Mathematics Colloquia and Seminars

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Description

I will present two projects connected by a common theme: identifying algebraic and combinatorial structure that makes difficult high-dimensional problems explicit and computable. The first project develops a chamber-based framework for computing volumes and higher moments of slices and slabs of polyhedral norm balls, with a particular focus on cubes. Using combinatorial classification, triangulations, and polynomial integration over simplices via Brion's formula, the project obtains chamber-wise rational formulas and gives a polynomial-time algorithm in fixed dimension. Computationally, this work recovers known formulas in dimensions $2$ and $3$, and gives a complete classification of the volume formulas for slices and slabs of the $4$-cube, together with higher-moment computations in dimensions $2$, $3$, and $4$. The second project studies symmetric overcomplete tensor decomposition for generic order-$3$ tensors through the Koszul-Young flattening method. The main result gives rank bounds under which the flattening detects the tensor rank and enables efficient decomposition. This tensor perspective also motivates future work on heterogeneous multi-reference alignment, with potential applications to cryo-EM. Together, these projects illustrate my broader research goal of using algebraic, geometric, and combinatorial methods to obtain rigorous and computable results in applied mathematics.