Jordy Greenblatt

PhD Student, UCLA Department of Mathematics


If you're bored, try to find all 9 rings (one of them is tricky)

Interests:
Real Analysis, Harmonic Analysis, and Functional Analysis

Teaching:

Math 32A (Calculus of Several Variables), TA for Rostyslav Kozhan, Fall 2011
Math 32A, TA for Robert Greene, Winter 2012
Math 33B (Differential Equations), TA for Ryan Reich, Spring 2012
Math 32A, TA for Rostyslav Kozhan, Fall 2012
Math 61 (Discrete Structures), TA for Igor Pak, Fall 2012
Math 106 (History of Mathematics), TA for William Duke, Winter 2013
Math 61, TA for David Gieseker, Winter 2013
Math 33A (Linear Algebra and Applications), TA for Olga Radko, Spring 2013
Math 115A (Linear Algebra), TA for Don Blasius, Spring 2013

My Spring 2013 office hours are...
33A: M 12:30-1:30
115: T 12-1:30

My policy is that each class gets priority in its respective office hour time slot but, if there are not too many student from that class, I may be able to attend to students from the other class.

Qualifying Exams:

Completed

Basic: Passed Fall 2011
Analysis: Passed Spring 2012
Algebra: Passed Fall 2012
French Language: Passé Automne 2012

In the process of studying, I wrote up solutions for almost every relevant algebra problem (there is no longer a linear algebra section so I skipped those) from the following qualifying exams: S12, S11, F11, S10, F10, S09, F09, S08, F08, S07, F07, S06. The only two I didn't complete from those exams were S1007 and F09C1, but I gave some direction for each. The problems are organized by topic and in the order I listed the exams above within each topic document. Let me know if you see a substantive error and I will try to fix it. I owe special thanks to the following people for giving me a lot of help with this project along the way: Richard Elman, Paul Balmer, Andy Soffer, Tianyu Wu, Andreas Aaserud, David Clyde, and Scott Garrabrant. I also borrowed some solutions from Jed Yang's old solutions found here. Jed's writeups go further back than mine chronologically but mine end later. Also, most of my writeups of overly detailed in the interest of being as self-contained as possible; they are probably in much more detail than you want to use on an actual qualifying exam.

Category Theory
Commutative Ring/Module Theory
Galois/Field Theory
Group Theory
Representation/Noncommutative Ring/ModuleTheory

Talks:

"Presentation on Singular Value Decomposition and Schatten-Classes," UCLA Participating Analysis Seminar, 2/14/12 and 2/21/12

Presentation on Singular Value Decomposition and Schatten-Classes Notes

I'm sure there are still plenty of mistakes so, if you happen to find one, please send me an email. I will be grateful, not offended. Also, I know the inner product brackets are a little unsightly but I have no desire to go back through 50 pages for a small aesthetic detail.

I will be giving another talk for the UCLA Participating Analysis Seminar in February 2013 on Hardy Spaces and Harmonic Functions on the Upper Half Space.

Discoveries:

In the course of preparing my talks I discovered a couple oringal proofs. To the best of my knowledge, they have never appeared elsewhere.

In my February 2012 Schatten-class presentation, I found that the standard proof of a key duality theorem for Schatten-Classes uses holomorphic functional calculus that I had not covered earlier in the talk. My proof avoids these techniques and replaces them with more basic analysis tools. For greater context, the proof appears in slightly more detail on pages 43-47 of the notes for the talk. While the notes below are self-contained, they assume familiarity with the singular value decomposition and the definition of a Schatten-class, Theorem 1.2 and Definition 2.9 respectively in the full notes.

Schatten-Class Theory Lemma

In my February 2013 Hardy Space presentation, I found that an inequality that appears in Eli Stein's Harmonic Analysis (Chapter II, Section 2.5) that, while sufficient for the relevant theory, is not optimal. Below is a self-contained proof of the optimal bound.

Tent Theory Lemma

For those who are interested in logic, mathematical history, and generally weird stuff:

Here is an email correspondence with my neighbor back in New Jersey about the wonderous mysteries of set theory after she sent me this article.

It is not intended for people with a pure math background...

The Hilbert Hotel article is very nicely explained. However, it actually gets a hell of a lot weirder. For instance:

(1) There are infinitely many "classes" (called cardinalities) of infinity (like the 2 he mentioned, countable sets and "continuum cardinality," i.e. that of the real numbers) that are "larger" than the real numbers (i.e. if you had a hotel with a room for each real number and a bus came with a higher cardinality of passengers, there would be no way to accommodate them).

(2) The original set theory bible was written by a philosopher/mathematician name Frege around the turn of the 20th century. He worked under the seemingly obvious assumption that if you can describe a set, then it exists. Then Betrand Russell (a graduate student at the time, I believe!) sent him a letter that described a set (call it R) of "all sets that don't contain themselves." Well, if the set exists, the question is, does it contain itself? If it does, then we're in trouble because we built R so that none of its members contain themselves. If it doesn't contain itself, then it meets the admission criterion for R so it should contain itself. So we're up the creek either way. Don't be put off if you have trouble with that idea; keep in mind that it's so weird that none of the titans of 19th Century math thought of it. The post script is that Frege published his set theory bible with a preface that said something to the effect of "this book is basically wrong, but there's hopefully something useful in it." The modern name for Frege's approach to set theory is "naive set theory" (like, there are textbooks with that name).

(3) Once two mathematicians, Zermelo and Fraenkel (Fraenkel was the head of the department at Hebrew University actually), put set theory on a firm logical basis, a question arose that plagued math for many years. Is there an infinite cardinality bigger than the natural numbers but smaller than the real numbers? The assumption that there doesn't exist such a cardinality is called the continuum hypothesis. It wasn't proved until the 60s that there isn't an answer! That is, you can just say "I accept all of Zermelo and Fraenkel's assumptions but also I DON'T want there to be a cardinality between the natural numbers and the reals" and it would be consistent but you could just as easily instead say "I accept all of Zermelo and Fraenkel's assumptions but also I DO want there to be a cardinality between the natural numbers and the reals" and it would still be consistent.

(4) Also around the turn of the century, a French mathematician named Lebesgue came up with a theory for "measuring" sets, but the weird thing is that it really didn't look much like set theory. The idea is that if you measure the interval from 0 to 1, it should have measure 1, but a single point should have measure 0. If you combine two sets with no overlap, their measures should add. But if you combine all of the real numbers (each of which on its own has measure 0), it gives you the whole number line, which should have infinite measure. It is not too hard to show that all countable sets have measure 0, but what about uncountable sets? In fact, the same Georg Cantor mentioned in the article came up with an idea for a set that was uncountable but also had zero measure. The idea is that you start with the interval from 0 to 1 (including 0 and 1) and then, in the first step you cut out the middle third, leaving 1/3 and 2/3 (and everything below 1/3 and above 2/3) in the set. Then you're left with two intervals of length 1/3 and you cut out their middle thirds the same way. Doing this infinitely many times yields the "Cantor middle thirds set" which has some really weird properties, including being uncountable but having measure 0.

(5) Zermelo and Faenkel made an assumption called the "axiom of choice" which, while consistent with the other assumptions, is "independent" of them (that is if you don't want it to be true, that's perfectly fine with the other assumptions) and has some really weird consequences. Some of them are very useful, but two of them are that there is a subset of the real numbers that can't be measured in a useful way and that, if you have a solid ball, you can cut it up into 7 pieces (they're not solid pieces, but 7 crazy subsets) and rotate them, move them, and put them back together to get two solid balls equal in size to the original ball! This last one is called the "Banach-Tarski Paradox" and is the origin of one of the relatively few math jokes I like. Q: "What's an anagram for Banach-Tarski?" A: "Banach-Tarski Banach-Tarski."

Those are the weirdest set theory results I can think of but it goes on from there. It is a surprisingly deep field considering how bare bones it seems to just study sets.


If you have too much time on your hands, here's some of my favorite humor writing:

From the Onion:

Maybe just
a little morbid.
This is a bit lighter.

From Jack Handey:

In case the PhD doesn't pan out.
The title says it all.

From yours truly:

My 15 Minutes.

From my friend River:

I suspect this is autobiographical.
If you know me well enough, you should know that my passion for history is only exceeded by my passion for unnecessarily long titles.
Reading this thing will make you happy.
I thought people would stop complaining about Hipsters once I left college. I was mistaken.

I don't know this guy:

but I hope he's right.

If You Need Me:
Office: MS 2954
Department of Mathematics
University of California, Los Angeles
Los Angeles, CA 90055-1555
Email: jsg66@math...