## Organizers

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CHAMP is a weekly online seminar series; its main goal is to give graduate students and other early career researchers on the job market a platform to give the 50-minute version of their research talk. We will have 50 minute talks on zoom every Tuesday at 3:30 pm eastern time, followed by 10 minutes of questions and a 30-minute tea room. We will be recording talks, and posting them here. Each speaker will also record a 3 to 5 minute elevator pitch to showcase their research. The speakers are all young researchers in search of an academic position.

You can watch past talks and elevator pitch videos in our YouTube channel.

You can watch past talks and elevator pitch videos in our YouTube channel.

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## 2021/2022 academic year

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## CHAMPS academic year 2021-2022 speakers

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Monday, February 21, 2021, 3 pm US eastern time

Abstract: 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.

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## Past talks

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>>>>>>> 3128809ed0bcc56572650e0742ee4ad74f57da10Monday, February 14, 2021, 3 pm US eastern time

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.

Monday, February 7, 2021, 3 pm US eastern time

Abstract: Given a graded triple \( (M,R,I) \) in a positive characteristic \(p\) and for any complex number \(y\), we show that the limit

\(\underset{n \to \infty}{\lim}(\frac{1}{p^n})^{\text{dim}(M)}\sum \limits_{j= -\infty}^{\infty}\lambda \left( (\frac{M}{I^{[p^n]}M})_j\right)e^{-iyj/p^n}\)
exists. This limit as a function in the complex \(y\) captures the Hilbert-Kunz multiplicity of the triple \( (M,R,I) \). We name this limiting function the *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.

Monday, January 24, 2021, 3 pm US eastern time

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.

Monday, December 6, 2021, 3 pm US eastern time

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.

Monday, November 29, 2021, 3 pm US eastern time

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.

Monday, November 22, 2021, 3 pm US eastern time

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.

Monday, November 15, 2021, 3 pm US eastern time

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.

Monday, November 8, 2021, 3 pm US eastern time

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.

Monday, November 1, 2021, 3 pm US eastern time

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 \(i\neq j\).

Monday, October 25, 2021, 3 pm US eastern time

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.

Monday, October 18, 2021, 3 pm US eastern time

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.

Monday, October 11, 2021, 3 pm US eastern time

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.

Monday, October 4, 2021, 3 pm US eastern time

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.

Monday, September 27, 2021, 3 pm US eastern time

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.

Monday, September 20, 2021, 3 pm US eastern time

(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.

Monday, September, 2021, 3 pm US eastern time

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.

Monday, September 6, 2021, 3 pm US eastern time

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.

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## Academic year 2020‐2021

Tuesday, March 9, 2021, 3:30 pm US eastern time

Title: Base Change Along the Frobenius Endomorphism And The Gorenstein Property

Abstract:

Let \(R\) be a local ring of positive characteristic and \(X\) a complex with nonzero finitely generated homology and finite injective dimension. We prove that if derived base change of \(X\) via the Frobenius (or more generally, via a contracting) endomorphism has finite injective dimension then \(R\) is Gorenstein.

Tuesday, March 2, 2021, 3:30 pm eastern time

Some criteria for detecting large, small, and Golod homomorphisms

Abstract:

This talk consists of two parts. In the first part, I will focus on large homomorphisms introduced by Levin in 1978. I will address some known examples and characterization of large homomorphisms in terms of Koszul homologies over complete intersection and Golod local rings. Next, I will discuss large homomorphisms from or to a Koszul algebra and its consequences. In the second part, I will focus on a special class of local rings namely minimal intersections. Recall that a local ring \(R\) is a minimal intersection if R (or the completion of R) has a presentation \(R=Q/(I+J)\) with Q a local ring and \(Q/I\) and \(Q/J\) Tor-independent as \(Q\)-modules. I will discuss the large and smallness of the natural map from \(Q\) to \(Q/I\) and its consequences on computing the Poincaré series of \(R\).

Tuesday, February 23, 2021, 3:30 pm eastern time

Koszul property of symbolic powers cover ideals

Abstract:

For cover ideals we are motivated by the results of Villarreal showing that whiskering a graph results in a Cohen-Macaulay graph which, in turn, implies the cover ideal of the whiskered graph has linear resolution. Later it was shown that whiskering \(S\subsetneq V(G)\) resulted in the cover ideal of the graph whiskered at \(S\), \(J(G\cup W(S))\), being sequentially Cohen-Macaulay and therefore Koszul. In '16, Fakhari introduced a graph construction \(G_k\) that corresponds to the symbolic power of the cover ideal \(J(G)^{(k)}\). Using this construction and the whiskering technique we will establish conditions on \(S\) such that \(J(G\cup W(S))^{(k)}\) is Koszul for all \(k\).

Tuesday, February 16, 2021, 3:30 pm eastern

Classification of Totally Acyclic Complexes over Local Gorenstein Rings

Abstract:

Let \(Q\) be a commutative local ring to which we associate the subcategory \(\textrm{Ktac}(Q)\) of the homotopy category of \(Q\)-complexes, consisting of the totally acyclic complexes. Assume further that \(Q\) is a Henselian Gorenstein ring with a surjective ring homomorphism \(Q \xrightarrow{\phi} R\) and \(\textrm{pd}_Q(R) < \infty\). We will use the indecomposable objects of \(Q\), and the idea of approximations to classify totally acyclic complexes over \(R\) using our defined notion of Arnold-tuples.

Tuesday, February 9, 2021, 3:30 pm eastern

Stable Harbourne—Huneke Containment and Bounds on Waldschmidt Constant

Abstract: The study of the degree of a homogeneous polynomial vanishing on given points with multiplicities is always intriguing. Nagata raised the following fundamental question:

Q: Given a finite set of points \(X= \{P_1,\dots P_s \} \subset \mathbf{P}_{\mathbb{C}}^N \) what is the minimal degree, \(\alpha_x(m) \) of a hyper-surface that passes through the points with multiplicity at least \(m\)?

Chudnovsky provided a conjectural answer to the above question which was generalized by Demailly. Both conjectures have equivalent statements involving a lower bound of the Waldschmidt constant of the ideal defining points. Harbourne and Huneke gave containment conjectures involving the symbolic and the ordinary powers of the ideals, which implies Chudnovsky's conjecture and Demailly's conjecture respectively.

We study the stable versions of the containment conjectures and consequently, we prove Chudnovsky and Demailly's conjecture for a large number of general points. In this talk, I will introduce all these conjectures and the tools that we used. I will also present the results from our joint work with Eloísa Grifo, Tài Huy Hà, and Thái Thành Nguyễn.

Tuesday, February 2nd, 2021, 3:30 pm eastern

Finding maximal Cohen Macaulay and reflexive modules

Abstract:

Maximal Cohen Macaulay and Reflexive modules are both very classical objects and their properties have been studied extensively. However, their existence and/or ubiquity is far from clear. In this talk, I will consider these questions and provide answers in certain settings.

Tuesday, January 26, 2021, 3:30 pm eastern

Geometric equations for matroid varieties

Abstract:

Each point \(x\) in Gr\((r,n)\) corresponds to an \(r \times n\) matrix \(A_x\) which gives rise to a matroid \(M_x\) on its columns. Gel'fand, Goresky, MacPherson, and Serganova showed that the sets \(\left\{y \in \right.\)Gr\(\left.(r,n)\, |\, M_y = M_x\right\}\) form a stratification of Gr\((r,n)\) with many beautiful properties. However, results of Mnëv and Sturmfels show that these strata can be quite complicated, and in particular may have arbitrary singularities. We study the ideals \(I_x\) of matroid varieties, the Zariski closures of these strata. We construct several classes of examples based on theorems from projective geometry and describe how the Grassmann-Cayley algebra may be used to derive non-trivial elements of \(I_x\) geometrically when the combinatorics of the matroid is sufficiently rich.

Tuesday, January 19, 2021, 3:30 pm eastern

How valuation rings behave like non-noetherian regular rings

Abstract:

Valuation rings have wide-ranging applications in algebra, arithmetic and geometry. These highly non-noetherian objects became popular in commutative algebra and algebraic geometry when Zariski used them to study normality and resolution of singularities. In recent years there has been a renewed interest in valuation rings because of their foundational roles in various approaches to rigid geometry such as Berkovich and Perfectoid spaces as well as their applications in K-stability and moduli problems. In this talk we will examine commutative and homological algebraic properties of valuation rings. We will show that despite being non-noetherian, valuation rings share striking similarities with regular rings. This talk is based on joint works with Karen Smith and Benjamin Antieau.

Tuesday, January 12, 2021, 3:30 pm eastern

Perturbing Ideals in Arbitrary Noetherian Local Rings and the \(\mathfrak{m}\)-adic Continuity of Hilbert-Kunz Multiplicity

Abstract:

In 2018, Polstra and Smirnov showed that the Hilbert-Kunz multiplicity of F-finite CM local rings exhibits a remarkable kind of \(\mathfrak{m}\)-adic continuity. In this talk I will discuss techniques that can be used to extend their result to arbitrary F-finite local rings. These methods also have applications to Hilbert-Samuel multiplicity and general questions of \(\mathfrak{m}\)-adic stability in equal characteristic Noetherian local rings.

Tuesday, January 5, 2021, 3:30 pm eastern

Splines and a counter-example to the Schenck-Stiller ''\(2r+1\)" conjecture.

Abstract:

To approximate a function over a region, it is useful to consider a subdivision of the region and then approximate the function by a piecewise polynomial. In this talk, I would like to talk about what we know about splines, commutative algebra tools we use to study this subject, conjectures on splines and a counter-example to the Schenck-Stiller ''\(2r+1\)" conjecture. This talk is based on joint work with Mike Stillman and Hal Schenck.

December 16, 2020, 3 pm eastern

Cohen-Macaulay property of the fiber cone of modules

Abstract:

Let R be a Noetherian local ring and let E be a finite R-module. The fiber cone of E is the graded algebra F(E) defined by tensoring the Rees algebra R(E) with the residue field of R. In 2003 Simis, Ulrich and Vasconcelos showed that the study of the Cohen-Macaulay property of the Rees algebra R(E) can be reduced to the case of Rees algebras of ideals, by means of the so called generic Bourbaki ideals. The Cohen-Macaulay property of Rees algebras and fiber cones are usually unrelated. However, in this talk I will show that sometimes generic Bourbaki ideals can effectively be used in order to study the Cohen-Macaulay property of the fiber cone F(E) as well, and provide classes of modules whose fiber cone is Cohen-Macaulay. The talk is based on a preprint available at https://arxiv.org/abs/2011.08453.

December 9, 2020, 3 pm eastern

Complete intersections and the cotangent complex

Abstract: The cotangent complex is an important but difficult to understand object in commutative algebra. For a homomorphism \(\varphi: R\to S\) of commutative noetherian rings, this is a complex \(L_{\varphi} = L_0\leftarrow L_1\leftarrow \cdots\) of free \(S\)-modules. Inside it you can find the Kähler differentials, the conormal module, the Koszul homology, and it has a lot to say about deformation theory. For complete intersection maps one can completely write down \(L_{\varphi}\), but otherwise it's extremely complicated. When it was introduced by Quillen, he conjectured (for maps of finite flat dimension) that if \(\varphi\) is not complete intersection then \(L_{\varphi}\) must go on forever. This was proven by Avramov in 1999. I will explain how to exploit the connection between the cotangent complex and Hochschild cohomology to get a new proof, and how to simultaneously prove a conjecture of Vasconcelos on the conormal module. This is joint work with Srikanth Iyengar.

December 2, 2020

F-purity deforms in \(\mathbb{Q}\)-Gorenstein rings

Abstract:

We positively settle a long-standing question in the theory of prime characteristic singularities: If \( (R,\mathfrak{m},k) \) is a local \( \mathbb{Q} \)-Gorenstein ring of prime characteristic \(p>0\) which admits a non-zero-divisor \(f\) so that \(R/(f)\) is normal and \(F\)-pure, is \(R\) necessarily \(F\)-pure? The origins of the question date back to work of Fedder in the early 1980's where it was shown that \(F\)-purity deforms in Gorenstein rings but fails to deform in rings which are not \(\mathbb{Q}\)-Gorenstein. This talk is based on joint work with Austyn Simpson.

November 18, 2020

Using totally acyclic complexes to extend work of Buchweitz into a non-affine setting

Abstract:

In the 1950s, Auslander, Buchsbaum, and Serre set the stage for systematically using homological dimensions to understand ring structure with their elegant characterization of regular local rings in terms of projective dimension. This was followed by the characterization of Gorenstein local rings in terms of “G-dimension” by Auslander and Bridger, which in turn inspired the notions of Gorenstein projective, injective, and flat modules. Such modules are defined in terms of "totally acyclic" complexes. In this talk I will outline recent joint work with Christensen and Estrada where we propose a general notion of total acyclicity that unifies these classic notions and introduces the new “Gorenstein flat cotorsion” modules. This abstraction allows us to extend work of Buchweitz—involving equivalences between homotopy, stable, and singularity categories—into the non-affine setting, thus completing a project initiated by Murfet and Salarian.

November 11, 2020

The Critical and Cocritical Degrees of a Totally Acyclic Complex

Abstract:

It is widely known that minimal free resolutions of a module over a complete intersection ring have nice patterns that arise in their betti sequences. In the late 1990's Avramov, Gasharov and Peeva defined a new class of \(R\)-modules (those with finite CI dimension) that would exhibit similar patterns in their free resolutions. In doing so, they additionally defined the notion of critical degree for an \(R\)-module, which describes exactly when such patterns arise in the betti sequence. In this talk, I will define a ``naive'' extension of critical degree to the category of totally acyclic complexes, \(\mathbf{K_{tac}}(R)\), where \(R\) is a commutative local ring. After discussing some basic properties, we will then explore an alternative extension and investigate the relationship between the two definitions.

October 26, 2020 (note unusual date)

Random Monomial Ideals

Abstract:

Random and Probabilistic techniques have a long history across a variety of fields, but their use in commutative algebra has been comparatively limited. Random monomial ideals, as inspired by results in random graphs and random simplicial complexes, are a unique perspective that allows us to study the asymptotic behavior of ideals. I discuss a pair of models for random monomial ideals, and a collection of results for these models including work with Caytlin Booms and Daniel Erman as well as work with Lily Silverstein and Dane Wilburne.

October 28, 2020

Bernstein-Sato polynomials on singular rings in positive characteristic

Abstract:

Given a smooth \(\mathbb{C}\)-algebra \(R\) and an ideal \(\mathfrak{a} \subseteq R\), an invariant from \(D\)-module theory known as the Bernstein-Sato polynomial of \(\mathfrak{a}\) quantifies the singularities of the pair \( (R, \mathfrak{a}) \). Recently, two generalizations of this construction were given: in one direction, Àlvarez-Montaner, Huneke, and Núñez-Betancourt showed that Bernstein-Sato polynomials can still be defined in some settings where \(R\) has singularities; in the other direction, work of Mustaţă, Bitoun and myself shows that we can also define Bernstein-Sato polynomials in positive characteristic. In this talk, I present joint work with J. Jeffries and L. Núñez-Betancourt in which we show that the combined generalization is possible. Namely, we show that Bernstein-Sato polynomials exist in positive characteristic when \(R\) has mild singularities (direct summand or graded F-finite representation type).

October 21, 2020

Characterizing Cohen-Macaulay power edge ideals of trees

Abstract:

Every electric power system can be modeled by a graph \(G\) whose vertices represent electrical buses and whose edges represent power lines. A*phasor measurement unit* (PMU) is a monitor that can be placed at a bus to observe the voltage at that bus as well as the current and phase through all incident power lines. The problem of monitoring an entire electric power system using the fewest number of PMUs is closely related to the well-known vertex covering and dominating set problems in graph theory.

In this talk, we will give an overview of the PMU placement problem and its connections to commutative ring theory. Specifically, we will define the*power edge ideal* \(I^P_G\) of a graph \(G\) as a way to generate polynomial rings with desired algebraic properties using graphs of electric power grids. Finally, we will classify the trees \(G\) for which \(I^P_G\) is Cohen-Macaulay and prove that every such ideal is also a complete intersection.

This project is joint work with Michael Cowen, Alan Hahn, Frank Moore, and Sean Sather-Wagstaff.

October 14, 2020

A Study of Colength in Dimension One

Abstract:

We define and study an invariant of any module over a local one dimensional analytically unramified Noetherian domain whose integral closure is a DVR. We shall discuss a key property of this invarant. If time permits, we will briefly venture into trace ideals and into reflexive ideals.

October 7, 2020

The closed support problem over a complete intersection ring

Abstract:

Local cohomology modules are (typically) very large algebraic objects that encode rich geometric information about the structure of a commutative ring. These modules are rarely finitely generated, but sometimes still exhibit remarkable finiteness properties. For example, the local cohomology of a smooth algebra over a field will always have a finite set of associated primes. This property can fail for complete intersection rings (even in codimension 1), but independent results of Hochster and Núñez-Betancourt (2017) or Katzman and Zhang (2017) have shown that at least in characteristic \(p>0\), the local cohomology of a hypersurface ring will still have Zariski closed support. It remains open whether this property holds in arbitrary codimension. In this talk, I will present my results on the local cohomology of a parameter ideal illustrating an obstruction to straightforwardly generalizing existing hypersurface strategies. I will then present joint work with Eric Canton on a possible alternative route of attack in higher codimension, involving a novel Frobenius-compatible simplicial complex of local cohomology modules.

September 30, 2020

The "size" of an ideal

Abstract:

Hochster and Huneke defined quasilength for any \(I\)-torsion module, generalizing the notion of length to any non-maximal ideal \(I\). Based on quasilength, we develop a new numerical invariant for ideals, called "size". It is invariant up to taking radicals and bounded between the arithmetic rank and height of the ideal. We will present some results in low dimensions and discuss a lot of open questions related to "size" and asymptotic behaviors of quasilength.

September 23, 2020

Ulrich modules do not always exist

Abstract: Ulrich modules were introduced by Bernd Ulrich in 1984 and has since been a very active area of research. The existence of Ulrich modules for complete local domains have powerful applications. For example, existence implies Lech's conjecture: given a flat local map of local rings from R to S, the Hilbert-Samuel multiplicity of S is at least the Hilbert-Samuel multiplicity of R. Until recently, it was unknown if there were any counterexamples to the existence of Ulrich modules for (complete) local domains. In this talk, I will introduce the notion of Ulrich modules and discuss implications of their existence. Then I will give a counterexample to the existence of Ulrich modules for (complete) local domains.

September 16, 2020

DG-Structures on Minimal Free Resolutions of Fiber Products

Abstract:

A construction of Tate shows that every algebra over a ring \(R\) possess a DG-algebra resolution over \(R\). These resolutions are not always minimal and Avramov even shows that certain algebras cannot have a minimal resolution with a DG-algebra structure. In the first half of this talk, I give an explicit construction of the minimal resolution of the fiber product \(k[\underline{x}]/\mathcal{I} \times_k k[\underline{y}]/\mathcal{J}\) over \(k[\underline{x},\underline{y}]\) where \(\mathcal{I} \subseteq \langle \underline{x} \rangle^2\) and \(\mathcal{J} \subseteq \langle \underline{y} \rangle^2\). In the second half, I show how to put different DG-structures on these minimal free resolutions.

September 10, 2020, 11 am eastern

Title: Polarizations of Powers of Graded Maximal Ideals

Abstract:

Given a monomial ideal, one can "polarize" it to a square-free monomial ideal that has all of the same homological invariants as the original one. Many commutative algebraists are familiar with the use of the "standard" polarization, but the first use of a nonstandard polarization was by Nagel and Reiner in the 2000s, who used the "box polarization" to produce a minimal cellular resolution for strongly stable ideals. This leads to the natural question: what other ways are there to polarize a monomial ideal, and what other applications might there be for these non-standard polarizations? In this talk, I will give a complete combinatorial characterization of all possible polarizations of powers of the graded maximal ideal in a polynomial ring. I will also give a combinatorial description of their Alexander duals and discuss applications of polarizations to commutative algebra, algebraic geometry, and combinatorics. This is joint work with Gunnar Fløystad and Henning Lohne.

September 2, 2020

Tate-like Complexes and Their Applications to DG Algebras

Abstract:

In this talk, I will talk about so-called Tate-like complexes. These are defined as a quotient of the tensor product complex and encompass more well-known complexes such as the "symmetric square" complex, commonly used to induce DG-products on resolutions (associative up to homotopy). After introducing necessary notation and terminology, I will talk about how these complexes have made a tacit appearance in previous work of others, and give a brief overview on how these complexes can be used to endow the length 4 "big from small" construction of Kustin and Miller with the structure of an associative DG algebra.

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### Organizers

Eloísa Grifo (University of Nebraska — Lincoln)
Emeritus organizer: Keri Sather-Wagstaff (Clemson University)

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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.

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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

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.

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Pinches Dirnfield, University of Utah

Title: Base Change Along the Frobenius Endomorphism And The Gorenstein Property

Abstract:

Let \(R\) be a local ring of positive characteristic and \(X\) a complex with nonzero finitely generated homology and finite injective dimension. We prove that if derived base change of \(X\) via the Frobenius (or more generally, via a contracting) endomorphism has finite injective dimension then \(R\) is Gorenstein.

Mohsen Gheibi, University Of Texas At Arlington

Some criteria for detecting large, small, and Golod homomorphisms

Abstract:

This talk consists of two parts. In the first part, I will focus on large homomorphisms introduced by Levin in 1978. I will address some known examples and characterization of large homomorphisms in terms of Koszul homologies over complete intersection and Golod local rings. Next, I will discuss large homomorphisms from or to a Koszul algebra and its consequences. In the second part, I will focus on a special class of local rings namely minimal intersections. Recall that a local ring \(R\) is a minimal intersection if R (or the completion of R) has a presentation \(R=Q/(I+J)\) with Q a local ring and \(Q/I\) and \(Q/J\) Tor-independent as \(Q\)-modules. I will discuss the large and smallness of the natural map from \(Q\) to \(Q/I\) and its consequences on computing the Poincaré series of \(R\).

Joseph Skelton, Tulane University

Koszul property of symbolic powers cover ideals

Abstract:

For cover ideals we are motivated by the results of Villarreal showing that whiskering a graph results in a Cohen-Macaulay graph which, in turn, implies the cover ideal of the whiskered graph has linear resolution. Later it was shown that whiskering \(S\subsetneq V(G)\) resulted in the cover ideal of the graph whiskered at \(S\), \(J(G\cup W(S))\), being sequentially Cohen-Macaulay and therefore Koszul. In '16, Fakhari introduced a graph construction \(G_k\) that corresponds to the symbolic power of the cover ideal \(J(G)^{(k)}\). Using this construction and the whiskering technique we will establish conditions on \(S\) such that \(J(G\cup W(S))^{(k)}\) is Koszul for all \(k\).

Tyler Anway, University of Texas at Arlington

Classification of Totally Acyclic Complexes over Local Gorenstein Rings

Abstract:

Let \(Q\) be a commutative local ring to which we associate the subcategory \(\textrm{Ktac}(Q)\) of the homotopy category of \(Q\)-complexes, consisting of the totally acyclic complexes. Assume further that \(Q\) is a Henselian Gorenstein ring with a surjective ring homomorphism \(Q \xrightarrow{\phi} R\) and \(\textrm{pd}_Q(R) < \infty\). We will use the indecomposable objects of \(Q\), and the idea of approximations to classify totally acyclic complexes over \(R\) using our defined notion of Arnold-tuples.

Sankhaneel Bisui, Tulane University

Stable Harbourne—Huneke Containment and Bounds on Waldschmidt Constant

Abstract: The study of the degree of a homogeneous polynomial vanishing on given points with multiplicities is always intriguing. Nagata raised the following fundamental question:

Q: Given a finite set of points \(X= \{P_1,\dots P_s \} \subset \mathbf{P}_{\mathbb{C}}^N \) what is the minimal degree, \(\alpha_x(m) \) of a hyper-surface that passes through the points with multiplicity at least \(m\)?

Chudnovsky provided a conjectural answer to the above question which was generalized by Demailly. Both conjectures have equivalent statements involving a lower bound of the Waldschmidt constant of the ideal defining points. Harbourne and Huneke gave containment conjectures involving the symbolic and the ordinary powers of the ideals, which implies Chudnovsky's conjecture and Demailly's conjecture respectively.

We study the stable versions of the containment conjectures and consequently, we prove Chudnovsky and Demailly's conjecture for a large number of general points. In this talk, I will introduce all these conjectures and the tools that we used. I will also present the results from our joint work with Eloísa Grifo, Tài Huy Hà, and Thái Thành Nguyễn.

Prashanth Sridhar, University of Kansas

Finding maximal Cohen Macaulay and reflexive modules

Abstract:

Maximal Cohen Macaulay and Reflexive modules are both very classical objects and their properties have been studied extensively. However, their existence and/or ubiquity is far from clear. In this talk, I will consider these questions and provide answers in certain settings.

Ashley Wheeler, Mount Holyoke College

Geometric equations for matroid varieties

Abstract:

Each point \(x\) in Gr\((r,n)\) corresponds to an \(r \times n\) matrix \(A_x\) which gives rise to a matroid \(M_x\) on its columns. Gel'fand, Goresky, MacPherson, and Serganova showed that the sets \(\left\{y \in \right.\)Gr\(\left.(r,n)\, |\, M_y = M_x\right\}\) form a stratification of Gr\((r,n)\) with many beautiful properties. However, results of Mnëv and Sturmfels show that these strata can be quite complicated, and in particular may have arbitrary singularities. We study the ideals \(I_x\) of matroid varieties, the Zariski closures of these strata. We construct several classes of examples based on theorems from projective geometry and describe how the Grassmann-Cayley algebra may be used to derive non-trivial elements of \(I_x\) geometrically when the combinatorics of the matroid is sufficiently rich.

Rankeya Datta, University of Illinois at Chicago

How valuation rings behave like non-noetherian regular rings

Abstract:

Valuation rings have wide-ranging applications in algebra, arithmetic and geometry. These highly non-noetherian objects became popular in commutative algebra and algebraic geometry when Zariski used them to study normality and resolution of singularities. In recent years there has been a renewed interest in valuation rings because of their foundational roles in various approaches to rigid geometry such as Berkovich and Perfectoid spaces as well as their applications in K-stability and moduli problems. In this talk we will examine commutative and homological algebraic properties of valuation rings. We will show that despite being non-noetherian, valuation rings share striking similarities with regular rings. This talk is based on joint works with Karen Smith and Benjamin Antieau.

Nick Cox-Steib, University of Missouri

Perturbing Ideals in Arbitrary Noetherian Local Rings and the \(\mathfrak{m}\)-adic Continuity of Hilbert-Kunz Multiplicity

Abstract:

In 2018, Polstra and Smirnov showed that the Hilbert-Kunz multiplicity of F-finite CM local rings exhibits a remarkable kind of \(\mathfrak{m}\)-adic continuity. In this talk I will discuss techniques that can be used to extend their result to arbitrary F-finite local rings. These methods also have applications to Hilbert-Samuel multiplicity and general questions of \(\mathfrak{m}\)-adic stability in equal characteristic Noetherian local rings.

Beihui Yuan, Cornell University

Splines and a counter-example to the Schenck-Stiller ''\(2r+1\)" conjecture.

Abstract:

To approximate a function over a region, it is useful to consider a subdivision of the region and then approximate the function by a piecewise polynomial. In this talk, I would like to talk about what we know about splines, commutative algebra tools we use to study this subject, conjectures on splines and a counter-example to the Schenck-Stiller ''\(2r+1\)" conjecture. This talk is based on joint work with Mike Stillman and Hal Schenck.

Alessandra Costantini, University of California, Riverside

Cohen-Macaulay property of the fiber cone of modules

Abstract:

Let R be a Noetherian local ring and let E be a finite R-module. The fiber cone of E is the graded algebra F(E) defined by tensoring the Rees algebra R(E) with the residue field of R. In 2003 Simis, Ulrich and Vasconcelos showed that the study of the Cohen-Macaulay property of the Rees algebra R(E) can be reduced to the case of Rees algebras of ideals, by means of the so called generic Bourbaki ideals. The Cohen-Macaulay property of Rees algebras and fiber cones are usually unrelated. However, in this talk I will show that sometimes generic Bourbaki ideals can effectively be used in order to study the Cohen-Macaulay property of the fiber cone F(E) as well, and provide classes of modules whose fiber cone is Cohen-Macaulay. The talk is based on a preprint available at https://arxiv.org/abs/2011.08453.

Benjamin Briggs, University of Utah

Complete intersections and the cotangent complex

Abstract: The cotangent complex is an important but difficult to understand object in commutative algebra. For a homomorphism \(\varphi: R\to S\) of commutative noetherian rings, this is a complex \(L_{\varphi} = L_0\leftarrow L_1\leftarrow \cdots\) of free \(S\)-modules. Inside it you can find the Kähler differentials, the conormal module, the Koszul homology, and it has a lot to say about deformation theory. For complete intersection maps one can completely write down \(L_{\varphi}\), but otherwise it's extremely complicated. When it was introduced by Quillen, he conjectured (for maps of finite flat dimension) that if \(\varphi\) is not complete intersection then \(L_{\varphi}\) must go on forever. This was proven by Avramov in 1999. I will explain how to exploit the connection between the cotangent complex and Hochschild cohomology to get a new proof, and how to simultaneously prove a conjecture of Vasconcelos on the conormal module. This is joint work with Srikanth Iyengar.

Thomas Polstra, University of Virginia

F-purity deforms in \(\mathbb{Q}\)-Gorenstein rings

Abstract:

We positively settle a long-standing question in the theory of prime characteristic singularities: If \( (R,\mathfrak{m},k) \) is a local \( \mathbb{Q} \)-Gorenstein ring of prime characteristic \(p>0\) which admits a non-zero-divisor \(f\) so that \(R/(f)\) is normal and \(F\)-pure, is \(R\) necessarily \(F\)-pure? The origins of the question date back to work of Fedder in the early 1980's where it was shown that \(F\)-purity deforms in Gorenstein rings but fails to deform in rings which are not \(\mathbb{Q}\)-Gorenstein. This talk is based on joint work with Austyn Simpson.

Peder Thompson, Norwegian University of Science and Technology

Using totally acyclic complexes to extend work of Buchweitz into a non-affine setting

Abstract:

In the 1950s, Auslander, Buchsbaum, and Serre set the stage for systematically using homological dimensions to understand ring structure with their elegant characterization of regular local rings in terms of projective dimension. This was followed by the characterization of Gorenstein local rings in terms of “G-dimension” by Auslander and Bridger, which in turn inspired the notions of Gorenstein projective, injective, and flat modules. Such modules are defined in terms of "totally acyclic" complexes. In this talk I will outline recent joint work with Christensen and Estrada where we propose a general notion of total acyclicity that unifies these classic notions and introduces the new “Gorenstein flat cotorsion” modules. This abstraction allows us to extend work of Buchweitz—involving equivalences between homotopy, stable, and singularity categories—into the non-affine setting, thus completing a project initiated by Murfet and Salarian.

Rebekah Aduddell, University of Texas at Arlington

The Critical and Cocritical Degrees of a Totally Acyclic Complex

Abstract:

It is widely known that minimal free resolutions of a module over a complete intersection ring have nice patterns that arise in their betti sequences. In the late 1990's Avramov, Gasharov and Peeva defined a new class of \(R\)-modules (those with finite CI dimension) that would exhibit similar patterns in their free resolutions. In doing so, they additionally defined the notion of critical degree for an \(R\)-module, which describes exactly when such patterns arise in the betti sequence. In this talk, I will define a ``naive'' extension of critical degree to the category of totally acyclic complexes, \(\mathbf{K_{tac}}(R)\), where \(R\) is a commutative local ring. After discussing some basic properties, we will then explore an alternative extension and investigate the relationship between the two definitions.

Jay Yang, University of Minnesota

Random Monomial Ideals

Abstract:

Random and Probabilistic techniques have a long history across a variety of fields, but their use in commutative algebra has been comparatively limited. Random monomial ideals, as inspired by results in random graphs and random simplicial complexes, are a unique perspective that allows us to study the asymptotic behavior of ideals. I discuss a pair of models for random monomial ideals, and a collection of results for these models including work with Caytlin Booms and Daniel Erman as well as work with Lily Silverstein and Dane Wilburne.

Eamon Quinlan-Gallego, University of Michigan

Bernstein-Sato polynomials on singular rings in positive characteristic

Abstract:

Given a smooth \(\mathbb{C}\)-algebra \(R\) and an ideal \(\mathfrak{a} \subseteq R\), an invariant from \(D\)-module theory known as the Bernstein-Sato polynomial of \(\mathfrak{a}\) quantifies the singularities of the pair \( (R, \mathfrak{a}) \). Recently, two generalizations of this construction were given: in one direction, Àlvarez-Montaner, Huneke, and Núñez-Betancourt showed that Bernstein-Sato polynomials can still be defined in some settings where \(R\) has singularities; in the other direction, work of Mustaţă, Bitoun and myself shows that we can also define Bernstein-Sato polynomials in positive characteristic. In this talk, I present joint work with J. Jeffries and L. Núñez-Betancourt in which we show that the combined generalization is possible. Namely, we show that Bernstein-Sato polynomials exist in positive characteristic when \(R\) has mild singularities (direct summand or graded F-finite representation type).

James Gossell, Clemson University

Characterizing Cohen-Macaulay power edge ideals of trees

Abstract:

Every electric power system can be modeled by a graph \(G\) whose vertices represent electrical buses and whose edges represent power lines. A

In this talk, we will give an overview of the PMU placement problem and its connections to commutative ring theory. Specifically, we will define the

This project is joint work with Michael Cowen, Alan Hahn, Frank Moore, and Sean Sather-Wagstaff.

Sarasij Maitra, University of Virginia

A Study of Colength in Dimension One

Abstract:

We define and study an invariant of any module over a local one dimensional analytically unramified Noetherian domain whose integral closure is a DVR. We shall discuss a key property of this invarant. If time permits, we will briefly venture into trace ideals and into reflexive ideals.

Monica Lewis, University of Michigan

The closed support problem over a complete intersection ring

Abstract:

Local cohomology modules are (typically) very large algebraic objects that encode rich geometric information about the structure of a commutative ring. These modules are rarely finitely generated, but sometimes still exhibit remarkable finiteness properties. For example, the local cohomology of a smooth algebra over a field will always have a finite set of associated primes. This property can fail for complete intersection rings (even in codimension 1), but independent results of Hochster and Núñez-Betancourt (2017) or Katzman and Zhang (2017) have shown that at least in characteristic \(p>0\), the local cohomology of a hypersurface ring will still have Zariski closed support. It remains open whether this property holds in arbitrary codimension. In this talk, I will present my results on the local cohomology of a parameter ideal illustrating an obstruction to straightforwardly generalizing existing hypersurface strategies. I will then present joint work with Eric Canton on a possible alternative route of attack in higher codimension, involving a novel Frobenius-compatible simplicial complex of local cohomology modules.

Zhan Jiang, University of Michigan

The "size" of an ideal

Abstract:

Hochster and Huneke defined quasilength for any \(I\)-torsion module, generalizing the notion of length to any non-maximal ideal \(I\). Based on quasilength, we develop a new numerical invariant for ideals, called "size". It is invariant up to taking radicals and bounded between the arithmetic rank and height of the ideal. We will present some results in low dimensions and discuss a lot of open questions related to "size" and asymptotic behaviors of quasilength.

Farrah Yhee, University of Michigan

Ulrich modules do not always exist

Abstract: Ulrich modules were introduced by Bernd Ulrich in 1984 and has since been a very active area of research. The existence of Ulrich modules for complete local domains have powerful applications. For example, existence implies Lech's conjecture: given a flat local map of local rings from R to S, the Hilbert-Samuel multiplicity of S is at least the Hilbert-Samuel multiplicity of R. Until recently, it was unknown if there were any counterexamples to the existence of Ulrich modules for (complete) local domains. In this talk, I will introduce the notion of Ulrich modules and discuss implications of their existence. Then I will give a counterexample to the existence of Ulrich modules for (complete) local domains.

Hugh Geller, Clemson University

DG-Structures on Minimal Free Resolutions of Fiber Products

Abstract:

A construction of Tate shows that every algebra over a ring \(R\) possess a DG-algebra resolution over \(R\). These resolutions are not always minimal and Avramov even shows that certain algebras cannot have a minimal resolution with a DG-algebra structure. In the first half of this talk, I give an explicit construction of the minimal resolution of the fiber product \(k[\underline{x}]/\mathcal{I} \times_k k[\underline{y}]/\mathcal{J}\) over \(k[\underline{x},\underline{y}]\) where \(\mathcal{I} \subseteq \langle \underline{x} \rangle^2\) and \(\mathcal{J} \subseteq \langle \underline{y} \rangle^2\). In the second half, I show how to put different DG-structures on these minimal free resolutions.

Ayah Almousa, Cornell University

Title: Polarizations of Powers of Graded Maximal Ideals

Abstract:

Given a monomial ideal, one can "polarize" it to a square-free monomial ideal that has all of the same homological invariants as the original one. Many commutative algebraists are familiar with the use of the "standard" polarization, but the first use of a nonstandard polarization was by Nagel and Reiner in the 2000s, who used the "box polarization" to produce a minimal cellular resolution for strongly stable ideals. This leads to the natural question: what other ways are there to polarize a monomial ideal, and what other applications might there be for these non-standard polarizations? In this talk, I will give a complete combinatorial characterization of all possible polarizations of powers of the graded maximal ideal in a polynomial ring. I will also give a combinatorial description of their Alexander duals and discuss applications of polarizations to commutative algebra, algebraic geometry, and combinatorics. This is joint work with Gunnar Fløystad and Henning Lohne.

Keller VandeBogert, University of South Carolina

Tate-like Complexes and Their Applications to DG Algebras

Abstract:

In this talk, I will talk about so-called Tate-like complexes. These are defined as a quotient of the tensor product complex and encompass more well-known complexes such as the "symmetric square" complex, commonly used to induce DG-products on resolutions (associative up to homotopy). After introducing necessary notation and terminology, I will talk about how these complexes have made a tacit appearance in previous work of others, and give a brief overview on how these complexes can be used to endow the length 4 "big from small" construction of Kustin and Miller with the structure of an associative DG algebra.

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