This project will be organized by Steven Sam. In 2016, Mark Walker proved the weak Buchsbaum-Eisenbud-Horrocks Conjecture by incorporating Adams operations on chain complexes [1]. This proof underscored the significance of allowing functorial operations on chain complexes, and it led to the development of the Macaulay2 package ChainComplexOperations by David Eisenbud, which implemented functions for computing \(\bigwedge^2\) and \(\text{Sym}_2\) of chain complexes. This project will significantly expand this line of thinking by allowing one to apply a Schur functors \(\textbf{S}_{\lambda}\) to a chain complex. Beyond the connection with Walker's groundbreaking work, these operations are fundamental in homological commutative algebra. This will further strengthen Macaulay2's already impressive computational tools in homological commutative aglebra.

References

1. Mark Walker, Total Betti numbers of modules of finite projective dimension, Ann. of Math. (2) 186 (2017), no. 2, 641-646. MR3702675 doi:10.4007/annals.2017.186.2.6

This project will be organized by John Wiltshire-Gordon . FI-modules have become objects of interest in recent years, with applications to representation stability (see [1,2,3,4,5] to name a few recent publications). This project aims to produce the first algorithms for computing with FI-modules. An FI-module may be described by an FI-matrix over the cateogry \(\text{FI}\). This project will implement two basic operations on FI-matrices: finding generators for kernels, and solving the equation \(AX=B\). These operations can be used to compute \(\text{Tor}\) for FI-modules, and also Hilbert series, and those will be implemented as well. The algorithm over \(\mathbb Q\) relies on the quiver computation of Sam-Snowden, and the algorithm over \(\mathbb Z\) relies on the Castelnuovo-Mumford regularity result of Church-Ellenberg [1]. This package will allow researchers to begin experimenting with FI-modules, with the hope of finding further generalizations of results like Church-Ellenberg's regularity result.

References

1. Thomas Church and Jordan S. Ellenberg, Homology of FI-modules, Geom. Topol. 21 (2017), no. 4, 2373–2418. MR3654111 doi:10.2140/gt.2017.21.2373.
2. Thomas Church, Jordan S. Ellenberg, and Benson Farb, FI-modules and stability for representations of symmetric groups, Duke Math. J. 164 (2015), no. 9, 1833–1910. MR3357185 doi:10.1215/00127094-3120274.
3. Thomas Church, Jordan S. Ellenberg, Benson Farb, and Rohit Nagpal, FI-modules over Noetherian rings, Geom. Topol. 18 (2014), no. 5, 2951–2984. MR3285226 doi:10.2140/gt.2014.18.2951.
4. Rita Jiménez Rolland, The cohomology of \(\mathcal{M}_{0,n}\) as an FI-module, Configuration spaces, Springer INdAM Ser., vol. 14, Springer, [Cham], 2016, pp. 313–323. MR3615738 doi:10.1007/978.3.319.31580.
5. Jennifer C. H. Wilson, FIW-modules and constraints on classical Weyl group characters, Math. Z. 281 (2015), no. 1-2, 1–42. MR3384861 doi:10.1007/s00209.015.1473.0.

This project will develop code for performing computations related to algebraic Kakeya problems over finite fields, including: sum-sets, slice rank, and the degeneration techniques of Ellenberg-Erman.

This project will be organized by Lily Silverstein, Dane Wilburne, and Jay Yang. Two separate groups produced recent work on models for random monomial ideals [1,2]. A current collaborative effort of these two groups aims to better understand critical region phenomena between the dimension jumps in the model of [1]. Early experiments suggest that, in the limit, the ideals in these critical regions behave like analogues of fractals, and that we should think of limiting dimension as being somewhere strictly between two integers. To better understand this phenomena, and to open the door for further experimentation of other groups of researchers, this project will develop a package around the random monomial ideals of [1]. The basic functionality has already been implemented, so this package will be primarily interested in producing a framework for experimenting with these ideals.

References

1. Jesús A. De Loera, Sonja Petrovíc, Lily Silverstein, Despina Stasi, and Dane Wilburne, Random Monomial Ideals. arXiv:1701.07130.
2. Daniel Erman and Jay Yang, Random flag complexes and asymptotic syzygies. arXiv:1706.01488.

The Points package is an existing M2 package routines useful for studying points in affine and projective spaces. The goal of this project is to extend functionality to the Points package in two different directions. The first involves extending the currents functionalities of the package, especially the creation of random sets of points, to any normal toric variety. The second goal would allow one to specify multiplicities at the various points (e.g. allowing random sets of “fat points”).

This project will develop code for working with virtual resolutions, as defined in the recent paper of Berkesch Zamaere-Erman-Smith. This will also include adding functionality for better displaying multigraded Betti tables.