Amin Idelhaj

How are all of you doing? Especially Ted, it's been a while :)

While I stumble here less than before, I must occasionally keep the nerdity in check

Hello

I totally forgot about wordle

Whoops

How's everything going?

Hey everyone

I heard there's a party in here, how are you guys doing?

Yeah never a good time

But so it goes. Thanks

Doing alright, bit sick unfortunately

How's it going Astyx

How are you doing?

Hey everyone!

Also Thorgott careful there's a random surface behind you

Anyway I gotta get going for now but yeah it was good catching up, I should ask you more about the stuff you've been doing as well! :)

Another possibility would be expander graphs/complexes

By doing "linear programming on h in the trace formula"

But one possibility would be estimating the volume of a surface whose lambda_1 is at least some T

There's sorta two directions at the moment regarding my possible research area. I originally was gonna be looking at bounding spherical functions but that just wasn't doing it for me. Not sure if it's because I'm just not into the problem or because of burnout holdover or what

But yeah I tend to think of Ramanujan graphs as being analogues of what we hope arithmetic surfaces are, namely that in each case the spectrum of the Laplacian is controlled by that of the universal cover

It seems like this kinda thing is hard to compute

As for examples, I don't know any myself and I'm not even sure what's in general known. iirc we don't know an infinite family of arithmetic surfaces whose lambda_1 is bounded below by 1/4

This is for a general surface

ONly for arithmetic surfaces

And one of the results presented in this conference was that for $\varepsilon > 0$, the probability that a random surface in $\mathcal{M}_g$ has $\lambda_1 < \frac{3}{16} - \varepsilon$ goes to $0$ as $g\to\infty$

Selberg was able to prove that it's at least 3/16

And this set the stage for some cool stuff. One common theme for hyperbolic surfaces is the Selberg eigenvalue conjecture, that for Gamma a congruence subgroup of SL(2,Z), the smallest non-zero eigenvalue of the Laplacian on Gamma\H is at least 1/4

Second did go into some detail on Mirzakhani's work on integrating on moduli space

But yeah it started off talking about why you can speak of a "random surface", first talk just kinda stated that volume of moduli space is finite

Hahahahaha

But a lot of what I did go to was p sick

It was a bit packed (6 hours a day of talks) and I was fairly fatigued so I missed a good bit of stuff and am a bit salty as a result

And yeah there's some weird connections in this way. I went to this conference about a month and a half ago about Laplacians on random graphs and surfaces

Because the (p+1)-regular tree is basically "p-adic hyperbolic plane" in a way

I'm not quite familiar with the term "hyperbolic graph" as such, in general I actually tend to think of k-regular graphs as being analogous to hyperbolic surfaces

The proof of the trace formula basically boils down to, these two expressions are both different ways of writing the trace of a certain integral operator

Where they construct Ramanujan graphs

Hmm, I've seen something similar in Lubotzky-Phillips-Sarnak

But yeah stuff like Weyl Law, prime geodesic theorem, and bounds you get in arithmetic quantum chaos (here you want to consider Hecke operators as well, and you get an "amplified trace formula") are all applications where you're sorta choosing g and h in a clever way. Some applications (big in Langlands) are where you vary the group. These I don't understand in the least lol

Yup

Prime geodesics are basically those that "don't wind around"

Is that "sum of eigenvalues = sum of diagonals" is similar. LHS = spectral side, RHS = geometric side :P

And my quasi-joke except the more time goes on the more I wonder how much of a joke this

FWIW, the LHS of this expression is called the "spectral side" of the trace formula, and the RHS is called the geometric side

@BalarkaSen Yeah, I don't know the details but there's basically the "prime geodesic theorem" that's proven using this business

It's also analysis and it's also hyperbolic geometry

This is number theory Thorgott

Yup!

So now depending on your choice of $g$ and $h$ you can use this to do different things. For instance, if you want to prove the Weyl law, you want to choose $g$ so that its support lies in $[-R,R]$ where $R$ is less than the length of any geodesic on $S$. You'd hope to then be able to $h$ to be characteristic of $[-X,X]$, but obv $h$ must have Paley-Wiener type, so you need to massage it a fair bit