My name is Jacob Zoromski, and I am a PhD candidate in Mathematics at the University of Notre Dame, expected to graduate May 2025. I study commutative algebra and algebraic geometry, and I have a special interest in using representation theory, combinatorics, and computation to study free resolutions. My advisor is Claudiu Raicu.

email: jzoromsk@nd.edu

Publications:

• MultiRegeneration for Polynomial System Solving with Colin Crowley, Jose Israel Rodriguez, and Jacob Weiker (link).

  We demonstrate our implementation of a continuation method as described by Hauenstein and Rodriguez for solving polynomials systems. Given a sequence of (multi)homogeneous polynomials, the software multiregeneration outputs the respective (multi)degree in a wide range of cases and partial multidegree in all others. We use Python for the file processing, while Bertini is needed for the continuation. Moreover, parallelization options and several strategies for solving structured polynomial systems are available.

On the arXiv:

• Equivariant Syzygies of the Ideal of 2 x 2 Permanents of a 2 x n Matrix (link).

  We describe the equivariant syzygies of the ideal of $2 \times 2$ permanents of a generic $2 \times n$ matrix under its natural symmetric and torus group actions. Our proof gives us a new method of finding the Betti numbers of this ideal, which were first described by Gesmundo, Huang, Schenck, and Weyman.

• Monomial Cycles in Koszul Homology (link).

  In this paper we study monomial cycles in Koszul homology over a monomial ring. The main result is that a monomial cycle is a boundary precisely when the monomial representing that cycle is contained in an ideal we introduce called the boundary ideal. As a consequence, we obtain necessary ideal-theoretic conditions for a monomial ideal to be Golod. We classify Golod monomial ideals in four variables in terms of these conditions. We further apply these conditions to symmetric monomial ideals, allowing us to classify Golod ideals generated by the permutations of one monomial. Lastly, we show that a class of ideals with linear quotients admit a basis for Koszul homology consisting of monomial cycles. This class includes the famous case of stable monomial ideals as well as new cases, such as symmetric shifted ideals.

Talks:

• Invited Speaker – AMS Spring Central Sectional Meeting Special Session on “Algebraic and Homological Perspectives with a view towards Geometry”, University of Kansas, 3/29/2025
• "Monomial Cycles in Koszul Homology". Poster presentation, ALGECOM XXIV, University of Michigan, 11/09/2024, Poster can be found here
• "A History of Calculus for Instructors". Graduate Student Seminar, University of Notre Dame, 11/27/2023, Slides can be found here
• "Ideals With Linear Quotients". Graduate Student Algebra Seminar, University of Notre Dame, 4/24/2023
• "The History of Calculus". Graduate Student Seminar, University of Notre Dame, 1/24/2023
• "Infinite Minimal Free Resolutions". Graduate Student Seminar, University of Notre Dame, 10/3/2022
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