Activities This Week

A union closed family F is a family of sets, so that for any two sets A,B in F, A$\cup$B is also on F. Frankl conjectured in 1979 that for any union-closed family F of subsets of [n], there is some element i $\in$ [n] that appears in at least half the members of F.

The hydrogen atom system is one of the most thoroughly studied examples of a quantum mechanical system. It can be fully solved, and the main reason why is its (hidden) symmetry. In this talk I shall explain how the symmetries of the Schrödinger equation for the hydrogen atom, both visible and hidden, give rise to an example in the recently developed theory of algebraic families of Harish-Chandra modules. I will show how the algebraic structure of these symmetries completely determines the spectrum of the Schrödinger operator and sheds new light on the quantum nature of the system. No prior knowledge on quantum mechanics or representation theory will be assumed.

Tensor categories are abelian k-linear monoidal categories modeled on the representation categories of affine (super)group schemes over k. Deligne gave very succinct intrinsic criteria for a tensor category to be equivalent to such a representation category, over fields k of characteristic zero. These descriptions are known to fail badly in prime characteristics. In this talk, I will present analogues in prime characteristic of these intrinsic criteria. Time permitting, I will comment on the link with a recent conjecture of V. Ostrik which aims to extend Deligne’s work in a different direction.

Let $\{p_j\}_{j=1}^\infty$ denote the set of prime numbers in increasing order, let $\Omega_N\subset \mathbb{N}$ denote the set of positive integers with no prime factor larger than $p_N$ and let $P_N$ denote the probability measure on $\Omega_N$ which gives to each $n\in\Omega_N$ a probability proportional to $\frac{1}{n}$. This measure is in fact the distribution of the random integer $I_N\in\Omega_N$ defined by $I_N=\prod_{j=1}^Np_j^{X_{p_j}}$, where $\{X_{p_j}\}_{j=1}^\infty$ are independent random variables and $X_{p_j}$ is distributed as Geom$(1-\frac{1}{p_j})$. We show that $\frac{\log n}{\log N}$ under $P_N$ converges weakly to the Dickman distribution. As a corollary, we recover a classical result from classical multiplicative number theory—Mertens’ formula, which states that $\sum_{n\in\Omega_N}\frac{1}{n}\sim e^\gamma\log N$ as $N\to\infty$.

Let $D_{\text{nat}}(A)$ denote the natural density of $A\subset\mathbb{N}$, if it exists, and let $D_{\text{log-indep}}(A)=\lim_{N\to\infty}P_N(A\cap\Omega_N)$ denote the density of $A$ arising from $\{P_N\}_{N=1}^\infty$, if it exists. We show that the two densities coincide on a natural algebra of subsets of $\mathbb{N}$. We also show that they do not agree on the sets of $n^\frac{1}{s}$- smooth numbers $\{n\in\mathbb{N}: p^+(n)\le n^\frac{1}{s}\}$, $s>1$, where $p^+(n)$ is the largest prime divisor of $n$. This last consideration concerns distributions involving the Dickman function. We also consider the sets of $n^\frac{1}{s}$- rough numbers ${n\in\mathbb{N}:p^-(n)\ge n^{\frac{1}{s}}}$, $s>1$, where $p^-(n)$ is the smallest prime divisor of $n$. We show that the probabilities of these sets, under the uniform distribution on $[N]={1,\ldots, N}$ and under the $P_N$-distribution on $\Omega_N$, have the same asymptotic decay profile as functions of $s$, although their rates are necessarily different. This profile involves the Buchstab function. We also prove a new representation for the Buchstab function.