Abstracts

Week of April 28, 2024

April 29, 2024
Group Actions, Geometry and Dynamics No seminar 4:00pm -
May 1, 2024
Applied Mathematics Moore machines duality 1:00pm -
LOM 214
Let q ≥ 2 be an integer. A Moore machine M is the data of
(1) two finite sets A (the set of states) and B (the output alphabet),
(2) an element i ∈ A (the initial state),
(3) a mapping h from {1, 2, … , q} to B
(4) a mapping from A×{1, 2, … , q} to A (the transition function).
The last mapping can be viewed as follows: for each a ∈ A there are q
arrows labeled 1, 2,… , q stemming from a and pointing to some state.
Then each word w on the alphabet {1, 2, … , q} defines a path starting
from the initial state i and ending at some state i · w. So, feeding the
machine M with w gives the output h(i · w).
We define the dual of a Moore machine. It appears that the bidual
of a machine M is equivalent to M and is minimal in that sense that
it has the least number of states among the equivalent machines.
This gives a new proof of the existence and uniqueness of the minimal
machine and provides an algorithm to construct it.
May 2, 2024
Special Guest Lecture Strong-weak symmetry and quantum modularity of resurgent topological strings 2:00pm -
KT 801

Quantizing the mirror curve to a toric Calabi-Yau threefold gives rise to quantum operators whose fermionic spectral traces produce factorially divergent series in the Planck constant and its inverse. These are captured by the Nekrasov-Shatashvili and standard topological strings via the TS/ST correspondence. In this talk, I will discuss the resurgence of these dual asymptotic series and present an exact solution for the spectral trace of local P^2. A full-fledged strong-weak symmetry is at play, exchanging the perturbative/nonperturbative contributions to the holomorphic and anti-holomorphic blocks in the factorization of the spectral trace. This relies on a network of relations connecting the dual regimes and building upon the analytic properties of the L-functions with coefficients given by the Stokes constants and the q-series acting as their generating functions. Finally, I will mention how these results fit into a broader paradigm linking resurgence and quantum modularity. This talk is based on arXiv:2212.10606, 2404.10695, and 2404.11550.