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► Monotonicity of Minkowski Averages
Nathaniel Kenschaft, Zubin Narayan

Advisors: Martin Rapaport and Tomasz Tkocz
Abstract: This project explored two interconnected problems in convex geometry and analysis. First, the students analyzed a new proof of the classical Borell--Brascamp--Lieb inequality, a fundamental result that generalizes the Brunn--Minkowski and Prékopa--Leindler inequalities. Second, they investigated a conjecture concerning the monotonicity of Minkowski averages, which is known to hold in dimension one and to fail in sufficiently high dimensions, but remains unresolved in dimensions $2 \leq n \leq 11$.

The students examined structural properties of iterated Minkowski sums, explored functional analogues of the conjecture, and studied related geometric quantities such as the Hausdorff distance and the Schneider convexity index.

In addition, they implemented a computational model to approximate iterated Minkowski sums and visualize their growth directly in dimension~2, and investigated approaches to the conjecture via discretizations on $\mathbb{Z}^2$ as a potential direction to eventually prove the conjecture. Finally, they wrote a short report documenting their progress, methods, and ideas for further work on this open problem.

► New Interpretations of $q$-Ballot Numbers
Alex Wittig, Will Schremmer, Corey Predella

Advisor: Irina Gheorghiciuc
Abstract: A Ballot path is a finite lattice path in the $xy$-plane that starts at the origin, consists of diagonal up steps $(1,1)$ and diagonal down steps $(1,-1)$, and never goes below the $x$-axis. The Ballot number $B(a,b)$ is the number of Ballot paths that end in the point $(a+b,a-b)$. In this project we considered two different polynomial generalizations of the Ballot number, $B_q(n,r)$ and $B^{(2)}_q(n,r)$, both of which matter in the combinatorial interpretation of the Super Catalan polynomial.

Symmetric Lattice of Dyck Paths (length 6)
Smaller area paths are on top

We found a new interpretation of $B_q(n,r)$ in therms of the inversion statistics $inv(w)$, and a less elegant interpretation of $B^{(2)}_q(n,r)$ in terms of the major $maj(w)$ and $des(w)$ statistics.