Half-June

So today wasn’t really my day, overall. Generally speaking, I woke up feeling congested and nasty, I spent the whole day with a migraine, and I was only not-lethargic for about four hours total overall.

Unsurprisingly, then, I realllly couldn’t force my brain to do any real math. For that reason, I completely avoided reading new things and instead typed up the expository analysis entry during the middle part of the afternoon. I ended the night doing some solutions for Hatcher – Chapter 0 of which I’m hoping to knock out soon to begin Chapter 1 – and drawing (really really) poor diagrams in MSPaint. I’ve emailed Dr. Sjamaar from Cornell to ask how he gets his diagrams drawn, but thus far have heard nothing back.

Pleeeeeeease don’t leave me hangin’, Dr. Sjamaar: My blog is evidence that I’m in desperate need of your resource knowledge!

Anyway, it’s almost 2am and I’m about to call it a night.

Auf Wiedersehen.

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Online reading seminar for Zhang’s “bounded gaps between primes”

Dr. Terence Tao has arranged for an online reading seminar to go through Dr. Yitang Zhang’s recent proof of the “Bounded Gaps Conjecture.” To say that this is a wonderful opportunity to pick something valuable up regarding a field that’s very hot right now in the research community would be the ultimate understatement.

What's new

In a recent paper, Yitang Zhang has proven the following theorem:

Theorem 1 (Bounded gaps between primes) There exists a natural number $latex {H}&fg=000000$ such that there are infinitely many pairs of distinct primes $latex {p,q}&fg=000000$ with $latex {|p-q| \leq H}&fg=000000$.

Zhang obtained the explicit value of $latex {70,000,000}&fg=000000$ for $latex {H}&fg=000000$. A polymath project has been proposed to lower this value and also to improve the understanding of Zhang’s results; as of this time of writing, the current “world record” is $latex {H = 4,802,222}&fg=000000$ (and the link given should stay updated with the most recent progress.

Zhang’s argument naturally divides into three steps, which we describe in reverse order. The last step, which is the most elementary, is to deduce the above theorem from the following weak version of the Dickson-Hardy-Littlewood (DHL) conjecture for some $latex {k_0}&fg=000000$:

Theorem 2 ($latex {DHL[k_0,2]}&fg=000000$) Let $latex {{\mathcal H}}&fg=000000$ be an…

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Algebraic Geometry Observation I: Algebraic Varieties

In order to define any algebraic geometry structures (a sheaf, for example), one has to first understand what an algebraic variety is. And thus:

Observation I. It’s damn-near impossible to find someone who gives a straightforward definition of an algebraic variety straight off the bat.

Instead, most authors tend to define an affine algebraic variety – first as the common zero set of a collection \{F_i\}_{i\in I} of complex polynomials in \mathbb{C}^n and later as a “variety that can be embedded in affine space as a Zariski-closed set” (Smith et. al., An Invitation to Algebraic Variety). Then, half a book later or more (it’s on page 144 of the aforementioned book), it’s said that an (abstract) algebraic variety is a topological space with an open cover consisting of sets homeomorphic to affine algebraic varieties which are glued together by so-called transition functions that are morphisms in the category of affine algebraic varieties.

This of course requires knowledge of category theory, the Zariski topology, etc. etc.

As of now, this 30+ minutes of searching has gotten me through about 3/4 of a page in Chris Elliott’s online manuscript concerning D-modules.

le sigh

CW Complex Video Checklist

This is really just a link dump for me to use later when I need to rewatch stuff on CW complexes. To make sure credit is given where due, these videos are from the YouTube channel of Harpreet Bedi.

  1. Motivation w/ Torus
  2. Definition, etc.
  3. Examples, Attachment Maps
  4. Subcomplex Stuff

Also, for later, I should consult this video for talks about suspensions, etc., of topological spaces.