Réunion d'hiver SMC 2025
Toronto, 5 - 8 decembre 2025
Org: Giulia Gaggero (McMaster University), Mahrud Sayrafi (Fields/McMaster University) et Adam Van Tuyl (McMaster University)
[PDF]
- KIERAN BHASKARA, McMaster University
$h$-polynomials and the GVD property of toric ideals of graphs [PDF]
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Abstract: Geometrically vertex decomposable (GVD) ideals form a class of ideals that generalize the Stanley-Reisner ideals of vertex decomposable simplicial complexes. It has been shown that some special families of toric ideals of graphs are GVD, but a complete characterization remains unknown. At the same time, several studies have investigated the algebraic invariants of toric ideals of graphs. One such invariant, the $h$-polynomial, has been explicitly described for only a small number of graph classes. In this talk, we describe a new family of graphs whose toric ideals are GVD. We then illustrate how the GVD property allows us to compute an explicit formula for the $h$-polynomials of this family. This talk is based on joint work with Higashitani and Shibu Deepthi, and separate joint work with Van Tuyl and Zotine.
- DENYS BULAVKA, Dalhousie University
A Hilton-Milner theorem for exterior algebras [PDF]
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A set family F is pairwise-intersecting if every pair of its members intersect. In 1960, Erdős, Ko, and Rado gave an upper-bound on the size of a pairwise-intersecting family of k-sets coming from a ground set of size n. Moreover, they characterized the families achieving the upper-bound. These are families whose members all share exactly one element, so called trivial families. Later, Hilton and Milner provided the next best upper-bound for pairwise-intersecting families that are not trivial.
There are several generalizations of the above results. We will focus on the case when the set family is replaced with a subspace of the exterior algebra. In this scenario intersection is replaced with the wedge product, being pairwise-intersecting with self-annihilating and being trivial with being annihilated by a 1-form. Scott and Wilmer, and Woodroofe gave an upper-bound on the dimension of self-annihilating subspaces of the exterior algebra. In the current work we show that the better upper-bound coming from Hilton and Milner's theorem holds for non-trivial self-annihilating subspaces.
This is a joint work with Francesca Gandini and Russ Woodroofe.
- SUSAN COOPER, University of Manitoba
Decomposing Star Configuration Ideals [PDF]
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Geometrically vertex decomposable ideals have a recursive structure that allows one to study the original ideal by investigating two related ideals, each of which is in one less variable. This underlying structure can be exploited to study important invariants of the ideal. Examples of ideals that are geometrically vertex decomposable include Schubert determinantal ideals and toric ideals of bipartite graphs. In this talk, we determine when the defining ideal of a star configuration in projective space is geometrically vertex decomposable. This is joint work with E. Guardo, E. Marangone, and A. Van Tuyl.
- SARA FARIDI, Dalhousie University
Extremal and D-Extremal ideals and their algebraic properties [PDF]
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Extremal ideals are square-free monomial ideals which model or bound the algebraic properties of all square-free monomial ideals and their powers. A refinement of this concept is the notion of D-extremal ideals, which also takes into account the divisibility relations between the generators of the ideals. This talk will be an overview of of what extremal ideals and what kinds of questions can be answered by them.
- SELVI KARA, Bryn Mawr College
Algebraic Study of Polarized Neural Ideals [PDF]
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Neural codes, collections of binary strings representing the firing patterns of neurons, can be studied algebraically through their associated neural ideals. These ideals encode receptive field relationships among neurons in a stimulus space. However, neural ideals are not necessarily monomial (not even homogeneous) and do not yield standard graded invariants. We study their polarization, yielding squarefree monomial ideals that preserve combinatorial structure. Focusing on these polar neural ideals, we compute and bound their graded Betti numbers, projective dimensions, and regularity. In particular, we prove an upper bound on projective dimension for polar neural ideals on n neurons and classify when this bound is attained. This is joint work with Ellie Lew.
- GRAHAM KEIPER, The University of Catania
Symbolic Powers of Toric Ideals [PDF]
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This talk will discuss some recent joint work with Giuseppe Favacchio relating to symbolic powers of toric ideals. We will go over the necessary background on toric ideals as well as symbolic powers. We will then discuss two new results useful in the computation of symbolic powers of toric ideals. The first result will involve a novel method of computing the symbolic powers of toric ideals via tensors and the second result will show how the symbolic powers of toric ideals can be computed in terms of a particular saturation.
- IRESHA MADDUWE, Dalhousie University
Reconstruction Conjecture on Homological Invariants of Cameron-Walker Graphs [PDF]
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We show that key homological invariants of edge ideals of Cameron Walker graphs, such as regularity, and depth can be reconstructed from its vertex deleted subgraphs. In addition, we discuss the reconstruction of the top-degree Betti numbers of these edge ideals. Furthermore, we speak on the reconstruction of the lattice points of the edge ideals of Cameron Walker graphs such as $(\text{reg}(R/I),\text{deg h}(R/I))$ and $(\text{depth}(R/I),\text{dim}(R/I))$ using the lattice points of their vertex-deleted subgraphs.
- HASAN MAHMOOD, Dalhousie University
Simplicial Resolutions of Powers of Monomial Ideals [PDF]
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Given a monomial ideal \( I \) minimally generated by \( q \) monomials, we construct a simplicial complex \( \mathbb{M}_q^2 \) that supports a free resolution of the square \( I^2 \). We also define a natural subcomplex \( \mathbb{M}^2(I) \), depending on the specific generators of \( I \), that likewise supports a resolution of \( I^2 \). Our framework yields new bounds on the projective dimension of second powers of monomial ideals and provides sharper bounds for the Betti numbers of \( I^2 \) compared to those obtained from the Taylor resolution.
Furthermore, we introduce {permutation ideal} \( \mathcal{T}_q \), generated by \( q \) monomials, and prove that for any monomial ideal \( I \) with \( q \) generators, \( \beta(I^2) \;\leq\; \beta(\mathcal{T}_q^2). \) We also show that the simplicial complex \( \mathbb{M}_q^2 \) supports the {minimal} resolution of \( \mathcal{T}_q^2 \), and that \( \mathbb{M}_q^2 \) in fact coincides with the Scarf complex of \( \mathcal{T}_q^2 \). Finally, we present analogous constructions and results for third and higher powers of general monomial ideals. This is joint work with {Susan Cooper} and {Sara Faridi}, with additional related work in progress with {Sara Faridi} and {Chau Trung}.
- EMANUELA MARANGONE, University of Manitoba
Weighted Veronese Rings [PDF]
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For a standard-graded polynomial ring R, the d-Veronese subring is generated as a k-algebra by degree d monomials, is Koszul, and its defining ideal is quadratic, binomial, and determinantal. In this talk, I will discuss what happens when we start instead with a non-standard graded polynomial ring.
In joint work with A. Seceleanu, L. Fiorindo, B. Chase, T. de Holleben, S. Singh, T. Nguyen, S. Bisui, we show that in the two-variable case, these weighted Veronese rings preserve many of these properties: they are Cohen–Macaulay, Koszul, and have a determinantal presentation. Moreover, their Hilbert function and Betti numbers depend only on the number and degrees of the generators. In contrast, in three or more variables, these properties no longer hold in general.
- GREGORY G. SMITH, Queen's University
Cellular free resolutions for normalizations of toric ideals [PDF]
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The theory of cellular resolutions provides a concrete way to construct free resolutions of monomial ideals by relating them to cell complexes. While Bayer–Sturmfels formulate an analogue for binomial ideals, this theory is less fully developed. In this talk, we will expand the framework for cellular free resolutions and give explicit free resolutions for the normalization of a toric ideal. This is based on joint work with Christine Berkesch, Lauren Cranton Heller, and Jay Yang.
- JANET VASSILEV, University of New Mexico
Patterns in differential powers of ideals in affine semigroup rings [PDF]
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Let $R$ be a two-dimensional normal affine semigroup ring, one of whose facets lies on the $x$-axis, the other lying within the first quadrant. We will discuss how the slope of the second facet determines patterns for the differential powers of ideals whose radical is the canonical ideal.
- DHARM VEER, Dalhousie University
Binomial ideals associated to polycubes. [PDF]
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A unit cube in $\mathbb{R}^3$ is a set of the form $\{(x,y,z)\in\mathbb{R}^3 : a \le x \le a+1,\ b \le y \le b+1,\ c \le z \le c+1\}$, where $(a,b,c)\in\mathbb{N}^3$.
In this talk, we associate a binomial ideal to a collection of unit cubes. We discuss the algebraic invariants of this ideal when the collection of cells forms a polycube. For a certain class of polycubes, we prove that the associated quotient ring is Koszul, and we characterize when this quotient ring is Cohen–Macaulay by studying its initial ideal.
- JAY YANG, Vanderbilt University
Controlling Homology in Virtual Resolutions of Monomial Ideals [PDF]
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Virtual Resolutions are certain complexes of free modules over the Cox ring of a toric variety. These complexes are exactly those that sheafify to a free resolution of sheaves. However, a virtual resolution may have homology in higher homological degrees. In recent work, I describe a construction of labeled cell complexes describing virtual resolutions with higher homology and how a simple subdivision operation on the cell complex can often yield a virtual resolution of the same module without higher homology.
- SHAH RASHAN ZAMIR, Tulane University
On the algebraic properties of the Böröczky configuration [PDF]
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The Böröczky configuration of lines and (multiple) points exhibits extremal behavior in commutative algebra and combinatorics. Examples of this appear in the context of the containment problem for ordinary and symbolic powers of ideals and the proof of the Dirac-Motzkin conjecture by Green and Tao. This paper studies the algebraic properties of Böröczky configurations of arbitrary size. Our results compute the Waldschmit constant of the defining ideal of these configurations. Moreover, we use the weighted projective plane P(1,2,3) to give an upper bound for the degree of the minimal generators of their ideal. Finally, this construction is applied to an elliptic curve in the projective plane to give a new counterexample to the aforementioned containment problem.