# First AMS Grad Student Blog

So I’ve decided to write for the AMS Grad student blog, and so I will be posting there once a month. My first post, about applying for a NSF Graduate Research Fellowship, went live today!! You should check it out:

Welcome. As this is my first post for this blog, an introduction is probably in order. I’m David, but most people call me DJ, and I am a second year graduate student at the University of Wisconsin – Madison. My research focuses on the intersection of commutative algebra, algebraic geometry, and number theory. Outside of mathematics I enjoy watching and playing sports (Go Blue!) as well as long walks on the beach. (Kidding I’d prefer to be playing volleyball on the beach.)

As Mathew mentioned in a previous blog post the National Science Foundation Graduate Research Fellowship Program (NSF-GRFP) is an amazing program that, “Recognizes and supports outstanding graduate students in NSF-supported science, technology, engineering, and mathematics disciplines who are pursuing research-based Master’s and doctoral degrees at accredited United States institutions.” [1] I have a bit of experience with the Graduate Research Fellowship (GRF) application, having applied to the NSF-GRFP twice, and am currently supported by the NSF-GRFP. Thus, since the deadline for this is fast approaching I figured I’d share a bit of advice regarding the application process. I should say that I am by no means an expert on the NSF-GRFP, and so all of this advice should be taken with a, possibly large, grain of salt.

# 2015 AG Institute in Utah: Day 14

(For background on the conference and my time in Utah see my previous updates: Day 1,  Day 3, Day 7.)

Week two of three is now officially in the books, and I have exactly seven six days left in Salt Lake. The second week was a little more laid back with a bit less exploring. Instead I spent most of my free time playing basketball at the gym, talking to other participants, and studying. Oh and on Wednesday a couple other Wisconsin grad students and I found a decent bar downtown!!

First good beer in SLC.

I wonder how well Mormons take this…

I like the Food Pairings…

Friday turned out to be Pioneer Day, which celebrates the arrival of the Mormons in the Salt Lake Valley. According to a local it is bigger than the 4th of July, which based on the number of things that were closed on a Friday just might be true. (As an aside can anyone else think of another local holiday — state or city/town — that is widely observed and results in things shutting down?)

Foothills

The talks this week focused on derived algebraic geometry, mirror symmetric, tropical geometry and so I was pretty far outside of my comfort zone. Thus, these summaries are going to be a bit shorter than previously. However, here are a few things I found interesting/learned:

• Uniform Bounds on Rational Points (Joseph Rabinoff): Given a curve $C$ over a the rational numbers how many rational points does $C$ have? Of course this question, and many similar questions, have been intensely studied by numerous people over the years. In his talk Joseph Rabinoff, speaking about joint work with Eric Katz and David Zureick-Brow, discussed recent work on this question. In particular, he presented the following really cool theorem:

Theorem (KRZ-B): If $C$ is a smooth curve of genus $g$ over $\mathbb{Q}$ satifying some additional conditions then:

$\#C(\mathbb{Q})\leq 76g^2-82g+22$.

Notice that in this theorem the bound only depends on the genus of the curve, nothing else! The precise statement of their theorem, which is vastly more general can be found in their paper. In the paper they also address a similar question called the Mumford-Manin conjecture. Once again providing uniform bounds for a class of specific curves.

• Generalized Fields (Jacob Lurie): The idea behind recent Breakthrough Prize winner Jacob Lurie’s three plenary lectures is that algebraic topology seems to provide a way to generalize many of the algebraic structures we know and love. To illustrate this concept Jacob spent his lectures focusing on the example of generalized fields. A generalized field is an associative ring spectrum $E$ such that every module over $E^*(\{x\})$ is free. (A ring spectrum, at least in my mind is a cohomology theory, which spits out rings. Although by Brown Representability you might be supposed to think about this as a sequence of topological spaces. Not really sure on this…) This definition is intended to mimic definition of skew field as an associative ring all of whose modules are free.

One can also transfer other algebraic notions from algebra to algebraic topology; often to surprising results. For example, you can define the characteristic of a generalized field, and instead of there being just two flavors — zero and p — there turns out to be generalized fields of intermediate characteristics. Jacob went on to discuss how one can do other algebraic things over these fields of intermediate characteristic — representation theory & roots of unity — and how the results differ from the results we know over regular fields.

All of this was very interesting and exciting, but did leave me somewhat unsure of whether the analogy between algebra and topology is a deep connection or simply a useful mental crutch for understanding these new ideas. (Possibly both??) That said this almost certainly stems from a lack of understanding on my part. Regardless his talks were extremely enjoyable and down to Earth. I would high recommend them if only as a glimpse to a new area of math that seems to potentially have a lot of promise. You can find his slides here, and videos of the talks should be posted shortly.

Tom Bridgeland also gave a series of excellent talks on stability conditions, but I am not sure I understood them quite well enough to write about. So if you’re interested check out his slides and video when posted.

PS: The afternoon talks are hosted in the business school, and on Friday I found something called the “Leadership Lounge” which appears to be a lounge for business (grad) students complete with multiple ping pong tables and a foozball table.

#buzzwords

I am starting to think the lives of a math grad student and a business grad student are very different… (Also what exactly makes a lounge “leadership”?)

# Life Updates – (The Decision)

So as I promised in the last post here is a quick rundown of all the exciting things that have happened in the last couple months!

• The Wait is Over: I got offers from graduate schools! Without going into all of the details I received multiple exceptionally generous offers from a many amazing places. This made deciding on where I was going pretty difficult, although I feel I had an easier time of it than others, but I am so blessed to have been given all of these choices. Thanks to all those who have helped me through the last year; I wouldn’t be where I am without you.

• The Decision: I decided where I am going to graduate school!! But being the closest thing I’ll probably ever get to my own The Decision, I am going to drag the suspense out as long as possible… (or just scroll to the bottom).

• Final Finals: I took my last ever finals as an undergraduate. It turns out that one of my professors had decided to offer the final in-class before the scheduled day on the syllabus. Needless to say I missed this change, and so ended up not making the in-class final! (You’d think that by my senior year I would have figured this whole school thing out.) Luckily the professor was understanding and allowed me, and the other students who missed the change (I wasn’t the only one), to take the final on the originally scheduled day, and everything worked out. Also in the end I wrote, what I thought, was a pretty interesting essay on the role technology has played in the ascendancy of presidential power over congress. (Maybe I’ll write about this someday.)

• Graduation: I graduated from the University of Michigan!  For me this marked to closing of an amazing four years full of great friends, great classes, and great experiences many of which I would have never imagined I’d have when I arrived on campus as a freshperson. That said graduation itself was a great time fill with FOUR different ceremonies over two days; one of which included spending fours hours in forty degree rain, ok that part was not to fun. Overall it was a nice way to end my time at Michigan, and start my next adventure at….

So without further ado: I have decide to talk my talents (whatever amount I have) to the University of Wisconsin in Madison, WI!