We are happy to present Ruiwen Dong’s talk titled “Identity Problem for Unitriangular Matrices of Dimension Four”.

The Identity Problem in semigroup algorithmic theory asks the
following: Given as input a set of square matrices $G = {A_1, A_2, …,
A_k}$, does the semigroup generated by G contain the identity
matrix $I$? The Identity Problem has been shown to be undecidable even
when restricted to matrices of low dimensions. For example, Bell et
al. showed its undecidability for matrices in $SL(4, Z)$. Some
decidability results for the Identity Problem include its NP-completeness
for matrices in $SL(2, Z)$, as well as its PTIME decidability in the
Heisenberg group $H_3$. In this talk, we show a decidability result of the
Identity Problem for matrices in the group $UT(4, Z)$ of unitriangular
integer matrices of dimension four. Some of the techniques used for this
result may be generalized to tackle higher dimensional unipotent
groups.

### Bio

Ruiwen Dong is currently a PhD student in Computer Science at the
University of Oxford. He obtained his MsC from Ecole Polytechnique,
France, and his BsC from Peking University, China. His main areas
of interests include semigroup algorithmic problems, rings and algebra, as
well as symbolic computing.