Georgia Institute of Technology College of Sciences

I’ll describe some recent work (joint with Jeff Kahn and Jinyoung Park) on the "Second" Kahn--Kalai Conjecture (KKC2), the original conjecture on graph containment in $G_{n,p}$ that motivated what is now the Park--Pham Theorem (PPT). KKC2 says that $p_c(H)$, the threshold for containing a graph $H$ in $G_{n,p}$, satisfies $p_c(H) < O(p_E(H) log n)$, where $p_E(H)$ is the smallest p such that the expected number of copies of any subgraph of $H$ is at least one. Thus, according to KKC2, the simplest expectation considerations predict $p_c(H)$ up to a log factor. This serves as a refinement of PPT in the restricted case of graph containment in $G_{n,p}$. Our main result is that $p_c(H) < O(p_E(H) log^3(n))$. This last statement follows (via PPT) from our results on a completely deterministic graph theory problem about maximizing subgraph counts under sparsity constraints. 

The Walsh-quantized baker's maps are models for quantum chaos on the torus. We show that for all baker's map scaling factors D\ge2 except for D=4, typically (in the sense of Haar measure on the eigenspaces, which are degenerate) the distribution of the matrix element fluctuations for a randomly chosen eigenbasis looks Gaussian in the semiclassical limit N\to\infty, with variance given in terms of classical baker's map correlations. This determines the precise rate of convergence in the quantum ergodic theorem for these eigenbases. The presence of the classical correlations highlights that these eigenstates, while random, have microscopic correlations that differentiate them from Haar random vectors. For the single value D=4, the Gaussianity of the matrix element fluctuations depends on the values of the classical observable on a fractal subset of the torus.

Kakeya sets are compact subsets of $\mathbb{R}^n$ that contain a unit line segment pointing in every direction and the Kakeya conjecture states that such sets must have Hausdorff dimension $n$. The property of stickiness was first discovered by by Katz-Laba-Tao in their 1999 breakthrough paper on the Kakeya problem. Then Wang-Zahl formalized the definition of a sticky Kakeya set as a subclass of general Kakeya sets in 2022. Sticky Kakeya sets played an important role as Wang and Zahl solved the Kakeya conjecture for  $\mathbb{R}^3$ in a major recent development.
The planebrush method is a geometric argument by Katz-Zahl which gives the current best bound of 3.059 for Hausdorff dimension of Kakeya sets in $\mathbb{R}^4$. Our new result shows that sticky Kakeya sets in $\mathbb{R}^4$ have dimension 3.25. The planebrush argument when combined with the sticky hypothesis gives us this better bound. 

Many real life problems rely on understanding the solutions to a system of polynomial equations. In this talk, I will outline some of these applications and how tools from algebraic geometry can provide answers to relevant engineering questions.