Abstract: 

This talk consists of two parts. 

First, we revisit the combinatorial formulation of the Ekeland-Hofer-Zehnder (EHZ) capacity for simplices due to Haim-Kislev, highlighting its trace/max-structure and its connection to maximum acyclic subgraph and feedback-arc-set problems. 

Second, we develop a nonsmooth Riemannian optimization framework on $SL(2n,\mathbb{R})$ that operates directly on simplices: Clarke subgradients of the max-trace objective are projected to the tangent space $T_X SL(2n,\mathbb{R})$ and updated via retractions, enabling efficient line-search methods despite the nonsmooth structure. 

We present numerical results for simplices up to dimension $14$, including examples in $\mathbb{R}^6$ with higher systolic ratios than previously known and families outperforming the standard simplex for $2n \le 12$.