How do some Ws on transcripts affect PhD admissions in mathematics? I am a senior who is graduating in December with a Bachelor's Degree in Pure math. I would really love to apply to the PhD program. However, I have been really depressed because even though I have a good GPA (3.85), and I have gotten mostly A's in all of my math classes (I have taken math classes up to Topology, measure theory, and Algebra), I feel like this is not good enough since I have 8 W's [withdrawals--no grade given] on my transcript. The horrible part is that 6 of these W's are on math courses. All those W's occurred during my first year at my university. The second year up to now I haven't gotten any W's.

My question is: How horrible (by the Committee who selects candidates for the program) is a W seen on a transcript? Is there anything I can do about it?

I feel like I have a big hole on my transcript, and it has been chasing me throughout my undergrad years even though I managed to hide myself from this monster and continued to take a lot of math courses, earning A's (and two B's) in all of them.
Your situation doesn't sound so bad to me at all.

First of all, though perhaps I shouldn't, I will admit that many graduate admissions personnel do not thoroughly scour the transcripts.  They look at the GPA and they look at the courses taken.  Often this information is given separately on the application, so a look at the actual transcript may only be required if there is something missing there.  So there's a tip for you: if you are asked for separate information about courses taken, definitely be sure to list it.  (If on the other hand you are asked to list the courses taken together with the grades you got, it seems to me that you are ethically obligated to list the W's.)

More than this though, W's occupy a sort of nether region in academic grading.  In my university we have WP (withdrew passing) and WF (withdrew failing).  Only the latter affects your grade, and my understanding is that WP is not meant to be a stigmatic mark at all.  (In fact undergraduate students are limited to 4 WP's over the course of their careers.)  Unless the registrar puts an asterisk next to your 3.85 GPA and says warning: there were some W's! then in at least one very official sense the W's are not being counted.

More good news: all of the W's are in your first year.  That is exactly the sort of localized difficulty followed by dramatic improvement that admissions committees are looking for.  

I think it would be a good idea to use your personal statement to briefly address the W's.  I'm thinking of one or two sentences which acknowledge that they exist, say a few vague words of excuse (e.g. "time to adjust to a new academic environment", "personal difficulties long since resolved"; nothing too specific or gory), and especially: point out how nicely you've moved on.  If you feel like you can use the W's as part of a larger depiction of a crescendo of academic accomplishment, you might try that, but that's a more "advanced technique", so to speak.

I think it is quite likely that the average effect this will have on your application is little or none.  Honestly, to me you sound significantly more guilty / apprehensive about a minor issue long since resolved than you need to be.  I forgive you!  Please don't hesitate to apply to all the PhD programs you're interested in. (If you like algebra/number theory/geometry/topology, please consider UGA.)