. So, I was expecting the grading to be done on Friday. Seems logical, right? Apparently not. I have to come in on SUNDAY to grade! Sundays are my day. With the exception of an hour and a half on Thursdays, Sunday's are the ONLY day that I can spend a few hours doing things for me, where what I do benefits MY wellbeing. Anyone who's been through graduate school knows how important it is to do this every so often. So, I go to church on Sundays and spent time with a few of the women afterwards doing something relaxing, this week we're having a potluck lunch. So, I was expecting to be at church until mid-afternoon. Now, while I am allowed to show up late (and I will), I'm expected to be there by 2pm at the latest. Which means I'd have to leave church at 1:30pm, and that is hardly mid-afternoon. It is so frustrating! So, I decided that I'm not coming in on Friday. Which means now I have to go out and buy some candy for Halloween because I was planning to be in hiding all night but now that I'm not coming into work, I'll be home. Fabulous.

So, you can see my frustration. But, now I'm off to go tackle Lang for another round of torture in that respect. I just finished my torturous round of Analysis - so it should be another fun day.

Sorry for the unnecessary ranting...

Quick Update

2008-10-27T10:52:00.003-04:00

Still alive here in Baltimore...

Last week was insane.

My Analysis midterm on Monday went horribly, but there are rumors that grades in grad school don't count, there's a massive curve, and everyone gets A's. So, hopefully the people that got 70's on the midterm will be in the A range, so I can be in the low B range (yes, I did that poorly). Although, my professor did say, "You guys did pretty well, nobody bombed the exam." I don't see how I don't classify as "bombing" the exam though.

Anyway, I spent all day/night on Monday doing my Algebraic Topology Assignment, which I had been forced to put off due to my midterm. Trying to decipher Van Kampen's theorem and deconstruct topological spaces is not the best thing to do when you're frazzled.

And in similar fashion, I spent all day Tuesday doing my Analysis assignment that was due on Wednesday (Lebesgue Integrable functions, and Lp Spaces) and Wednesday doing my Algebra assignment that was due on Thursday (Rings, Ideals).

This week will be much of the same. I am spending today studying for my Algebra exam, although my professor told me outright that most people do not do well on the midterm, so to not worry about it. Tomorrow will be spent doing my Analysis assignment that is due on Wednesday, and Wednesday will be spent doing the Algebra assignment that is due on Thursday. Then I can rest. At least for a few hours.

*sigh* Until next time...

Midterms

2008-10-16T09:10:00.003-04:00

I apologize for the shortness of this entry, along with the lack of entries in recent days.

It's midterm season for my graduate courses. And, they've really loaded up on the workload. So, my schedule for the next two weeks or so?

Algebra Assignment Due Today
Analysis Midterm on Monday
Analysis Assignment Hypothetically Due Monday ("It will help you on the exam.")
Algebraic Topology Assignment Due Tuesday
Algebra Assignment Due Thursday
Calculus Review Session to Run on Monday
Algebra Midterm on Tuesday

After all that - it should be a big sigh of relief for at least a week or so. Just have to push through for a week and a half! Until then, I probably won't be writing all that much, but I may, you never know!

Lang?! Why, Yes.

2008-10-09T09:57:00.002-04:00

I was quite amused by the two comments on my prior post. Both remarked on how seemingly crazy it is to try and learn/teach from Lang's Algebra. Lucky for me, that is exactly what I'm doing. It would not be so bad if I had a really solid background in the topic, but alas, I don't. My one semester of basic algebra is not really cutting it. Doing my algebra homework usually requires a lot of reading into other textbooks (i.e. Fraleigh and Dummit/Foote, among others). It is fun when I occasionally run into the problem that I'm trying to prove, worked out in other textbooks. Of course, I am not so naive as to copy the proofs from them. That is both illegal and unethical; and I could probably get kicked out of graduate school for doing such. However, if I'm looking up things in other textbooks, I've usually given it a fair amount of work and thought, so what I typically do is read the proofs in the other books, internalize it, understand what they're saying, and then go back and write my proof from scratch.

So, my response to the Lang issue? Probably not the greatest to learn from, but the exercises are excellent in the sense that they encompass a great deal of material in a single question and really force you to think outside of just a 'simple' question such as "Prove that Alt(5) is simple." (sorry, bad math joke).

I finished the assignment, forced through the grunt work and the tears. I was up until 2am last night, and was in the office by 9am, but I am done, with even 15 minutes or so to spare. It's a good day.

I also commented on my last post that I wanted to make a note about my teaching. I think I'm getting better at it. Practice makes perfect, or so they say! My students just finished the beginnings of doing "real derivatives" as they are calling it, i.e. power rule, product rule, quotient rule and chain rule ... as opposed to the good ol' formal definition: f'(x)=[lim h --> 0] (f(x+h)-f(x))/h. They are much happier too. My class on Tuesday is my most question based group. They are a little more needy than my Thursday class. But, we started all this derivative stuff, and we just did loads of problems. I have them for about 50 minutes (although this past Tuesday was more like 45, because I forgot my markers and had to run, and yes, I legit ran, back to my office to grab them, and then didn't want to force out another in depth chain rule problem in the remaining 2 minutes of class). One of my fellow first years sits in on my Tuesday class, and usually he has something to say or asks a question during section, but there was nothing really to say about Tuesdays class, which i was happy about. And, I even did that section without notes or anything since I had so many assignments due. If there was any week that I could "wing it", this was the week to do it, and it all turned out well. I typically don't use my notes anyway, so I actually think I did better, because I wasn't constantly going back to my notes to find out where I should go next. I was content. Hopefully today's class goes just as well!!

Finally, I can breathe easy for a few days. :-) No assignments due until Wednesday, barring any assignment from Algebraic Topology today that he makes due on Tuesday.

Permutations and Orbits

2008-10-08T09:41:00.002-04:00

Well, it turns out that this week hasn't been so bad. I think I'm getting a cold, but that's the worst of it! My best friend is coming down to visit me this weekend, so I guess I've just been extra motivated to get my work done.

Analysis was handed in on Monday.

Monday all day (and night) was spent doing Algebraic Topology. And, what we discovered about that, was that it lends itself to pictures quite nicely. So, Tim and I stayed at the office until 1:15am on Tuesday morning (the assignment was due at noon on Tuesday). While we were there very late, it wasn't so bad. We drew pictures of tori (plural for torus??) to figure out whether the Borsuk-Ulam Theorem held (we found that it doesn't - consider a plan perpendicular to the axis of rotation through the center of the torus. the projection of the torus onto this plane does not allow any f(x,y)=f((-x),(-y))). I enjoyed that problem. And there were some other interesting problems which we worked out, and we got it done, by some stroke of luck.

And yesterday we all started Algebra when we got back from classes and teaching. There are 6 problems on the assignment, and I seeme to have gotten through the first 3 pretty well, and I got through the 4th one, but I have to go back and redo that one - I had a misconception of the definition of what it means for two permutations to be conjugate. And then just the last two, which are the big problems, unfortunately, but I have all day to do them. It should be a good day. I'm hoping to have the scratch work for those last few problems done before I head to my Real Variables lecture at 1:30, and then to have the writeup of the final draft done (since I never EVER hand in my first draft/scratch work) before wine and cheese this afternoon at 4:30. It should be good.

The assignment is based on permuations and orbits, which I think I may have been taught the computations of incorrectly, or that there are at least three different ways to do them, according to the three books that I have. I have to get that straightened out. I broke the ice with my Algebra professor yesterday since I was having some issues on figuring out what exactly I needed to prove, i.e. distinguishing between which "if" statements they were simply giving me and which they wanted me to prove. So I went to go and get that cleared up, and my professor ended up explaining to me the whoel problem, which was nice, because I'm not sure that I would have done it the way he had explained it, so yay for that. But, I got to tell him about my worries being in the graduate program here, and he completely understood. I told him that my algebra prior to arriving at JHU consisted on one very basic course, and he asked which book I had used. And when I told him that I had used Fraleigh, he said, "Oh, so this book [Lang's Algebra rev. 3rd ed.] is pretty difficult for you then." And I very much so agreed, but the fact that I've done relatively well on the homeworks so far and I haven't completely sunk yet is a good indication. He did tell me though that the problems of chapter 1 are the hardest, and it should get easier as the semester goes on. I also think that's because we have more of a mathematical maturity. I can tell that my proof writing, while good before, is continuing to get better (which is should, considering I write nearly 25 proofs a week, outside of what we prove in class). And, just deciphering problems and figuring out how to go from given to conclusion seems to be working out pretty well and the skill is getting better. Hard work, obviously, but it's getting there.

But, I must have been super motivated this week - especially since I have the most amazing weekend ever coming up including a trip to DC, football, and we'll see what else ... all with my best friend who's coming down to visit for the long weekend. Thank you COLUMBUS!

**Note to self: Next entry should include something on my teaching thus far.**

Sinking

2008-10-06T10:51:00.001-04:00

It's going to be another rough day ... well, week actually. I'm handing in an Analysis assignment this afternoon that I am far from okay with. There was six problems on the assignment, and the first three I think I did okay. And there were parts on the latter three that I think were okay as well, but there were some parts where I waved my hand and wrote something just to write something.

I don't have a choice at this point, I have an Algebraic Topology assignment due tomorrow morning, which I just started today (because I've been working on Analysis), and don't you know I can't those problems either. So, I get to bust my rear end all day today and pray that nobody comes to my office hours, and hope that I don't pass out due to stress and exhaustion. So is the life of a graduate student, I suppose.

And, when that's all said and done, I have an Algebra assignment due on Thursday that I haven't started, because of the other two classes. So much for being in a groove, right? I really thought I was getting there. Apparently not so much.

On the 7 Dwarfs Scale, I am currently Dopey, Bashful, Sleeping, Grumpy, and Sneezy. All 5 that I have the capacity to be right now. (Happy and Doc come later... much much later... I hope.)

Measurability

2008-10-03T09:08:00.002-04:00

Well, we've started measurability (i.e. the art of sets and functions being measurable) in Real Variables last week and into this week, and presumably for the rest of the semester. This stuff actually makes some sort of logical sense to me. Not completely, but it seems to be a lot of set theory (at least with the set measure) and a lot of, what my professor would call, 'following your nose'. In other words, it is all logical, and you just have to know how to follow the steps. I seem to think that's most of the proofs, but there's no real trick as of yet. There were a few slightly illogical proofs in the sense that we randomply were summing up 1/k, but I think that's just where my lack of trust in proving using 'fudge factors' comes in.

So, my goals for the weekend include doing the homework on measurable functions. And, also my algebraic topology homework which is dealing with the fundamental group, which should be ... interesting. But, that's where I'm at, and hopefully not where I'll be in a few days!

Solvable Groups and Normal Subgroups

2008-10-01T09:42:00.002-04:00

I think it's clear that I'm on an Algebra kick right now.

Well, I'm still working on the problems I mentioned in yesterday's post. I did prove that any group of order (p^2)q, and any group of order 2pq, where p and q are distinct primes (odd primes in the latter case, although it occurs to me that it wouldn't matter much, since if there exists one or both of p,q=2, then it resolves down to the (p^2)q case, or just simply a p-group).

It seems silly, but I'm still working on proving those last few cases of orders less than 60 being solvable. My goal with those, which I'm a bit more comfortable with, is to find a normal p-Sylow subgroup (H) in each (i.e. a unique p-Sylow subgroup), and thus show that it is then solvable, and that G/H is also solvable, and then G will be solvable. But, alas, it's a bit harder to me than it looks. Although, I was also doing this stuff until 12:15am in the library. So, the last I looked at it, I probably wasn't exactly of clear mind. But then again, some would argue that that's when the best of mathematics comes out of a person - when they're not thinking 100% clearly and such. It's kind of like when you are just about to fall asleep and you had been pondering a question (I do it mostly with math, obviously, but I'm sure it applies to all disciplines) and just as you're lying there the answer comes to you, as clear as day. Let's hope that happens today with me! (Although, not while I'm about to fall asleep - that would be bad.

Normal Groups, p-Sylow Groups, Solvable Group ... Oh My!

2008-09-30T08:57:00.003-04:00

So, I've been studying a lot of Algebra lately. Currently we're working on Sylow groups and just lat class we started proving 'solvability' and 'simplicity' of groups using the Sylow theorems. And, shocker, since I've never seen this stuff before, I have double the work of not only trying to understand the concepts of these theorems and properties, but also learning the basics of the concepts. Which is fine, I'm doing okay with it - surprisingly. Right now, I'm working on proving that any group with order less than 60 is solvable. I've gotten it down to about 15 groups, whose orders don't follow the form of a p-group (i.e. have order p^n for some p prime) or the form pq, where p and q are distinct primes. I have a hint from a fellow first year to consider looking at the theorem that we have which stats that G is solvable if and only if H and G/H are solvable (where H is a normal subgroup of G). So, that's my current status.

We just started measurable functions in my Real Variables class. That stuff seems to be going okay, at least for right now. I didn't freak out of the homework that was due yesterday which is a massive bonus. I know I botched one of the proofs; the one proving that even under a continuous mapping, you can map a measurable set into a nonmeasurable set. We did this using the Cantor Lebesgue function, the function that is commonly used in tandom to describe the devil's staircase (i.e. maps intervals of [0,1] which have been removed forming the Cantor set to constants, with linear mappings on intervals that remain). So, I recognized most of what I needed to, but there was this one part actually showing the nonmeasurability of a subset of [0,1]. I just used the theorem we had to state that such a subset exists. So, I'm not completely and utterly distraught over that error. And the second error that I'm pretty sure I made, was in proving that if x is an element of the Cantor set, x can be written in base 3 using only 0's and 1's in its ternary expansion. That one was just a really sloppy proof. I was not proud of that one at all. But, on lighter notes! I think I pretty much nailed the other 5 problems in the set, so yay for that!

And Algebraic Topology, I needn't talk about. As bad as it is on my part, I haven't touched it in quite some time. That'll need to change, because our first assignment is due a week from today, and none of us know anything that we've been doing in that class. So, we'll be having an Algebraic Topology Library party, I'm sure. These are always fun. We just grab any book we think might be useful to us, get a big table, and go at it. It's one of those typical things you see in the movies (which I never thought actually happened) with the mass chaos of students with every book opened all across the table. It's quite funny actually.

Well, on that note, I must get to work. I think I'll top thinking about that proof that any group of order less than 60 is solvable for the moment and move on. It's only the first of 10 problems. And maybe the proof of it will come a bit later - I love when that happens. You don't understand part of a proof, and then BAM! The proof is completely laid out in your head. It's a wonderful experience, and I hope any mathematician reading this has had that joy. And I'm sure it applies to other disciplines as well in some way or another.

*UPDATE: So, I just went back to do the next problem in my problem set for Algebra, and came across two more things (that I will need to prove now) that help me in getting through to the goal of proving that any group of order less than 60 is solvable! Namely, now I need to prove that any group of order (p^2)q, where p and q are distinct primes is solvable, and that any group of order 2pq is solvable, where p and q are distinct odd primes. So, now I'm down to groups of order 36, 40, 48, and 56. Back to work!

Pushing Through

2008-09-23T21:23:00.002-04:00

No - this is not related to anything math, technically anyway!

I just wanted to post so people know that I'm still here, and still working through the degree. Those that follow both of my blogs know that I've been having a bit of confidence issues lately. I'm still dealing with that, but I know I can do this, so I just have to buckle down and put everything else aside and do it.

I handed in my Analysis homework yesterday, with half the problems almost entirely proved with magic and hand waving, which, if you're in my position as a graduate student on the Ph.D. track, this is simply not allowed. The philosophy in general is that if you don't have a Ph.D. you're not allowed to use the words "clearly", "obvious", "trivial", or "commonly known" in your assignments. Therefore, I know I'll be seeing a lot of red on my paper when I get it back. But, I had no other choice. It was either that, or not hand in anything - which was simply not an option.

I took about 24 hours off from problem sets, to let my brain form back to a semi-solid mass, rather than liquid mush. And, now I have buckled back down to begin my Algebra assignment ... 12 problems long (most with multiple parts) due 10:30am on Thursday, so 37 hours to go. And, considering that I'm having trouble deciphering the first problem - it should be a very interesting 37 hours.

On a brighter note - for some of my readers anyway - I will be gradually moving over to a new blog site. I've determined that since I can't (or don't know how to) input latex codes into here and I want to use the blog as a tracking of my mathematical maturity, questions, comments, etc.; I have decided to move to a new site.

The new site will be here. So, while there's nothing there more than a title right now, with time I'll be making that switch. For now, all major posts will be here, as I still need to brush up on my latex coding, and get used to the new site.

Now, to either drown in or tackle Algebra - I'm hoping for the latter.

Sickly Blog

2008-09-21T09:19:00.000-04:00

So - you may have noticed that my prior two blog entries have combined into one incoherent mess. Please note that if you click on the actual titles of the entries, you can read them (with minor errors - the latest entry before this one is mostly in italics). I will try and troubleshoot this at the next convenient time, which may be in December - sorry.

Bounded Variation and Rectifiable Curves

2008-09-21T08:56:00.004-04:00

I have two words...

Severely Struggling.

I have worked on these problems for hours - and I am not kidding. I must have spent more than 6 hours in the library on Friday alone. And out of the 8 problems that I was assigned, I have gotten 2 done - and even those, I'm really not sure about - but I have an answer, so I can't dwell on those right now.

One of the problems, we know that V[f; a+e,b] is finite and less than or equal to G' where G' is abelian is a homomorphism. Fine. But then, we need to show that f 'factors through' G/(G^c) where G^c is the commutator group. We had also shown previously that G^c is normal.

So, we defined the canonical map from G --> G/(G^c) and then found another map f*: G/(G^c) --> G' and we showed that f* is well-defined and a homomorphism. So hopefully, that is generally what 'factors through' means. Our text had never used the term before, and there are very few cases of it in the literature that we found.

If anyone can shed light onto this phrase, it'd be helpful. And if anyone knows how to either input latex into this thing or how to go about using math text/symbols ... italics can only go so far.

Another Week Ahead

2008-09-15T08:45:00.002-04:00

Well, I think I successfully made it through my first Real Variables assignment. We'll see what how I feel about it after I hand it in/it gets graded, but I think I gave it my best shot.

I was in the office til 10pm on Friday night working through a solution with one of the other first years. He's really helped me a lot, and it's only the second week!

Now, I get to fight through an Algebra assignment, which should be interesting. I still don't know which branch I like better. Algebra is nice because, at least at this level, the definitions and a basic knowledge of how things work will get you pretty far. Analysis sometimes requires this intuition, which I haven't fully developed yet, but I relate to the material better. I found myself getting hung up on this one problem of the assignment because I could not just let the case where a function had a discontinuity go. I eventually forced myself to and convinced myself that it wouldn't matter if the function had a discontinuity in it, and I'm pretty sure it'll all work out, but it took hours to get past it.

That will probably be one of my biggest problems throughout this degree process, although it may be one of my best qualities as a mathematician, I'm not sure. I have a tendency to nit-pick the smallest details - i.e. discontinuities. Which is great when writing a proof or trying to develop your own research. However, there is a very fine line between nit-picking and being obsessed - which I believe I travel between - I sit on that line and depending on the problem or task at hand, I either fall to the side of nit-picky or to the side of obsessed. I need to learn to just step back and look at the larger picture sometimes, rather than these small cases (unless necessary of course).

Thanks for all of your comments - I'm learning a lot from those that are posting, and I appreciate it! That's one thing I love about the field of mathematics, at least in my experience, it's rare that those in the field hoard their knowledge.

Hawaiian Earring

2008-09-12T10:47:00.000-04:00

This is the one thing that stuck in my mind after the two back-to-back classes yesterday. Essentially, it's the set of circles centered at (1/n,0) with radius 1/n for all n in the natural numbers (the counting numbers, n=1,2,3,...). It's actually a quite beautiful thing. Take a peak! (Thank you wikipedia!)

Now, the one thing that got me during class was that my professor contradicted himself. He first made the point that this is very similar to the wedge of circles (but that it is not homeomorphic - their topologies are very different). The difference in these two structures? Well, visually, we can see that the Hawaiian Earring are circles within circles, adjoined at one vertex. The wedge of circles (commonly referred to as the rose) are a ring of circles adjoined at one vertex. So, there's that difference. It doesn't seem that the rose contains strictly the circles of the same radius, so the size of the circles does not seem to matter.

Then it got confusing. A point was made that the Hawaiian Earring (X) is compact, and that it is not a CW-complex. But, then he went on to say that one vertex and infinitely many edges implies that the space is not compact. Yet, X has infinitely many edges and one vertex, so why is it compact?

After some research, it turns out that X is compact. And, I'm pretty certain that the rose is not compact, although I still haven't found any literature on it - I'll be hitting the library later on.

'Easy' Homework

2008-09-11T09:19:00.000-04:00

So, my Real Variables professor gave us a rather large homework assignment which is due on Monday. It has about 6 or 7 problems on it (if I remember off the top of my head), which is a lot but he said that it should be an easy assignment for us, so it shouldn't take too long. Is more than 2 hours on one problem too long?

I feel as if it shouldn't be that hard. I know what I have to do for most of the proofs, it's just a matter of writing it out and making it work. We have to prove such theorems as the Bolzano-Weierstrass theorem in R^n and the Tietze extension theorem for the real line. And there are other problems like showing that a function is Riemann integrable iff certain characteristics are satisfied, and showing that if E is relatively open with respect to an interval I, then E can be written as a countable union of nonoverlapping intervals.

It seems straight forward enough. I know what I need to asssume, andd I know what I have to prove. How hard could it be? This is all essentially review of an undergraduate course in Real Analysis anyway, right? Wrong. I'd like to think that I had a solid undergraduate background. And, for the most part, I think that I do because what I do know, I know really well as opposed to knowing a whole lot, but in not much detail.

I got my first comment yesterday about how I work really hard, and it's true. I come into the office at 9am every morning (usually a bit earlier) and I don't leave until usually at least 7pm. In fact, the other night, I was here until 9pm. I think I'll start keeping track of how much time I put into my studies and responsibilities here. It might be interesting. But, for now - I'm off to prove some "introductory" problems in Real Variables, or maybe I'll read for Algebra. I haven't decided yet.

Pushout Squares

2008-09-10T09:09:00.000-04:00

Algebraic Topology

I'm still not quite sure I understand the concept of pushout squares and adjunction spaces, but what I got from the lecture yesterday seems like it's an interesting topic.

Essentially, if you have two mappings i and f, where i: A --> X and f: A --> B, and then two more mappings g and j s.t. (such that) j o f = g o i (thus it commutes) and it is universal, i.e. given another (any) space Z, there exist yet two more functions h: X --> Z and k: B --> Z s.t. k o f = h o i, then there exists a unique map m: Y --> Z s.t. m o g = h and m o j = k (i.e. new maps commute). Then Y is a pushout of i and f.

I know, that seems a bit weird. It's easier to see with a diagram, which I cannot post here.

Actually, click here for a diagram, compliments of wikipedia. They use different letters here, but I think I've defined the mappings above well enough that if you want, you can replace the letters.

At any rate, we proved many propositions based on this idea of pushouts and adjunction spaces. Hopefully, with a bit more time, I'll be able to get a better grasp on this.

One things that I'm still not grasping as well as I'd like to be is the idea of CW-complexes (cell-complexes) and the idea of n-cells. We tried to get an idea from another grad student in our office yesterday, but the only thing I really got (which, granted, is more than I got from the textbook) is that it's generally all about boundaries and the n-cells are within the boundaries enforced through the cell-complexes. And, I'm still not sure if I'm explaining it and thus understanding it correctly. But, that is the point of this blog, right?

Background

2008-09-09T09:01:00.000-04:00

Those reading this, and I myself obviously, already know most of what I'm doing and where I am. But, for those that don't - I'm now on my first full week of graduate school at Johns Hopkins University, pursuing a Ph.D. in Mathematics. It's hard to believe that I'm actually here. Most knew that I'd get into graduate school somewhere, in fact, most believed that I'd get into most of the places that I applied. However, I think few of us thought that I'd get into Johns Hopkins. I'm still now sure that I belong here, but I'll make the best of it while I'm here regardless. Since I've been given the opportunity to be here, I might as well get the best education as I can, right?

So, that's what I'm doing. I'm currently in the Fall 2008 semester and I'm taking three courses.

110.601 (Algebra I) is not regarding the algebra that most took in 8th or 9th grade. This course, according to the syllabus, will go through Groups, Rings and Fields, Modules, Extensions, and Noetherian Rings (among other topics, if time permits). The first lecture of this course scared the living daylights out of me. He went through ALL of Chapter 1 in Serge Lang's "Algebra" and while I understood about half of what he was talking about - I didn't understand about half of what he was talking about. It's a scary thought to feel inferior. I'm coming from an undergraduate program where I was assumed to be the top of every math course I registered for, and if I didn't understand something, it meant that it wasn't clearly explained by the professor. Now, I'm at the bottom of the courses and I think that most of the professors here believe that I will sink in due time. I'm out to prove them wrong. I spent the morning yesterday trying to teach myself how to find the center of a group, I'm still not sure that I'm comfortable with it. I ran through three textbooks, and the internet, and I'm still not sure I found a suitable example for illustration, but I suppose it'll get better with time. The professor made note that he would be going back through Chapter 1 in more detail (and a tad slower, and more methodically), so hopefully he will touch on some of the points I'm not sure about.

110.605 (Real Variables) should be a good course. We'll be going through 9 chapters of our textbook and then one chapter of a book he wrote regarding Fourier Analysis on R^n. This first lecture I understood nearly completely. There were a few things that I don't quite have a visual for yet, but I'll get there, no doubt. I'm looking forward to this course.

110.615 (Algebraic Topology) I think will be my favorite course. We started the first lecture talking about homotopy and homotopy types, contractible sets, and other such "elementary" topics. Eventually, we will take this course into Homology and Cohomology. It assumes that we have a solid background in topology, and while most of us have taken a topology course, I don't think any of us feel comfortable saying that we have a solid background in it. We'll figure it out though.

So that's really all for the moment. I now must go and read the last 8 pages or so of my Algebra reading so I can follow today's lecture.