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We introduce IRIS, a geometric and heuristic-based scoring system for evaluating mathematical conjectures and theorems expressed as linear inequalities over numerical invariants. The IRIS score reflects multiple dimensions of significance—including sharpness, diversity, difficulty, and novelty—and enables the principled ranking of conjectures by their structural importance. As a tool for fully automated discovery, IRIS supports the generation and prioritization of high-value conjectures. We demonstrate its utility through case studies in convex geometry and graph theory, showing that IRIS can assist in both rediscovery of known results and proposal of novel, nontrivial conjectures. 

We study the structure of the set of all possible affine hyperplane sections of a convex polytope. We present two different cell decompositions of this set, induced by hyperplane arrangements. Using our decomposition, we bound the number of possible combinatorial types of sections and craft algorithms that compute optimal sections of the polytope according to various combinatorial and metric criteria, including sections that maximize the number of -dimensional faces, maximize the volume, and maximize the integral of a polynomial. Our optimization algorithms run in polynomial time in fixed dimension, but the same problems show computational complexity hardness otherwise. Our tools can be extended to intersection with halfspaces and projections onto hyperplanes. Finally, we present several experiments illustrating our theorems and algorithms on famous polytopes. 

Light microscopy methods have continued to advance allowing for unprecedented analysis of various cell types in tissues including the brain. Although the functional state of some cell types such as microglia can be determined by morphometric analysis, techniques to perform robust, quick, and accurate measurements have not kept pace with the amount of imaging data that can now be generated. Most of these image segmentation tools are further burdened by an inability to assess structures in three-dimensions. Despite the rise of machine learning techniques, the nature of some biological structures prevents the training of several current day implementations. Here we present PrestoCell, a novel use of persistence-based clustering to segment cells in light microscopy images, as a customized Python-based tool that leverages the free multidimensional image viewer Napari. In evaluating and comparing PrestoCell to several existing tools, including 3DMorph, Omipose, and Imaris, we demonstrate that PrestoCell produces image segmentations that rival these solutions. In particular, our use of cell nuclei information resulted in the ability to correctly segment individual cells that were interacting with one another to increase accuracy. These benefits are in addition to the simplified graphically based user refinement of cell masks that does not require expensive commercial software licenses. We further demonstrate that PrestoCell can complete image segmentation in large samples from light sheet microscopy, allowing quantitative analysis of these large datasets. As an open-source program that leverages freely available visualization software, with minimum computer requirements, we believe that PrestoCell can significantly increase the ability of users without data or computer science expertise to perform complex image analysis.