# Activities This Week

## Colloquium

### Symmetries of the hydrogen atom and algebraic families

*Dec 18, 14:30-15:30, 2018*,
**Math -101**

#### Speaker

**Eyal Subag** (*Penn State*)

#### Abstract

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.

## Combinatorics Seminar

### TBA

*Dec 18, 15:45-16:45, 2018*,
**201**

#### Speaker

**Ilan Karpas**

## AGNT

### Tensor categories in positive characteristic

*Dec 19, 15:10-16:25, 2018*,
**-101**

#### Speaker

**Kevin Coulembier** (*University of Sydney*)

#### Abstract

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.

## BGU Probability and Ergodic Theory (PET) seminar

### A Natural probabilistic model on the integers and its relation to Dickman-type distributions and Buchstab’s function

*Dec 20, 11:00-12:00, 2018*,
**-101**

#### Speaker

**Ross Pinsky** (*Technion*)

#### Abstract

Let denote the set of prime numbers in increasing order, let denote the set of positive integers with no prime factor larger than and
let denote the probability measure on which gives to each a probability proportional to .
This measure is in fact the distribution of the random integer defined by , where
are independent random variables and is distributed as Geom.
We show that under converges weakly to the *Dickman distribution*. As a corollary, we recover a classical result from classical multiplicative number theory—*Mertens’
formula*, which states that as .

Let $D_{\text{nat}}(A)$ denote the natural density of $A\subset\mathbb{N}$, if it exists, and let denote the
density of $A$ arising from , 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 - *smooth numbers* , $s>1$, where 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.