Abstract: Ehrhart Theory is about understanding how the number of lattice points of a rational polytope changes under dilation. This information is commonly captured in two ways: the Ehrhart (quasi)-polynomial, and the Ehrhart series. This project begins by understanding how we can compute these objects in Macaulay2 with: Polyhedra, Normaliz, or other packages and libraries. We will then consider two (if time permits) generalisations of Ehrhart theory: weighted and equivariant.

Weighted Ehrhart theory counts each lattice point with multiplicity governed by some fixed (quasi)-polynomial. The first goal is to produce effective code for computing weighted volumes, quasi-polynomials, and series in this setting. Another topical generalisation is equivariant Ehrhart theory in which the polytope has a distinguished set of symmetries. The second goal is to develop code for computing the equivariant analogue of the Ehrhart series and equivariant volumes for polytopes that are invariant under cyclic groups.

Abstract: This project will develop homological algebra tools for working over the exterior algebra. Since this standard-graded skew-commutative algebra is injective as a module over itself, there are effective algorithms for constructing injective resolutions of modules and complexes over the exterior algebra. We will start by implementing these methods in Macaulay2. Additionally, we plan to produce counterparts to the Koszul complex (also known as the Priddy complex), horseshoe resolutions, connecting homomorphisms, and maps extending along quasi-isomorphism. The overarching goal is to create injective analogues of the existing methods in Macaulay2 for working with free complexes over a polynomial ring. Time permitting, we would like to enhance routines (especially functoriality) related to the Bernstein-Gelfand-Gelfand correspondence.

Abstract: We will implement variants of the "elimination template" technique, popularized in the computer vision literature, for computing complex zeros within families of zero-dimensional polynomial ideals. These solvers are constructed in an offline stage by modular Groebner basis methods, then run in online mode by row-reducing a certain submatrix of the classical Macaulay matrix (the "template") and extracting the eigenvectors of an appropriate multiplication matrix. After implementing a basic version, we will test additional strategies for constructing better (=smaller) templates, such as those involving syzygies, saturation, and heuristics for multiplication matrix bases.

Suggested reading: Efficient Solvers for Minimal Problems by Syzygy-based Reduction

Abstract: In this project, we will develop interval-based data types in Macaulay2. Interval-based computations are used in certified computations to study the behavior of a function over a region, and for approximate coefficients when a constant only known up to some error.

Macaulay2 includes an implementation of basic real intervals (RRi). The main goals of this project are to extend the existing implementation of real intervals to complex intervals and to introduce implementations of higher dimensional interval arithmetic (in both real and complex spaces), called ball arithmetic.

All of these steps will require work at all levels of Macaulay2, with a heavy emphasis on development in the back-end of the system. The original introduction of basic real intervals took a considerable amount of time and work to complete. It is expected that, even with this template to follow, the proposed extensions will also take a considerable amount of time and code to complete.

Abstract: The A1-Brouwer degree refines the classical topological Brouwer degree, taking values in the Grothendieck-Witt ring of non-degenerate symmetric bilinear forms over a field. This refinement not only preserves the classical Brouwer degree as the rank of the bilinear form, but also encodes additional arithmetic information specific to the field. A1-Brouwer degrees have gained prominence in the rapidly growing field of A1-enumerative geometry, which seeks to establish a framework for enumerative geometry over fields that are not necessarily algebraically closed. Recent work by Brazelton, McKean, and Pauli provided explicit algebraic formulae for computing A1-degrees, which were subsequently incorporated into a Macaulay2 package by Borisov et al. This project aims to extend the functionality of this package by implementing traces for non-rational points, unstable local A1-degrees, and additional tools for computation in A1-homotopy theory, including but not limited to motivic residual intersection formulae, motivic Euler characteristics, and tropical methods for A1-enumerative geometry, depending on participant interest.

Abstract: In equivariant algebraic topology, Mackey functors play the role of abelian groups. Computations with Mackey functors have been central in resolving various problems in algebraic topology, including the Kervaire invariant one problem. In non-equivariant topology, computations are aided by the use of software, e.g. for spectral sequence computations. In equivariant algebraic topology, no such software exists, and current computations are all done by hand.

The goal of this project is to encode the theory of Mackey functors over a cyclic group of prime (power) order, and subsequently to develop methods to compute free resolutions, Ext, Tor, and box products for such Mackey functors. The proposed software would dramatically improve the computational capabilities of mathematicians working in equivariant homotopy theory, representation theory, and equivariant algebra more broadly.