**Combinatorics and Probability of Quantum Oscillations **

Dr. Alexander Moll

(University of Massachusetts Boston)

**Abstract:
**The discovery of ``quantization'' and ``uncertainty'' at the quantum scale prompted many new research directions in mathematics. In my research, I use probability theory to model these two phenomena in dynamical systems. For many systems, this modeling leads naturally to probabilistic methods in enumerative combinatorics. In this talk, I will illustrate the concepts of ``quantization'' and ``uncertainty'' in the example of the harmonic oscillator. After quantization, we will see that the number distribution of this system in a coherent state is a Poisson distribution. This probabilistic result will allow us to calculate Stirling numbers of the second kind S(n,k) which count the number of ways to partition the set {1,2,..., n} into k non-empty subsets. In this framework, our ability to compute S(n,k) recursively can be traced to the fact that the oscillatory solutions of Newton's equation x''(t) = - ω^2 x(t) do not exhibit chaos. At the end of the talk, I will outline recent results for oscillations of quantum nonlinear waves and applications to combinatorics and probability that suggest much to be explored.

**Biographical Sketch:**

Alexander Moll is a mathematician working in probability theory and mathematical physics. His research focuses on the interplay between randomness, geometry, and dynamics in the theory of quantum solitons. After growing up in New Orleans, Louisiana, he eventually earned his Ph.D. from M.I.T. with Alexei Borodin, became the first Robert T. Seeley Visiting Assistant Professor at UMass Boston, and will join Reed College as an Assistant Professor in Fall.

**Date and time: **

Saturday, April 30, 2022

12:45 PM—2:30 PM

**Zoom Meeting Link:**

https://umassboston.zoom.us/j/95930920889