## Improvements on Lower Bounds of Sphere Packing Densities in High Dimensions

*Date Written: February 20, 2023; Last Modified: February 20, 2023*

I gave this talk on December 2, 2022 at the UCLA Analysis Participating Seminar; you can find a link to the Zoom recording there. The talk was largely based on Professor Akshay Venkatesh’s paper.

This was my first math talk (ever!), and I learned a lot from the experience — this page is meant to document that. Overall, the talk went quite well (I think), but there are things that don’t fit in talks as well as criticisms and afterthoughts from after the talk.

### Venkatesh’s “Dispersive” Technique

In another talk about exponential sums, I was introduced to something called Linnik’s dispersion method. This technique is almost completely unrelated to the sphere packing problem and Venkatesh’s reasoning, but I found an interesting and rather illuminating pattern that I hadn’t noticed throughout the rest of my mathematical studies.

Linnik’s dispersion method addresses the number of solutions to additive problems like \(a+b=n\) (where \(a\), \(b\), and \(n\) are integers subject to certain constraints), and in the aforementioned talk, it was applied to a system of such equations. Rather than considering or constructing solutions one at a time, the dispersion method considers all possible values of \(a\), \(b\), and \(n\), then uses probabilistic estimates alongside the discrete nature of the problem to produce nontrivial estimates of the number of such solutions.

Venkatesh’s argument vaguely resembles this technique when you take away the problem statement (this sentence sounds profoundly stupid, now that I’ve written it down). Rather than constructing explicit packings, Venkatesh considered an entire space of possible sphere packings, then used a probabilistic estimate alongside the discrete nature of these packings to produce a nontrivial lower bound on the optimal sphere packing density.

I think this is significant because I haven’t seen this type of argument elsewhere. Perhaps it’s because I haven’t really formally studied probability theory, where these kinds of arguments may be more natural, or perhaps it’s because I haven’t been aware of its presence elsewhere.

Either way, I wanted to bring this up here because I didn’t think this remark was appropriate for the talk. Do you know of any other applications of this kind of argument?

### Feedback

I bothered two attendees for feedback; they were both very kind and gave good constructive criticism!

First, I was advised to build some geometric intuition for the problem by either drawing figures or providing graphics. This was specifically regarding the lattice \(\mathbb Z\left[\mu_n\right]\) in the cyclotomic field \(\mathbb Q\left[\mu_n\right]\), which looks familiar to algebraic number theorists but less so to analysts (the audience of the talk).

In general, capturing geometric intuition was a challenging aspect of the talk I should have spent more time thinking about. Perhaps briefly describing \(\mathbb Q\left[\mu_3\right]\), which is a simple 2-dimensional \(\mathbb Q\)-vector space, may have been a good choice of a simple example. In fact, the later construction \(\mathbb Q\left[\mu_3\right]\otimes_{\mathbb Q}\mathbb R\) should be isomorphic to \(\mathbb C\), adding another level of familiarity.

The second piece of criticism said to spend some more time on the simpler arguments of Minkowski before delving into how it was used in Venkatesh’s argument. I did have about 5-10 minutes of extra time to spare, and I think this would have been a very way to spend them.

I think this could have been avoided by rehearsing more. I did a practise run using slides, but I ended up deciding that a blackboard talk was more appropriate because of the derivations and equations and whatnot. Doing another practise run to check my timing after that change would have benefitted a lot, though that would have taken a lot more time and energy than I think I had at the time.

### Preparing for Talks

Something I learned in the process of giving this talk is how much time it takes to put it together, and it particularly makes me wonder how professors prepare an entire quarter or semester’s worth of material. Finding the right level of depth and good motivations and good examples is very, very time-consuming, even for a short, self-contained talk like this one.

Is this a skill that gets easier over time? I think part of what took so long was deciphering Venkatesh’s paper. That’s not to say that it was a bad paper — on the contrary, it was very thorough and contained rich, insightful commentary. Instead, I lacked a lot of background, and filling in those holes probably contributed to this difficulty.

I had to miss the participating seminar for Winter 2023 because I had to take GEs (Genuinely Evil classes). Hopefully I can give another talk next quarter!