New paper on arxiv, with my Rutgers colleague Swee Hong Chan and Igor Pak of UCLA: arxiv.org/abs/2411.18782 We study numbers of spanning trees of graphs, answering a 55-year old question of Sedlacek. A graph is just a bunch of vertices connected by some edges. Here’s a pic of all the connected (that is, spanning all the vertices) graphs on 4 vertices. A “spanning tree” is a subgraph (drop some edges) that is still connected but has no cycles (no paths through the graph that come back to where they started). For a given graph, G, write τ(G) for how many spanning trees it has. This is a fascinating quantity that’s been studied since (at least) 1847, when it appeared in Kirchhoff’s work on resistance in electrical circuits (this is where he proved the "matrix-tree theorem", that τ can be computed as the determinant of a minor of the graph Laplacian). By happenstance, this quantity appears all over the place. In evolutionary biology, τ counts the number of possible phylogenic trees connecting sets of species based on genetic or evolutionary data. In the arithmetic of Ramanujan graphs (of Lubotzky-Phillips-Sarnak), τ coincides with class numbers of certain function fields. Etc etc. In the 60’s, Sedlacek started to ask a number of very natural questions about what values τ can take in different families, ordered by the number of vertices. For the size-4 graphs in the image, we see that the set of values taken by τ is: 1, 3, 4, 8, and 16. Of particular interest to us is the family of planar graphs. It’s easy to show from Euler characteristic that τ takes *at most* exponentially many values. Does the set of values also grow *at least* exponentially? Sedlacek could prove that they grow at least quadratically. Skipping a lot of intermediate results, the previous best growth rate was due to Stong, who showed in 2022 that the number of distinct values of τ on planar graphs with n vertices grows at least like exp(C n^{2/3}). One of the difficulties here is that there are only exponentially many planar graphs to begin with, and τ can have exponentially large multiplicity (for example, there are exponentially many trees, for whom τ is of course always 1). Nevertheless, we’re able in this paper to resolve the question (at least qualitatively), that τ indeed grows exponentially. In fact we show much more, namely, it looks like the set of values of τ will eventually be just *all* (sufficiently large) numbers, in an exponential range; we prove a positive proportion and outline how to prove density 1 (if you just want to show exponential growth, even less is needed). The method of proof is really fun, too, and happens in three steps: • We reduce the graph question to a question in Diophantine geometry, that smells a lot like Zaremba’s conjecture. • A decade ago, Bourgain and I made some progress on that problem, and that work (which has been extended now by Frolenkov, Huang, and especially Kan) applies to this context. It turns out that a key role is played by the Hausdorff dimension of a certain fractal, which needs to overcome a threshold value. The Diophantine problem is thus reduced to one in Thermodynamics. • To resolve the Thermodynamics, we have to rigorously compute this dimension, and (very luckily for us!) it turns out to indeed exceed the threshold, but just barely, in the hundredths place. This was a really fun project to work on. For more details, please see the paper.

Really pleased to finally share my preprint: Diffusion Geometry (1/4) Diffusion geometry defines Riemannian geometry for data and probability spaces! arxiv.org/abs/2405.10858

Our little cutie is out: A counterexample to the singular Weinstein conjecture sciencedirect.com/science/ar…

On Thursday, November 14th, we are honored to host a lecture from Haynes Miller from MIT. Professor Miller will discuss the history of the proof of the Sullivan conjecture and its influence on subsequent research. Register for attendance at: discord.gg/TxKbTUb2wC

Deeply honored to share that I’ve been awarded the Professur Gauss 2025! 🙏 Taking on Carl Friedrich Gauss' chair at the Göttingen Academy of Sciences is a true privilege, recognizing contributions in fields like astronomy, geophysics, math, and physics🌟 adw-goe.de/preise/kategorien…

A few weeks ago, we had our first All-Members meeting of the Association for Mathematical Research @AMathRes, where we described what we've been up to over the last year. Here's a summary of the meeting.

I totally identify with the description of deep moments in mathematical research coined by Voisin in this @QuantaMagazine article "The Rest of the World Disappears". Germans probably have a word for this... Die Deutschen müssen dafür ein Wort haben, aber ich kenne es nicht!

A crisp 3D printed Roelofs nested knot. I also think of these as polyknots because they are analogous to Holden polylinks. #mathart #3dprinting @rinusroelofs #knots #knottheory

Congratulations Yakov Eliashberg and Claire Voisin for this prize!👏👏👏👏

How long will it be before a computer can get a gold medal in the International Mathematical Olympiad? Now that there is a big prize at stake, maybe not as long as it would otherwise have been ...

Explore the addition table for adding some simple orders A B. #InfinitelyMore #PanoramaOfLogic

My sunny days in California are over. Lovely math conversations, Colloquium at USC, and conference at @mathmoves. Inspiring sunlight and palm trees in LA, and mesmerizing views from SLmaths, I'll miss you!Talks available at CAMS youtube channel and: slmath.org/workshops/1054/sc…

new project in formalizing mathematics! (Prime Number Theorem in Lean)

The Crafoord Prize 2024 – Mathematics💡 This year, the Crafoord Prize in Mathematics is awarded to Claire Voisin, Institut de Mathématiques de Jussieu, @CNRS, France “for outstanding contributions to complex and algebraic geometry, including Hodge theory, algebraic cycles, and hyperkähler geometry”. Read more: kva.se/en/news/this-years-cr… #crafoordprize #Mathematics

Excited to share our work analyzing extracellular matrix structures in lung cancer using applied topology! biorxiv.org/content/10.1101/…

The first Issue of the Journal of the Association for Mathematical Research @AMathRes is live! jamathr.org/index.php/jamr/i… There are two papers, one by Gromov, and one by Entov-Verbitsky. I hope you enjoy reading high-quality Diamond Open Access mathematics!