Speaker
Andrea Rosana, MPI MiS Leipzig
Abstract
The degree of an algebraic variety is one of its fundamental algebraic invariants. At first sight, the volume seems instead to be of a rather different nature. In the first part of the talk we show that these two notions coincide when an appropriate metric is chosen. We explain how this result can be used for degree computations and discuss the advantages, challenges and limitations of this approach compared with algebraic and numerical ones. In the second part, we apply this technique to the case of Tensor Train varieties. Via a recursive application of the coarea formula, we obtain a combinatorial expression for their degrees. Finally, we show that our algorithmic implementation of this strategy outperforms existing symbolic and numerical methods.
This is joint work with Otto T.P. Schmidt (https://arxiv.org/abs/2606.11847).