## CHAMPS academic year 2021-2022 speakers

Vinh Nguyen, Purdue University

Lech's Inequality for Generalized MultiplicitiesAbstract: The Hilbert-Samuel multiplicity is an important invariant of m-primary ideals in Noetherian local rings. The multiplicity of parameter ideals, in some sense, measures the singularity of the ring. Singularities are detected based on how much the multiplicity of an ideal differs from its colength. For instance, under mild assumptions, a local ring is Cohen-Macaulay if and only if the colength of a parameter ideal equals its multiplicity. A bound for the multiplicity of an ideal is of interest. Lech proved an upper bound which involves the colength of the ideal. A recent improvement to Lech's bound is due to Huneke, Smirnov, and Validashti. They improved it by giving an analogous bound for the multiplicity of a given ideal times the maximal ideal. In this talk I will give a brief overview of the different generalizations of Hilbert-Samuel multiplicity. I will then talk about Lech type bounds for these generalized multiplicities. This is joint work with Kelsey Walters.

Tim Tribone, Syracuse University

Branched covers and matrix factorizations

Abstract: A matrix factorization of an element \(f\) in a regular local ring consists of a pair of square matrices whose product is \(f\) times an identity matrix of the appropriate size. These objects were introduced by Eisenbud in 1980 to study free resolutions of modules over the hypersurface ring defined by \(f\). In this talk, we will consider a generalization to factorizations by more than two matrices. We will present extensions of two theorems of Knörrer regarding what I will refer to as the \(d\)-fold branched cover of the hypersurface defined by \(f\). Portions of this work are joint with Graham Leuschke.

Alapan Mukhopadhyay, University of Michigan

Frobenius-Poincaré function and Hilbert-Kunz multiplicityAbstract: Given a graded triple \( (M,R,I) \) in a positive characteristic \(p\) and for any complex number \(y\), we show that the limit

*Frobenius-Poincaré function*of \( (M,R,I) \). We establish that Frobenius-Poincaré functions are holomorphic everywhere on the complex plane. We shall discuss properties of Frobenius-Poincare functions, give examples and describe these functions in terms of the sequence of graded Betti numbers of \( \frac{M}{I^{[p^n]}M}\). We conclude by mentioning some questions on the structure and properties on Frobenius-Poincaré functions.

Kevin Harris, University of Texas at Arlington

Decompositions of modules over subalgebras of truncated polynomial rings

Abstract: We investigate how modules decompose over principal subalgebras of certain truncated polynomial rings. In particular, we will investigate how a module de- composition may (or may not) change when we decompose over different principal subalgebras. Varying decompositions are related to the notion of rank varieties. Finally, we will examine how one might extend the notion of rank varieties to more general truncated polynomial rings and investigate the Clebsch-Gordan problem for truncated polynomial rings in one variable.

Thái Thành Nguyễn, Tulane University

Newton-Okounkov bodies, Rees algebra, and analytic spread of graded families of monomial ideals

Abstract: Newton-Okounkov bodies are convex sets associated to algebro-geometric objects, that was first introduced by Okounkov in order to show the log-concavity of the degrees of algebraic varieties. In special cases, Newton-Okounkov bodies associated to graded families of ordinary powers and symbolic powers of a monomial ideal are Newton polyhedron and symbolic polyhedron of the ideal. Studying these polyhedra can be beneficial to the study of relation between ordinary powers, integral closure powers and symbolic powers of a monomial ideal as well as its algebraic invariants. In this talk, I will survey some known results in this subject and present our results on characterizing Noetherian property, computing and bounding the analytic spread of a graded family of monomial ideals and some related invariants through the associated Newton-Okounkov body. This is based on joint work with Tài Huy Hà.

Vaibhav Pandey, University of Utah

Are natural embeddings of determinantal rings split?

Abstract: Over an infinite field, a generic determinantal ring is the fixed subring of an action of the general linear group on a polynomial ring; this is the natural embedding of the title. If the field has characteristic zero, the general linear group is linearly reductive, and it follows that the invariant ring is a split subring of the polynomial ring. We determine if the natural embedding is split in the case of a field of positive characteristic.

Time permitting, we will address the corresponding question for Pfaffian and symmetric determinantal rings. This is ongoing joint work with Mel Hochster, Jack Jeffries, and Anurag Singh.

Amy Huang, Texas A&M

Tensor Ranks and Matrix Multiplication Complexity

Abstract: Tensors are just multi-dimensional arrays. Notions of ranks and border rank abound in the literature. Tensor decompositions also have a lot of application in data analysis, physics, and other areas of science. I will try to give a colloquium-style talk surveying my recent two results about tensor ranks and their application to matrix multiplication complexity. The first result relates different notions of tensor ranks to polynomials of vanishing Hessian. The second one computes the border rank of \(3 \times 3\) permanent. I will also briefly discuss the newest technique we used to achieve our results: border apolarity.

Shuai Wei, Clemson University

Cohen-Macaulay type of weighted edge ideals and \(r\)-path ideals

Abstract: We investigate the Cohen-Macaulay property of several special classes of monomial ideals that are important for graph theory and combinatorics. Then we compute the Cohen-Macaulay type of these ideals combinatorially.

Sara Jones, University of Arkansas

Lyubeznik Numbers of Unmixed Edge Ideals

Abstract: Lyubeznik numbers, defined in terms of local cohomology, are invariants of local rings that are able to detect many algebraic and geometric properties. Notably they recognize topological behaviors of various structures associated to rings. We will discuss computations of these numbers in the case of unmixed edge ideals by giving a completely combinatorial construction which realizes the connectedness information captured by these numbers.

Michael DeBellevue, University of Nebraska — Lincoln

Graded Deviations, Rigidity, and Koszulness

Abstract: The graded deviations \(\varepsilon_{ij}(R)\) of a graded ring \(R\) record the number of algebra generators of a differential graded algebra resolution of the residue field of \(R\). Vanishing of deviations encodes properties of the ring: for example, \(\varepsilon_{ij}(R)= 0\) for \(i \geqslant 3\) if and only if \( R \) is complete intersection and, provided \( R \) is standard graded, \(\varepsilon_{ij}(R)\) whenever \( i \neq j \) implies \( R \) is Koszul. We extend this fact by showing that if \( \varepsilon_{ij}(R)=0 \) whenever \( j, i \geqslant 3\), then \( R \) is a quotient of a Koszul algebra by a regular sequence. This answers a conjecture by Ferraro. The ordinary deviations \( \varepsilon_{i}(R) \) enjoy a rigidity property: if \(\varepsilon_{i}(R)=0\) for \( i \gg 0\), then \(R\) is a complete intersection and \( \varepsilon_{i}(R)=0 \) for \( i \geqslant 3\). We will also discuss recent work for proving the analogous fact in the graded case: that if \( \varepsilon_{ij}(R)=0\) for \(i\) sufficiently large and \( i\neq j\), then \( \varepsilon_{ij}(R)=0 \) for all odd \( i \geqslant 3\) and \(i\neq j\).

Todd Morra, Clemson University

Minimal Differential Graded Algebra Resolutions of Certain Stanley-Reisner Rings

Abstract: Stanley-Reisner rings possess rich algebraic information about the rings encoded as combinatorial information in simplicial complexes. We define certain Stanley-Reisner rings which are Cohen-Macaulay, admit canonical modules, and have an easily computed Cohen-Macaulay type. Based on work by D'Alí, Fløystad, and Hematbakhsh, these Stanley-Reisner rings also give rise to an explicit, multi-graded finite free resolution. In this talk we showcase an explicit differential graded algebra structure for these resolutions.

Matt Weaver, Purdue University

The Equations Defining Rees Algebras of Ideals of Hypersurface Rings

Abstract: The defining equations of Rees algebras provide a natural pathway to study the blowup algebras. However, a minimal generating set of the defining ideal is rarely understood outside of a few classes of ideals. Moreover, most of these results only pertain to ideals of polynomial rings. With this, it is an interesting question as to what the defining equations of the Rees algebra are for an ideal outside of this setting. In this talk we consider ideals of codimension two of hypersurface rings and the equations defining their Rees algebras. By introducing the modified Jacobian dual, we apply a recursive algorithm with this matrix and produce a minimal generating set of the defining ideal.

Yevgeniya Tarasova, Purdue University

Residual Intersections of Determinantal Ideals of \(2\times n\) Matrices

Abstract: In this talk we prove that \(n\)-residual intersections of ideals generated by \(2\times 2\) minors of generic \(2\times n\) matrices can be written as a sum of links.

Francesca Gandini, Kalamazoo College

Noether's Degree Bound in the Exterior Algebra

Abstract: A famous result of Noether states that in characteristic zero the maximal degree of a minimal generating invariant is bounded above by the order of the group. Our work establishes that the same bound holds for invariant skew polynomials in the exterior algebra. Our approach to the problem relies on a theorem of Derksen that connects invariant theory to the study of ideals of subspace arrangements. We adapt this approach to the exterior algebra context and reduce the problem to establishing a bound on the Castelnuovo-Mumford regularity of intersections of linear ideals in the exterior algebra.

Patricia Klein, University of Minnesota

Bumpless pipe dreams encode Gröbner geometry of Schubert polynomials

Abstract: Knutson and Miller established a connection between the anti-diagonal Gröbner degenerations of matrix Schubert varieties and the pre-existing combinatorics of pipe dreams. They used this correspondence to give a geometrically-natural explanation for the appearance of the combinatorially-defined Schubert polynomials as representatives of Schubert classes. Recently, Hamaker, Pechenik, and Weigandt conjectured a similar connection between diagonal degenerations of matrix Schubert varieties and bumpless pipe dreams, newer combinatorial objects introduced by Lam, Lee, and Shimozono. We prove this conjecture in full generality. The proof provides tools for assessing the Cohen--Macaulayness of equidimensional unions of matrix Schubert varieties, of which alternating sign matrix varieties are an important example. This talk is based on joint work with Anna Weigandt.

Justin Chen, ICERM/Georgia Tech

(Differential) Primary decomposition of modules

Abstract: Primary decomposition is an indispensable tool in commutative algebra, both theoretically and computationally in practice. While primary decomposition of ideals is ubiquitous, the case for general modules is less well-known. I will give a comprehensive exposition of primary decomposition for modules, starting with a gentle review of practical symbolic algorithms, leading up to recent developments including differential primary decomposition and numerical primary decomposition. Based on joint works with Yairon Cid-Ruiz, Marc Harkonen, Robert Krone, and Anton Leykin.

Janet Page, University of Michigan

Title: Extremal Singularities in Positive Characteristic

Abstract: What is the most singular possible (reduced) hypersurface in positive characteristic? One answer to this question comes from finding a lower bound on an invariant called the F-pure threshold of a polynomial in terms of its degree. In this talk, I'll introduce a new class of hypersurfaces which obtain a minimal F-pure threshold and discuss some of their surprising geometric properties. They are cut out by polynomials that we call Frobenius forms, which have a rich algebraic structure coming from the fact that they have a matrix factorization mirroring the theory of quadratic forms. Further, we fully classify them and show that there are only finitely many of them up to a linear change of coordinates in any bounded degree and number of variables.

Luigi Ferraro, Texas Tech University

The homotopy Lie algebra of a Tor-independent tensor product

Abstract: We investigate a pair of surjective local ring maps \(S_1\leftarrow R\to S_2\) between local commutative rings and their relation to the canonical projection \(R\to S_1\otimes_R S_2\), where \(S_1,S_2\) are Tor-independent over \(R\). The main result asserts a structural connection between the homotopy Lie algebra of \(S\), denoted \(\pi(S)\), in terms of those of \(R,S_1\) and \(S_2\), where \(S=S_1\otimes_R S_2\). Namely, \(\pi(S)\) is the pullback of (restricted) Lie algebras along the maps \(\pi(S_i)\to \pi(R)\) in a wide variety cases, including when the maps above have residual characteristic zero. Consequences to the main theorem include structural results on André—Quillen cohomology, stable cohomology, and Tor algebras, as well as an equality relating the Poincaré series of the common residue field of \(R,S_1,S_2\) and \(S\), and that the map \(R\to S\) can never be Golod.