publications

2026

1. Exact Volumes of Semi-Algebraic Convex Bodies

   Lakshmi Ramesh, and Nicolas Weiss

   2026

   Abs arXiv Code

   We compute the volumes of convex bodies that are given by inequalities of concave polynomials. These volumes are found to arbitrary precision thanks to the representation of periods by linear differential equations. Our approach rests on work of Lairez, Mezzarobba, and Safey El Din. We present a novel method to identify the relevant critical values. Convexity allows us to reduce the required number of creative telescoping steps by an exponential factor. We provide an implementation based on the ore_algebra package in SageMath. This is applied to a problem in geometric statistics, where the convex body is an intersection of \ell_p-balls.

2. Complexity of Equal 0-Surgeries

   Tetsuya Abe, Marc Kegel, and Nicolas Weiss

   New York J. Math., 2026

   Abs arXiv

   We say that two knots are friends if they share the same 0-surgery. Two friends with different sliceness status would provide a counterexample to the 4-dimensional smooth Poincare conjecture. Here we create a census of all friends with small crossing numbers c and tetrahedral complexities t, and compute their smooth 4-genera. In particular, we compute the minimum of c(K)+c(K’) and of t(K)+t(K’) among all friends K and K’. Along the way, we classify all 0-surgeries of prime knots of at most 15 crossings. Moreover, we determine for many friends in our census if their traces are equivalent or not. For that, we develop a new obstruction for two traces being homeomorphic coming from symmetry-exceptional slopes of hyperbolic knots. This is enough to also determine the minimum value of c(K)+c(K’) among all friends K and K’ whose traces are not homeomorphic.

2025

1. Connection Matrices in Macaulay2

   Paul Görlach, Joris Koefler, Anna-Laura Sattelberger, Mahrud Sayrafi, Hendrik Schroeder, Nicolas Weiss, and Francesca Zaffalon

   2025

   Abs arXiv

   In this article, we describe the theoretical foundations of the Macaulay2 package ConnectionMatrices and explain how to use it. For a left ideal in the Weyl algebra that is of finite holonomic rank, we implement the computation of the encoded system of linear PDEs in connection form with respect to an elimination term order that depends on a chosen positive weight vector. We also implement the gauge transformation for carrying out a change of basis over the field of rational functions. We demonstrate all implemented algorithms with examples.